"The infamous fifty integrals" are the problems given by Kevin Charlwood in his article "Integration on Computer Algebra Systems" (The Electronic Journal of Mathematics and Technology, 2008). In this article Charlwood considers indefinite integrals and the ability of three computer algebra systems to evaluate them in closed-form, appealing only to the class of real, elementary functions.
Charlwood writes: "Although these systems have been widely available for many years and have undergone major enhancements in new versions, it is interesting to note that there are still indefinite integrals that escape the capacity of these systems to provide antiderivatives."
Charlwood puts forward 50 problems. The problems were discussed in the newsgroup sci.math.symbolic (The Charlwood Fifty) in 2013 and the optimal antiderivatives given here emanate from this discussion. They owe their current form mostly to the analysis of Martin Clicliclic and Albert Rich.
Albert Rich also compared the results produced by 6 symbolic integrators on the Charlwood's test-suite (Rubi, FriCAS, Mathematica, Derive, Maple and Maxima): Charlwood Fifty test results. Rich's results show FriCAS as the second best integrator (after Rubi).
We added a second test suite which was also discussed on sci.math.symbolic in the thread "An independent integration test suite". It is based on the Russian book "Integration of Functions" (Интегрирование функций) published by A. F. Timofeev (А.Ф. Тимофеев) in 1948 which provides many integration examples. Here we use the antiderivatives as given by Martin Clicliclic.
Below we give a Sage/Maxima (5.10) implementation of the test suites (which can also be used as a blueprint for a Sage/SymPy test).
We present the problems in the form
sage: Charlwood_problem(n)
integrand : ...
antideriv : ...
result : ...
The table below compares Sage's results with the optimal antiderivatives as compiled by Albert Rich in Charlwood's Integration Problems. Like Rich we do not allow domain restrictions on variables.
You can use the table to jump to the problems.
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| Failed, returns unevaluated (7) | Run time error (9) |
| Failed, needs additional assumptions (5) | Wrong result (6) |
| Computes longer than a minute (1) | Not elementary (6) |
| (An underlined index indicates a messy solution at least twice as long as necessary.) | |
If you want to check our results and do not have Sage installed you can try the test cases online at the Sage Cell Server. For example let's take test case 3 from the Charlwood suite. Enter:
Or try instead of the algorithm='maxima' the algorithm='sympy':
For further cross checking you can try your luck with Wolfram's Mathematica Online Integrator. Enter:
Mathematica's answer is a little bit messy, but it is a valid answer. A good answer gives FriCAS 1.2.1.
It is a pitty that there is no option 'fricas' in the selection box of Sage's integration algorithm.
(As at Sep 04 2013)
S A G E / M A X I M A - Charlwood's integrals
$$ \int \! \log\left(x\right) \arcsin\left(x\right) {dx} $$
[1] diffs back: False integrand: log(x)*arcsin(x) antideriv: (log(x) - 1)*x*arcsin(x) + sqrt(-x^2 + 1)*log(x) - 2*sqrt(-x^2 + 1) + arctanh(sqrt(-x^2 + 1)) maxima : (x*log(x) - x)*arcsin(x) + sqrt(-x^2 + 1)*log(x) - 2*sqrt(-x^2 + 1) + log(2*sqrt(-x^2 + 1)/abs(x) + 2/abs(x))
$$ \int \! \frac{x \arcsin\left(x\right)}{\sqrt{-x^{2} + 1}} {dx} $$
[2] diffs back: True integrand: x*arcsin(x)/sqrt(-x^2 + 1) antideriv: -sqrt(-x^2 + 1)*arcsin(x) + x maxima : -sqrt(-x^2 + 1)*arcsin(x) + x
$$ \int \!-\arcsin \left( -\sqrt {x+1}+\sqrt {x} \right) {dx} $$
[3] RuntimeError executing code in Maxima: `quotient' by `zero' integrand: arcsin(sqrt(x + 1) - sqrt(x)) antideriv: -1/8*(8*x + 3)*arcsin(-sqrt(x + 1) + sqrt(x)) + 1/8*(3*sqrt(x + 1) + sqrt(x))*sqrt(sqrt(x + 1)*sqrt(x) - x)*sqrt(2)
$$ \int \! \log\left(\sqrt{x^{2} + 1} x + 1\right) {dx} $$
[4] diffs back: True, answer contains integral integrand: log(sqrt(x^2 + 1)*x + 1) antideriv: x*log(sqrt(x^2 + 1)*x + 1) - sqrt(2*sqrt(5) - 2)*arctanh(sqrt(sqrt(5) + 2)*(x + sqrt(x^2 + 1))) + sqrt(2*sqrt(5) + 2)*arctan(sqrt(sqrt(5) - 2)*(x + sqrt(x^2 + 1))) - 2*x maxima : x*log(sqrt(x^2 + 1)*x + 1) - x + 1/2*arctan(x) - integrate(1/2*(2*x^6 + 3*x^4 - x^2 - 1)/(x^6 + 2*x^4 + 2*x^2 + 2*sqrt(x^2 + 1)*(x^3 + x) + 1), x)
$$ \int \!{\frac { \left( \cos \left( x \right) \right) ^{2}}{\sqrt { \left( \cos \left( x \right) \right) ^{4}+ \left( \cos \left( x \right) \right) ^{2}+1}}}{dx} $$
[5] RuntimeError executing code in Maxima: sign: argument cannot be imaginary integrand: cos(x)^2/sqrt(cos(x)^4 + cos(x)^2 + 1) antideriv: 1/3*x + 1/3*arctan((cos(x)^2 + 1)*sin(x)*cos(x)/(sqrt(cos(x)^4 + cos(x)^2 + 1)*cos(x)^2 + 1))
$$ \int \!\tan \left( x \right) \sqrt {1+ \left( \tan \left( x \right) \right) ^{4}}{dx} $$
[6] RuntimeError executing code in Maxima: sign: argument cannot be imaginary integrand: sqrt(tan(x)^4 + 1)*tan(x) antideriv: -1/2*sqrt(2)*arctanh(-1/2*(tan(x)^2 - 1)*sqrt(2)/sqrt(tan(x)^4 + 1)) + 1/2*sqrt(tan(x)^4 + 1) - 1/2*arcsinh(tan(x)^2)
$$ \int \! \frac{\tan\left(x\right)}{\sqrt{\sec\left(x\right)^{3} + 1}} {dx} $$
[7] diffs back: True integrand: tan(x)/sqrt(sec(x)^3 + 1) antideriv: -2/3*arctanh(sqrt(sec(x)^3 + 1)) maxima : 1/3*log(sqrt(1/cos(x)^3 + 1) - 1) - 1/3*log(sqrt(1/cos(x)^3 + 1) + 1)
$$ \int \!\sqrt { \left( \tan \left( x \right) \right) ^{2}+2\,\tan \left( x \right) +2}{dx} $$
[8] FAILED, computes longer than a minute integrand: sqrt(tan(x)^2 + 2*tan(x) + 2) antideriv: -sqrt(1/2*sqrt(5) - 1/2)*arctanh(1/2*(sqrt(sqrt(5) - 1)*tan(x) + sqrt(sqrt(5) + 1))*sqrt(2)/sqrt((tan(x) + 2)*tan(x) + 2)) + sqrt(1/2*sqrt(5) + 1/2)*arctan(1/2*(sqrt(sqrt(5) + 1)*tan(x) - sqrt(sqrt(5) - 1))*sqrt(2)/sqrt((tan(x) + 2)*tan(x) + 2)) + arcsinh(tan(x) + 1)
$$ \int \! \sin\left(x\right) \arctan\left(\sqrt{\sec\left(x\right) - 1}\right) {dx} $$
[9] diffs back: True integrand: sin(x)*arctan(sqrt(sec(x) - 1)) antideriv: -cos(x)*arctan(sqrt(sec(x) - 1)) + 1/2*sqrt(sec(x) - 1)*cos(x) + 1/2*arctan(sqrt(sec(x) - 1)) maxima : -cos(x)*arctan(sqrt(-(cos(x) - 1)/cos(x))) - 1/2*sqrt(-(cos(x) - 1)/cos(x))/((cos(x) - 1)/cos(x) - 1) + 1/2*arctan(sqrt(-(cos(x) - 1)/cos(x)))
$$ \int \! \frac{x^{3} e^{\arcsin\left(x\right)}}{\sqrt{-x^{2} + 1}} {dx} $$
[10] FAILED! Unevaluated! integrand: x^3*e^arcsin(x)/sqrt(-x^2 + 1) antideriv: -1/10*(3*sqrt(-x^2 + 1)*x^2 + 3*sqrt(-x^2 + 1) - x^3 - 3*x)*e^arcsin(x) maxima : integrate(x^3*e^arcsin(x)/sqrt(-x^2 + 1), x)
$$ \int \! \frac{x \log\left(x^{2} + 1\right) \log\left(x + \sqrt{x^{2} + 1}\right)}{\sqrt{x^{2} + 1}} {dx} $$
[11] diffs back: False, answer contains integral integrand: x*log(x^2 + 1)*log(x + sqrt(x^2 + 1))/sqrt(x^2 + 1) antideriv: (sqrt(x^2 + 1)*log(x + sqrt(x^2 + 1)) - x)*log(x^2 + 1) - 2*sqrt(x^2 + 1)*log(x + sqrt(x^2 + 1)) + 4*x - 2*arctan(x) maxima : -1/2*(2*sqrt(x^2 + 1) + log(sqrt(x^2 + 1) - 1) - log(sqrt(x^2 + 1) + 1))*log(x^2 + 1) + 1/2*log(x^2 + 1)*log(sqrt(x^2 + 1) - 1) - log(sqrt(x^2 + 1) + 1)*log(-sqrt(x^2 + 1)) + ((x^2 + 1)*log(x^2 + 1) - 2*x^2 - 2)*log(x + sqrt(x^2 + 1))/sqrt(x^2 + 1) + 4*sqrt(x^2 + 1) - 1/2*(log(x^2 + 1) - 2)/x + arctan(x) - 2*arcsinh(1/abs(x)) + polylog(2, -sqrt(x^2 + 1) + 1) - polylog(2, sqrt(x^2 + 1) + 1) - integrate(1/2*(log(x^2 + 1) - 2)/(2*x^4 + 2*sqrt(x^2 + 1)*x^3 + x^2), x)
$$ \int \! \arctan\left(x + \sqrt{-x^{2} + 1}\right) {dx} $$
[12] diffs back: True, answer contains integral integrand: arctan(x + sqrt(-x^2 + 1)) antideriv: x*arctan(x + sqrt(-x^2 + 1)) - 1/4*sqrt(3)*arctan(1/3*(2*x^2 - 1)*sqrt(3)) + 1/4*sqrt(3)*arctan((sqrt(3)*x - 1)/sqrt(-x^2 + 1)) + 1/4*sqrt(3)*arctan((sqrt(3)*x + 1)/sqrt(-x^2 + 1)) - 1/8*log(x^4 - x^2 + 1) - 1/2*arcsin(x) - 1/4*arctanh(sqrt(-x^2 + 1)*x) maxima : x*arctan(sqrt(-x + 1)*sqrt(x + 1) + x) - 1/4*log(x - 1) - 1/4*log(x + 1) - integrate(1/2*(x^3 - 2*x)/(sqrt(-x + 1)*sqrt(x + 1)*(x^3 - x) + x^2 - 1), x)
$$ \int \! \frac{x \arctan\left(x + \sqrt{-x^{2} + 1}\right)}{\sqrt{-x^{2} + 1}} {dx} $$
[13] diffs back: False, WRONG RESULT! integrand: x*arctan(x + sqrt(-x^2 + 1))/sqrt(-x^2 + 1) antideriv: -sqrt(-x^2 + 1)*arctan(x + sqrt(-x^2 + 1)) - 1/4*sqrt(3)*arctan(1/3*(2*x^2 - 1)*sqrt(3)) + 1/4*sqrt(3)*arctan((sqrt(3)*x - 1)/sqrt(-x^2 + 1)) + 1/4*sqrt(3)*arctan((sqrt(3)*x + 1)/sqrt(-x^2 + 1)) + 1/8*log(x^4 - x^2 + 1) - 1/2*arcsin(x) + 1/4*arctanh(sqrt(-x^2 + 1)*x) maxima : -sqrt(-x + 1)*sqrt(x + 1)*arctan(sqrt(-x + 1)*sqrt(x + 1) + x) - integrate(-x/((x - 1)*(x + 1) - x^2 - 2*x*e^(1/2*log(-x + 1) + 1/2*log(x + 1)) - 1), x)
$$ \int \! \frac{\arcsin\left(x\right)}{\sqrt{-x^{2} + 1} + 1} {dx} $$
[14] FAILED! Unevaluated! integrand: arcsin(x)/(sqrt(-x^2 + 1) + 1) antideriv: 1/2*arcsin(x)^2 + (sqrt(-x^2 + 1) - 1)*arcsin(x)/x - log(sqrt(-x^2 + 1) + 1) maxima : integrate(arcsin(x)/(sqrt(-x^2 + 1) + 1), x)
$$ \int \! \frac{\log\left(x + \sqrt{x^{2} + 1}\right)}{{\left(-x^{2} + 1\right)}^{\frac{3}{2}}} {dx} $$
[15] FAILED! Unevaluated! integrand: log(x + sqrt(x^2 + 1))/(-x^2 + 1)^(3/2) antideriv: x*log(x + sqrt(x^2 + 1))/sqrt(-x^2 + 1) - 1/2*arcsin(x^2) maxima : integrate(log(x + sqrt(x^2 + 1))/(-x^2 + 1)^(3/2), x)
$$ \int \! \frac{\arcsin\left(x\right)}{{\left(x^{2} + 1\right)}^{\frac{3}{2}}} {dx} $$
[16] diffs back: True integrand: arcsin(x)/(x^2 + 1)^(3/2) antideriv: x*arcsin(x)/sqrt(x^2 + 1) - 1/2*arcsin(x^2) maxima : x*arcsin(x)/sqrt(x^2 + 1) - 1/2*arcsin(x^2)
$$ \int \! \frac{\log\left(x + \sqrt{x^{2} - 1}\right)}{{\left(x^{2} + 1\right)}^{\frac{3}{2}}} {dx} $$
[17] FAILED! Unevaluated! integrand: log(x + sqrt(x^2 - 1))/(x^2 + 1)^(3/2) antideriv: x*log(x + sqrt(x^2 - 1))/sqrt(x^2 + 1) - 1/2*arccosh(x^2) maxima : integrate(log(x + sqrt(x^2 - 1))/(x^2 + 1)^(3/2), x)
$$ \int \! \frac{\log\left(x\right)}{\sqrt{x^{2} - 1} x^{2}} {dx} $$
[18] diffs back: True integrand: log(x)/(sqrt(x^2 - 1)*x^2) antideriv: sqrt(x^2 - 1)*log(x)/x + sqrt(x^2 - 1)/x - arctanh(x/sqrt(x^2 - 1)) maxima : sqrt(x^2 - 1)*log(x)/x + sqrt(x^2 - 1)/x - log(2*x + 2*sqrt(x^2 - 1))
$$ \int \! \frac{\sqrt{x^{3} + 1}}{x} {dx} $$
[19] diffs back: True integrand: sqrt(x^3 + 1)/x antideriv: 2/3*sqrt(x^3 + 1) - 2/3*arctanh(sqrt(x^3 + 1)) maxima : 2/3*sqrt(x^3 + 1) + 1/3*log(sqrt(x^3 + 1) - 1) - 1/3*log(sqrt(x^3 + 1) + 1)
$$ \int \! \frac{x \log\left(x + \sqrt{x^{2} - 1}\right)}{\sqrt{x^{2} - 1}} {dx} $$
[20] diffs back: True integrand: x*log(x + sqrt(x^2 - 1))/sqrt(x^2 - 1) antideriv: sqrt(x^2 - 1)*log(x + sqrt(x^2 - 1)) - x maxima : sqrt(x^2 - 1)*log(x + sqrt(x^2 - 1)) - x
$$ \int \! \frac{x^{3} \arcsin\left(x\right)}{\sqrt{-x^{4} + 1}} {dx} $$
[21] diffs back: True integrand: x^3*arcsin(x)/sqrt(-x^4 + 1) antideriv: 1/4*sqrt(x^2 + 1)*x - 1/2*sqrt(-x^4 + 1)*arcsin(x) + 1/4*arcsinh(x) maxima : 1/4*sqrt(x^2 + 1)*x - 1/2*sqrt(-x^4 + 1)*arcsin(x) + 1/4*arcsinh(x)
$$ \int \! \frac{x^{3} {\rm arcsec}\left(x\right)}{\sqrt{x^{4} - 1}} {dx} $$
[22] diffs back: False, WRONG RESULT! integrand: x^3*arcsec(x)/sqrt(x^4 - 1) antideriv: 1/2*sqrt(x^4 - 1)*arcsec(x) - 1/2*sqrt(x^4 - 1)/(sqrt(-1/x^2 + 1)*x) + 1/2*arctanh(sqrt(-1/x^2 + 1)*x/sqrt(x^4 - 1)) maxima : 1/2*sqrt(x^4 - 1)*arcsec(x) - 1/2*sqrt(x^2 + 1) + 1/2*arcsinh(1/abs(x))
$$ \int \! \frac{x \log\left(x + \sqrt{x^{2} + 1}\right) \arctan\left(x\right)}{\sqrt{x^{2} + 1}} {dx} $$
[23] diffs back: False integrand: x*log(x + sqrt(x^2 + 1))*arctan(x)/sqrt(x^2 + 1) antideriv: sqrt(x^2 + 1)*log(x + sqrt(x^2 + 1))*arctan(x) - x*arctan(x) - 1/2*log(x + sqrt(x^2 + 1))^2 + 1/2*log(x^2 + 1) maxima : (sqrt(x^2 + 1)*log(x + sqrt(x^2 + 1)) - x)*arctan(x) + 1/2*log(x + sqrt(x^2 + 1))^2 - log(x + sqrt(x^2 + 1))*arcsinh(x) + 1/2*log(x^2 + 1)
$$ \int \! \frac{x \log\left(\sqrt{-x^{2} + 1} + 1\right)}{\sqrt{-x^{2} + 1}} {dx} $$
[24] diffs back: True integrand: x*log(sqrt(-x^2 + 1) + 1)/sqrt(-x^2 + 1) antideriv: sqrt(1 - x^2) - (1 + sqrt(1 - x^2))*log(1 + sqrt(1 - x^2)) maxima : -(sqrt(-x^2 + 1) + 1)*log(sqrt(-x^2 + 1) + 1) + sqrt(-x^2 + 1) + 1
$$ \int \! \frac{x \log\left(x + \sqrt{x^{2} + 1}\right)}{\sqrt{x^{2} + 1}} {dx} $$
[25] diffs back: True integrand: x*log(x + sqrt(x^2 + 1))/sqrt(x^2 + 1) antideriv: sqrt(x^2 + 1)*log(x + sqrt(x^2 + 1)) - x maxima : sqrt(x^2 + 1)*log(x + sqrt(x^2 + 1)) - x
$$ \int \! \frac{x \log\left(x + \sqrt{-x^{2} + 1}\right)}{\sqrt{-x^{2} + 1}} {dx} $$
[26] diffs back: False, answer contains integral integrand: x*log(x + sqrt(-x^2 + 1))/sqrt(-x^2 + 1) antideriv: -sqrt(-x^2 + 1)*log(x + sqrt(-x^2 + 1)) + 1/2*sqrt(2)*arctanh(sqrt(2)*x) - 1/2*sqrt(2)*arctanh(sqrt(-x^2 + 1)*sqrt(2)) + sqrt(-x^2 + 1) maxima : (x^2 - 1)*log(sqrt(-x + 1)*sqrt(x + 1) + x)/(sqrt(-x + 1)*sqrt(x + 1)) + sqrt(-x^2 + 1) - log(2*sqrt(-x^2 + 1)/abs(x) + 2/abs(x)) - integrate(1/(sqrt(-x + 1)*sqrt(x + 1)*x + x^2), x)
$$ \int \! \frac{\log\left(x\right)}{\sqrt{-x^{2} + 1} x^{2}} {dx} $$
[27] diffs back: True integrand: log(x)/(sqrt(-x^2 + 1)*x^2) antideriv: -sqrt(-x^2 + 1)*log(x)/x - sqrt(-x^2 + 1)/x - arcsin(x) maxima : -sqrt(-x^2 + 1)*log(x)/x - sqrt(-x^2 + 1)/x - arcsin(x)
$$ \int \! \frac{x \arctan\left(x\right)}{\sqrt{x^{2} + 1}} {dx} $$
[28] diffs back: True integrand: x*arctan(x)/sqrt(x^2 + 1) antideriv: sqrt(x^2 + 1)*arctan(x) - arcsinh(x) maxima : sqrt(x^2 + 1)*arctan(x) - arcsinh(x)
$$ \int \!{\frac {\arctan \left( x \right) }{{x}^{2}\sqrt {-{x}^{2}+1}}}{dx} $$
[29] RuntimeError executing code in Maxima: sign: argument cannot be imaginary integrand: arctan(x)/(sqrt(-x^2 + 1)*x^2) antideriv: sqrt(2)*arctanh(1/2*sqrt(-x^2 + 1)*sqrt(2)) - sqrt(-x^2 + 1)*arctan(x)/x - arctanh(sqrt(-x^2 + 1))
$$ \int \!{\frac {x\arctan \left( x \right) }{\sqrt {-{x}^{2}+1}}}{dx} $$
[30] RuntimeError executing code in Maxima: sign: argument cannot be imaginary integrand: x*arctan(x)/sqrt(-x^2 + 1) antideriv: -sqrt(-x^2 + 1)*arctan(x) + sqrt(2)*arctan(sqrt(2)*x/sqrt(-x^2 + 1)) - arcsin(x)
$$ \int \! \frac{\arctan\left(x\right)}{\sqrt{x^{2} + 1} x^{2}} {dx} $$
[31] diffs back: False integrand: arctan(x)/(sqrt(x^2 + 1)*x^2) antideriv: -sqrt(x^2 + 1)*arctan(x)/x - arctanh(sqrt(x^2 + 1)) maxima : -sqrt(x^2 + 1)*arctan(x)/x - arcsinh(1/abs(x))
$$ \int \! \frac{\arcsin\left(x\right)}{\sqrt{-x^{2} + 1} x^{2}} {dx} $$
[32] diffs back: True integrand: arcsin(x)/(sqrt(-x^2 + 1)*x^2) antideriv: -sqrt(-x^2 + 1)*arcsin(x)/x + log(x) maxima : -sqrt(-x^2 + 1)*arcsin(x)/x + log(x)
$$ \int \! \frac{x \log\left(x\right)}{\sqrt{x^{2} - 1}} {dx} $$
[33] diffs back: False integrand: x*log(x)/sqrt(x^2 - 1) antideriv: sqrt(x^2 - 1)*log(x) - sqrt(x^2 - 1) + arctan(sqrt(x^2 - 1)) maxima : sqrt(x^2 - 1)*log(x) - sqrt(x^2 - 1) - arcsin(1/abs(x))
$$ \int \! \frac{\log\left(x\right)}{\sqrt{x^{2} + 1} x^{2}} {dx} $$
[34] diffs back: True integrand: log(x)/(sqrt(x^2 + 1)*x^2) antideriv: -sqrt(x^2 + 1)*log(x)/x - sqrt(x^2 + 1)/x + arcsinh(x) maxima : -sqrt(x^2 + 1)*log(x)/x - sqrt(x^2 + 1)/x + arcsinh(x)
$$ \int \! \frac{x {\rm arcsec}\left(x\right)}{\sqrt{x^{2} - 1}} {dx} $$
[35] diffs back: False, WRONG RESULT! integrand: x*arcsec(x)/sqrt(x^2 - 1) antideriv: -sqrt(-1/x^2 + 1)*x*log(x)/sqrt(x^2 - 1) + sqrt(x^2 - 1)*arcsec(x) maxima : sqrt(x^2 - 1)*arcsec(x) - log(x)
$$ \int \! \frac{x \log\left(x\right)}{\sqrt{x^{2} + 1}} {dx} $$
[36] diffs back: False integrand: x*log(x)/sqrt(x^2 + 1) antideriv: sqrt(x^2 + 1)*log(x) - sqrt(x^2 + 1) + arctanh(sqrt(x^2 + 1)) maxima : sqrt(x^2 + 1)*log(x) - sqrt(x^2 + 1) + arcsinh(1/abs(x))
$$ \int \! \frac{\sin\left(x\right)}{\sin\left(x\right)^{2} + 1} {dx} $$
[37] diffs back: True integrand: sin(x)/(sin(x)^2 + 1) antideriv: -1/2*sqrt(2)*arctanh(1/2*sqrt(2)*cos(x)) maxima : 1/4*sqrt(2)*log(-(sqrt(2) - cos(x))/(sqrt(2) + cos(x)))
$$ \int \! -\frac{x^{2} + 1}{{\left(x^{2} - 1\right)} \sqrt{x^{4} + 1}} {dx} $$
[38] FAILED! Unevaluated! integrand: -(x^2 + 1)/((x^2 - 1)*sqrt(x^4 + 1)) antideriv: 1/2*sqrt(2)*arctanh(sqrt(2)*x/sqrt(x^4 + 1)) maxima : -integrate((x^2 + 1)/((x^2 - 1)*sqrt(x^4 + 1)), x)
$$ \int \! -\frac{x^{2} - 1}{{\left(x^{2} + 1\right)} \sqrt{x^{4} + 1}} {dx} $$
[39] FAILED! Unevaluated! integrand: -(x^2 - 1)/((x^2 + 1)*sqrt(x^4 + 1)) antideriv: 1/2*sqrt(2)*arctan(sqrt(2)*x/sqrt(x^4 + 1)) maxima : -integrate((x^2 - 1)/((x^2 + 1)*sqrt(x^4 + 1)), x)
$$ \int \! \frac{\log\left(\sin\left(x\right)\right)}{\sin\left(x\right) + 1} {dx} $$
[40] diffs back: True integrand: log(sin(x))/(sin(x) + 1) antideriv: -log(sin(x))*cos(x)/(sin(x) + 1) - x - arctanh(cos(x)) maxima : -2*log(2*sin(x)/((cos(x) + 1)*(sin(x)^2/(cos(x) + 1)^2 + 1)))/(sin(x)/(cos(x) + 1) + 1) - log(sin(x)^2/(cos(x) + 1)^2 + 1) + 2*log(sin(x)/(cos(x) + 1)) - 2*arctan(sin(x)/(cos(x) + 1))
$$ \int \!\ln \left( \sin \left( x \right) \right) \sqrt {1+\sin \left( x \right) }{dx} $$
[41] FAILED, Computes longer than a minute integrand: sqrt(sin(x) + 1)*log(sin(x)) antideriv: -2*log(sin(x))*cos(x)/sqrt(sin(x) + 1) + 4*cos(x)/sqrt(sin(x) + 1) - 4*arctanh(cos(x)/sqrt(sin(x) + 1))
$$ \int \! \frac{\sec\left(x\right)}{\sqrt{\sec\left(x\right)^{4} - 1}} {dx} $$
[42] FAILED! Unevaluated! integrand: sec(x)/sqrt(sec(x)^4 - 1) antideriv: -1/2*sqrt(2)*arctanh(1/2*sqrt(sec(x)^4 - 1)*sqrt(2)*cos(x)*cot(x)) maxima : integrate(sec(x)/sqrt(sec(x)^4 - 1), x)
$$ \int \! \frac{\tan\left(x\right)}{\sqrt{\tan\left(x\right)^{4} + 1}} {dx} $$
[43] diffs back: False, WRONG RESULT! integrand: tan(x)/sqrt(tan(x)^4 + 1) antideriv: -1/4*sqrt(2)*arctanh(-1/2*(tan(x)^2 - 1)*sqrt(2)/sqrt(tan(x)^4 + 1)) maxima : -1/4*sqrt(2)*arcsinh(2*sin(x)^2 - 1)
$$ \int \! \frac{\sin\left(x\right)}{\sqrt{-\sin\left(x\right)^{6} + 1}} {dx} $$
[44] diffs back: False, WRONG RESULT! integrand: sin(x)/sqrt(-sin(x)^6 + 1) antideriv: 1/6*sqrt(3)*arctanh(1/2*(sin(x)^2 + 1)*sqrt(3)*cos(x)/sqrt(-sin(x)^6 + 1)) maxima : 1/6*sqrt(3)*arcsinh(-sqrt(3) + 2*sqrt(3)/cos(x)^2)
$$ \int \!\sqrt {\sqrt {\sec \left( x \right) +1}-\sqrt {\sec \left( x \right) -1}}{dx} $$
[45] RuntimeError executing code in Maxima: sign: argument cannot be imaginary integrand: sqrt(-sqrt(sec(x) - 1) + sqrt(sec(x) + 1)) antideriv: sqrt(sec(x) - 1)*sqrt(sec(x) + 1)*(sqrt(sqrt(2) - 1)*arctan(-1/2*sqrt(2*sqrt(2) - 2)*(sqrt(sec(x) - 1) - sqrt(sec(x) + 1) + sqrt(2))/sqrt(-sqrt(sec(x) - 1) + sqrt(sec(x) + 1))) + sqrt(sqrt(2) - 1)*arctanh(-sqrt(2*sqrt(2) + 2)*sqrt(-sqrt(sec(x) - 1) + sqrt(sec(x) + 1))/(sqrt(sec(x) - 1) - sqrt(sec(x) + 1) - sqrt(2))) - sqrt(sqrt(2) + 1)*arctan(-1/2*sqrt(2*sqrt(2) + 2)*(sqrt(sec(x) - 1) - sqrt(sec(x) + 1) + sqrt(2))/sqrt(-sqrt(sec(x) - 1) + sqrt(sec(x) + 1))) - sqrt(sqrt(2) + 1)*arctanh(-sqrt(2*sqrt(2) - 2)*sqrt(-sqrt(sec(x) - 1) + sqrt(sec(x) + 1))/(sqrt(sec(x) - 1) - sqrt(sec(x) + 1) - sqrt(2))))*sqrt(2)*cot(x)
$$ \int \! x \log\left(x^{2} + 1\right) \arctan\left(x\right)^{2} {dx} $$
[46] diffs back: True integrand: x*log(x^2 + 1)*arctan(x)^2 antideriv: -(log(x^2 + 1) - 3)*x*arctan(x) + 1/2*((x^2 + 1)*log(x^2 + 1) - x^2 - 3)*arctan(x)^2 + 1/4*(log(x^2 + 1) - 6)*log(x^2 + 1) maxima : 1/2*((x^2 + 1)*log(x^2 + 1) - x^2 - 1)*arctan(x)^2 - (x*log(x^2 + 1) - 3*x + 2*arctan(x))*arctan(x) + 1/4*log(x^2 + 1)^2 + arctan(x)^2 - 3/2*log(x^2 + 1)
$$ \int \! \arctan\left(\sqrt{x^{2} + 1} x\right) {dx} $$
[47] diffs back: True, answer contains integral integrand: arctan(sqrt(x^2 + 1)*x) antideriv: x*arctan(sqrt(x^2 + 1)*x) + 1/2*sqrt(3)*arctanh(sqrt(x^2 + 1)*sqrt(3)/(x^2 + 2)) + 1/2*arctan(sqrt(x^2 + 1)/x^2) maxima : x*arctan(sqrt(x^2 + 1)*x) - integrate((2*x^3 + x)*e^(1/2*log(x^2 + 1))/((x^2 + 1)*(x^4 + x^2) + x^2 + 1), x)
$$ \int \! \arctan\left(\sqrt{x + 1} - \sqrt{x}\right) {dx} $$
[48] diffs back: True integrand: arctan(sqrt(x + 1) - sqrt(x)) antideriv: (x + 1)*arctan(sqrt(x + 1) - sqrt(x)) + 1/2*sqrt(x) maxima : x*arctan(sqrt(x + 1) - sqrt(x)) + 1/2*e^(1/2*log(x)) - 1/2*arctan(e^(1/2*log(x)))
$$ \int \! \arcsin\left(\frac{x}{\sqrt{-x^{2} + 1}}\right) {dx} $$
[49] diffs back: False integrand: arcsin(x/sqrt(-x^2 + 1)) antideriv: x*arcsin(x/sqrt(-x^2 + 1)) + arctan(sqrt(-2*x^2 + 1)) maxima : x*arcsin(x/sqrt(-x^2 + 1)) - 1/2*(-2*I*x^2 + I)/sqrt(2*x^2 - 1) - 1/2*I*sqrt(2*x^2 - 1) - 1/2*I*log(sqrt(2*x^2 - 1) - 1) + 1/2*I*log(sqrt(2*x^2 - 1) + 1)
$$ \int \! \arctan\left(\sqrt{-x^{2} + 1} x\right) {dx} $$
[50] diffs back: True, answer contains integral integrand: arctan(sqrt(-x^2 + 1)*x) antideriv: x*arctan(sqrt(-x^2 + 1)*x) + sqrt(1/2*sqrt(5) - 1/2)*arctanh(sqrt(-x^2 + 1)*sqrt(1/2*sqrt(5) - 1/2)) - sqrt(1/2*sqrt(5) + 1/2)*arctan(sqrt(-x^2 + 1)*sqrt(1/2*sqrt(5) + 1/2)) maxima : x*arctan(sqrt(-x + 1)*sqrt(x + 1)*x) - integrate(-(2*x^3 - x)*e^(1/2*log(-x + 1) + 1/2*log(x + 1))/((x - 1)*(x + 1)*(x^4 - x^2) - x^2 + 1), x)
S A G E / M A X I M A - Timofeev Chapter 1
$$ \int \! -\frac{1}{b^{2} x^{2} - a^{2}} {dx} $$
[1] diffs back : True : Solution is too complex! integrand : -1/(b^2*x^2 - a^2) antideriv : arctanh(b*x/a)/(a*b) maxima : -1/2*log(b*x - a)/(a*b) + 1/2*log(b*x + a)/(a*b)
$$ \int \! \frac{1}{b^{2} x^{2} + a^{2}} {dx} $$
[2] diffs back : True integrand : 1/(b^2*x^2 + a^2) antideriv : arctan(b*x/a)/(a*b) maxima : arctan(b*x/a)/(a*b)
$$ \int \! \sec\left(2 \, a x\right) {dx} $$
[3] diffs back : True integrand : sec(2*a*x) antideriv : 1/2*log(tan(1/4*pi + a*x))/a maxima : 1/2*log(tan(2*a*x) + sec(2*a*x))/a
$$ \int \! \frac{1}{4 \, \sin\left(\frac{1}{3} \, x\right)} {dx} $$
[4] diffs back : True : Solution is too complex! integrand : 1/4/sin(1/3*x) antideriv : 3/4*log(tan(1/6*x)) maxima : 3/8*log(cos(1/3*x) - 1) - 3/8*log(cos(1/3*x) + 1)
$$ \int \! \frac{1}{\cos\left(\frac{3}{4} \, \pi - 2 \, x\right)} {dx} $$
[5] diffs back : True : Solution is too complex! integrand : 1/cos(3/4*pi - 2*x) antideriv : 1/2*log(tan(1/8*pi - x)) maxima : -1/4*log(sin(-3/4*pi + 2*x) - 1) + 1/4*log(sin(-3/4*pi + 2*x) + 1)
$$ \int \! \tan\left(x\right) \sec\left(x\right) {dx} $$
[6] diffs back : True integrand : tan(x)*sec(x) antideriv : sec(x) maxima : 1/cos(x)
$$ \int \! \csc\left(x\right) \cot\left(x\right) {dx} $$
[7] diffs back : True integrand : csc(x)*cot(x) antideriv : -csc(x) maxima : -1/sin(x)
$$ \int \! \frac{\tan\left(x\right)}{\sin\left(2 \, x\right)} {dx} $$
[8] diffs back : True : Solution is too complex! integrand : tan(x)/sin(2*x) antideriv : 1/2*tan(x) maxima : sin(2*x)/(sin(2*x)^2 + cos(2*x)^2 + 2*cos(2*x) + 1)
$$ \int \! \frac{1}{\cos\left(x\right) + 1} {dx} $$
[9] diffs back : True integrand : 1/(cos(x) + 1) antideriv : tan(1/2*x) maxima : sin(x)/(cos(x) + 1)
$$ \int \! -\frac{1}{\cos\left(x\right) - 1} {dx} $$
[10] diffs back : True integrand : -1/(cos(x) - 1) antideriv : -cot(1/2*x) maxima : -(cos(x) + 1)/sin(x)
$$ \int \! -\frac{\sin\left(x\right)}{b \cos\left(x\right) - a} {dx} $$
[11] diffs back : True integrand : -sin(x)/(b*cos(x) - a) antideriv : log(-b*cos(x) + a)/b maxima : log(b*cos(x) - a)/b
$$ \int \! \frac{\cos\left(x\right)}{b^{2} \sin\left(x\right)^{2} + a^{2}} {dx} $$
[12] diffs back : True integrand : cos(x)/(b^2*sin(x)^2 + a^2) antideriv : arctan(b*sin(x)/a)/(a*b) maxima : arctan(b*sin(x)/a)/(a*b)
$$ \int \! -\frac{\cos\left(x\right)}{b^{2} \sin\left(x\right)^{2} - a^{2}} {dx} $$
[13] diffs back : True : Solution is too complex! integrand : -cos(x)/(b^2*sin(x)^2 - a^2) antideriv : arctanh(b*sin(x)/a)/(a*b) maxima : -1/2*log(b*sin(x) - a)/(a*b) + 1/2*log(b*sin(x) + a)/(a*b)
$$ \int \! \frac{\sin\left(2 \, x\right)}{b^{2} \sin\left(x\right)^{2} + a^{2}} {dx} $$
[14] diffs back : True integrand : sin(2*x)/(b^2*sin(x)^2 + a^2) antideriv : log(b^2*sin(x)^2 + a^2)/b^2 maxima : log(b^2*sin(x)^2 + a^2)/b^2
$$ \int \! \frac{\sin\left(2 \, x\right)}{b^{2} \sin\left(x\right)^{2} - a^{2}} {dx} $$
[15] diffs back : True integrand : sin(2*x)/(b^2*sin(x)^2 - a^2) antideriv : log(-b^2*sin(x)^2 + a^2)/b^2 maxima : log(b^2*sin(x)^2 - a^2)/b^2
$$ \int \! \frac{\sin\left(2 \, x\right)}{b^{2} \cos\left(x\right)^{2} + a^{2}} {dx} $$
[16] diffs back : True integrand : sin(2*x)/(b^2*cos(x)^2 + a^2) antideriv : -log(b^2*cos(x)^2 + a^2)/b^2 maxima : -log(b^2*cos(x)^2 + a^2)/b^2
$$ \int \! \frac{\sin\left(2 \, x\right)}{b^{2} \cos\left(x\right)^{2} - a^{2}} {dx} $$
[17] diffs back : True integrand : sin(2*x)/(b^2*cos(x)^2 - a^2) antideriv : -log(-b^2*cos(x)^2 + a^2)/b^2 maxima : -log(b^2*cos(x)^2 - a^2)/b^2
$$ \int \! -\frac{1}{\cos\left(x\right)^{2} - 4} {dx} $$
[18] diffs back : True integrand : -1/(cos(x)^2 - 4) antideriv : 1/6*(x + arctan(sin(x)*cos(x)/(2*sqrt(3) - cos(x)^2 + 4)))*sqrt(3) maxima : 1/6*sqrt(3)*arctan(2/3*sqrt(3)*tan(x))
$$ \int \! \frac{e^{x}}{e^{\left(2 \, x\right)} - 1} {dx} $$
[19] diffs back : True : Solution is too complex! integrand : e^x/(e^(2*x) - 1) antideriv : -arctanh(e^x) maxima : 1/2*log(e^x - 1) - 1/2*log(e^x + 1)
$$ \int \! \frac{1}{x \log\left(x\right)} {dx} $$
[20] diffs back : True integrand : 1/(x*log(x)) antideriv : log(log(x)) maxima : log(log(x))
$$ \int \! \frac{1}{{\left(\log\left(x\right)^{2} + 1\right)} x} {dx} $$
[21] diffs back : True integrand : 1/((log(x)^2 + 1)*x) antideriv : arctan(log(x)) maxima : arctan(log(x))
$$ \int \! -\frac{1}{{\left(\log\left(x\right) - 1\right)} x} {dx} $$
[22] diffs back : True integrand : -1/((log(x) - 1)*x) antideriv : -log(-log(x) + 1) maxima : -log(log(x) - 1)
$$ \int \! \frac{1}{{\left(\log\left(\frac{x}{a}\right) + 1\right)} x} {dx} $$
[23] diffs back : True integrand : 1/((log(x/a) + 1)*x) antideriv : log(log(x/a) + 1) maxima : log(log(x/a) + 1)
$$ \int \! \frac{{\left(x - \sqrt{x} + 1\right)}^{2}}{x^{2}} {dx} $$
[24] diffs back : True integrand : (x - sqrt(x) + 1)^2/x^2 antideriv : x - 4*sqrt(x) + 4/sqrt(x) - 1/x + 3*log(x) maxima : x - 4*sqrt(x) + (4*sqrt(x) - 1)/x + 3*log(x)
$$ \int \! -\frac{{\left(x^{\frac{2}{3}} - 2\right)} {\left(x + \sqrt{x}\right)}}{x^{\frac{3}{2}}} {dx} $$
[25] diffs back : True integrand : -(x^(2/3) - 2)*(x + sqrt(x))/x^(3/2) antideriv : -6/7*x^(7/6) - 3/2*x^(2/3) + 4*sqrt(x) + 2*log(x) maxima : -6/7*x^(7/6) - 3/2*x^(2/3) + 4*sqrt(x) + 2*log(x)
$$ \int \! \frac{2 \, x - 1}{2 \, x + 3} {dx} $$
[26] diffs back : True integrand : (2*x - 1)/(2*x + 3) antideriv : x - 2*log(2*x + 3) maxima : x - 2*log(2*x + 3)
$$ \int \! \frac{2 \, x - 5}{3 \, x^{2} - 2} {dx} $$
[27] diffs back : True integrand : (2*x - 5)/(3*x^2 - 2) antideriv : 5/6*sqrt(6)*arctanh(1/2*sqrt(6)*x) + 1/3*log(-3*x^2 + 2) maxima : -5/12*sqrt(6)*log((3*x - sqrt(6))/(3*x + sqrt(6))) + 1/3*log(3*x^2 - 2)
$$ \int \! \frac{2 \, x - 5}{3 \, x^{2} + 2} {dx} $$
[28] diffs back : True integrand : (2*x - 5)/(3*x^2 + 2) antideriv : -5/6*sqrt(6)*arctan(1/2*sqrt(6)*x) + 1/3*log(3*x^2 + 2) maxima : -5/6*sqrt(6)*arctan(1/2*sqrt(6)*x) + 1/3*log(3*x^2 + 2)
$$ \int \! \sin\left(\frac{1}{4} \, x\right) \sin\left(x\right) {dx} $$
[29] diffs back : False integrand : sin(1/4*x)*sin(x) antideriv : 2/3*sin(3/4*x) - 2/5*sin(5/4*x) maxima : 2/3*sin(3/4*x) - 2/5*sin(5/4*x)
$$ \int \! \cos\left(3 \, x\right) \cos\left(4 \, x\right) {dx} $$
[30] diffs back : True integrand : cos(3*x)*cos(4*x) antideriv : 1/14*sin(7*x) + 1/2*sin(x) maxima : 1/14*sin(7*x) + 1/2*sin(x)
$$ \int \! \tan\left(a - x\right) \tan\left(x\right) {dx} $$
[31] diffs back : True : Solution is too complex! integrand : tan(a - x)*tan(x) antideriv : -x + log(tan(a)*tan(x) + 1)/tan(a) maxima : ((sin(2*a)^2 + cos(2*a)^2 - 1)*arctan2(sin(2*a) + sin(2*x), cos(2*a) + cos(2*x)) - (sin(2*a)^2 + cos(2*a)^2 - 1)*arctan2(sin(2*x), cos(2*x) + 1) + (sin(2*a)^2 + cos(2*a)^2 - 2*cos(2*a) + 1)*x + log(sin(2*x)^2 + cos(2*x)^2 + 2*cos(2*x) + 1)*sin(2*a) - log(sin(2*a)^2 + 2*sin(2*a)*sin(2*x) + sin(2*x)^2 + cos(2*a)^2 + 2*cos(2*a)*cos(2*x) + cos(2*x)^2)*sin(2*a))/(sin(2*a)^2 + cos(2*a)^2 - 2*cos(2*a) + 1)
$$ \int \! \sin\left(x\right)^{2} {dx} $$
[32] diffs back : True integrand : sin(x)^2 antideriv : -1/2*sin(x)*cos(x) + 1/2*x maxima : 1/2*x - 1/4*sin(2*x)
$$ \int \! \cos\left(x\right)^{2} {dx} $$
[33] diffs back : True integrand : cos(x)^2 antideriv : 1/2*sin(x)*cos(x) + 1/2*x maxima : 1/2*x + 1/4*sin(2*x)
$$ \int \! \sin\left(x\right) \cos\left(x\right)^{3} {dx} $$
[34] diffs back : True integrand : sin(x)*cos(x)^3 antideriv : -1/4*cos(x)^4 maxima : -1/4*cos(x)^4
$$ \int \! \frac{\cos\left(x\right)^{3}}{\sin\left(x\right)^{4}} {dx} $$
[35] diffs back : True integrand : cos(x)^3/sin(x)^4 antideriv : 1/sin(x) - 1/3/sin(x)^3 maxima : 1/3*(3*sin(x)^2 - 1)/sin(x)^3
$$ \int \! \frac{1}{\sin\left(x\right)^{2} \cos\left(x\right)^{2}} {dx} $$
[36] diffs back : True integrand : 1/(sin(x)^2*cos(x)^2) antideriv : tan(x) - cot(x) maxima : -1/tan(x) + tan(x)
$$ \int \! \cot\left(\frac{3}{4} \, x\right)^{2} {dx} $$
[37] diffs back : True integrand : cot(3/4*x)^2 antideriv : -x - 4/3*cot(3/4*x) maxima : -x - 4/3/tan(3/4*x)
$$ \int \! {\left(\tan\left(2 \, x\right) + 1\right)}^{2} {dx} $$
[38] diffs back : True integrand : (tan(2*x) + 1)^2 antideriv : -log(cos(2*x)) + 1/2*tan(2*x) maxima : log(sec(2*x)) + 1/2*tan(2*x)
$$ \int \! {\left(\tan\left(x\right) - \cot\left(x\right)\right)}^{2} {dx} $$
[39] diffs back : True integrand : (tan(x) - cot(x))^2 antideriv : -4*x + tan(x) - cot(x) maxima : -4*x - 1/tan(x) + tan(x)
$$ \int \! {\left(\tan\left(x\right) - \sec\left(x\right)\right)}^{2} {dx} $$
[40] diffs back : True integrand : (tan(x) - sec(x))^2 antideriv : -x + 2*tan(-1/4*pi + 1/2*x) maxima : -x - 2/cos(x) + 2*tan(x)
$$ \int \! \frac{\sin\left(x\right)}{\sin\left(x\right) + 1} {dx} $$
[41] diffs back : True : Solution is too complex! integrand : sin(x)/(sin(x) + 1) antideriv : x + tan(1/4*pi - 1/2*x) maxima : 2/(sin(x)/(cos(x) + 1) + 1) + 2*arctan(sin(x)/(cos(x) + 1))
$$ \int \! -\frac{\cos\left(x\right)}{\cos\left(x\right) - 1} {dx} $$
[42] diffs back : True : Solution is too complex! integrand : -cos(x)/(cos(x) - 1) antideriv : -x - cot(1/2*x) maxima : -(cos(x) + 1)/sin(x) - 2*arctan(sin(x)/(cos(x) + 1))
$$ \int \! {\left(e^{\left(\frac{1}{2} \, x\right)} - 1\right)}^{3} e^{\left(-\frac{1}{2} \, x\right)} {dx} $$
[43] diffs back : True integrand : (e^(1/2*x) - 1)^3*e^(-1/2*x) antideriv : 3*x + 2*e^(-1/2*x) - 6*e^(1/2*x) + e^x maxima : 3*x + 2*e^(-1/2*x) - 6*e^(1/2*x) + e^x
$$ \int \! \frac{1}{x^{2} - 6 \, x + 5} {dx} $$
[44] diffs back : True integrand : 1/(x^2 - 6*x + 5) antideriv : 1/4*log((x - 5)/(x - 1)) maxima : 1/4*log(x - 5) - 1/4*log(x - 1)
$$ \int \! \frac{x^{2}}{x^{6} - 6 \, x^{3} + 13} {dx} $$
[45] diffs back : True integrand : x^2/(x^6 - 6*x^3 + 13) antideriv : 1/6*arctan(1/2*x^3 - 3/2) maxima : 1/6*arctan(1/2*x^3 - 3/2)
$$ \int \! \frac{x + 2}{x^{2} - 4 \, x - 1} {dx} $$
[46] diffs back : True integrand : (x + 2)/(x^2 - 4*x - 1) antideriv : 4/5*sqrt(5)*arctanh(-1/5*(x - 2)*sqrt(5)) + 1/2*log(-x^2 + 4*x + 1) maxima : 2/5*sqrt(5)*log((x - sqrt(5) - 2)/(x + sqrt(5) - 2)) + 1/2*log(x^2 - 4*x - 1)
$$ \int \! \frac{1}{{\left(x + 1\right)}^{\frac{1}{3}} + 1} {dx} $$
[47] diffs back : True integrand : 1/((x + 1)^(1/3) + 1) antideriv : 3/2*(x + 1)^(2/3) - 3*(x + 1)^(1/3) + 3*log((x + 1)^(1/3) + 1) maxima : 3/2*(x + 1)^(2/3) - 3*(x + 1)^(1/3) + 3*log((x + 1)^(1/3) + 1)
$$ \int \! \frac{1}{{\left(a x + b\right)} \sqrt{x}} {dx} $$
[48] FAILED, needs additional assumptions, assume(a*b>0) integrand: 1/((a*x + b)*sqrt(x)) antideriv: 2*arctan(sqrt(a)*sqrt(x)/sqrt(b))/(sqrt(a)*sqrt(b))
$$ \int \! \sqrt{x^{2} + 1} x^{3} {dx} $$
[49] diffs back : True integrand : sqrt(x^2 + 1)*x^3 antideriv : 1/15*sqrt(x^2 + 1)*(3*x^4 + x^2 - 2) maxima : 1/5*(x^2 + 1)^(3/2)*x^2 - 2/15*(x^2 + 1)^(3/2)
$$ \int \! \frac{x}{\sqrt{a^{4} - x^{4}}} {dx} $$
[50] diffs back : True integrand : x/sqrt(a^4 - x^4) antideriv : 1/2*arctan(x^2/sqrt(a^4 - x^4)) maxima : -1/2*arctan(sqrt(a^4 - x^4)/x^2)
$$ \int \! \frac{1}{\sqrt{-a^{2} + x^{2}} x} {dx} $$
[51] FAILED, needs additional assumptions, assume(a>0) integrand: 1/(sqrt(-a^2 + x^2)*x) antideriv: arctan(sqrt(-a^2 + x^2)/a)/a
$$ \int \! \frac{1}{\sqrt{a^{2} - x^{2}} x} {dx} $$
[52] FAILED, needs additional assumptions, assume(a>0) integrand: 1/(sqrt(a^2 - x^2)*x) antideriv: -arctanh(sqrt(a^2 - x^2)/a)/a
$$ \int \! \frac{1}{\sqrt{a^{2} + x^{2}} x} {dx} $$
[53] FAILED, needs additional assumptions, assume(a>0) integrand: 1/(sqrt(a^2 + x^2)*x) antideriv: -arctanh(a/sqrt(a^2 + x^2))/a
$$ \int \! \frac{1}{\sqrt{-x^{2} + x + 2}} {dx} $$
[54] diffs back : True integrand : 1/sqrt(-x^2 + x + 2) antideriv : arcsin(2/3*x - 1/3) maxima : -arcsin(-2/3*x + 1/3)
$$ \int \! \frac{1}{\sqrt{3 \, x^{2} - 4 \, x + 5}} {dx} $$
[55] diffs back : True integrand : 1/sqrt(3*x^2 - 4*x + 5) antideriv : 1/3*sqrt(3)*arcsinh(1/11*(3*x - 2)*sqrt(11)) maxima : 1/3*sqrt(3)*arcsinh(1/11*(3*x - 2)*sqrt(11))
$$ \int \! \frac{1}{\sqrt{-x^{2} + x}} {dx} $$
[56] diffs back : True integrand : 1/sqrt(-x^2 + x) antideriv : arcsin(2*x - 1) maxima : arcsin(2*x - 1)
$$ \int \! \frac{2 \, x + 1}{\sqrt{-x^{2} + x + 2}} {dx} $$
[57] diffs back : True integrand : (2*x + 1)/sqrt(-x^2 + x + 2) antideriv : -2*sqrt(-x^2 + x + 2) + 2*arcsin(2/3*x - 1/3) maxima : -2*sqrt(-x^2 + x + 2) - 2*arcsin(-2/3*x + 1/3)
$$ \int \! \frac{1}{\sqrt{-x^{2} + x + 2} x} {dx} $$
[58] diffs back : False integrand : 1/(sqrt(-x^2 + x + 2)*x) antideriv : -1/2*sqrt(2)*arctanh(2*sqrt(-x^2 + x + 2)*sqrt(2)/(x + 4)) maxima : -1/2*sqrt(2)*log(2*sqrt(-x^2 + x + 2)*sqrt(2)/abs(x) + 4/abs(x) + 1)
$$ \int \! \frac{1}{{\left(x - 2\right)} \sqrt{-x^{2} + x + 2}} {dx} $$
[59] diffs back : True integrand : 1/((x - 2)*sqrt(-x^2 + x + 2)) antideriv : 2/3*sqrt(-x^2 + x + 2)/(x - 2) maxima : 2/3*sqrt(-x^2 + x + 2)/(x - 2)
$$ \int \! -\frac{3 \, \sin\left(x\right) + 2}{{\left(\cos\left(x\right) - 1\right)} \sin\left(x\right)} {dx} $$
[60] diffs back : True integrand : -(3*sin(x) + 2)/((cos(x) - 1)*sin(x)) antideriv : (3*sin(x) + 1)/(cos(x) - 1) - arctanh(cos(x)) maxima : -1/2*(cos(x) + 1)^2/sin(x)^2 - 3*(cos(x) + 1)/sin(x) + log(sin(x)/(cos(x) + 1))
$$ \int \! \frac{1}{3 \, \cos\left(x\right)^{2} + 2} {dx} $$
[61] diffs back : True integrand : 1/(3*cos(x)^2 + 2) antideriv : 1/10*(x - arctan(3*sin(x)*cos(x)/(sqrt(10) + 3*cos(x)^2 + 2)))*sqrt(10) maxima : 1/10*sqrt(10)*arctan(1/5*sqrt(10)*tan(x))
$$ \int \! -\frac{\tan\left(x\right) - 1}{\sin\left(2 \, x\right)} {dx} $$
[62] diffs back : True : Solution is too complex! integrand : -(tan(x) - 1)/sin(2*x) antideriv : 1/2*log(tan(x)) - 1/2*tan(x) maxima : -sin(2*x)/(sin(2*x)^2 + cos(2*x)^2 + 2*cos(2*x) + 1) + 1/4*log(cos(2*x) - 1) - 1/4*log(cos(2*x) + 1)
$$ \int \! -\frac{\tan\left(x\right)^{2} + 1}{\tan\left(x\right)^{2} - 1} {dx} $$
[63] diffs back : True integrand : -(tan(x)^2 + 1)/(tan(x)^2 - 1) antideriv : 1/2*log(-(tan(x) + 1)/(tan(x) - 1)) maxima : -1/2*log(tan(x) - 1) + 1/2*log(tan(x) + 1)
$$ \int \! {\left(a^{2} - 4 \, \cos\left(x\right)^{2}\right)}^{\frac{3}{4}} \sin\left(2 \, x\right) {dx} $$
[64] diffs back : True integrand : (a^2 - 4*cos(x)^2)^(3/4)*sin(2*x) antideriv : 1/7*(a^2 - 4*cos(x)^2)^(7/4) maxima : 1/7*(a^2 - 4*cos(x)^2)^(7/4)
$$ \int \! \frac{\sin\left(2 \, x\right)}{{\left(a^{2} - 4 \, \sin\left(x\right)^{2}\right)}^{\frac{1}{3}}} {dx} $$
[65] diffs back : True integrand : sin(2*x)/(a^2 - 4*sin(x)^2)^(1/3) antideriv : -3/8*(a^2 - 4*sin(x)^2)^(2/3) maxima : -3/8*(a^2 - 4*sin(x)^2)^(2/3)
$$ \int \! \frac{1}{\sqrt{a^{2 \, x} - 1}} {dx} $$
[66] diffs back : True integrand : 1/sqrt(a^(2*x) - 1) antideriv : arctan(sqrt(a^(2*x) - 1))/log(a) maxima : arctan(sqrt(a^(2*x) - 1))/log(a)
$$ \int \! \frac{e^{\left(\frac{1}{2} \, x\right)}}{\sqrt{e^{x} - 1}} {dx} $$
[67] diffs back : True integrand : e^(1/2*x)/sqrt(e^x - 1) antideriv : 2*log(sqrt(e^x - 1) + e^(1/2*x)) maxima : 2*log(2*sqrt(e^x - 1) + 2*e^(1/2*x))
$$ \int \! \frac{\arctan\left(x\right)^{n}}{x^{2} + 1} {dx} $$
[68] FAILED, needs additional assumptions, assume(n+1>0) integrand: arctan(x)^n/(x^2 + 1) antideriv: arctan(x)^(n + 1)/(n + 1)
$$ \int \! \frac{\arcsin\left(\frac{x}{a}\right)^{\frac{3}{2}}}{\sqrt{a^{2} - x^{2}}} {dx} $$
[69] FAILED, computes more than 1 minute integrand: arcsin(x/a)^(3/2)/sqrt(a^2 - x^2) antideriv: 2/5*sqrt(-x^2/a^2 + 1)*a*arcsin(x/a)^(5/2)/sqrt(a^2 - x^2)
$$ \int \! \frac{1}{\sqrt{-x^{2} + 1} \arccos\left(x\right)^{3}} {dx} $$
[70] diffs back : True integrand : 1/(sqrt(-x^2 + 1)*arccos(x)^3) antideriv : 1/2/arccos(x)^2 maxima : 1/2/arccos(x)^2
$$ \int \! x \log\left(x\right)^{2} {dx} $$
[71] diffs back : True integrand : x*log(x)^2 antideriv : 1/4*(2*log(x)^2 - 2*log(x) + 1)*x^2 maxima : 1/4*(2*log(x)^2 - 2*log(x) + 1)*x^2
$$ \int \! \frac{\log\left(x\right)}{x^{5}} {dx} $$
[72] diffs back : True integrand : log(x)/x^5 antideriv : -1/16*(4*log(x) + 1)/x^4 maxima : -1/4*log(x)/x^4 - 1/16/x^4
$$ \int \! x^{2} \log\left(\frac{x - 1}{x}\right) {dx} $$
[73] diffs back : True integrand : x^2*log((x - 1)/x) antideriv : 1/3*x^3*log((x - 1)/x) - 1/6*(x + 2)*x - 1/3*log(x - 1) maxima : 1/3*x^3*log((x - 1)/x) - 1/6*x^2 - 1/3*x - 1/3*log(x - 1)
$$ \int \! \cos\left(x\right)^{5} {dx} $$
[74] diffs back : True integrand : cos(x)^5 antideriv : 1/15*(3*cos(x)^4 + 4*cos(x)^2 + 8)*sin(x) maxima : 1/5*sin(x)^5 - 2/3*sin(x)^3 + sin(x)
$$ \int \! \sin\left(x\right)^{2} \cos\left(x\right)^{4} {dx} $$
[75] diffs back : True integrand : sin(x)^2*cos(x)^4 antideriv : 1/6*sin(x)^3*cos(x)^3 + 1/8*sin(x)^3*cos(x) - 1/16*sin(x)*cos(x) + 1/16*x maxima : 1/48*sin(2*x)^3 + 1/16*x - 1/64*sin(4*x)
$$ \int \! \frac{1}{\sin\left(x\right)^{5}} {dx} $$
[76] diffs back : True integrand : sin(x)^(-5) antideriv : -3/8*cos(x)/sin(x)^2 - 1/4*cos(x)/sin(x)^4 - 3/8*arctanh(cos(x)) maxima : 1/8*(3*cos(x)^3 - 5*cos(x))/(cos(x)^4 - 2*cos(x)^2 + 1) + 3/16*log(cos(x) - 1) - 3/16*log(cos(x) + 1)
$$ \int \! e^{\left(-x\right)} \sin\left(x\right) {dx} $$
[77] diffs back : True integrand : e^(-x)*sin(x) antideriv : -1/2*(sin(x) + cos(x))*e^(-x) maxima : -1/2*(sin(x) + cos(x))*e^(-x)
$$ \int \! e^{\left(2 \, x\right)} \sin\left(3 \, x\right) {dx} $$
[78] diffs back : True integrand : e^(2*x)*sin(3*x) antideriv : 1/13*(2*sin(3*x) - 3*cos(3*x))*e^(2*x) maxima : 1/13*(2*sin(3*x) - 3*cos(3*x))*e^(2*x)
$$ \int \! a^{x} \cos\left(x\right) {dx} $$
[79] diffs back : True integrand : a^x*cos(x) antideriv : (log(a)*cos(x) + sin(x))*a^x/(log(a)^2 + 1) maxima : (e^(x*log(a))*log(a)*cos(x) + e^(x*log(a))*sin(x))/(log(a)^2 + 1)
$$ \int \! \cos\left(\log\left(x\right)\right) {dx} $$
[80] diffs back : True integrand : cos(log(x)) antideriv : 1/2*(sin(log(x)) + cos(log(x)))*x maxima : 1/2*(sin(log(x)) + cos(log(x)))*x
$$ \int \! \log\left(\cos\left(x\right)\right) \sec\left(x\right)^{2} {dx} $$
[81] diffs back : True : Solution is too complex! integrand : log(cos(x))*sec(x)^2 antideriv : log(cos(x))*tan(x) - x + tan(x) maxima : -2*log(-(sin(x)^2/(cos(x) + 1)^2 - 1)/(sin(x)^2/(cos(x) + 1)^2 + 1))*sin(x)/((cos(x) + 1)*(sin(x)^2/(cos(x) + 1)^2 - 1)) - 2*sin(x)/((cos(x) + 1)*(sin(x)^2/(cos(x) + 1)^2 - 1)) - 2*arctan(sin(x)/(cos(x) + 1))
$$ \int \! x \tan\left(x\right)^{2} {dx} $$
[82] diffs back : True : Solution is too complex! integrand : x*tan(x)^2 antideriv : -1/2*x^2 + x*tan(x) + log(cos(x)) maxima : -1/2*(x^2*sin(2*x)^2 + x^2*cos(2*x)^2 + 2*x^2*cos(2*x) - (sin(2*x)^2 + cos(2*x)^2 + 2*cos(2*x) + 1)*log(sin(2*x)^2 + cos(2*x)^2 + 2*cos(2*x) + 1) + x^2 - 4*x*sin(2*x))/(sin(2*x)^2 + cos(2*x)^2 + 2*cos(2*x) + 1)
$$ \int \! \frac{\arcsin\left(x\right)}{x^{2}} {dx} $$
[83] diffs back : False integrand : arcsin(x)/x^2 antideriv : -arcsin(x)/x - arctanh(sqrt(-x^2 + 1)) maxima : -arcsin(x)/x - log(2*sqrt(-x^2 + 1)/abs(x) + 2/abs(x))
$$ \int \! \arcsin\left(x\right)^{2} {dx} $$
[84] diffs back : True integrand : arcsin(x)^2 antideriv : x*arcsin(x)^2 + 2*sqrt(-x^2 + 1)*arcsin(x) - 2*x maxima : x*arcsin(x)^2 + 2*sqrt(-x^2 + 1)*arcsin(x) - 2*x
$$ \int \! \frac{x^{2} \arctan\left(x\right)}{x^{2} + 1} {dx} $$
[85] diffs back : True integrand : x^2*arctan(x)/(x^2 + 1) antideriv : x*arctan(x) - 1/2*arctan(x)^2 - 1/2*log(x^2 + 1) maxima : (x - arctan(x))*arctan(x) + 1/2*arctan(x)^2 - 1/2*log(x^2 + 1)
$$ \int \! \arccos\left(\sqrt{\frac{x}{x + 1}}\right) {dx} $$
[86] diffs back : True : Solution is too complex! integrand : arccos(sqrt(x/(x + 1))) antideriv : (x + 1)*(sqrt(x/(x + 1))*sqrt(1/(x + 1)) + arccos(sqrt(x/(x + 1)))) maxima : -arccos(sqrt(x/(x + 1)))/(x/(x + 1) - 1) - 1/2*sqrt(-x/(x + 1) + 1)/(sqrt(x/(x + 1)) - 1) - 1/2*sqrt(-x/(x + 1) + 1)/(sqrt(x/(x + 1)) + 1)
S A G E / M A X I M A - Timofeev Chapter 3
$$ \int \! \frac{1}{{\left(x - 2\right)}^{3} {\left(x + 1\right)}^{2}} {dx} $$
[1] diffs back: True integrand: 1/((x - 2)^3*(x + 1)^2) antideriv: 1/18*(2*x^2 - 5*x - 1)/((x - 2)^2*(x + 1)) + 1/27*log((x - 2)/(x + 1)) maxima : 1/18*(2*x^2 - 5*x - 1)/(x^3 - 3*x^2 + 4) + 1/27*log(x - 2) - 1/27*log(x + 1)
$$ \int \! \frac{1}{{\left(x + 2\right)}^{3} {\left(x + 3\right)}^{4}} {dx} $$
[2] diffs back: True integrand: 1/((x + 2)^3*(x + 3)^4) antideriv: 1/6*(60*x^4 + 630*x^3 + 2450*x^2 + 4175*x + 2627)/((x + 2)^2*(x + 3)^3) + 10*log((x + 2)/(x + 3)) maxima : 1/6*(60*x^4 + 630*x^3 + 2450*x^2 + 4175*x + 2627)/(x^5 + 13*x^4 + 67*x^3 + 171*x^2 + 216*x + 108) + 10*log(x + 2) - 10*log(x + 3)
$$ \int \! \frac{x^{5}}{{\left(x + 3\right)}^{2}} {dx} $$
[3] diffs back: True integrand: x^5/(x + 3)^2 antideriv: 243/(x + 3) + 1/4*x^4 - 2*x^3 + 27/2*x^2 - 108*x + 405*log(x + 3) maxima : -108*x + 243/(x + 3) + 1/4*x^4 - 2*x^3 + 27/2*x^2 + 405*log(x + 3)
$$ \int \! \frac{x}{2 \, x^{2} + 6 \, x + 3} {dx} $$
[4] diffs back: True integrand: x/(2*x^2 + 6*x + 3) antideriv: 1/2*sqrt(3)*arctanh(1/3*(2*x + 3)*sqrt(3)) + 1/4*log(-2*x^2 - 6*x - 3) maxima : -1/4*sqrt(3)*log((2*x - sqrt(3) + 3)/(2*x + sqrt(3) + 3)) + 1/4*log(2*x^2 + 6*x + 3)
$$ \int \! \frac{2 \, x - 3}{{\left(2 \, x^{2} + 6 \, x + 3\right)}^{3}} {dx} $$
[5] diffs back: True integrand: (2*x - 3)/(2*x^2 + 6*x + 3)^3 antideriv: 1/3*sqrt(3)*arctanh(1/3*(2*x + 3)*sqrt(3)) - 1/4*(8*x^3 + 36*x^2 + 44*x + 13)/(2*x^2 + 6*x + 3)^2 maxima : -1/6*sqrt(3)*log((2*x - sqrt(3) + 3)/(2*x + sqrt(3) + 3)) - 1/4*(8*x^3 + 36*x^2 + 44*x + 13)/(4*x^4 + 24*x^3 + 48*x^2 + 36*x + 9)
$$ \int \! \frac{x - 1}{{\left(x^{2} + 5 \, x + 4\right)}^{2}} {dx} $$
[6] diffs back: True integrand: (x - 1)/(x^2 + 5*x + 4)^2 antideriv: 1/9*(7*x + 13)/(x^2 + 5*x + 4) + 7/27*log((x + 1)/(x + 4)) maxima : 1/9*(7*x + 13)/(x^2 + 5*x + 4) + 7/27*log(x + 1) - 7/27*log(x + 4)
$$ \int \! \frac{1}{{\left(x^{2} + 3 \, x + 2\right)}^{5}} {dx} $$
[7] diffs back: True integrand: (x^2 + 3*x + 2)^(-5) antideriv: 1/12*(2*x + 3)*(420*(x^2 + 3*x + 2)^3 - 70*(x^2 + 3*x + 2)^2 + 14*x^2 + 42*x + 25)/(x^2 + 3*x + 2)^4 + 70*log((x + 1)/(x + 2)) maxima : 1/12*(840*x^7 + 8820*x^6 + 38920*x^5 + 93450*x^4 + 131768*x^3 + 109116*x^2 + 49176*x + 9315)/(x^8 + 12*x^7 + 62*x^6 + 180*x^5 + 321*x^4 + 360*x^3 + 248*x^2 + 96*x + 16) + 70*log(x + 1) - 70*log(x + 2)
$$ \int \! \frac{1}{{\left(2 \, x^{2} - 6 \, x + 7\right)}^{2} x^{3}} {dx} $$
[8] diffs back: True integrand: 1/((2*x^2 - 6*x + 7)^2*x^3) antideriv: 234/60025*sqrt(5)*arctan(1/5*(2*x - 3)*sqrt(5)) - 2/1715*(9*x - 41)/(2*x^2 - 6*x + 7) - 12/343/x - 1/98/x^2 - 40/2401*log(2*x^2 - 6*x + 7) + 80/2401*log(x) maxima : 234/60025*sqrt(5)*arctan(1/5*(2*x - 3)*sqrt(5)) - 1/3430*(276*x^3 - 814*x^2 + 630*x + 245)/(2*x^4 - 6*x^3 + 7*x^2) - 40/2401*log(2*x^2 - 6*x + 7) + 80/2401*log(x)
$$ \int \! \frac{x^{9}}{{\left(x^{2} + 3 \, x + 2\right)}^{5}} {dx} $$
[9] diffs back: True integrand: x^9/(x^2 + 3*x + 2)^5 antideriv: -1/24*(25*x^8 + 35292*x^7 + 369950*x^6 + 1632276*x^5 + 3919731*x^4 + 5527800*x^3 + 4578216*x^2 + 2063520*x + 390960)/(x^2 + 3*x + 2)^4 - 1471*log(x + 1) + 1472*log(x + 2) maxima : -1/12*(17496*x^7 + 184200*x^6 + 813888*x^5 + 1955853*x^4 + 2759400*x^3 + 2286008*x^2 + 1030560*x + 195280)/(x^8 + 12*x^7 + 62*x^6 + 180*x^5 + 321*x^4 + 360*x^3 + 248*x^2 + 96*x + 16) - 1471*log(x + 1) + 1472*log(x + 2)
$$ \int \! \frac{{\left(2 \, x + 1\right)}^{2}}{{\left(2 \, x^{2} + 5 \, x + 3\right)}^{5}} {dx} $$
[10] diffs back: True integrand: (2*x + 1)^2/(2*x^2 + 5*x + 3)^5 antideriv: 31/6*(4*x + 5)*(120*(2*x^2 + 5*x + 3)^2 - 20*x^2 - 50*x - 29)/(2*x^2 + 5*x + 3)^3 - 1/4*(10*x + 11)/(2*x^2 + 5*x + 3)^4 + 2480*log((x + 1)/(2*x + 3)) maxima : 1/12*(238080*x^7 + 2083200*x^6 + 7757440*x^5 + 15934000*x^4 + 19495776*x^3 + 14209160*x^2 + 5712464*x + 977397)/(16*x^8 + 160*x^7 + 696*x^6 + 1720*x^5 + 2641*x^4 + 2580*x^3 + 1566*x^2 + 540*x + 81) + 2480*log(x + 1) - 2480*log(2*x + 3)
$$ \int \! -\frac{{\left(b x^{2} - a\right)}^{3}}{x^{7}} {dx} $$
[11] diffs back: True integrand: -(b*x^2 - a)^3/x^7 antideriv: -b^3*log(x) - 3/2*a*b^2/x^2 + 3/4*a^2*b/x^4 - 1/6*a^3/x^6 maxima : -1/2*b^3*log(x^2) - 1/12*(18*a*b^2*x^4 - 9*a^2*b*x^2 + 2*a^3)/x^6
$$ \int \! \frac{x^{13}}{{\left(a^{4} + x^{4}\right)}^{5}} {dx} $$
[12] diffs back: True integrand: x^13/(a^4 + x^4)^5 antideriv: -1/768*(15*a^12 + 55*a^8*x^4 + 73*a^4*x^8 - 15*x^12)*x^2/((a^4 + x^4)^4*a^4) + 5/256*arctan(x^2/a^2)/a^6 maxima : -1/768*(15*a^12*x^2 + 55*a^8*x^6 + 73*a^4*x^10 - 15*x^14)/(a^20 + 4*a^16*x^4 + 6*a^12*x^8 + 4*a^8*x^12 + a^4*x^16) + 5/256*arctan(x^2/a^2)/a^6
S A G E / M A X I M A - Timofeev Chapter 7
$$ \int \! x^{2} \cos\left(x\right)^{5} {dx} $$
[1] diffs back: True integrand: x^2*cos(x)^5 antideriv: 5/8*(x^2 - 2)*sin(x) + 5/432*(9*x^2 - 2)*sin(3*x) + 1/2000*(25*x^2 - 2)*sin(5*x) + 5/72*x*cos(3*x) + 1/200*x*cos(5*x) + 5/4*x*cos(x) maxima : 5/8*(x^2 - 2)*sin(x) + 5/432*(9*x^2 - 2)*sin(3*x) + 1/2000*(25*x^2 - 2)*sin(5*x) + 5/72*x*cos(3*x) + 1/200*x*cos(5*x) + 5/4*x*cos(x)
$$ \int \! x^{3} \sin\left(x\right)^{3} {dx} $$
[2] diffs back: True integrand: x^3*sin(x)^3 antideriv: 9/4*(x^2 - 2)*sin(x) - 1/108*(9*x^2 - 2)*sin(3*x) - 3/4*(x^3 - 6*x)*cos(x) + 1/36*(3*x^3 - 2*x)*cos(3*x) maxima : 9/4*(x^2 - 2)*sin(x) - 1/108*(9*x^2 - 2)*sin(3*x) - 3/4*(x^3 - 6*x)*cos(x) + 1/36*(3*x^3 - 2*x)*cos(3*x)
$$ \int \! x^{2} \sin\left(x\right)^{6} {dx} $$
[3] diffs back: True integrand: x^2*sin(x)^6 antideriv: 5/48*x^3 - 15/128*(2*x^2 - 1)*sin(2*x) + 3/512*(8*x^2 - 1)*sin(4*x) - 1/3456*(18*x^2 - 1)*sin(6*x) - 15/64*x*cos(2*x) + 3/128*x*cos(4*x) - 1/576*x*cos(6*x) maxima : 5/48*x^3 - 15/128*(2*x^2 - 1)*sin(2*x) + 3/512*(8*x^2 - 1)*sin(4*x) - 1/3456*(18*x^2 - 1)*sin(6*x) - 15/64*x*cos(2*x) + 3/128*x*cos(4*x) - 1/576*x*cos(6*x)
$$ \int \! x^{2} \sin\left(x\right)^{2} \cos\left(x\right) {dx} $$
[4] diffs back: True integrand: x^2*sin(x)^2*cos(x) antideriv: 1/3*x^2*sin(x)^3 - 1/18*x*cos(3*x) + 1/2*x*cos(x) + 1/54*sin(3*x) - 1/2*sin(x) maxima : 1/4*(x^2 - 2)*sin(x) - 1/108*(9*x^2 - 2)*sin(3*x) - 1/18*x*cos(3*x) + 1/2*x*cos(x)
$$ \int \!{\frac { \left( \cos \left( x \right) \right) ^{4}x}{ \left( \sin \left( x \right) \right) ^{2}}}{dx} $$
[5] RuntimeError: in Maxima: `quotient' by `zero' integrand: x*cos(x)^4/sin(x)^2 antideriv: -1/2*(2/sin(x) + sin(x))*x*cos(x) - 3/4*x^2 + 1/4*sin(x)^2 + log(sin(x))
$$ \int \! \frac{x \sin\left(x\right)^{3}}{\cos\left(x\right)^{4}} {dx} $$
[6] diffs back: True integrand: x*sin(x)^3/cos(x)^4 antideriv: -1/3*(3/cos(x) - 1/cos(x)^3)*x - 1/6*sin(x)/cos(x)^2 + 5/6*arctanh(sin(x)) maxima : -1/12*(48*x*sin(2*x)*sin(3*x) + 12*(6*x*cos(x) - sin(x))*cos(2*x) + 16*(3*x*cos(2*x) + x)*cos(3*x) + 12*(6*x*sin(x) + cos(x))*sin(2*x) + 12*(4*x*cos(3*x) + 6*x*cos(x) - sin(x))*cos(4*x) + 12*(4*x*sin(3*x) + 6*x*sin(x) + cos(x))*sin(4*x) + 4*(18*x*sin(2*x) + 18*x*sin(4*x) + 3*cos(2*x) + 3*cos(4*x) + 1)*sin(5*x) + 4*(4*x*cos(3*x) + 6*x*cos(5*x) + 6*x*cos(x) + sin(5*x) - sin(x))*cos(6*x) + 12*(6*x*cos(2*x) + 6*x*cos(4*x) + 2*x - sin(2*x) - sin(4*x))*cos(5*x) + 4*(4*x*sin(3*x) + 6*x*sin(5*x) + 6*x*sin(x) - cos(5*x) + cos(x))*sin(6*x) + 5*(6*(3*cos(2*x) + 1)*cos(4*x) + 2*(3*cos(2*x) + 3*cos(4*x) + 1)*cos(6*x) + 6*(sin(2*x) + sin(4*x))*sin(6*x) + 9*sin(2*x)^2 + 18*sin(2*x)*sin(4*x) + 9*sin(4*x)^2 + sin(6*x)^2 + 9*cos(2*x)^2 + 9*cos(4*x)^2 + cos(6*x)^2 + 6*cos(2*x) + 1)*log(sin(x)^2 + cos(x)^2 - 2*sin(x) + 1) - 5*(6*(3*cos(2*x) + 1)*cos(4*x) + 2*(3*cos(2*x) + 3*cos(4*x) + 1)*cos(6*x) + 6*(sin(2*x) + sin(4*x))*sin(6*x) + 9*sin(2*x)^2 + 18*sin(2*x)*sin(4*x) + 9*sin(4*x)^2 + sin(6*x)^2 + 9*cos(2*x)^2 + 9*cos(4*x)^2 + cos(6*x)^2 + 6*cos(2*x) + 1)*log(sin(x)^2 + cos(x)^2 + 2*sin(x) + 1) + 24*x*cos(x) - 4*sin(x))/(6*(3*cos(2*x) + 1)*cos(4*x) + 2*(3*cos(2*x) + 3*cos(4*x) + 1)*cos(6*x) + 6*(sin(2*x) + sin(4*x))*sin(6*x) + 9*sin(2*x)^2 + 18*sin(2*x)*sin(4*x) + 9*sin(4*x)^2 + sin(6*x)^2 + 9*cos(2*x)^2 + 9*cos(4*x)^2 + cos(6*x)^2 + 6*cos(2*x) + 1)
$$ \int \! \frac{x \sin\left(x\right)}{\cos\left(x\right)^{3}} {dx} $$
[7] diffs back: True integrand: x*sin(x)/cos(x)^3 antideriv: 1/2*x/cos(x)^2 - 1/2*tan(x) maxima : (4*x*sin(2*x)^2 + 4*x*cos(2*x)^2 + (2*x*cos(2*x) + sin(2*x))*cos(4*x) + (2*x*sin(2*x) - cos(2*x) - 1)*sin(4*x) + 2*x*cos(2*x) - sin(2*x))/(2*(2*cos(2*x) + 1)*cos(4*x) + 4*sin(2*x)^2 + 4*sin(2*x)*sin(4*x) + sin(4*x)^2 + 4*cos(2*x)^2 + cos(4*x)^2 + 4*cos(2*x) + 1)
$$ \int \! \frac{x \sin\left(x\right)^{3}}{\cos\left(x\right)} {dx} $$
[8] diffs back: False integrand: x*sin(x)^3/cos(x) antideriv: 1/2*I*x^2 - 1/2*x*log(sin(2*x)^2 + cos(2*x)^2 + 2*cos(2*x) + 1) + 1/4*x*cos(2*x) - I*x*arctan2(sin(2*x), cos(2*x) + 1) - 1/8*sin(2*x) + 1/2*I*polylog(2, -e^(2*I*x)) maxima : 1/2*I*x^2 - 1/2*x*log(sin(2*x)^2 + cos(2*x)^2 + 2*cos(2*x) + 1) + 1/4*x*cos(2*x) - I*x*arctan2(sin(2*x), cos(2*x) + 1) - 1/8*sin(2*x) + 1/2*I*polylog(2, -e^(2*I*x))
$$ \int \! \frac{x \sin\left(x\right)^{3}}{\cos\left(x\right)^{3}} {dx} $$
[9] diffs back: False integrand: x*sin(x)^3/cos(x)^3 antideriv: -1/2*I*x^2 + 1/2*x*log(sin(2*x)^2 + cos(2*x)^2 + 2*cos(2*x) + 1) + I*x*arctan2(sin(2*x), cos(2*x) + 1) + 1/2*x/cos(x)^2 - 1/2*tan(x) - 1/2*I*polylog(2, -e^(2*I*x)) maxima : (-I*x^2*sin(4*x) - x^2*cos(4*x) + (-2*I*x^2 + 4*x - 2*I)*sin(2*x) - (2*x^2 + 4*I*x + 2)*cos(2*x) + (-2*I*sin(2*x) - I*sin(4*x) - 2*cos(2*x) - cos(4*x) - 1)*polylog(2, -e^(2*I*x)) + (4*I*x*sin(2*x) + 2*I*x*sin(4*x) + 4*x*cos(2*x) + 2*x*cos(4*x) + 2*x)*arctan2(sin(2*x), cos(2*x) + 1) + (2*x*sin(2*x) + x*sin(4*x) - 2*I*x*cos(2*x) - I*x*cos(4*x) - I*x)*log(sin(2*x)^2 + cos(2*x)^2 + 2*cos(2*x) + 1) - x^2 - 2)/(4*sin(2*x) + 2*sin(4*x) - 4*I*cos(2*x) - 2*I*cos(4*x) - 2*I)
$$ \int \! \frac{2 \, x + \sin\left(2 \, x\right)}{{\left(x \sin\left(x\right) + \cos\left(x\right)\right)}^{2}} {dx} $$
[10] diffs back: True integrand: (2*x + sin(2*x))/(x*sin(x) + cos(x))^2 antideriv: -2*cos(x)/(x*sin(x) + cos(x)) maxima : -2*(2*x*sin(2*x) + sin(2*x)^2 + cos(2*x)^2 + 2*cos(2*x) + 1)/((x^2 + 1)*sin(2*x)^2 + (x^2 + 1)*cos(2*x)^2 - 2*(x^2 - 1)*cos(2*x) + x^2 + 4*x*sin(2*x) + 1)
$$ \int \! \frac{x^{2}}{{\left(x \cos\left(x\right) - \sin\left(x\right)\right)}^{2}} {dx} $$
[11] diffs back: True integrand: x^2/(x*cos(x) - sin(x))^2 antideriv: (x*sin(x) + cos(x))/(x*cos(x) - sin(x)) maxima : 2*((x^2 - 1)*sin(2*x) + 2*x*cos(2*x))/((x^2 + 1)*sin(2*x)^2 + (x^2 + 1)*cos(2*x)^2 + 2*(x^2 - 1)*cos(2*x) + x^2 - 4*x*sin(2*x) + 1)
R É S U M É
"...nobody understands how Maxima's integration code works.
I can't wait until we have our symbolic integration code."
William Stein (2010)
S A G E - S O U R C E
Charlwood = dict() Charlwood[ 1] = (arcsin(x)*log(x),-2*sqrt(1-x^2)+arctanh(sqrt(1-x^2))-x*arcsin(x)*(1-log(x))+sqrt(1-x^2)*log(x)) Charlwood[ 2] = (x*arcsin(x)/sqrt(1-x^2),x-sqrt(1-x^2)*arcsin(x)) Charlwood[ 3] = (arcsin(sqrt(x+1)-sqrt(x)),((sqrt(x)+3*sqrt(1+x))*sqrt(-x+sqrt(x)*sqrt(1+x)))/(4*sqrt(2))-(3/8+x)*arcsin(sqrt(x)-sqrt(1+x))) Charlwood[ 4] = (log(1+x*sqrt(1+x^2)),-2*x+sqrt(2*(1+sqrt(5)))*arctan(sqrt(-2+sqrt(5))*(x+sqrt(1+x^2)))-sqrt(2*(-1+sqrt(5)))*arctanh(sqrt(2+sqrt(5))*(x+sqrt(1+x^2)))+x*log(1+x*sqrt(1+x^2))) Charlwood[ 5] = (cos(x)^2/sqrt(cos(x)^4+cos(x)^2+1),x/3+(1/3)*arctan((cos(x)*(1+cos(x)^2)*sin(x))/(1+cos(x)^2*sqrt(1+cos(x)^2+cos(x)^4)))) Charlwood[ 6] = (tan(x)*sqrt(1+tan(x)^4),(-(1/2))*arcsinh(tan(x)^2)-arctanh((1-tan(x)^2)/(sqrt(2)*sqrt(1+tan(x)^4)))/sqrt(2)+(1/2)*sqrt(1+tan(x)^4)) Charlwood[ 7] = (tan(x)/sqrt(1+sec(x)^3),(-(2/3))*arctanh(sqrt(1+sec(x)^3))) Charlwood[ 8] = (sqrt(tan(x)^2+2*tan(x)+2),arcsinh(1+tan(x))+sqrt((1/2)*(1+sqrt(5)))*arctan((-sqrt(-1+sqrt(5))+sqrt(1+sqrt(5))*tan(x))/(sqrt(2)*sqrt(2+tan(x)*(2+tan(x)))))-sqrt((1/2)*(-1+sqrt(5)))*arctanh((sqrt(1+sqrt(5))+sqrt(-1+sqrt(5))*tan(x))/(sqrt(2)*sqrt(2+tan(x)*(2+tan(x)))))) Charlwood[ 9] = (sin(x)*arctan(sqrt(sec(x)-1)),(1/2)*arctan(sqrt(-1+sec(x)))-arctan(sqrt(-1+sec(x)))*cos(x)+(1/2)*cos(x)*sqrt(-1+sec(x))) Charlwood[10] = (x^3*exp(arcsin(x))/sqrt(1-x^2),(1/10)*exp(arcsin(x))*(3*x+x^3-3*sqrt(1-x^2)-3*x^2*sqrt(1-x^2))) Charlwood[11] = ((x*log(1+x^2)*log(x+sqrt(1+x^2)))/sqrt(1+x^2),4*x-2*arctan(x)-2*sqrt(1+x^2)*log(x+sqrt(1+x^2))+log(1+x^2)*(-x+sqrt(1+x^2)*log(x+sqrt(1+x^2)))) Charlwood[12] = (arctan(x+sqrt(1-x^2)),-(arcsin(x)/2)+(1/4)*sqrt(3)*arctan((-1+sqrt(3)*x)/sqrt(1-x^2))+(1/4)*sqrt(3)*arctan((1+sqrt(3)*x)/sqrt(1-x^2))-(1/4)*sqrt(3)*arctan((-1+2*x^2)/sqrt(3))+x*arctan(x+sqrt(1-x^2))-(1/4)*arctanh(x*sqrt(1-x^2))-(1/8)*log(1-x^2+x^4)) Charlwood[13] = (x*arctan(x+sqrt(1-x^2))/sqrt(1-x^2),-(arcsin(x)/2)+(1/4)*sqrt(3)*arctan((-1+sqrt(3)*x)/sqrt(1-x^2))+(1/4)*sqrt(3)*arctan((1+sqrt(3)*x)/sqrt(1-x^2))-(1/4)*sqrt(3)*arctan((-1+2*x^2)/sqrt(3))-sqrt(1-x^2)*arctan(x+sqrt(1-x^2))+(1/4)*arctanh(x*sqrt(1-x^2))+(1/8)*log(1-x^2+x^4)) Charlwood[14] = (arcsin(x)/(1+sqrt(1-x^2)),((-1+sqrt(1-x^2))*arcsin(x))/x+arcsin(x)^2/2-log(1+sqrt(1-x^2))) Charlwood[15] = (log(x+sqrt(1+x^2))/(1-x^2)^(3/2),(-(1/2))*arcsin(x^2)+(x*log(x+sqrt(1+x^2)))/sqrt(1-x^2)) Charlwood[16] = (arcsin(x)/(1+x^2)^(3/2),(x*arcsin(x))/sqrt(1+x^2)-arcsin(x^2)/2) Charlwood[17] = (log(x+sqrt(x^2-1))/(1+x^2)^(3/2),(-(1/2))*arccosh(x^2)+(x*log(x+sqrt(-1+x^2)))/sqrt(1+x^2)) Charlwood[18] = (log(x)/(x^2*sqrt(x^2-1)),sqrt(-1+x^2)/x-arctanh(x/sqrt(-1+x^2))+(sqrt(-1+x^2)*log(x))/x) Charlwood[19] = (sqrt(1+x^3)/x,(2*sqrt(1+x^3))/3-(2/3)*arctanh(sqrt(1+x^3))) Charlwood[20] = (x*log(x+sqrt(x^2-1))/sqrt(x^2-1),-x+sqrt(-1+x^2)*log(x+sqrt(-1+x^2))) Charlwood[21] = (x^3*(arcsin(x)/sqrt(1-x^4)),(1/4)*x*sqrt(1+x^2)-(1/2)*sqrt(1-x^4)*arcsin(x)+arcsinh(x)/4) Charlwood[22] = (x^3*(arcsec(x)/sqrt(x^4-1)),-(sqrt(-1+x^4)/(2*sqrt(1-1/x^2)*x))+(1/2)*sqrt(-1+x^4)*arcsec(x)+(1/2)*arctanh((sqrt(1-1/x^2)*x)/sqrt(-1+x^4))) Charlwood[23] = (x*arctan(x)*log(x+sqrt(1+x^2))/sqrt(1+x^2),(-x)*arctan(x)+(1/2)*log(1+x^2)+sqrt(1+x^2)*arctan(x)*log(x+sqrt(1+x^2))-(1/2)*log(x+sqrt(1+x^2))^2) Charlwood[24] = (x*log(1+sqrt(1-x^2))/sqrt(1-x^2),sqrt(1-x^2)-(1+sqrt(1-x^2))*log(1+sqrt(1-x^2))) Charlwood[25] = (x*log(x+sqrt(1+x^2))/sqrt(1+x^2),-x+sqrt(1+x^2)*log(x+sqrt(1+x^2))) Charlwood[26] = (x*log(x+sqrt(1-x^2))/sqrt(1-x^2),sqrt(1-x^2)+arctanh(sqrt(2)*x)/sqrt(2)-arctanh(sqrt(2)*sqrt(1-x^2))/sqrt(2)-sqrt(1-x^2)*log(x+sqrt(1-x^2))) Charlwood[27] = (log(x)/(x^2*sqrt(1-x^2)),-(sqrt(1-x^2)/x)-arcsin(x)-(sqrt(1-x^2)*log(x))/x) Charlwood[28] = (x*arctan(x)/sqrt(1+x^2),-arcsinh(x)+sqrt(1+x^2)*arctan(x)) Charlwood[29] = (arctan(x)/(x^2*sqrt(1-x^2)),-((sqrt(1-x^2)*arctan(x))/x)-arctanh(sqrt(1-x^2))+sqrt(2)*arctanh(sqrt(1-x^2)/sqrt(2))) Charlwood[30] = (x*arctan(x)/sqrt(1-x^2),-arcsin(x)-sqrt(1-x^2)*arctan(x)+sqrt(2)*arctan((sqrt(2)*x)/sqrt(1-x^2))) Charlwood[31] = (arctan(x)/(x^2*sqrt(1+x^2)),-((sqrt(1+x^2)*arctan(x))/x)-arctanh(sqrt(1+x^2))) Charlwood[32] = (arcsin(x)/(x^2*sqrt(1-x^2)),-((sqrt(1-x^2)*arcsin(x))/x)+log(x)) Charlwood[33] = (x*log(x)/sqrt(x^2-1),-sqrt(-1+x^2)+arctan(sqrt(-1+x^2))+sqrt(-1+x^2)*log(x)) Charlwood[34] = (log(x)/(x^2*sqrt(1+x^2)),-(sqrt(1+x^2)/x)+arcsinh(x)-(sqrt(1+x^2)*log(x))/x) Charlwood[35] = (x*arcsec(x)/sqrt(x^2-1),sqrt(x^2-1)*arcsec(x)-(sqrt(1-1/x^2)*x*log(x))/sqrt(x^2-1)) Charlwood[36] = (x*log(x)/sqrt(1+x^2),-sqrt(1+x^2)+arctanh(sqrt(1+x^2))+sqrt(1+x^2)*log(x)) Charlwood[37] = (sin(x)/(1+sin(x)^2),-(arctanh(cos(x)/sqrt(2))/sqrt(2))) Charlwood[38] = ((1+x^2)/((1-x^2)*sqrt(1+x^4)),(1/sqrt(2))*arctanh(sqrt(2)*(x/sqrt(1+x^4)))) Charlwood[39] = ((1-x^2)/((1+x^2)*sqrt(1+x^4)),arctan((sqrt(2)*x)/sqrt(1+x^4))/sqrt(2)) Charlwood[40] = (log(sin(x))/(1+sin(x)),-x-arctanh(cos(x))-(cos(x)*log(sin(x)))/(1+sin(x))) Charlwood[41] = (log(sin(x))*sqrt(1+sin(x)),(4*cos(x))/sqrt(1+sin(x))-(2*cos(x)*log(sin(x)))/sqrt(1+sin(x))-4*arctanh(cos(x)/sqrt(1+sin(x)))) Charlwood[42] = (sec(x)/sqrt(sec(x)^4-1),-(arctanh((cos(x)*cot(x)*sqrt(sec(x)^4-1))/sqrt(2))/sqrt(2))) Charlwood[43] = (tan(x)/sqrt(1+tan(x)^4),-(arctanh((1-tan(x)^2)/(sqrt(2)*sqrt(1+tan(x)^4)))/(2*sqrt(2)))) Charlwood[44] = (sin(x)/sqrt(1-sin(x)^6),arctanh((sqrt(3)*cos(x)*(1+sin(x)^2))/(2*sqrt(1-sin(x)^6)))/(2*sqrt(3))) Charlwood[45] = (sqrt(sqrt(sec(x)+1)-sqrt(sec(x)-1)),sqrt(2)*(sqrt(-1+sqrt(2))*arctan((sqrt(-2+2*sqrt(2))*(-sqrt(2)-sqrt(-1+sec(x))+sqrt(1+sec(x))))/(2*sqrt(-sqrt(-1+sec(x))+sqrt(1+sec(x)))))-sqrt(1+sqrt(2))*arctan((sqrt(2+2*sqrt(2))*(-sqrt(2)-sqrt(-1+sec(x))+sqrt(1+sec(x))))/(2*sqrt(-sqrt(-1+sec(x))+sqrt(1+sec(x)))))-sqrt(1+sqrt(2))*arctanh((sqrt(-2+2*sqrt(2))*sqrt(-sqrt(-1+sec(x))+sqrt(1+sec(x))))/(sqrt(2)-sqrt(-1+sec(x))+sqrt(1+sec(x))))+sqrt(-1+sqrt(2))*arctanh((sqrt(2+2*sqrt(2))*sqrt(-sqrt(-1+sec(x))+sqrt(1+sec(x))))/(sqrt(2)-sqrt(-1+sec(x))+sqrt(1+sec(x)))))*cot(x)*sqrt(-1+sec(x))*sqrt(1+sec(x))) Charlwood[46] = (x*log(x^2+1)*arctan(x)^2,(-x)*arctan(x)*(-3+log(1+x^2))+(1/4)*(-6+log(1+x^2))*log(1+x^2)+(1/2)*arctan(x)^2*(-3-x^2+(1+x^2)*log(1+x^2))) Charlwood[47] = (arctan(x*sqrt(1+x^2)),(1/2)*arctan(sqrt(1+x^2)/x^2)+x*arctan(x*sqrt(1+x^2))+(1/2)*sqrt(3)*arctanh((sqrt(3)*sqrt(1+x^2))/(2+x^2))) Charlwood[48] = (arctan(sqrt(x+1)-sqrt(x)),sqrt(x)/2+(1+x)*arctan(sqrt(1+x)-sqrt(x))) Charlwood[49] = (arcsin(x/sqrt(1-x^2)),x*arcsin(x/sqrt(1-x^2))+arctan(sqrt(1-2*x^2))) Charlwood[50] = (arctan(x*sqrt(1-x^2)),x*arctan(x*sqrt(1-x^2))-sqrt((1/2)*(1+sqrt(5)))*arctan(sqrt((1/2)*(1+sqrt(5)))*sqrt(1-x^2))+sqrt((1/2)*(-1+sqrt(5)))*arctanh(sqrt((1/2)*(-1+sqrt(5)))*sqrt(1-x^2)))
a = var('a'); b = var('b'); n = var('n')
Timofeev = dict()
# chapter 1
Timofeev[101] = (1/(a^2-b^2*x^2), 1/a/b*arctanh(b*x/a))
Timofeev[102] = (1/(a^2+b^2*x^2), 1/a/b*arctan(b*x/a))
Timofeev[103] = (sec(2*a*x), 1/(2*a)*ln(tan(pi/4+a*x)))
Timofeev[104] = (1/4/sin(1/3*x), 3/4*ln(tan(x/6)))
Timofeev[105] = (1/cos(3/4*pi-2*x), 1/2*ln(tan(pi/8-x)))
Timofeev[106] = (sec(x)*tan(x), sec(x))
Timofeev[107] = (csc(x)*cot(x), -csc(x))
Timofeev[108] = (tan(x)/sin(2*x), 1/2*tan(x))
Timofeev[109] = (1/(1+cos(x)), tan(1/2*x))
Timofeev[110] = (1/(1-cos(x)), -cot(1/2*x))
Timofeev[111] = (sin(x)/(a-b*cos(x)), 1/b*ln(a-b*cos(x)))
Timofeev[112] = (cos(x)/(a^2+b^2*sin(x)^2), 1/a/b*arctan(b*sin(x)/a))
Timofeev[113] = (cos(x)/(a^2-b^2*sin(x)^2), 1/a/b*arctanh(b*sin(x)/a))
Timofeev[114] = (sin(2*x)/(a^2+b^2*sin(x)^2), 1/b^2*ln(a^2+b^2*sin(x)^2))
Timofeev[115] = (sin(2*x)/(b^2*sin(x)^2-a^2), 1/b^2*ln(a^2-b^2*sin(x)^2))
Timofeev[116] = (sin(2*x)/(cos(x)^2*b^2+a^2), -1/b^2*ln(cos(x)^2*b^2+a^2))
Timofeev[117] = (sin(2*x)/(cos(x)^2*b^2-a^2), -1/b^2*ln(a^2-cos(x)^2*b^2))
Timofeev[118] = (1/(4-cos(x)^2), sqrt(3)/6*(atan(sin(x)*cos(x)/(2*sqrt(3)+4 -cos(x)^2))+x))
Timofeev[119] = (exp(x)/(exp(2*x)-1), -arctanh(exp(x)))
Timofeev[120] = (1/x/ln(x), ln(ln(x)))
Timofeev[121] = (1/x/(1+ln(x)^2), arctan(ln(x)))
Timofeev[122] = (1/x/(1-ln(x)), -ln(1-ln(x)))
Timofeev[123] = (1/x/(1+ln(x/a)), ln(1+ln(x/a)))
Timofeev[124] = ((1-x^(1/2)+x)^2/x^2, 3*ln(x)+x-4*x^(1/2)+4/x^(1/2)-1/x )
Timofeev[125] = ((2-x^(2/3))*(x+x^(1/2))/x^(3/2), 2*ln(x)-6/7*x^(7/6)-3/2*x^(2/3)+4*x^(1/2))
Timofeev[126] = ((2*x-1)/(2*x+3), x-2*ln(2*x+3))
Timofeev[127] = ((2*x-5)/(3*x^2-2), 1/3*ln(2-3*x^2)+5/6*6^(1/2)*arctanh(1/2*6^(1/2)*x))
Timofeev[128] = ((2*x-5)/(3*x^2+2), 1/3*ln(3*x^2+2)-5/6*6^(1/2)*arctan(1/2*6^(1/2)*x))
Timofeev[129] = (sin(x)*sin(1/4*x), 2/3*sin(3/4*x)-2/5*sin(5/4*x))
Timofeev[130] = (cos(3*x)*cos(4*x), 1/14*sin(7*x)+1/2*sin(x))
Timofeev[131] = (tan(x)*tan(a-x), 1/tan(a)*ln(1+tan(a)*tan(x))-x)
Timofeev[132] = (sin(x)^2, 1/2*x-1/2*sin(x)*cos(x))
Timofeev[133] = (cos(x)^2, 1/2*x+1/2*sin(x)*cos(x))
Timofeev[134] = (sin(x)*cos(x)^3, -1/4*cos(x)^4)
Timofeev[135] = (cos(x)^3/sin(x)^4, 1/sin(x)-1/3/sin(x)^3)
Timofeev[136] = (1/sin(x)^2/cos(x)^2, tan(x)-cot(x))
Timofeev[137] = (cot(3/4*x)^2, -4/3*cot(3/4*x)-x)
Timofeev[138] = ((1+tan(2*x))^2, 1/2*tan(2*x)-ln(cos(2*x)))
Timofeev[139] = ((tan(x)-cot(x))^2, tan(x)-cot(x)-4*x)
Timofeev[140] = ((tan(x)-sec(x))^2, 2*tan(x/2-pi/4)-x)
Timofeev[141] = (sin(x)/(1+sin(x)), tan(pi/4-x/2)+x)
Timofeev[142] = (cos(x)/(1-cos(x)), -cot(1/2*x)-x)
Timofeev[143] = ((exp(1/2*x)-1)^3*exp(-1/2*x), -6*exp(1/2*x)+2*exp(-1/2*x)+exp(x)+3*x)
Timofeev[144] = (1/(x^2-6*x+5), 1/4*ln((x-5)/(x-1)))
Timofeev[145] = (x^2/(x^6-6*x^3+13), 1/6*arctan(1/2*x^3-3/2))
Timofeev[146] = ((x+2)/(x^2-4*x-1), 1/2*ln(1+4*x-x^2)+4/sqrt(5)*atanh((2-x)/sqrt(5)))
Timofeev[147] = (1/(1+(x+1)^(1/3)), 3/2*(x+1)^(2/3)-3*(x+1)^(1/3)+3*ln(1+(x+1)^(1/3)))
Timofeev[148] = (1/(a*x+b)/x^(1/2), 2/a^(1/2)/b^(1/2)*arctan(a^(1/2)*x^(1/2)/b^(1/2)))
Timofeev[149] = (x^3*(x^2+1)^(1/2), 1/15*(3*x^4+x^2-2)*(x^2+1)^(1/2))
Timofeev[150] = (x/(a^4-x^4)^(1/2), 1/2*atan(x^2/sqrt(a^4-x^4)))
Timofeev[151] = (1/x/(x^2-a^2)^(1/2), 1/a*arctan((x^2-a^2)^(1/2)/a))
Timofeev[152] = (1/x/(a^2-x^2)^(1/2), -1/a*arctanh((a^2-x^2)^(1/2)/a))
Timofeev[153] = (1/x/(a^2+x^2)^(1/2), -1/a*arctanh(a/(a^2+x^2)^(1/2)))
Timofeev[154] = (1/(-x^2+x+2)^(1/2), arcsin(2/3*x-1/3))
Timofeev[155] = (1/(3*x^2-4*x+5)^(1/2), 1/sqrt(3)*asinh((3*x-2)/sqrt(11)))
Timofeev[156] = (1/(-x^2+x)^(1/2), arcsin(2*x-1))
Timofeev[157] = ((2*x+1)/(-x^2+x+2)^(1/2), 2*arcsin(2/3*x-1/3)-2*(-x^2+x+2)^(1/2))
Timofeev[158] = (1/x/(2+x-x^2)^(1/2), -1/sqrt(2)*atanh(2*sqrt(2)*sqrt(2+x-x^2)/(4+x)))
Timofeev[159] = (1/(x-2)/(-x^2+x+2)^(1/2), 2*(-x^2+x+2)^(1/2)/(3*x-6))
Timofeev[160] = ((2+3*sin(x))/sin(x)/(1-cos(x)), -atanh(cos(x))+(3*sin(x)+1)/(cos(x)-1))
Timofeev[161] = (1/(2+3*cos(x)^2), 1/10*10^(1/2)*(x-arctan(3*sin(x)*cos(x)/(10^(1/2)+2+3*cos(x)^2))))
Timofeev[162] = ((1-tan(x))/sin(2*x), 1/2*(ln(tan(x))-tan(x)))
Timofeev[163] = ((1+tan(x)^2)/(1-tan(x)^2), 1/2*ln((1+tan(x))/(1-tan(x))))
Timofeev[164] = ((a^2-4*cos(x)^2)^(3/4)*sin(2*x), 1/7*(a^2-4*cos(x)^2)^(7/4))
Timofeev[165] = (sin(2*x)/(a^2-4*sin(x)^2)^(1/3), -3/8*(a^2-4*sin(x)^2)^(2/3))
Timofeev[166] = (1/(a^(2*x)-1)^(1/2), 1/ln(a)*arctan((a^(2*x)-1)^(1/2)) )
Timofeev[167] = (exp(1/2*x)/(exp(x)-1)^(1/2), 2*ln((exp(x)-1)^(1/2)+exp(1/2*x)))
Timofeev[168] = (arctan(x)^n/(x^2+1), 1/(n+1)*arctan(x)^(n+1))
Timofeev[169] = (arcsin(x/a)^(3/2)/(a^2-x^2)^(1/2), 2/5*a/(a^2-x^2)^(1/2)*(1-x^2/a^2)^(1/2)*arcsin(x/a)^(5/2))
Timofeev[170] = (1/arccos(x)^3/(-x^2+1)^(1/2), 1/2/arccos(x)^2)
Timofeev[171] = (ln(x)^2*x, 1/2*x^2*(ln(x)^2-ln(x)+1/2))
Timofeev[172] = (ln(x)/x^5, -1/16*(4*ln(x)+1)/x^4)
Timofeev[173] = (x^2*ln((x-1)/x), 1/3*x^3*ln((x-1)/x)-1/3*ln(x-1)-1/6*x*(x+2))
Timofeev[174] = (cos(x)^5, 1/15*sin(x)*(3*cos(x)^4+4*cos(x)^2+8))
Timofeev[175] = (cos(x)^4*sin(x)^2, 1/6*sin(x)^3*cos(x)^3+1/8*sin(x)^3*cos(x)-1/16*sin(x)*cos(x)+1/16*x)
Timofeev[176] = (1/sin(x)^5, -3/8*atanh(cos(x))-3*cos(x)/(8*sin(x)^2)-cos(x)/(4*sin(x)^4))
Timofeev[177] = (sin(x)*exp(-x), -1/2*(cos(x)+sin(x))*exp(-x))
Timofeev[178] = (exp(2*x)*sin(3*x), 1/13*exp(2*x)*(2*sin(3*x)-3*cos(3*x)))
Timofeev[179] = (a^x*cos(x), a^x/(ln(a)^2+1)*(ln(a)*cos(x)+sin(x)))
Timofeev[180] = (cos(ln(x)), 1/2*x*(cos(ln(x))+sin(ln(x))))
Timofeev[181] = (sec(x)^2*ln(cos(x)), tan(x)*ln(cos(x))+tan(x)-x)
Timofeev[182] = (x*tan(x)^2, ln(cos(x))+x*tan(x)-1/2*x^2)
Timofeev[183] = (arcsin(x)/x^2, -asin(x)/x-atanh(sqrt(1-x^2)))
Timofeev[184] = (arcsin(x)^2, x*arcsin(x)^2+2*arcsin(x)*(-x^2+1)^(1/2)-2*x)
Timofeev[185] = (x^2*arctan(x)/(x^2+1), x*arctan(x)-1/2*arctan(x)^2-1/2*ln(x^2+1))
Timofeev[186] = (arccos((x/(x+1))^(1/2)), (x+1)*(arccos((x/(x+1))^(1/2))+(1/(x+1))^(1/2)*(x/(x+1))^(1/2)))
# chapter 3
Timofeev[301] = (1/((x-2)^3*(x+1)^2), (2*x^2-5*x-1)/(18*(x+1)*(x-2)^2)+1/27*log((x-2)/(x+1)) )
Timofeev[302] = (1/((x+2)^3*(x+3)^4), (60*x^4+630*x^3+2450*x^2+4175*x+2627)/(6*(x+2)^2*(x+3)^3)+10*log((x+2)/(x+3)) )
Timofeev[303] = (x^5/(3+x)^2, 1/4*x^4-2*x^3+27/2*x^2-108*x+243/(x+3)+405*log(x+3) )
Timofeev[304] = (x/(3+6*x+2*x^2), 1/4*log(-(3+6*x+2*x^2))+sqrt(3)/2*arctanh((3+2*x)/sqrt(3)) )
Timofeev[305] = ((2*x-3)/(3+6*x+2*x^2)^3, -(8*x^3+36*x^2+44*x+13)/(4*(2*x^2+6*x+3)^2)+1/sqrt(3)*arctanh((3+2*x)/sqrt(3)) )
Timofeev[306] = ((x-1)/(x^2+5*x+4)^2, (7*x+13)/(9*(x^2+5*x+4))+7/27*log((x+1)/(x+4)) )
Timofeev[307] = (1/(x^2+3*x+2)^5, (2*x+3)/(4*(x^2+3*x+2)^4)*(-1+14/3*(x^2+3*x+2)-70/3*(x^2+3*x+2)^2+140*(x^2+3*x+2)^3)+70*log((x+1)/(x+2)) )
Timofeev[308] = (1/(x^3*(7-6*x+2*x^2)^2), -1/(98*x^2)-12/(343*x)+2*(41-9*x)/(1715*(7-6*x+2*x^2))+80/2401*log(x)-40/2401*log(7-6*x+2*x^2)+234*sqrt(5)/60025*arctan((2*x-3)/sqrt(5)) )
Timofeev[309] = (x^9/(x^2+3*x+2)^5, -(25*x^8+35292*x^7+369950*x^6+1632276*x^5+3919731*x^4+5527800*x^3+4578216*x^2+2063520*x+390960)/(24*(x^2+3*x+2)^4)+1472*log(x+2)-1471*log(x+1) )
Timofeev[310] = ((1+2*x)^2/(3+5*x+2*x^2)^5, -(11+10*x)/(4*(2*x^2+5*x+3)^4)+31*(5+4*x)/(6*(2*x^2+5*x+3)^3)*(1-10*(2*x^2+5*x+3)+120*(2*x^2+5*x+3)^2)+2480*log((x+1)/(2*x+3)) )
Timofeev[311] = ((a-b*x^2)^3/x^7, -a^3/(6*x^6)+3*a^2*b/(4*x^4)-3*a*b^2/(2*x^2)-b^3*log(x) )
Timofeev[312] = (x^13/(a^4+x^4)^5, x^2*(15*x^12-73*a^4*x^8-55*a^8*x^4-15*a^12)/(768*a^4*(x^4+a^4)^4)+5/(256*a^6)*arctan(x^2/a^2) )
# chapter 7
Timofeev[701] = (x^2*cos(x)^5, 1/200*x*cos(5*x)+(1/80*x^2-1/1000)*sin(5*x)+5/72*x*cos(3*x)+(5/48*x^2-5/216)*sin(3*x)+5/4*x*cos(x)+(5/8*x^2-5/4)*sin(x) )
Timofeev[702] = (x^3*sin(x)^3, 1/12*(x^3-2/3*x)*cos(3*x)-1/12*(x^2-2/9)*sin(3*x)-3/4*(x^3-6*x)*cos(x)+9/4*(x^2-2)*sin(x) )
Timofeev[703] = (x^2*sin(x)^6, 5/48*x^3-1/192*(x^2-1/18)*sin(6*x)-1/576*x*cos(6*x)+3/64*(x^2-1/8)*sin(4*x)+3/128*x*cos(4*x)-15/64*(x^2-1/2)*sin(2*x)-15/64*x*cos(2*x) )
Timofeev[704] = (x^2*sin(x)^2*cos(x), 1/3*x^2*sin(x)^3-1/18*x*cos(3*x)+1/54*sin(3*x)+1/2*x*cos(x)-1/2*sin(x) )
Timofeev[705] = (x*cos(x)^4/sin(x)^2, -x*cos(x)*(1/2*sin(x)+1/sin(x))+1/4*sin(x)^2+log(sin(x))-3/4*x^2 )
Timofeev[706] = (x*sin(x)^3/cos(x)^4, x*(1/(3*cos(x)^3)-1/cos(x))-sin(x)/(6*cos(x)^2)+5/6*arctanh(sin(x)) )
Timofeev[707] = (x*sin(x)/cos(x)^3, x/(2*cos(x)^2)-1/2*tan(x) )
Timofeev[708] = (x*sin(x)^3/cos(x), 1/4*x*cos(2*x)-1/8*sin(2*x)+integrate(x*tan(x),x) )
Timofeev[709] = (x*sin(x)^3/cos(x)^3, x/(2*cos(x)^2)-1/2*tan(x)-integrate(x*tan(x),x) )
Timofeev[710] = ((2*x+sin(2*x))/(x*sin(x)+cos(x))^2, -2*cos(x)/(x*sin(x)+cos(x)) )
Timofeev[711] = ((x/(x*cos(x)-sin(x)))^2, (x*sin(x)+cos(x))/(x*cos(x)-sin(x)) )
def test(suite, problem, algo):
integrand, antideriv = suite[problem]
print [problem]
print "integrand :", integrand
print "antideriv :", antideriv
alarm(30) # set timeout
try:
solution = integrate(integrand, x, algorithm = algo)
print "result :", solution
if len(str(solution)) >= 2*len(str(antideriv)):
print "comment : Solution is messy."
if 'integrate' in str(solution):
print "comment : Unevaluated or not an elementary solution!"
if 'I' in str(solution):
print "comment : Not a real solution!"
diffsolution = solution.derivative(x)
if(simplify(integrand - diffsolution)):
print "comment : Diffs not back to zero. (Check use of abs?)"
except RuntimeError:
print "Sage : RuntimeError"
except KeyboardInterrupt:
print "Sage : Timed out! (30 s)"
except AttributeError:
print "Sage : AttributeError"
except ValueError:
print "Sage : Additional constraints? Assume?"
alarm(0)
print
def test_all(suite, algo):
for problem in suite.iteritems():
test(suite, problem[0], algo)
# Sample single tests test(Charlwood, 13, 'maxima') # [13] # integrand: -1/(b^2*x^2 - a^2) # antideriv: arctanh(b*x/a)/(a*b) # maxima : -1/2*log(b*x - a)/(a*b) + 1/2*log(b*x + a)/(a*b) test(Timofeev, 101, 'maxima') # [101] # integrand: x*arcsin(x)/sqrt(-x^2 + 1) # antideriv: -sqrt(-x^2 + 1)*arcsin(x) + x # maxima : -sqrt(-x^2 + 1)*arcsin(x) + x # Sample full tests test_all(Charlwood, 'maxima') test_all(Timofeev, 'maxima')
You can download the Sage work sheet from GitHub.