In [1]:
from time import clock  
from sympy import *
from __future__ import division
 
x = Symbol('x')
In [2]:
 
def asec(x):
    return acos(1/x)
In [3]:
 
Charlwood = dict()
Charlwood[ 1] = (asin(x)*log(x),-2*sqrt(1-x**2)+atanh(sqrt(1-x**2))-x*asin(x)*(1-log(x))+sqrt(1-x**2)*log(x))
Charlwood[ 2] = (x*asin(x)/sqrt(1-x**2),x-sqrt(1-x**2)*asin(x))
Charlwood[ 3] = (asin(sqrt(x+1)-sqrt(x)),((sqrt(x)+3*sqrt(1+x))*sqrt(-x+sqrt(x)*sqrt(1+x)))/(4*sqrt(2))-(Rational(3,8)+x)*asin(sqrt(x)-sqrt(1+x)))
Charlwood[ 4] = (log(1+x*sqrt(1+x**2)),-2*x+sqrt(2*(1+sqrt(5)))*atan(sqrt(-2+sqrt(5))*(x+sqrt(1+x**2)))-sqrt(2*(-1+sqrt(5)))*atanh(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+Rational(1,3)*atan((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),(-Rational(1,2))*asinh(tan(x)**2)-atanh((1-tan(x)**2)/(sqrt(2)*sqrt(1+tan(x)**4)))/sqrt(2)+Rational(1,2)*sqrt(1+tan(x)**4))
Charlwood[ 7] = (tan(x)/sqrt(1+sec(x)**3),(-Rational(2,3))*atanh(sqrt(1+sec(x)**3)))
Charlwood[ 8] = (sqrt(tan(x)**2+2*tan(x)+2),asinh(1+tan(x))+sqrt(Rational(1,2)*(1+sqrt(5)))*atan((-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)))*atanh((sqrt(1+sqrt(5))+sqrt(-1+sqrt(5))*tan(x))/(sqrt(2)*sqrt(2+tan(x)*(2+tan(x))))))
Charlwood[ 9] = (sin(x)*atan(sqrt(sec(x)-1)),Rational(1,2)*atan(sqrt(-1+sec(x)))-atan(sqrt(-1+sec(x)))*cos(x)+Rational(1,2)*cos(x)*sqrt(-1+sec(x)))
Charlwood[10] = (x**3*exp(asin(x))/sqrt(1-x**2),Rational(1,10)*exp(asin(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*atan(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] = (atan(x+sqrt(1-x**2)),-(asin(x)/2)+Rational(1,4)*sqrt(3)*atan((-1+sqrt(3)*x)/sqrt(1-x**2))+Rational(1,4)*sqrt(3)*atan((1+sqrt(3)*x)/sqrt(1-x**2))-Rational(1,4)*sqrt(3)*atan((-1+2*x**2)/sqrt(3))+x*atan(x+sqrt(1-x**2))-Rational(1,4)*atanh(x*sqrt(1-x**2))-Rational(1,8)*log(1-x**2+x**4))
Charlwood[13] = (x*atan(x+sqrt(1-x**2))/sqrt(1-x**2),-(asin(x)/2)+Rational(1,4)*sqrt(3)*atan((-1+sqrt(3)*x)/sqrt(1-x**2))+Rational(1,4)*sqrt(3)*atan((1+sqrt(3)*x)/sqrt(1-x**2))-Rational(1,4)*sqrt(3)*atan((-1+2*x**2)/sqrt(3))-sqrt(1-x**2)*atan(x+sqrt(1-x**2))+Rational(1,4)*atanh(x*sqrt(1-x**2))+Rational(1,8)*log(1-x**2+x**4))
Charlwood[14] = (asin(x)/(1+sqrt(1-x**2)),((-1+sqrt(1-x**2))*asin(x))/x+asin(x)**2/2-log(1+sqrt(1-x**2)))
Charlwood[15] = (log(x+sqrt(1+x**2))/(1-x**2)**Rational(3,2),(-Rational(1,2))*asin(x**2)+(x*log(x+sqrt(1+x**2)))/sqrt(1-x**2))
Charlwood[16] = (asin(x)/(1+x**2)**Rational(3,2),(x*asin(x))/sqrt(1+x**2)-asin(x**2)/2)
Charlwood[17] = (log(x+sqrt(x**2-1))/(1+x**2)**Rational(3,2),(-Rational(1,2))*acosh(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-atanh(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-Rational(2,3)*atanh(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*(asin(x)/sqrt(1-x**4)),Rational(1,4)*x*sqrt(1+x**2)-Rational(1,2)*sqrt(1-x**4)*asin(x)+asinh(x)/4)
Charlwood[22] = (x**3*(asec(x)/sqrt(x**4-1)),-(sqrt(-1+x**4)/(2*sqrt(1-1/x**2)*x))+Rational(1,2)*sqrt(-1+x**4)*asec(x)+Rational(1,2)*atanh((sqrt(1-1/x**2)*x)/sqrt(-1+x**4)))
Charlwood[23] = (x*atan(x)*log(x+sqrt(1+x**2))/sqrt(1+x**2),(-x)*atan(x)+Rational(1,2)*log(1+x**2)+sqrt(1+x**2)*atan(x)*log(x+sqrt(1+x**2))-Rational(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)+atanh(sqrt(2)*x)/sqrt(2)-atanh(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)-asin(x)-(sqrt(1-x**2)*log(x))/x)
Charlwood[28] = (x*atan(x)/sqrt(1+x**2),-asinh(x)+sqrt(1+x**2)*atan(x))
Charlwood[29] = (atan(x)/(x**2*sqrt(1-x**2)),-((sqrt(1-x**2)*atan(x))/x)-atanh(sqrt(1-x**2))+sqrt(2)*atanh(sqrt(1-x**2)/sqrt(2)))
Charlwood[30] = (x*atan(x)/sqrt(1-x**2),-asin(x)-sqrt(1-x**2)*atan(x)+sqrt(2)*atan((sqrt(2)*x)/sqrt(1-x**2)))
Charlwood[31] = (atan(x)/(x**2*sqrt(1+x**2)),-((sqrt(1+x**2)*atan(x))/x)-atanh(sqrt(1+x**2)))
Charlwood[32] = (asin(x)/(x**2*sqrt(1-x**2)),-((sqrt(1-x**2)*asin(x))/x)+log(x))
Charlwood[33] = (x*log(x)/sqrt(x**2-1),-sqrt(-1+x**2)+atan(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)+asinh(x)-(sqrt(1+x**2)*log(x))/x)
Charlwood[35] = (x*asec(x)/sqrt(x**2-1),sqrt(x**2-1)*asec(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)+atanh(sqrt(1+x**2))+sqrt(1+x**2)*log(x))
Charlwood[37] = (sin(x)/(1+sin(x)**2),-(atanh(cos(x)/sqrt(2))/sqrt(2)))
Charlwood[38] = ((1+x**2)/((1-x**2)*sqrt(1+x**4)),(1/sqrt(2))*atanh(sqrt(2)*(x/sqrt(1+x**4))))
Charlwood[39] = ((1-x**2)/((1+x**2)*sqrt(1+x**4)),atan((sqrt(2)*x)/sqrt(1+x**4))/sqrt(2))
Charlwood[40] = (log(sin(x))/(1+sin(x)),-x-atanh(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*atanh(cos(x)/sqrt(1+sin(x))))
Charlwood[42] = (sec(x)/sqrt(sec(x)**4-1),-(atanh((cos(x)*cot(x)*sqrt(sec(x)**4-1))/sqrt(2))/sqrt(2)))
Charlwood[43] = (tan(x)/sqrt(1+tan(x)**4),-(atanh((1-tan(x)**2)/(sqrt(2)*sqrt(1+tan(x)**4)))/(2*sqrt(2))))
Charlwood[44] = (sin(x)/sqrt(1-sin(x)**6),atanh((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))*atan((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))*atan((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))*atanh((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))*atanh((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)*atan(x)**2,(-x)*atan(x)*(-3+log(1+x**2))+Rational(1,4)*(-6+log(1+x**2))*log(1+x**2)+Rational(1,2)*atan(x)**2*(-3-x**2+(1+x**2)*log(1+x**2)))
Charlwood[47] = (atan(x*sqrt(1+x**2)),Rational(1,2)*atan(sqrt(1+x**2)/x**2)+x*atan(x*sqrt(1+x**2))+Rational(1,2)*sqrt(3)*atanh((sqrt(3)*sqrt(1+x**2))/(2+x**2)))
Charlwood[48] = (atan(sqrt(x+1)-sqrt(x)),sqrt(x)/2+(1+x)*atan(sqrt(1+x)-sqrt(x)))
Charlwood[49] = (asin(x/sqrt(1-x**2)),x*asin(x/sqrt(1-x**2))+atan(sqrt(1-2*x**2)))
Charlwood[50] = (atan(x*sqrt(1-x**2)),x*atan(x*sqrt(1-x**2))-sqrt(Rational(1,2)*(1+sqrt(5)))*atan(sqrt(Rational(1,2)*(1+sqrt(5)))*sqrt(1-x**2))+sqrt(Rational(1,2)*(-1+sqrt(5)))*atanh(sqrt(Rational(1,2)*(-1+sqrt(5)))*sqrt(1-x**2)))
In [ ]:
 
def test(suite, problem):
    
    integrand, antideriv = suite[problem]
    print [problem], "Charlwood"
    print "integrand :", integrand 
    print "antideriv :", antideriv
    
    try:
        start = clock()  
        solution = integrate(integrand, x)
        duration = clock() - start  
    
    except RuntimeError, e:
        print "integrator: RuntimeError:", e
        print 
        return
    except KeyboardInterrupt:
        print "time      :", round(clock() - start) 
        print "integrator: Timed out! \n"
        return
    except AttributeError, e:
        print "integrator: AttributeError:", e
        print
        return
    except ValueError, e: 
        print "integrator: ValueError:", e 
        print "comment   : Additional constraints? Assume? \n"
        return
    
    print "sympy     :", solution
          
    str_solution = str(solution)
    diffsolution = solution.diff(x) 
    matheq = 0 == simplify((integrand - diffsolution).expand())
    
    if len(str_solution) > 2*len(str(antideriv)):
        print "comment   : Solution is messy."
            
    if 'Integral' in str_solution: 
            print "comment   : Unevaluated or not an elementary solution!"
    else:
        if 'I' in str_solution: 
            print "comment   : Not a real solution!"  
            
    if 'Piecewise' in str_solution:
        print "comment   : Not a continous solution!"
    
    if not matheq:    
        print "comment   : Diffs not back to zero. (Check use of abs.)"
    
    print "time      :", round(duration)
    if duration > 60 :
        print "You are wasting my time!"
    print
 
def test_all(suite): 
    start = clock()  
    for problem in suite.iteritems(): 
        test(suite, problem[0])
    duration = clock() - start 
    print "total time:", round(duration)
In [5]:
 
test(Charlwood, 1)
[1] Charlwood
integrand : log(x)*asin(x)
antideriv : -x*(-log(x) + 1)*asin(x) + sqrt(-x**2 + 1)*log(x) - 2*sqrt(-x**2 + 1) + atanh(sqrt(-x**2 + 1))
sympy     : Integral(log(x)*asin(x), x)
comment   : Unevaluated or not an elementary solution!
time      : 9.0
In [6]:
 
test(Charlwood, 2)
[2] Charlwood
integrand : x*asin(x)/sqrt(-x**2 + 1)
antideriv : x - sqrt(-x**2 + 1)*asin(x)
sympy     : x - sqrt(-x**2 + 1)*asin(x)
time      : 0.0
In [7]:
 
test(Charlwood, 3)
[3] Charlwood
integrand : -asin(sqrt(x) - sqrt(x + 1))
antideriv : sqrt(2)*(sqrt(x) + 3*sqrt(x + 1))*sqrt(sqrt(x)*sqrt(x + 1) - x)/8 - (x + 3/8)*asin(sqrt(x) - sqrt(x + 1))
sympy     : Integral(-asin(sqrt(x) - sqrt(x + 1)), x)
comment   : Unevaluated or not an elementary solution!
time      : 8.0
In [8]:
 
test(Charlwood, 4)
[4] Charlwood
integrand : log(x*sqrt(x**2 + 1) + 1)
antideriv : x*log(x*sqrt(x**2 + 1) + 1) - 2*x + sqrt(2 + 2*sqrt(5))*atan(sqrt(-2 + sqrt(5))*(x + sqrt(x**2 + 1))) - sqrt(-2 + 2*sqrt(5))*atanh(sqrt(2 + sqrt(5))*(x + sqrt(x**2 + 1)))
time      : 75.0
integrator: Timed out!
In [9]:
 
test(Charlwood, 5)
[5] Charlwood
integrand : cos(x)**2/sqrt(cos(x)**4 + cos(x)**2 + 1)
antideriv : x/3 + atan((cos(x)**2 + 1)*sin(x)*cos(x)/(sqrt(cos(x)**4 + cos(x)**2 + 1)*cos(x)**2 + 1))/3
time      : 60.0
integrator: Timed out!
In [10]:
 
test(Charlwood, 6)
[6] Charlwood
integrand : sqrt(tan(x)**4 + 1)*tan(x)
antideriv : sqrt(tan(x)**4 + 1)/2 - asinh(tan(x)**2)/2 - sqrt(2)*atanh(sqrt(2)*(-tan(x)**2 + 1)/(2*sqrt(tan(x)**4 + 1)))/2
sympy     : Integral(sqrt(tan(x)**4 + 1)*tan(x), x)
comment   : Unevaluated or not an elementary solution!
time      : 1.0
In [11]:
 
test(Charlwood, 7)
[7] Charlwood
integrand : tan(x)/sqrt(sec(x)**3 + 1)
antideriv : -2*atanh(sqrt(sec(x)**3 + 1))/3
sympy     : Integral(tan(x)/sqrt(sec(x)**3 + 1), x)
comment   : Unevaluated or not an elementary solution!
time      : 1.0
In [12]:
 
test(Charlwood, 8)
[8] Charlwood
integrand : sqrt(tan(x)**2 + 2*tan(x) + 2)
antideriv : asinh(tan(x) + 1) + sqrt(1/2 + sqrt(5)/2)*atan(sqrt(2)*(sqrt(1 + sqrt(5))*tan(x) - sqrt(-1 + sqrt(5)))/(2*sqrt((tan(x) + 2)*tan(x) + 2))) - sqrt(-0.5 + 0.5*sqrt(5))*atanh(sqrt(2)*(sqrt(-1 + sqrt(5))*tan(x) + sqrt(1 + sqrt(5)))/(2*sqrt((tan(x) + 2)*tan(x) + 2)))
sympy     : Integral(sqrt(tan(x)**2 + 2*tan(x) + 2), x)
comment   : Unevaluated or not an elementary solution!
time      : 1.0
In [13]:
 
test(Charlwood, 9)
[9] Charlwood
integrand : sin(x)*atan(sqrt(sec(x) - 1))
antideriv : sqrt(sec(x) - 1)*cos(x)/2 - cos(x)*atan(sqrt(sec(x) - 1)) + atan(sqrt(sec(x) - 1))/2
time      : 62.0
integrator: Timed out!
In [14]:
 
test(Charlwood, 10)
[10] Charlwood
integrand : x**3*exp(asin(x))/sqrt(-x**2 + 1)
antideriv : (x**3 - 3*x**2*sqrt(-x**2 + 1) + 3*x - 3*sqrt(-x**2 + 1))*exp(asin(x))/10
sympy     : x**3*exp(asin(x))/10 - 3*x**2*sqrt(-x**2 + 1)*exp(asin(x))/10 + 3*x*exp(asin(x))/10 - 3*sqrt(-x**2 + 1)*exp(asin(x))/10
time      : 5.0
In [15]:
 
test(Charlwood, 11)
[11] Charlwood
integrand : x*log(x + sqrt(x**2 + 1))*log(x**2 + 1)/sqrt(x**2 + 1)
antideriv : 4*x + (-x + sqrt(x**2 + 1)*log(x + sqrt(x**2 + 1)))*log(x**2 + 1) - 2*sqrt(x**2 + 1)*log(x + sqrt(x**2 + 1)) - 2*atan(x)
time      : 61.0
integrator: Timed out!
In [16]:
 
test(Charlwood, 12)
[12] Charlwood
integrand : atan(x + sqrt(-x**2 + 1))
antideriv : x*atan(x + sqrt(-x**2 + 1)) - log(x**4 - x**2 + 1)/8 - asin(x)/2 - sqrt(3)*atan(sqrt(3)*(2*x**2 - 1)/3)/4 + sqrt(3)*atan((sqrt(3)*x - 1)/sqrt(-x**2 + 1))/4 + sqrt(3)*atan((sqrt(3)*x + 1)/sqrt(-x**2 + 1))/4 - atanh(x*sqrt(-x**2 + 1))/4
time      : 70.0
integrator: Timed out!
In [17]:
 
test(Charlwood, 13)
[13] Charlwood
integrand : x*atan(x + sqrt(-x**2 + 1))/sqrt(-x**2 + 1)
antideriv : -sqrt(-x**2 + 1)*atan(x + sqrt(-x**2 + 1)) + log(x**4 - x**2 + 1)/8 - asin(x)/2 - sqrt(3)*atan(sqrt(3)*(2*x**2 - 1)/3)/4 + sqrt(3)*atan((sqrt(3)*x - 1)/sqrt(-x**2 + 1))/4 + sqrt(3)*atan((sqrt(3)*x + 1)/sqrt(-x**2 + 1))/4 + atanh(x*sqrt(-x**2 + 1))/4
time      : 63.0
integrator: Timed out!
In [20]:
 
test(Charlwood, 14)
[14] Charlwood
integrand : asin(x)/(sqrt(-x**2 + 1) + 1)
antideriv : -log(sqrt(-x**2 + 1) + 1) + asin(x)**2/2 + (sqrt(-x**2 + 1) - 1)*asin(x)/x
sympy     : Integral(asin(x)/(sqrt(-x**2 + 1) + 1), x)
comment   : Unevaluated or not an elementary solution!
time      : 0.0
In [21]:
 
test(Charlwood, 15)
[15] Charlwood
integrand : log(x + sqrt(x**2 + 1))/(-x**2 + 1)**(3/2)
antideriv : x*log(x + sqrt(x**2 + 1))/sqrt(-x**2 + 1) - asin(x**2)/2
time      : 63.0
integrator: Timed out!
In [22]:
 
test(Charlwood, 16)
[16] Charlwood
integrand : asin(x)/(x**2 + 1)**(3/2)
antideriv : x*asin(x)/sqrt(x**2 + 1) - asin(x**2)/2
sympy     : Integral(asin(x)/(x**2 + 1)**(3/2), x)
comment   : Unevaluated or not an elementary solution!
time      : 48.0
In [23]:
 
test(Charlwood, 17)
[17] Charlwood
integrand : log(x + sqrt(x**2 - 1))/(x**2 + 1)**(3/2)
antideriv : x*log(x + sqrt(x**2 - 1))/sqrt(x**2 + 1) - acosh(x**2)/2
time      : 61.0
integrator: Timed out!
In [24]:
 
test(Charlwood, 18)
[18] Charlwood
integrand : log(x)/(x**2*sqrt(x**2 - 1))
antideriv : -atanh(x/sqrt(x**2 - 1)) + sqrt(x**2 - 1)*log(x)/x + sqrt(x**2 - 1)/x
sympy     : Integral(log(x)/(x**2*sqrt(x**2 - 1)), x)
comment   : Unevaluated or not an elementary solution!
time      : 41.0
In [25]:
 
test(Charlwood, 19)
[19] Charlwood
integrand : sqrt(x**3 + 1)/x
antideriv : 2*sqrt(x**3 + 1)/3 - 2*atanh(sqrt(x**3 + 1))/3
sympy     : 2*x**(3/2)/(3*sqrt(1 + x**(-3))) - 2*asinh(x**(-3/2))/3 + 2/(3*x**(3/2)*sqrt(1 + x**(-3)))
comment   : Diffs not back to zero. (Check use of abs.)
time      : 1.0
In [26]:
 
test(Charlwood, 20)
[20] Charlwood
integrand : x*log(x + sqrt(x**2 - 1))/sqrt(x**2 - 1)
antideriv : -x + sqrt(x**2 - 1)*log(x + sqrt(x**2 - 1))
time      : 59.0
integrator: Timed out!
In [27]:
 
test(Charlwood, 21)
[21] Charlwood
integrand : x**3*asin(x)/sqrt(-x**4 + 1)
antideriv : x*sqrt(x**2 + 1)/4 - sqrt(-x**4 + 1)*asin(x)/2 + asinh(x)/4
sympy     : Integral(x**3*asin(x)/sqrt(-x**4 + 1), x)
comment   : Unevaluated or not an elementary solution!
time      : 37.0
In [28]:
 
test(Charlwood, 22)
[22] Charlwood
integrand : x**3*acos(1/x)/sqrt(x**4 - 1)
antideriv : sqrt(x**4 - 1)*acos(1/x)/2 + atanh(x*sqrt(1 - 1/x**2)/sqrt(x**4 - 1))/2 - sqrt(x**4 - 1)/(2*x*sqrt(1 - 1/x**2))
time      : 62.0
integrator: Timed out!
In [29]:
 
test(Charlwood, 23)
[23] Charlwood
integrand : x*log(x + sqrt(x**2 + 1))*atan(x)/sqrt(x**2 + 1)
antideriv : -x*atan(x) + sqrt(x**2 + 1)*log(x + sqrt(x**2 + 1))*atan(x) - log(x + sqrt(x**2 + 1))**2/2 + log(x**2 + 1)/2
time      : 64.0
integrator: Timed out!
In [30]:
 
test(Charlwood, 24)
[24] Charlwood
integrand : x*log(sqrt(-x**2 + 1) + 1)/sqrt(-x**2 + 1)
antideriv : sqrt(-x**2 + 1) - (sqrt(-x**2 + 1) + 1)*log(sqrt(-x**2 + 1) + 1)
time      : 61.0
integrator: Timed out!
In [31]:
 
test(Charlwood, 25)
[25] Charlwood
integrand : x*log(x + sqrt(x**2 + 1))/sqrt(x**2 + 1)
antideriv : -x + sqrt(x**2 + 1)*log(x + sqrt(x**2 + 1))
time      : 64.0
integrator: Timed out!
In [32]:
 
test(Charlwood, 26)
[26] Charlwood
integrand : x*log(x + sqrt(-x**2 + 1))/sqrt(-x**2 + 1)
antideriv : -sqrt(-x**2 + 1)*log(x + sqrt(-x**2 + 1)) + sqrt(-x**2 + 1) + sqrt(2)*atanh(sqrt(2)*x)/2 - sqrt(2)*atanh(sqrt(2)*sqrt(-x**2 + 1))/2
time      : 66.0
integrator: Timed out!
In [33]:
 
test(Charlwood, 27)
[27] Charlwood
integrand : log(x)/(x**2*sqrt(-x**2 + 1))
antideriv : -asin(x) - sqrt(-x**2 + 1)*log(x)/x - sqrt(-x**2 + 1)/x
sympy     : Integral(log(x)/(x**2*sqrt(-x**2 + 1)), x)
comment   : Unevaluated or not an elementary solution!
time      : 41.0
In [34]:
 
test(Charlwood, 28)
[28] Charlwood
integrand : x*atan(x)/sqrt(x**2 + 1)
antideriv : sqrt(x**2 + 1)*atan(x) - asinh(x)
sympy     : x**2*atan(x)/sqrt(x**2 + 1) - asinh(x) + atan(x)/sqrt(x**2 + 1)
time      : 6.0
In [35]:
 
test(Charlwood, 29)
[29] Charlwood
integrand : atan(x)/(x**2*sqrt(-x**2 + 1))
antideriv : sqrt(2)*atanh(sqrt(2)*sqrt(-x**2 + 1)/2) - atanh(sqrt(-x**2 + 1)) - sqrt(-x**2 + 1)*atan(x)/x
sympy     : Integral(atan(x)/(x**2*sqrt(-x**2 + 1)), x)
comment   : Unevaluated or not an elementary solution!
time      : 25.0
In [36]:
 
test(Charlwood, 30)
[30] Charlwood
integrand : x*atan(x)/sqrt(-x**2 + 1)
antideriv : -sqrt(-x**2 + 1)*atan(x) - asin(x) + sqrt(2)*atan(sqrt(2)*x/sqrt(-x**2 + 1))
time      : 63.0
integrator: Timed out!
In [37]:
 
test(Charlwood, 31)
[31] Charlwood
integrand : atan(x)/(x**2*sqrt(x**2 + 1))
antideriv : -atanh(sqrt(x**2 + 1)) - sqrt(x**2 + 1)*atan(x)/x
time      : 67.0
integrator: Timed out!
In [38]:
 
test(Charlwood, 32)
[32] Charlwood
integrand : asin(x)/(x**2*sqrt(-x**2 + 1))
antideriv : log(x) - sqrt(-x**2 + 1)*asin(x)/x
sympy     : Integral(asin(x)/(x**2*sqrt(-x**2 + 1)), x)
comment   : Unevaluated or not an elementary solution!
time      : 0.0
In [39]:
 
test(Charlwood, 33)
[33] Charlwood
integrand : x*log(x)/sqrt(x**2 - 1)
antideriv : sqrt(x**2 - 1)*log(x) - sqrt(x**2 - 1) + atan(sqrt(x**2 - 1))
sympy     : Integral(x*log(x)/sqrt(x**2 - 1), x)
comment   : Unevaluated or not an elementary solution!
time      : 10.0
In [40]:
 
test(Charlwood, 34)
[34] Charlwood
integrand : log(x)/(x**2*sqrt(x**2 + 1))
antideriv : asinh(x) - sqrt(x**2 + 1)*log(x)/x - sqrt(x**2 + 1)/x
sympy     : Integral(log(x)/(x**2*sqrt(x**2 + 1)), x)
comment   : Unevaluated or not an elementary solution!
time      : 44.0
In [41]:
 
test(Charlwood, 35)
[35] Charlwood
integrand : x*acos(1/x)/sqrt(x**2 - 1)
antideriv : -x*sqrt(1 - 1/x**2)*log(x)/sqrt(x**2 - 1) + sqrt(x**2 - 1)*acos(1/x)
time      : 68.0
integrator: Timed out!
In [42]:
 
test(Charlwood, 36)
[36] Charlwood
integrand : x*log(x)/sqrt(x**2 + 1)
antideriv : sqrt(x**2 + 1)*log(x) - sqrt(x**2 + 1) + atanh(sqrt(x**2 + 1))
sympy     : Integral(x*log(x)/sqrt(x**2 + 1), x)
comment   : Unevaluated or not an elementary solution!
time      : 10.0
In [43]:
 
test(Charlwood, 37)
[37] Charlwood
integrand : sin(x)/(sin(x)**2 + 1)
antideriv : -sqrt(2)*atanh(sqrt(2)*cos(x)/2)/2
time      : 59.0
integrator: Timed out!
In [44]:
 
test(Charlwood, 38)
[38] Charlwood
integrand : (x**2 + 1)/((-x**2 + 1)*sqrt(x**4 + 1))
antideriv : sqrt(2)*atanh(sqrt(2)*x/sqrt(x**4 + 1))/2
sympy     : Integral((x**2 + 1)/((-x**2 + 1)*sqrt(x**4 + 1)), x)
comment   : Unevaluated or not an elementary solution!
time      : 1.0
In [45]:
 
test(Charlwood, 39)
[39] Charlwood
integrand : (-x**2 + 1)/((x**2 + 1)*sqrt(x**4 + 1))
antideriv : sqrt(2)*atan(sqrt(2)*x/sqrt(x**4 + 1))/2
sympy     : Integral((-x**2 + 1)/((x**2 + 1)*sqrt(x**4 + 1)), x)
comment   : Unevaluated or not an elementary solution!
time      : 1.0
In [46]:
 
test(Charlwood, 40)
[40] Charlwood
integrand : log(sin(x))/(sin(x) + 1)
antideriv : -x - atanh(cos(x)) - log(sin(x))*cos(x)/(sin(x) + 1)
time      : 73.0
integrator: Timed out!
In [47]:
 
test(Charlwood, 41)
[41] Charlwood
integrand : sqrt(sin(x) + 1)*log(sin(x))
antideriv : -4*atanh(cos(x)/sqrt(sin(x) + 1)) - 2*log(sin(x))*cos(x)/sqrt(sin(x) + 1) + 4*cos(x)/sqrt(sin(x) + 1)
sympy     : Integral(sqrt(sin(x) + 1)*log(sin(x)), x)
comment   : Unevaluated or not an elementary solution!
time      : 35.0
In [48]:
 
test(Charlwood, 42)
[42] Charlwood
integrand : sec(x)/sqrt(sec(x)**4 - 1)
antideriv : -sqrt(2)*atanh(sqrt(2)*sqrt(sec(x)**4 - 1)*cos(x)*cot(x)/2)/2
sympy     : Integral(sec(x)/sqrt(sec(x)**4 - 1), x)
comment   : Unevaluated or not an elementary solution!
time      : 0.0
In [49]:
 
test(Charlwood, 43)
[43] Charlwood
integrand : tan(x)/sqrt(tan(x)**4 + 1)
antideriv : -sqrt(2)*atanh(sqrt(2)*(-tan(x)**2 + 1)/(2*sqrt(tan(x)**4 + 1)))/4
sympy     : Integral(tan(x)/sqrt(tan(x)**4 + 1), x)
comment   : Unevaluated or not an elementary solution!
time      : 0.0
In [50]:
 
test(Charlwood, 44)
[44] Charlwood
integrand : sin(x)/sqrt(-sin(x)**6 + 1)
antideriv : sqrt(3)*atanh(sqrt(3)*(sin(x)**2 + 1)*cos(x)/(2*sqrt(-sin(x)**6 + 1)))/6
time      : 60.0
integrator: Timed out!
In [51]:
 
test(Charlwood, 45)
[45] Charlwood
integrand : sqrt(-sqrt(sec(x) - 1) + sqrt(sec(x) + 1))
antideriv : sqrt(2)*sqrt(sec(x) - 1)*sqrt(sec(x) + 1)*(sqrt(-1 + sqrt(2))*atan(sqrt(-2 + 2*sqrt(2))*(-sqrt(sec(x) - 1) + sqrt(sec(x) + 1) - sqrt(2))/(2*sqrt(-sqrt(sec(x) - 1) + sqrt(sec(x) + 1)))) - sqrt(1 + sqrt(2))*atan(sqrt(2 + 2*sqrt(2))*(-sqrt(sec(x) - 1) + sqrt(sec(x) + 1) - sqrt(2))/(2*sqrt(-sqrt(sec(x) - 1) + sqrt(sec(x) + 1)))) - sqrt(1 + sqrt(2))*atanh(sqrt(-2 + 2*sqrt(2))*sqrt(-sqrt(sec(x) - 1) + sqrt(sec(x) + 1))/(-sqrt(sec(x) - 1) + sqrt(sec(x) + 1) + sqrt(2))) + sqrt(-1 + sqrt(2))*atanh(sqrt(2 + 2*sqrt(2))*sqrt(-sqrt(sec(x) - 1) + sqrt(sec(x) + 1))/(-sqrt(sec(x) - 1) + sqrt(sec(x) + 1) + sqrt(2))))*cot(x)
sympy     : Integral(sqrt(-sqrt(sec(x) - 1) + sqrt(sec(x) + 1)), x)
comment   : Unevaluated or not an elementary solution!
time      : 17.0
In [52]:
 
test(Charlwood, 46)
[46] Charlwood
integrand : x*log(x**2 + 1)*atan(x)**2
antideriv : -x*(log(x**2 + 1) - 3)*atan(x) + (log(x**2 + 1)/4 - 3/2)*log(x**2 + 1) + (-x**2 + (x**2 + 1)*log(x**2 + 1) - 3)*atan(x)**2/2
sympy     : x**2*log(x**2 + 1)*atan(x)**2/2 - x**2*atan(x)**2/2 - x*log(x**2 + 1)*atan(x) + 3*x*atan(x) + log(x**2 + 1)**2/4 + log(x**2 + 1)*atan(x)**2/2 - 3*log(x**2 + 1)/2 - 3*atan(x)**2/2
time      : 19.0
In [53]:
 
test(Charlwood, 47)
[47] Charlwood
integrand : atan(x*sqrt(x**2 + 1))
antideriv : x*atan(x*sqrt(x**2 + 1)) + atan(sqrt(x**2 + 1)/x**2)/2 + sqrt(3)*atanh(sqrt(3)*sqrt(x**2 + 1)/(x**2 + 2))/2
time      : 64.0
integrator: Timed out!
In [54]:
 
test(Charlwood, 48)
[48] Charlwood
integrand : -atan(sqrt(x) - sqrt(x + 1))
antideriv : sqrt(x)/2 - (x + 1)*atan(sqrt(x) - sqrt(x + 1))
time      : 64.0
integrator: Timed out!
In [55]:
 
test(Charlwood, 49)
[49] Charlwood
integrand : asin(x/sqrt(-x**2 + 1))
antideriv : x*asin(x/sqrt(-x**2 + 1)) + atan(sqrt(-2*x**2 + 1))
time      : 67.0
integrator: Timed out!
In [56]:
 
test(Charlwood, 50)
[50] Charlwood
integrand : atan(x*sqrt(-x**2 + 1))
antideriv : x*atan(x*sqrt(-x**2 + 1)) - sqrt(1/2 + sqrt(5)/2)*atan(sqrt(1/2 + sqrt(5)/2)*sqrt(-x**2 + 1)) + sqrt(-1/2 + sqrt(5)/2)*atanh(sqrt(-1/2 + sqrt(5)/2)*sqrt(-x**2 + 1))
time      : 67.0
integrator: Timed out!
In [1]:
x = Symbol('x')
a = Symbol('a')
b = Symbol('b')
n = Symbol('n')
In [2]:
 
Timofeev = dict()
# chapter 1
Timofeev[101] = (1/(a**2-b**2*x**2),  1/a/b*atanh(b*x/a))
Timofeev[102] = (1/(a**2+b**2*x**2),  1/a/b*atan(b*x/a))
Timofeev[103] = (sec(2*a*x), 1/(2*a)*ln(tan(pi/4+a*x)))
Timofeev[104] = (Rational(1,4)/sin(Rational(1,3)*x), Rational(3,4)*ln(tan(x/6)))
Timofeev[105] = (1/cos(Rational(3,4)*pi-2*x), Rational(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), Rational(1,2)*tan(x))
Timofeev[109] = (1/(1+cos(x)), tan(Rational(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*atan(b*sin(x)/a))
Timofeev[113] = (cos(x)/(a**2-b**2*sin(x)**2), 1/a/b*atanh(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), -atanh(exp(x)))
Timofeev[120] = (1/x/ln(x), ln(ln(x)))
Timofeev[121] = (1/x/(1+ln(x)**2), atan(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**Rational(1,2)+x)**2/x**2, 3*ln(x)+x-4*x**Rational(1,2)+4/x**Rational(1,2)-1/x )
Timofeev[125] = ((2-x**Rational(2,3))*(x+x**Rational(1,2))/x**Rational(3,2), 2*ln(x)-Rational(6,7)*x**Rational(7,6)-Rational(3,2)*x**Rational(2,3)+4*x**Rational(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), Rational(1,3)*ln(-3*x**2+2)+Rational(5,6)*6**Rational(1,2)*atanh(Rational(1,2)*6**Rational(1,2)*x) )
Timofeev[128] = ((2*x-5)/(3*x**2+2), Rational(1,3)*ln(3*x**2+2)-Rational(5,6)*6**Rational(1,2)*atan(Rational(1,2)*6**Rational(1,2)*x))
Timofeev[129] = (sin(x)*sin(Rational(1,4)*x), Rational(2,3)*sin(Rational(3,4)*x)-Rational(2,5)*sin(Rational(5,4)*x))
Timofeev[130] = (cos(3*x)*cos(4*x), Rational(1,14)*sin(7*x)+Rational(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, Rational(1,2)*x-Rational(1,2)*sin(x)*cos(x))
Timofeev[133] = (cos(x)**2, Rational(1,2)*x+Rational(1,2)*sin(x)*cos(x))
Timofeev[134] = (sin(x)*cos(x)**3, -Rational(1,4)*cos(x)**4)
Timofeev[135] = (cos(x)**3/sin(x)**4, 1/sin(x)-Rational(1,3)/sin(x)**3)
Timofeev[136] = (1/sin(x)**2/cos(x)**2, tan(x)-cot(x))
Timofeev[137] = (cot(Rational(3,4)*x)**2, -Rational(4,3)*cot(Rational(3,4)*x)-x)
Timofeev[138] = ((1+tan(2*x))**2, Rational(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(Rational(1,2)*x)-x)
Timofeev[143] = ((exp(Rational(1,2)*x)-1)**3*exp(-Rational(1,2)*x), -6*exp(Rational(1,2)*x)+2*exp(-Rational(1,2)*x)+exp(x)+3*x)
Timofeev[144] = (1/(x**2-6*x+5), Rational(1,4)*ln((x-5)/(x-1)))
Timofeev[145] = (x**2/(x**6-6*x**3+13), Rational(1,6)*atan(Rational(1,2)*x**3-Rational(3,2)))
Timofeev[146] = ((x+2)/(x**2-4*x-1), Rational(1,2)*ln(1+4*x-x**2)+4/sqrt(5)*atanh((2-x)/sqrt(5)))
Timofeev[147] = (1/(1+(x+1)**Rational(1,3)), Rational(3,2)*(x+1)**Rational(2,3)-3*(x+1)**Rational(1,3)+3*ln(1+(x+1)**Rational(1,3)))
Timofeev[148] = (1/(a*x+b)/x**Rational(1,2), 2/a**Rational(1,2)/b**Rational(1,2)*atan(a**Rational(1,2)*x**Rational(1,2)/b**Rational(1,2)))
Timofeev[149] = (x**3*(x**2+1)**Rational(1,2), Rational(1,15)*(3*x**4+x**2-2)*(x**2+1)**Rational(1,2))
Timofeev[150] = (x/(a**4-x**4)**Rational(1,2), 1/2*atan(x**2/sqrt(a**4-x**4)))
Timofeev[151] = (1/x/(x**2-a**2)**Rational(1,2), 1/a*atan((x**2-a**2)**Rational(1,2)/a))
Timofeev[152] = (1/x/(a**2-x**2)**Rational(1,2), -1/a*atanh((a**2-x**2)**Rational(1,2)/a))
Timofeev[153] = (1/x/(a**2+x**2)**Rational(1,2), -1/a*atanh(a/(a**2+x**2)**Rational(1,2)))
Timofeev[154] = (1/(-x**2+x+2)**Rational(1,2), asin(Rational(2,3)*x-Rational(1,3)))
Timofeev[155] = (1/(3*x**2-4*x+5)**Rational(1,2), 1/sqrt(3)*asinh((3*x-2)/sqrt(11)))
Timofeev[156] = (1/(-x**2+x)**Rational(1,2), asin(2*x-1))
Timofeev[157] = ((2*x+1)/(-x**2+x+2)**Rational(1,2), 2*asin(Rational(2,3)*x-Rational(1,3))-2*(-x**2+x+2)**Rational(1,2))
Timofeev[158] = (1/x/(2+x-x**2)**Rational(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), Rational(1,10)*10**Rational(1,2)*(x-atan(3*sin(x)*cos(x)/(10**Rational(1,2)+2+3*cos(x)**2))))
Timofeev[162] = ((1-tan(x))/sin(2*x), Rational(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)**Rational(3,4)*sin(2*x), Rational(1,7)*(a**2-4*cos(x)**2)**Rational(7,4))
Timofeev[165] = (sin(2*x)/(a**2-4*sin(x)**2)**Rational(1,3), -Rational(3,8)*(a**2-4*sin(x)**2)**Rational(2,3))
Timofeev[166] = (1/(a**(2*x)-1)**Rational(1,2), 1/ln(a)*atan((a**(2*x)-1)**Rational(1,2)) )
Timofeev[167] = (exp(Rational(1,2)*x)/(exp(x)-1)**Rational(1,2), 2*ln((exp(x)-1)**Rational(1,2)+exp(Rational(1,2)*x)))
Timofeev[168] = (atan(x)**n/(x**2+1), 1/(n+1)*atan(x)**(n+1))
Timofeev[169] = (asin(x/a)**Rational(3,2)/(a**2-x**2)**Rational(1,2), Rational(2,5)*a/(a**2-x**2)**Rational(1,2)*(1-x**2/a**2)**Rational(1,2)*asin(x/a)**Rational(5,2))
Timofeev[170] = (1/acos(x)**3/(-x**2+1)**Rational(1,2), 1/2/acos(x)**2)
Timofeev[171] = (ln(x)**2*x, Rational(1,2)*x**2*(ln(x)**2-ln(x)+Rational(1,2)))
Timofeev[172] = (ln(x)/x**5, -Rational(1,16)*(4*ln(x)+1)/x**4)
Timofeev[173] = (x**2*ln((x-1)/x), Rational(1,3)*x**3*ln((x-1)/x)-Rational(1,3)*ln(x-1)-Rational(1,6)*x*(x+2))
Timofeev[174] = (cos(x)**5, Rational(1,15)*sin(x)*(3*cos(x)**4+4*cos(x)**2+8))
Timofeev[175] = (cos(x)**4*sin(x)**2, Rational(1,6)*sin(x)**3*cos(x)**3+Rational(1,8)*sin(x)**3*cos(x)-Rational(1,16)*sin(x)*cos(x)+Rational(1,16)*x)
Timofeev[176] = (1/sin(x)**5, -Rational(3,8)*atanh(cos(x))-3*cos(x)/(8*sin(x)**2)-cos(x)/(4*sin(x)**4))
Timofeev[177] = (sin(x)*exp(-x), -Rational(1,2)*(cos(x)+sin(x))*exp(-x))
Timofeev[178] = (exp(2*x)*sin(3*x), Rational(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)), Rational(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)-Rational(1,2)*x**2)
Timofeev[183] = (asin(x)/x**2, -asin(x)/x-atanh(sqrt(1-x**2)))
Timofeev[184] = (asin(x)**2, x*asin(x)**2+2*asin(x)*(-x**2+1)**Rational(1,2)-2*x)
Timofeev[185] = (x**2*atan(x)/(x**2+1), x*atan(x)-Rational(1,2)*atan(x)**2-Rational(1,2)*ln(x**2+1))
Timofeev[186] = (acos((x/(x+1))**Rational(1,2)), (x+1)*(acos((x/(x+1))**Rational(1,2))+(1/(x+1))**(1/2)*(x/(x+1))**Rational(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)+Rational(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,  Rational(1,4)*x**4-2*x**3+Rational(27,2)*x**2-108*x+243/(x+3)+405*log(x+3) )
Timofeev[304] = (x/(3+6*x+2*x**2), Rational(1,4)*log(-(3+6*x+2*x**2))+sqrt(3)/2*atanh((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)*atanh((3+2*x)/sqrt(3)) )
Timofeev[306] = ((x-1)/(x**2+5*x+4)**2, (7*x+13)/(9*(x**2+5*x+4))+Rational(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+Rational(14,3)*(x**2+3*x+2)-Rational(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))+Rational(80,2401)*log(x)-Rational(40,2401)*log(7-6*x+2*x**2)+234*sqrt(5)/60025*atan((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)*atan(x**2/a**2) )
# chapter 7
Timofeev[701] = (x**2*cos(x)**5, Rational(1,200)*x*cos(5*x)+(Rational(1,80)*x**2-Rational(1,1000))*sin(5*x)+Rational(5,72)*x*cos(3*x)+(Rational(5,48)*x**2-Rational(5,216))*sin(3*x)+Rational(5,4)*x*cos(x)+(Rational(5,8)*x**2-Rational(5,4))*sin(x) )
Timofeev[702] = (x**3*sin(x)**3, Rational(1,12)*(x**3-Rational(2,3)*x)*cos(3*x)-Rational(1,12)*(x**2-Rational(2,9))*sin(3*x)-Rational(3,4)*(x**3-6*x)*cos(x)+Rational(9,4)*(x**2-2)*sin(x) )
Timofeev[703] = (x**2*sin(x)**6, Rational(5,48)*x**3-1/192*(x**2-Rational(1,18))*sin(6*x)-Rational(1,576)*x*cos(6*x)+Rational(3,64)*(x**2-Rational(1,8))*sin(4*x)+Rational(3,128)*x*cos(4*x)-Rational(15,64)*(x**2-Rational(1,2))*sin(2*x)-Rational(15,64)*x*cos(2*x) )
Timofeev[704] = (x**2*sin(x)**2*cos(x), Rational(1,3)*x**2*sin(x)**3-Rational(1,18)*x*cos(3*x)+Rational(1,54)*sin(3*x)+Rational(1,2)*x*cos(x)-Rational(1,2)*sin(x) )
Timofeev[705] = (x*cos(x)**4/sin(x)**2, -x*cos(x)*(Rational(1,2)*sin(x)+1/sin(x))+Rational(1,4)*sin(x)**2+log(sin(x))-Rational(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)+Rational(5,6)*atanh(sin(x)) )
Timofeev[707] = (x*sin(x)/cos(x)**3, x/(2*cos(x)**2)-Rational(1,2)*tan(x) )
Timofeev[708] = (x*sin(x)**3/cos(x), Rational(1,4)*x*cos(2*x)-Rational(1,8)*sin(2*x)+integrate(x*tan(x),x) )
Timofeev[709] = (x*sin(x)**3/cos(x)**3, x/(2*cos(x)**2)-Rational(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)) )
In [4]:
 
test_all(Timofeev)
[101] Timofeev
integrand : 1/(a**2 - b**2*x**2)
antideriv : atanh(b*x/a)/(a*b)
sympy     : -sqrt(1/(a**2*b**2))*log(-a**2*sqrt(1/(a**2*b**2)) + x)/2 + sqrt(1/(a**2*b**2))*log(a**2*sqrt(1/(a**2*b**2)) + x)/2
comment   : Solution is messy.
time      : 0.0

[102] Timofeev
integrand : 1/(a**2 + b**2*x**2)
antideriv : atan(b*x/a)/(a*b)
sympy     : -sqrt(-1/(a**2*b**2))*log(-a**2*sqrt(-1/(a**2*b**2)) + x)/2 + sqrt(-1/(a**2*b**2))*log(a**2*sqrt(-1/(a**2*b**2)) + x)/2
comment   : Solution is messy.
time      : 0.0

[103] Timofeev
integrand : sec(2*a*x)
antideriv : log(tan(a*x + pi/4))/(2*a)
sympy     : (-log(sin(2*a*x) - 1)/2 + log(sin(2*a*x) + 1)/2)/(2*a)
comment   : Solution is messy.
comment   : Diffs not back to zero. (Check use of abs.)
time      : 0.0

[104] Timofeev
integrand : 1/(4*sin(x/3))
antideriv : 3*log(tan(x/6))/4
sympy     : 3*log(cos(x/3) - 1)/8 - 3*log(cos(x/3) + 1)/8
comment   : Solution is messy.
time      : 0.0

[105] Timofeev
integrand : -1/cos(2*x + pi/4)
antideriv : log(cot(x + 3*pi/8))/2
sympy     : log(tan(x + pi/8) - 1)/2 - log(tan(x + pi/8) + 1)/2
comment   : Solution is messy.
comment   : Diffs not back to zero. (Check use of abs.)
time      : 0.0

[106] Timofeev
integrand : sec(x)*tan(x)
antideriv : sec(x)
sympy     : 1/cos(x)
comment   : Diffs not back to zero. (Check use of abs.)
time      : 0.0

[107] Timofeev
integrand : csc(x)*cot(x)
antideriv : -csc(x)
sympy     : -1/sin(x)
comment   : Diffs not back to zero. (Check use of abs.)
time      : 0.0

[108] Timofeev
integrand : tan(x)/sin(2*x)
antideriv : tan(x)/2
sympy     : Integral(tan(x)/sin(2*x), x)
comment   : Solution is messy.
comment   : Unevaluated or not an elementary solution!
time      : 0.0

[109] Timofeev
integrand : 1/(cos(x) + 1)
antideriv : tan(x/2)
sympy     : tan(x/2)
comment   : Diffs not back to zero. (Check use of abs.)
time      : 0.0

[110] Timofeev
integrand : 1/(-cos(x) + 1)
antideriv : -cot(0.5*x)
sympy     : -1/tan(x/2)
comment   : Diffs not back to zero. (Check use of abs.)
time      : 0.0

[111] Timofeev
integrand : sin(x)/(a - b*cos(x))
antideriv : log(a - b*cos(x))/b
sympy     : log(-a/b + cos(x))/b
time      : 0.0

[112] Timofeev
integrand : cos(x)/(a**2 + b**2*sin(x)**2)
antideriv : atan(b*sin(x)/a)/(a*b)
sympy     : -I*log(-I*a/b + sin(x))/(2*a*b) + I*log(I*a/b + sin(x))/(2*a*b)
comment   : Solution is messy.
comment   : Not a real solution!
time      : 1.0

[113] Timofeev
integrand : cos(x)/(a**2 - b**2*sin(x)**2)
antideriv : atanh(b*sin(x)/a)/(a*b)
sympy     : -log(-a/b + sin(x))/(2*a*b) + log(a/b + sin(x))/(2*a*b)
comment   : Solution is messy.
time      : 1.0

[114] Timofeev
integrand : sin(2*x)/(a**2 + b**2*sin(x)**2)
antideriv : log(a**2 + b**2*sin(x)**2)/b**2
sympy     : Integral(sin(2*x)/(a**2 + b**2*sin(x)**2), x)
comment   : Unevaluated or not an elementary solution!
time      : 15.0

[115] Timofeev
integrand : sin(2*x)/(-a**2 + b**2*sin(x)**2)
antideriv : log(a**2 - b**2*sin(x)**2)/b**2
sympy     : Integral(sin(2*x)/(-a**2 + b**2*sin(x)**2), x)
comment   : Unevaluated or not an elementary solution!
time      : 14.0

[116] Timofeev
integrand : sin(2*x)/(a**2 + b**2*cos(x)**2)
antideriv : -log(a**2 + b**2*cos(x)**2)/b**2
sympy     : Integral(sin(2*x)/(a**2 + b**2*cos(x)**2), x)
comment   : Unevaluated or not an elementary solution!
time      : 11.0

[117] Timofeev
integrand : sin(2*x)/(-a**2 + b**2*cos(x)**2)
antideriv : -log(a**2 - b**2*cos(x)**2)/b**2
sympy     : Integral(sin(2*x)/(-a**2 + b**2*cos(x)**2), x)
comment   : Unevaluated or not an elementary solution!
time      : 11.0

[118] Timofeev
integrand : 1/(-cos(x)**2 + 4)
antideriv : sqrt(3)*(x + atan(sin(x)*cos(x)/(-cos(x)**2 + 2*sqrt(3) + 4)))/6
sympy     : -sqrt(3)*I*log(tan(x/2) - sqrt(3)*I)/12 - sqrt(3)*I*log(tan(x/2) - sqrt(3)*I/3)/12 + sqrt(3)*I*log(tan(x/2) + sqrt(3)*I/3)/12 + sqrt(3)*I*log(tan(x/2) + sqrt(3)*I)/12
comment   : Solution is messy.
comment   : Not a real solution!
comment   : Diffs not back to zero. (Check use of abs.)
time      : 4.0

[119] Timofeev
integrand : exp(x)/(exp(2*x) - 1)
antideriv : -atanh(exp(x))
sympy     : log(exp(x) - 1)/2 - log(exp(x) + 1)/2
comment   : Solution is messy.
time      : 0.0

[120] Timofeev
integrand : 1/(x*log(x))
antideriv : log(log(x))
sympy     : log(log(x))
time      : 0.0

[121] Timofeev
integrand : 1/(x*(log(x)**2 + 1))
antideriv : atan(log(x))
sympy     : -I*log(log(x) - I)/2 + I*log(log(x) + I)/2
comment   : Solution is messy.
comment   : Not a real solution!
time      : 1.0

[122] Timofeev
integrand : 1/(x*(-log(x) + 1))
antideriv : -log(-log(x) + 1)
sympy     : -log(log(x) - 1)
time      : 0.0

[123] Timofeev
integrand : 1/(x*(log(x/a) + 1))
antideriv : log(log(x/a) + 1)
sympy     : log(-log(a) + log(x) + 1)
comment   : Diffs not back to zero. (Check use of abs.)
time      : 0.0

[124] Timofeev
integrand : (-sqrt(x) + x + 1)**2/x**2
antideriv : -4*sqrt(x) + x + 3*log(x) - 1/x + 4/sqrt(x)
sympy     : Integral((-sqrt(x) + x + 1)**2/x**2, x)
comment   : Unevaluated or not an elementary solution!
time      : 0.0

[125] Timofeev
integrand : (sqrt(x) + x)*(-x**(2/3) + 2)/x**(3/2)
antideriv : -6*x**(7/6)/7 - 3*x**(2/3)/2 + 4*sqrt(x) + 2*log(x)
sympy     : -6*x**(7/6)/7 - 3*x**(2/3)/2 + 4*sqrt(x) + 2*log(x)
time      : 7.0

[126] Timofeev
integrand : (2*x - 1)/(2*x + 3)
antideriv : x - 2*log(2*x + 3)
sympy     : x - 2*log(2*x + 3)
time      : 0.0

[127] Timofeev
integrand : (2*x - 5)/(3*x**2 - 2)
antideriv : log(-3*x**2 + 2)/3 + 5*sqrt(6)*atanh(sqrt(6)*x/2)/6
sympy     : (-5*sqrt(6)/12 + 1/3)*log(x - sqrt(6)/3) + (1/3 + 5*sqrt(6)/12)*log(x + sqrt(6)/3)
time      : 0.0

[128] Timofeev
integrand : (2*x - 5)/(3*x**2 + 2)
antideriv : log(3*x**2 + 2)/3 - 5*sqrt(6)*atan(sqrt(6)*x/2)/6
sympy     : log(x**2 + 2/3)/3 - 5*sqrt(6)*atan(sqrt(6)*x/2)/6
time      : 0.0

[129] Timofeev
integrand : sin(x/4)*sin(x)
antideriv : 2*sin(3*x/4)/3 - 2*sin(5*x/4)/5
sympy     : -16*sin(x/4)*cos(x)/15 + 4*sin(x)*cos(x/4)/15
time      : 1.0

[130] Timofeev
integrand : cos(3*x)*cos(4*x)
antideriv : sin(x)/2 + sin(7*x)/14
sympy     : -3*sin(3*x)*cos(4*x)/7 + 4*sin(4*x)*cos(3*x)/7
comment   : Solution is messy.
time      : 1.0

[131] Timofeev
integrand : tan(x)*tan(a - x)
antideriv : -x + log(tan(a)*tan(x) + 1)/tan(a)
sympy     : Integral(tan(x)*tan(a - x), x)
comment   : Unevaluated or not an elementary solution!
time      : 2.0

[132] Timofeev
integrand : sin(x)**2
antideriv : x/2 - sin(x)*cos(x)/2
sympy     : x/2 - sin(x)*cos(x)/2
time      : 0.0

[133] Timofeev
integrand : cos(x)**2
antideriv : x/2 + sin(x)*cos(x)/2
sympy     : x/2 + sin(x)*cos(x)/2
time      : 0.0

[134] Timofeev
integrand : sin(x)*cos(x)**3
antideriv : -cos(x)**4/4
sympy     : -cos(x)**4/4
time      : 0.0

[135] Timofeev
integrand : cos(x)**3/sin(x)**4
antideriv : 1/sin(x) - 1/(3*sin(x)**3)
sympy     : (3*sin(x)**2 - 1)/(3*sin(x)**3)
time      : 0.0

[136] Timofeev
integrand : 1/(sin(x)**2*cos(x)**2)
antideriv : tan(x) - cot(x)
sympy     : -2*cos(2*x)/sin(2*x)
comment   : Diffs not back to zero. (Check use of abs.)
time      : 0.0

[137] Timofeev
integrand : cot(3*x/4)**2
antideriv : -x - 4*cot(3*x/4)/3
sympy     : -x - 4*cos(3*x/4)/(3*sin(3*x/4))
comment   : Diffs not back to zero. (Check use of abs.)
time      : 0.0

[138] Timofeev
integrand : (tan(2*x) + 1)**2
antideriv : -log(cos(2*x)) + tan(2*x)/2
sympy     : log(tan(2*x)**2 + 1)/2 + tan(2*x)/2
time      : 0.0

[139] Timofeev
integrand : (tan(x) - cot(x))**2
antideriv : -4*x + tan(x) - cot(x)
sympy     : -4*x + tan(x) - 1/tan(x)
comment   : Diffs not back to zero. (Check use of abs.)
time      : 1.0

[140] Timofeev
integrand : (-sec(x) + tan(x))**2
antideriv : -x - 2*cot(x/2 + pi/4)
sympy     : -x + 2*sin(x)/cos(x) - 2/cos(x)
comment   : Diffs not back to zero. (Check use of abs.)
time      : 5.0

[141] Timofeev
integrand : sin(x)/(sin(x) + 1)
antideriv : x + cot(x/2 + pi/4)
sympy     : x*tan(x/2)/(tan(x/2) + 1) + x/(tan(x/2) + 1) + 2/(tan(x/2) + 1)
comment   : Solution is messy.
comment   : Diffs not back to zero. (Check use of abs.)
time      : 4.0

[142] Timofeev
integrand : cos(x)/(-cos(x) + 1)
antideriv : -x - cot(x/2)
sympy     : -x - 1/tan(x/2)
comment   : Diffs not back to zero. (Check use of abs.)
time      : 1.0

[143] Timofeev
integrand : (exp(x/2) - 1)**3*exp(-x/2)
antideriv : 3*x - 6*exp(x/2) + exp(x) + 2*exp(-x/2)
sympy     : 3*x - 6*exp(x/2) + exp(x) + 2*exp(-x/2)
time      : 1.0

[144] Timofeev
integrand : 1/(x**2 - 6*x + 5)
antideriv : log((x - 5)/(x - 1))/4
sympy     : log(x - 5)/4 - log(x - 1)/4
time      : 0.0

[145] Timofeev
integrand : x**2/(x**6 - 6*x**3 + 13)
antideriv : atan(x**3/2 - 3/2)/6
sympy     : atan(x**3/2 - 3/2)/6
time      : 0.0

[146] Timofeev
integrand : (x + 2)/(x**2 - 4*x - 1)
antideriv : log(-x**2 + 4*x + 1)/2 + 4*sqrt(5)*atanh(sqrt(5)*(-x + 2)/5)/5
sympy     : (-2*sqrt(5)/5 + 1/2)*log(x - 2 + sqrt(5)) + (1/2 + 2*sqrt(5)/5)*log(x - sqrt(5) - 2)
time      : 0.0

[147] Timofeev
integrand : 1/((x + 1)**(1/3) + 1)
antideriv : 3*(x + 1)**(2/3)/2 - 3*(x + 1)**(1/3) + 3*log((x + 1)**(1/3) + 1)
sympy     : 3*(x + 1)**(2/3)/2 - 3*(x + 1)**(1/3) + 3*log((x + 1)**(1/3) + 1)
time      : 0.0

[148] Timofeev
integrand : 1/(sqrt(x)*(a*x + b))
antideriv : 2*atan(sqrt(a)*sqrt(x)/sqrt(b))/(sqrt(a)*sqrt(b))
sympy     : Piecewise((2*I*acosh(sqrt(b)/(sqrt(a)*sqrt(x + b/a)))/(sqrt(a)*sqrt(b)), Abs(b/(a*(x + b/a))) > 1), (-2*asin(sqrt(b)/(sqrt(a)*sqrt(x + b/a)))/(sqrt(a)*sqrt(b)), True))
comment   : Solution is messy.
comment   : Not a real solution!
comment   : Not a continous solution!
comment   : Diffs not back to zero. (Check use of abs.)
time      : 1.0

[149] Timofeev
integrand : x**3*sqrt(x**2 + 1)
antideriv : sqrt(x**2 + 1)*(x**4/5 + x**2/15 - 2/15)
sympy     : x**4*sqrt(x**2 + 1)/5 + x**2*sqrt(x**2 + 1)/15 - 2*sqrt(x**2 + 1)/15
time      : 1.0

[150] Timofeev
integrand : x/sqrt(a**4 - x**4)
antideriv : 0.5*atan(x**2/sqrt(a**4 - x**4))
sympy     : Piecewise((-I*acosh(x**2/a**2)/2, Abs(x**4/a**4) > 1), (asin(x**2/a**2)/2, True))
comment   : Solution is messy.
comment   : Not a real solution!
comment   : Not a continous solution!
comment   : Diffs not back to zero. (Check use of abs.)
time      : 0.0

[151] Timofeev
integrand : 1/(x*sqrt(-a**2 + x**2))
antideriv : atan(sqrt(-a**2 + x**2)/a)/a
sympy     : Piecewise((I*acosh(a/x)/a, Abs(a**2/x**2) > 1), (-asin(a/x)/a, True))
comment   : Solution is messy.
comment   : Not a real solution!
comment   : Not a continous solution!
comment   : Diffs not back to zero. (Check use of abs.)
time      : 0.0

[152] Timofeev
integrand : 1/(x*sqrt(a**2 - x**2))
antideriv : -atanh(sqrt(a**2 - x**2)/a)/a
sympy     : Piecewise((-acosh(a/x)/a, Abs(a**2/x**2) > 1), (I*asin(a/x)/a, True))
comment   : Solution is messy.
comment   : Not a real solution!
comment   : Not a continous solution!
comment   : Diffs not back to zero. (Check use of abs.)
time      : 0.0

[153] Timofeev
integrand : 1/(x*sqrt(a**2 + x**2))
antideriv : -atanh(a/sqrt(a**2 + x**2))/a
sympy     : -asinh(a/x)/a
comment   : Diffs not back to zero. (Check use of abs.)
time      : 0.0

[154] Timofeev
integrand : 1/sqrt(-x**2 + x + 2)
antideriv : asin(2*x/3 - 1/3)
sympy     : Integral(1/sqrt(-x**2 + x + 2), x)
comment   : Unevaluated or not an elementary solution!
time      : 0.0

[155] Timofeev
integrand : 1/sqrt(3*x**2 - 4*x + 5)
antideriv : sqrt(3)*asinh(sqrt(11)*(3*x - 2)/11)/3
sympy     : Integral(1/sqrt(3*x**2 - 4*x + 5), x)
comment   : Unevaluated or not an elementary solution!
time      : 0.0

[156] Timofeev
integrand : 1/sqrt(-x**2 + x)
antideriv : asin(2*x - 1)
sympy     : Integral(1/sqrt(-x**2 + x), x)
comment   : Solution is messy.
comment   : Unevaluated or not an elementary solution!
time      : 0.0

[157] Timofeev
integrand : (2*x + 1)/sqrt(-x**2 + x + 2)
antideriv : -2*sqrt(-x**2 + x + 2) + 2*asin(2*x/3 - 1/3)
sympy     : Integral((2*x + 1)/sqrt(-x**2 + x + 2), x)
comment   : Unevaluated or not an elementary solution!
time      : 0.0

[158] Timofeev
integrand : 1/(x*sqrt(-x**2 + x + 2))
antideriv : -sqrt(2)*atanh(2*sqrt(2)*sqrt(-x**2 + x + 2)/(x + 4))/2
sympy     : Integral(1/(x*sqrt(-x**2 + x + 2)), x)
comment   : Unevaluated or not an elementary solution!
time      : 0.0

[159] Timofeev
integrand : (-x**2 + x + 2)**(-0.5)/(x - 2)
antideriv : 2*(-x**2 + x + 2)**0.5/(3*x - 6)
sympy     : Integral((-x**2 + x + 2)**(-0.5)/(x - 2), x)
comment   : Unevaluated or not an elementary solution!
time      : 1.0

[160] Timofeev
integrand : (3*sin(x) + 2)/((-cos(x) + 1)*sin(x))
antideriv : (3*sin(x) + 1)/(cos(x) - 1) - atanh(cos(x))
sympy     : log(tan(x/2)) - 3/tan(x/2) - 1/(2*tan(x/2)**2)
comment   : Diffs not back to zero. (Check use of abs.)
time      : 3.0

[161] Timofeev
integrand : 1/(3*cos(x)**2 + 2)
antideriv : sqrt(10)*(x - atan(3*sin(x)*cos(x)/(3*cos(x)**2 + 2 + sqrt(10))))/10
integrator: ValueError: logarithm of a negative number
comment   : Additional constraints? Assume? 

[162] Timofeev
integrand : (-tan(x) + 1)/sin(2*x)
antideriv : log(tan(x))/2 - tan(x)/2
sympy     : Integral((-tan(x) + 1)/sin(2*x), x)
comment   : Unevaluated or not an elementary solution!
time      : 1.0

[163] Timofeev
integrand : (tan(x)**2 + 1)/(-tan(x)**2 + 1)
antideriv : 0.5*log((tan(x) + 1)/(-tan(x) + 1))
sympy     : -log(tan(x) - 1)/2 + log(tan(x) + 1)/2
time      : 0.0

[164] Timofeev
integrand : (a**2 - 4*cos(x)**2)**(3/4)*sin(2*x)
antideriv : (a**2 - 4*cos(x)**2)**(7/4)/7
integrator: RuntimeError maximum recursion depth exceeded while calling a Python object

[165] Timofeev
integrand : sin(2*x)/(a**2 - 4*sin(x)**2)**(1/3)
antideriv : -3*(a**2 - 4*sin(x)**2)**(2/3)/8
sympy     : Integral(sin(2*x)/(a**2 - 4*sin(x)**2)**(1/3), x)
comment   : Unevaluated or not an elementary solution!
time      : 3.0

[166] Timofeev
integrand : 1/sqrt(a**(2*x) - 1)
antideriv : atan(sqrt(a**(2*x) - 1))/log(a)
sympy     : Integral(1/sqrt(a**(2*x) - 1), x)
comment   : Unevaluated or not an elementary solution!
time      : 0.0

[167] Timofeev
integrand : exp(x/2)/sqrt(exp(x) - 1)
antideriv : 2*log(sqrt(exp(x) - 1) + exp(x/2))
sympy     : Integral(exp(x/2)/sqrt(exp(x) - 1), x)
comment   : Unevaluated or not an elementary solution!
time      : 0.0

[168] Timofeev
integrand : atan(x)**n/(x**2 + 1)
antideriv : atan(x)**(n + 1)/(n + 1)
sympy     : Integral(atan(x)**n/(x**2 + 1), x)
comment   : Unevaluated or not an elementary solution!
time      : 101.0
You are wasting my time!

[169] Timofeev
integrand : asin(x/a)**(3/2)/sqrt(a**2 - x**2)
antideriv : 2*a*sqrt(1 - x**2/a**2)*asin(x/a)**(5/2)/(5*sqrt(a**2 - x**2))
sympy     : 0
comment   : Diffs not back to zero. (Check use of abs.)
time      : 149.0
You are wasting my time!

[170] Timofeev
integrand : 1/(sqrt(-x**2 + 1)*acos(x)**3)
antideriv : 0.5/acos(x)**2
sympy     : 1/(2*acos(x)**2)
time      : 7.0

[171] Timofeev
integrand : x*log(x)**2
antideriv : x**2*(log(x)**2 - log(x) + 1/2)/2
sympy     : x**2*log(x)**2/2 - x**2*log(x)/2 + x**2/4
time      : 0.0

[172] Timofeev
integrand : log(x)/x**5
antideriv : (-log(x)/4 - 1/16)/x**4
sympy     : -log(x)/(4*x**4) - 1/(16*x**4)
time      : 0.0

[173] Timofeev
integrand : x**2*log((x - 1)/x)
antideriv : x**3*log((x - 1)/x)/3 - x*(x + 2)/6 - log(x - 1)/3
sympy     : x**3*log(1 - 1/x)/3 - x**2/6 - x/3 - log(x - 1)/3
time      : 1.0

[174] Timofeev
integrand : cos(x)**5
antideriv : (3*cos(x)**4 + 4*cos(x)**2 + 8)*sin(x)/15
sympy     : sin(x)**5/5 - 2*sin(x)**3/3 + sin(x)
comment   : Diffs not back to zero. (Check use of abs.)
time      : 0.0

[175] Timofeev
integrand : sin(x)**2*cos(x)**4
antideriv : x/16 + sin(x)**3*cos(x)**3/6 + sin(x)**3*cos(x)/8 - sin(x)*cos(x)/16
sympy     : x/16 - sin(x)*cos(x)**5/6 + sin(x)*cos(x)**3/24 + sin(x)*cos(x)/16
time      : 0.0

[176] Timofeev
integrand : sin(x)**(-5)
antideriv : -3*atanh(cos(x))/8 - 3*cos(x)/(8*sin(x)**2) - cos(x)/(4*sin(x)**4)
sympy     : (3*cos(x)**3 - 5*cos(x))/(8*cos(x)**4 - 16*cos(x)**2 + 8) + 3*log(cos(x) - 1)/16 - 3*log(cos(x) + 1)/16
comment   : Diffs not back to zero. (Check use of abs.)
time      : 0.0

[177] Timofeev
integrand : exp(-x)*sin(x)
antideriv : (-sin(x)/2 - cos(x)/2)*exp(-x)
sympy     : -exp(-x)*sin(x)/2 - exp(-x)*cos(x)/2
time      : 0.0

[178] Timofeev
integrand : exp(2*x)*sin(3*x)
antideriv : (2*sin(3*x) - 3*cos(3*x))*exp(2*x)/13
sympy     : 2*exp(2*x)*sin(3*x)/13 - 3*exp(2*x)*cos(3*x)/13
time      : 0.0

[179] Timofeev
integrand : a**x*cos(x)
antideriv : a**x*(log(a)*cos(x) + sin(x))/(log(a)**2 + 1)
sympy     : a**x*log(a)*cos(x)/(log(a)**2 + 1) + a**x*sin(x)/(log(a)**2 + 1)
time      : 1.0

[180] Timofeev
integrand : cos(log(x))
antideriv : x*(sin(log(x)) + cos(log(x)))/2
sympy     : x*sin(log(x))/2 + x*cos(log(x))/2
time      : 0.0

[181] Timofeev
integrand : sec(x)**2*log(cos(x))
antideriv : -x + log(cos(x))*tan(x) + tan(x)
sympy     : Integral(sec(x)**2*log(cos(x)), x)
comment   : Unevaluated or not an elementary solution!
time      : 2809.0
You are wasting my time!

[182] Timofeev
integrand : x*tan(x)**2
antideriv : -x**2/2 + x*tan(x) + log(cos(x))
sympy     : -x**2/2 + x*tan(x) - log(tan(x)**2 + 1)/2
time      : 5.0

[183] Timofeev
integrand : asin(x)/x**2
antideriv : -atanh(sqrt(-x**2 + 1)) - asin(x)/x
sympy     : Integral(asin(x)/x**2, x)
comment   : Unevaluated or not an elementary solution!
time      : 1.0

[184] Timofeev
integrand : asin(x)**2
antideriv : x*asin(x)**2 - 2*x + 2*sqrt(-x**2 + 1)*asin(x)
sympy     : x*asin(x)**2 - 2*x + 2*sqrt(-x**2 + 1)*asin(x)
time      : 1.0

[185] Timofeev
integrand : x**2*atan(x)/(x**2 + 1)
antideriv : x*atan(x) - log(x**2 + 1)/2 - atan(x)**2/2
sympy     : x*atan(x) - log(x**2 + 1)/2 - atan(x)**2/2
time      : 4.0

[186] Timofeev
integrand : acos(sqrt(x/(x + 1)))
antideriv : (x + 1)*(sqrt(x/(x + 1))*(1/(x + 1))**0.5 + acos(sqrt(x/(x + 1))))
sympy     : Integral(acos(sqrt(x/(x + 1))), x)
comment   : Unevaluated or not an elementary solution!
time      : 239.0
You are wasting my time!

[701] Timofeev
integrand : x**2*cos(x)**5
antideriv : 5*x*cos(x)/4 + 5*x*cos(3*x)/72 + x*cos(5*x)/200 + (x**2/80 - 1/1000)*sin(5*x) + (5*x**2/48 - 5/216)*sin(3*x) + (5*x**2/8 - 5/4)*sin(x)
sympy     : 8*x**2*sin(x)**5/15 + 4*x**2*sin(x)**3*cos(x)**2/3 + x**2*sin(x)*cos(x)**4 + 16*x*sin(x)**4*cos(x)/15 + 104*x*sin(x)**2*cos(x)**3/45 + 298*x*cos(x)**5/225 - 4144*sin(x)**5/3375 - 1712*sin(x)**3*cos(x)**2/675 - 298*sin(x)*cos(x)**4/225
time      : 9.0

[702] Timofeev
integrand : x**3*sin(x)**3
antideriv : -(x**2/12 - 1/54)*sin(3*x) + (9*x**2/4 - 9/2)*sin(x) + (x**3/12 - x/18)*cos(3*x) - (3*x**3/4 - 9*x/2)*cos(x)
sympy     : -x**3*sin(x)**2*cos(x) - 2*x**3*cos(x)**3/3 + 7*x**2*sin(x)**3/3 + 2*x**2*sin(x)*cos(x)**2 + 14*x*sin(x)**2*cos(x)/3 + 40*x*cos(x)**3/9 - 122*sin(x)**3/27 - 40*sin(x)*cos(x)**2/9
time      : 4.0

[703] Timofeev
integrand : x**2*sin(x)**6
antideriv : 5*x**3/48 - 15*x*cos(2*x)/64 + 3*x*cos(4*x)/128 - x*cos(6*x)/576 - (0.00520833333333333*x**2 - 0.000289351851851852)*sin(6*x) + (3*x**2/64 - 3/512)*sin(4*x) - (15*x**2/64 - 15/128)*sin(2*x)
sympy     : 5*x**3*sin(x)**6/48 + 5*x**3*sin(x)**4*cos(x)**2/16 + 5*x**3*sin(x)**2*cos(x)**4/16 + 5*x**3*cos(x)**6/48 - 11*x**2*sin(x)**5*cos(x)/16 - 5*x**2*sin(x)**3*cos(x)**3/6 - 5*x**2*sin(x)*cos(x)**5/16 + 299*x*sin(x)**6/1152 + 35*x*sin(x)**4*cos(x)**2/384 - 125*x*sin(x)**2*cos(x)**4/384 - 245*x*cos(x)**6/1152 + 299*sin(x)**5*cos(x)/1152 + 25*sin(x)**3*cos(x)**3/54 + 245*sin(x)*cos(x)**5/1152
comment   : Solution is messy.
time      : 17.0

[704] Timofeev
integrand : x**2*sin(x)**2*cos(x)
antideriv : x**2*sin(x)**3/3 + x*cos(x)/2 - x*cos(3*x)/18 - sin(x)/2 + sin(3*x)/54
sympy     : x**2*sin(x)**3/3 + 2*x*sin(x)**2*cos(x)/3 + 4*x*cos(x)**3/9 - 14*sin(x)**3/27 - 4*sin(x)*cos(x)**2/9
time      : 2.0

[705] Timofeev
integrand : x*cos(x)**4/sin(x)**2
antideriv : -3*x**2/4 - x*(sin(x)/2 + 1/sin(x))*cos(x) + log(sin(x)) + sin(x)**2/4
sympy     : -3*x**2*tan(x/2)**5/(4*tan(x/2)**5 + 8*tan(x/2)**3 + 4*tan(x/2)) - 6*x**2*tan(x/2)**3/(4*tan(x/2)**5 + 8*tan(x/2)**3 + 4*tan(x/2)) - 3*x**2*tan(x/2)/(4*tan(x/2)**5 + 8*tan(x/2)**3 + 4*tan(x/2)) + 2*x*tan(x/2)**6/(4*tan(x/2)**5 + 8*tan(x/2)**3 + 4*tan(x/2)) + 6*x*tan(x/2)**4/(4*tan(x/2)**5 + 8*tan(x/2)**3 + 4*tan(x/2)) - 6*x*tan(x/2)**2/(4*tan(x/2)**5 + 8*tan(x/2)**3 + 4*tan(x/2)) - 2*x/(4*tan(x/2)**5 + 8*tan(x/2)**3 + 4*tan(x/2)) - 2*log(tan(x/2)**4 + 2*tan(x/2)**2 + 1)*tan(x/2)**5/(4*tan(x/2)**5 + 8*tan(x/2)**3 + 4*tan(x/2)) - 4*log(tan(x/2)**4 + 2*tan(x/2)**2 + 1)*tan(x/2)**3/(4*tan(x/2)**5 + 8*tan(x/2)**3 + 4*tan(x/2)) - 2*log(tan(x/2)**4 + 2*tan(x/2)**2 + 1)*tan(x/2)/(4*tan(x/2)**5 + 8*tan(x/2)**3 + 4*tan(x/2)) + 4*log(tan(x/2))*tan(x/2)**5/(4*tan(x/2)**5 + 8*tan(x/2)**3 + 4*tan(x/2)) + 8*log(tan(x/2))*tan(x/2)**3/(4*tan(x/2)**5 + 8*tan(x/2)**3 + 4*tan(x/2)) + 4*log(tan(x/2))*tan(x/2)/(4*tan(x/2)**5 + 8*tan(x/2)**3 + 4*tan(x/2)) + 4*tan(x/2)**3/(4*tan(x/2)**5 + 8*tan(x/2)**3 + 4*tan(x/2))
comment   : Solution is messy.
comment   : Diffs not back to zero. (Check use of abs.)
time      : 23.0

[706] Timofeev
integrand : x*sin(x)**3/cos(x)**4
antideriv : x*(-1/cos(x) + 1/(3*cos(x)**3)) - sin(x)/(6*cos(x)**2) + 5*atanh(sin(x))/6
sympy     : 4*x*tan(x/2)**6/(6*tan(x/2)**6 - 18*tan(x/2)**4 + 18*tan(x/2)**2 - 6) - 12*x*tan(x/2)**4/(6*tan(x/2)**6 - 18*tan(x/2)**4 + 18*tan(x/2)**2 - 6) - 12*x*tan(x/2)**2/(6*tan(x/2)**6 - 18*tan(x/2)**4 + 18*tan(x/2)**2 - 6) + 4*x/(6*tan(x/2)**6 - 18*tan(x/2)**4 + 18*tan(x/2)**2 - 6) - 5*log(tan(x/2) - 1)*tan(x/2)**6/(6*tan(x/2)**6 - 18*tan(x/2)**4 + 18*tan(x/2)**2 - 6) + 15*log(tan(x/2) - 1)*tan(x/2)**4/(6*tan(x/2)**6 - 18*tan(x/2)**4 + 18*tan(x/2)**2 - 6) - 15*log(tan(x/2) - 1)*tan(x/2)**2/(6*tan(x/2)**6 - 18*tan(x/2)**4 + 18*tan(x/2)**2 - 6) + 5*log(tan(x/2) - 1)/(6*tan(x/2)**6 - 18*tan(x/2)**4 + 18*tan(x/2)**2 - 6) + 5*log(tan(x/2) + 1)*tan(x/2)**6/(6*tan(x/2)**6 - 18*tan(x/2)**4 + 18*tan(x/2)**2 - 6) - 15*log(tan(x/2) + 1)*tan(x/2)**4/(6*tan(x/2)**6 - 18*tan(x/2)**4 + 18*tan(x/2)**2 - 6) + 15*log(tan(x/2) + 1)*tan(x/2)**2/(6*tan(x/2)**6 - 18*tan(x/2)**4 + 18*tan(x/2)**2 - 6) - 5*log(tan(x/2) + 1)/(6*tan(x/2)**6 - 18*tan(x/2)**4 + 18*tan(x/2)**2 - 6) - 2*tan(x/2)**5/(6*tan(x/2)**6 - 18*tan(x/2)**4 + 18*tan(x/2)**2 - 6) + 2*tan(x/2)/(6*tan(x/2)**6 - 18*tan(x/2)**4 + 18*tan(x/2)**2 - 6)
comment   : Solution is messy.
comment   : Diffs not back to zero. (Check use of abs.)
time      : 18.0

[707] Timofeev
integrand : x*sin(x)/cos(x)**3
antideriv : x/(2*cos(x)**2) - tan(x)/2
sympy     : x*tan(x/2)**4/(2*tan(x/2)**4 - 4*tan(x/2)**2 + 2) + 2*x*tan(x/2)**2/(2*tan(x/2)**4 - 4*tan(x/2)**2 + 2) + x/(2*tan(x/2)**4 - 4*tan(x/2)**2 + 2) + 2*tan(x/2)**3/(2*tan(x/2)**4 - 4*tan(x/2)**2 + 2) - 2*tan(x/2)/(2*tan(x/2)**4 - 4*tan(x/2)**2 + 2)
comment   : Solution is messy.
comment   : Diffs not back to zero. (Check use of abs.)
time      : 6.0

[708] Timofeev
integrand : x*sin(x)**3/cos(x)
antideriv : x*cos(2*x)/4 - sin(2*x)/8 + Integral(x*tan(x), x)
sympy     : Integral(x*sin(x)**3/cos(x), x)
comment   : Unevaluated or not an elementary solution!
time      : 10.0

[709] Timofeev
integrand : x*sin(x)**3/cos(x)**3
antideriv : x/(2*cos(x)**2) - tan(x)/2 - Integral(x*tan(x), x)
sympy     : Integral(x*sin(x)**3/cos(x)**3, x)
comment   : Unevaluated or not an elementary solution!
time      : 11.0

[710] Timofeev
integrand : (2*x + sin(2*x))/(x*sin(x) + cos(x))**2
antideriv : -2*cos(x)/(x*sin(x) + cos(x))
sympy     : Integral((2*x + sin(2*x))/(x*sin(x) + cos(x))**2, x)
comment   : Unevaluated or not an elementary solution!
time      : 921.0
You are wasting my time!

[711] Timofeev
integrand : x**2/(x*cos(x) - sin(x))**2
antideriv : (x*sin(x) + cos(x))/(x*cos(x) - sin(x))
sympy     : Integral(x**2/(x*cos(x) - sin(x))**2, x)
comment   : Unevaluated or not an elementary solution!
time      : 81.0
You are wasting my time!

[301] Timofeev
integrand : 1/((x - 2)**3*(x + 1)**2)
antideriv : log((x - 2)/(x + 1))/27 + (2*x**2 - 5*x - 1)/((x - 2)**2*(18*x + 18))
sympy     : (2*x**2 - 5*x - 1)/(18*x**3 - 54*x**2 + 72) + log(x - 2)/27 - log(x + 1)/27
time      : 5.0

[302] Timofeev
integrand : 1/((x + 2)**3*(x + 3)**4)
antideriv : 10*log((x + 2)/(x + 3)) + (60*x**4 + 630*x**3 + 2450*x**2 + 4175*x + 2627)/(6*(x + 2)**2*(x + 3)**3)
sympy     : (60*x**4 + 630*x**3 + 2450*x**2 + 4175*x + 2627)/(6*x**5 + 78*x**4 + 402*x**3 + 1026*x**2 + 1296*x + 648) + 10*log(x + 2) - 10*log(x + 3)
time      : 4.0

[303] Timofeev
integrand : x**5/(x + 3)**2
antideriv : x**4/4 - 2*x**3 + 27*x**2/2 - 108*x + 405*log(x + 3) + 243/(x + 3)
sympy     : x**4/4 - 2*x**3 + 27*x**2/2 - 108*x + 405*log(x + 3) + 243/(x + 3)
time      : 0.0

[304] Timofeev
integrand : x/(2*x**2 + 6*x + 3)
antideriv : log(-2*x**2 - 6*x - 3)/4 + sqrt(3)*atanh(sqrt(3)*(2*x + 3)/3)/2
sympy     : (-sqrt(3)/4 + 1/4)*log(x - sqrt(3)/2 + 3/2) + (1/4 + sqrt(3)/4)*log(x + sqrt(3)/2 + 3/2)
time      : 1.0

[305] Timofeev
integrand : (2*x - 3)/(2*x**2 + 6*x + 3)**3
antideriv : sqrt(3)*atanh(sqrt(3)*(2*x + 3)/3)/3 + (-8*x**3 - 36*x**2 - 44*x - 13)/(4*(2*x**2 + 6*x + 3)**2)
sympy     : -(8*x**3 + 36*x**2 + 44*x + 13)/(16*x**4 + 96*x**3 + 192*x**2 + 144*x + 36) - sqrt(3)*log(x - sqrt(3)/2 + 3/2)/6 + sqrt(3)*log(x + sqrt(3)/2 + 3/2)/6
time      : 1.0

[306] Timofeev
integrand : (x - 1)/(x**2 + 5*x + 4)**2
antideriv : (7*x + 13)/(9*x**2 + 45*x + 36) + 0.259259259259259*log((x + 1)/(x + 4))
sympy     : (7*x + 13)/(9*x**2 + 45*x + 36) + 7*log(x + 1)/27 - 7*log(x + 4)/27
time      : 1.0

[307] Timofeev
integrand : (x**2 + 3*x + 2)**(-5)
antideriv : (2*x + 3)*(14*x**2/3 + 14*x + 140*(x**2 + 3*x + 2)**3 - 70*(x**2 + 3*x + 2)**2/3 + 25/3)/(4*(x**2 + 3*x + 2)**4) + 70*log((x + 1)/(x + 2))
sympy     : (840*x**7 + 8820*x**6 + 38920*x**5 + 93450*x**4 + 131768*x**3 + 109116*x**2 + 49176*x + 9315)/(12*x**8 + 144*x**7 + 744*x**6 + 2160*x**5 + 3852*x**4 + 4320*x**3 + 2976*x**2 + 1152*x + 192) + 70*log(x + 1) - 70*log(x + 2)
time      : 1.0

[308] Timofeev
integrand : 1/(x**3*(2*x**2 - 6*x + 7)**2)
antideriv : (-18*x + 82)/(3430*x**2 - 10290*x + 12005) + 80*log(x)/2401 - 40*log(2*x**2 - 6*x + 7)/2401 + 234*sqrt(5)*atan(sqrt(5)*(2*x - 3)/5)/60025 - 12/(343*x) - 1/(98*x**2)
sympy     : 80*log(x)/2401 - 40*log(x**2 - 3*x + 7/2)/2401 + 234*sqrt(5)*atan(2*sqrt(5)*x/5 - 3*sqrt(5)/5)/60025 - (276*x**3 - 814*x**2 + 630*x + 245)/(6860*x**4 - 20580*x**3 + 24010*x**2)
time      : 3.0

[309] Timofeev
integrand : x**9/(x**2 + 3*x + 2)**5
antideriv : -1471*log(x + 1) + 1472*log(x + 2) + (-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)
sympy     : -(17496*x**7 + 184200*x**6 + 813888*x**5 + 1955853*x**4 + 2759400*x**3 + 2286008*x**2 + 1030560*x + 195280)/(12*x**8 + 144*x**7 + 744*x**6 + 2160*x**5 + 3852*x**4 + 4320*x**3 + 2976*x**2 + 1152*x + 192) - 1471*log(x + 1) + 1472*log(x + 2)
time      : 2.0

[310] Timofeev
integrand : (2*x + 1)**2/(2*x**2 + 5*x + 3)**5
antideriv : (-10*x - 11)/(4*(2*x**2 + 5*x + 3)**4) + (124*x + 155)*(-20*x**2 - 50*x + 120*(2*x**2 + 5*x + 3)**2 - 29)/(6*(2*x**2 + 5*x + 3)**3) + 2480*log((x + 1)/(2*x + 3))
sympy     : (238080*x**7 + 2083200*x**6 + 7757440*x**5 + 15934000*x**4 + 19495776*x**3 + 14209160*x**2 + 5712464*x + 977397)/(192*x**8 + 1920*x**7 + 8352*x**6 + 20640*x**5 + 31692*x**4 + 30960*x**3 + 18792*x**2 + 6480*x + 972) + 2480*log(x + 1) - 2480*log(x + 3/2)
time      : 1.0

[311] Timofeev
integrand : (a - b*x**2)**3/x**7
antideriv : -a**3/(6*x**6) + 3*a**2*b/(4*x**4) - 3*a*b**2/(2*x**2) - b**3*log(x)
sympy     : -b**3*log(x) - (2*a**3 - 9*a**2*b*x**2 + 18*a*b**2*x**4)/(12*x**6)
time      : 1.0

[312] Timofeev
integrand : x**13/(a**4 + x**4)**5
antideriv : x**2*(-15*a**12 - 55*a**8*x**4 - 73*a**4*x**8 + 15*x**12)/(768*a**4*(a**4 + x**4)**4) + 5*atan(x**2/a**2)/(256*a**6)
sympy     : -5*sqrt(-1/a**12)*log(-a**8*sqrt(-1/a**12) + x**2)/512 + 5*sqrt(-1/a**12)*log(a**8*sqrt(-1/a**12) + x**2)/512 + (-15*a**12*x**2 - 55*a**8*x**6 - 73*a**4*x**10 + 15*x**14)/(768*a**20 + 3072*a**16*x**4 + 4608*a**12*x**8 + 3072*a**8*x**12 + 768*a**4*x**16)
comment   : Solution is messy.
time      : 3.0

total time: 5504.0