Chapter 4
Listing of integrals solved by CAS which has no known antiderivatives

4.1 Test file Number [5]

4.1.1 Maxima

Integral number [145] \[ \int x \cos (k \csc (x)) \cot (x) \csc (x) \, dx \]

[C]   time = 0.222839 (sec), size = 240 ,normalized size = 21.82 \[ -\frac {{\left (x e^{\left (\frac {4 \, k \cos \left (2 \, x\right ) \cos \left (x\right )}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1} + \frac {4 \, k \sin \left (2 \, x\right ) \sin \left (x\right )}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}\right )} + x e^{\left (\frac {4 \, k \cos \left (x\right )}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}\right )}\right )} e^{\left (-\frac {2 \, k \cos \left (2 \, x\right ) \cos \left (x\right )}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1} - \frac {2 \, k \sin \left (2 \, x\right ) \sin \left (x\right )}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1} - \frac {2 \, k \cos \left (x\right )}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}\right )} \sin \left (\frac {2 \, {\left (k \cos \left (x\right ) \sin \left (2 \, x\right ) - k \cos \left (2 \, x\right ) \sin \left (x\right ) + k \sin \left (x\right )\right )}}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}\right )}{2 \, k} \]

[In]

integrate(x*cos(x)*cos(k/sin(x))/sin(x)^2,x, algorithm=""maxima"")

output

-1/2*(x*e^(4*k*cos(2*x)*cos(x)/(cos(2*x)^2 + sin(2*x)^2 - 2*cos(2*x) + 1) + 4* 
k*sin(2*x)*sin(x)/(cos(2*x)^2 + sin(2*x)^2 - 2*cos(2*x) + 1)) + x*e^(4*k*cos(x 
)/(cos(2*x)^2 + sin(2*x)^2 - 2*cos(2*x) + 1)))*e^(-2*k*cos(2*x)*cos(x)/(cos(2* 
x)^2 + sin(2*x)^2 - 2*cos(2*x) + 1) - 2*k*sin(2*x)*sin(x)/(cos(2*x)^2 + sin(2* 
x)^2 - 2*cos(2*x) + 1) - 2*k*cos(x)/(cos(2*x)^2 + sin(2*x)^2 - 2*cos(2*x) + 1) 
)*sin(2*(k*cos(x)*sin(2*x) - k*cos(2*x)*sin(x) + k*sin(x))/(cos(2*x)^2 + sin(2 
*x)^2 - 2*cos(2*x) + 1))/k
 

4.2 Test file Number [57]

4.2.1 Mathematica

Integral number [166] \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x} \, dx \]

[B]   time = 0.128514 (sec), size = 72 ,normalized size = 3.13 \[ \frac {x (f x)^m \left (-b n \, _3F_2\left (1,1+m,1+m;2+m,2+m;-\frac {e x}{d}\right )+(1+m) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{d (1+m)^2} \]

[In]

Integrate[((f*x)^m*(a + b*Log[c*x^n]))/(d + e*x),x]

output

(x*(f*x)^m*(-(b*n*HypergeometricPFQ[{1, 1 + m, 1 + m}, {2 + m, 2 + m}, -((e*x) 
/d)]) + (1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, -((e*x)/d)]*(a + b*Log[c*x^ 
n])))/(d*(1 + m)^2)
 

Integral number [167] \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx \]

[B]   time = 0.117558 (sec), size = 72 ,normalized size = 3.13 \[ \frac {x (f x)^m \left (-b n \, _3F_2\left (2,1+m,1+m;2+m,2+m;-\frac {e x}{d}\right )+(1+m) \operatorname {Hypergeometric2F1}\left (2,1+m,2+m,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{d^2 (1+m)^2} \]

[In]

Integrate[((f*x)^m*(a + b*Log[c*x^n]))/(d + e*x)^2,x]

output

(x*(f*x)^m*(-(b*n*HypergeometricPFQ[{2, 1 + m, 1 + m}, {2 + m, 2 + m}, -((e*x) 
/d)]) + (1 + m)*Hypergeometric2F1[2, 1 + m, 2 + m, -((e*x)/d)]*(a + b*Log[c*x^ 
n])))/(d^2*(1 + m)^2)
 

Integral number [168] \[ \int x (a+b x)^m \log \left (c x^n\right ) \, dx \]

[B]   time = 0.16101 (sec), size = 173 ,normalized size = 11.53 \[ \frac {(a+b x)^m \left (1+\frac {b x}{a}\right )^{-m} \left (-n \left (2 a b x \left (1+\frac {b x}{a}\right )^m+b^2 x^2 \left (1+\frac {b x}{a}\right )^m+a^2 \left (-1+\left (1+\frac {b x}{a}\right )^m\right )\right )+a b (2+m) n x \, _3F_2\left (1,1,-1-m;2,2;-\frac {b x}{a}\right )+\left (a b m x \left (1+\frac {b x}{a}\right )^m+b^2 (1+m) x^2 \left (1+\frac {b x}{a}\right )^m-a^2 \left (-1+\left (1+\frac {b x}{a}\right )^m\right )\right ) \log \left (c x^n\right )\right )}{b^2 (1+m) (2+m)} \]

[In]

Integrate[x*(a + b*x)^m*Log[c*x^n],x]

output

((a + b*x)^m*(-(n*(2*a*b*x*(1 + (b*x)/a)^m + b^2*x^2*(1 + (b*x)/a)^m + a^2*(-1 
 + (1 + (b*x)/a)^m))) + a*b*(2 + m)*n*x*HypergeometricPFQ[{1, 1, -1 - m}, {2, 
2}, -((b*x)/a)] + (a*b*m*x*(1 + (b*x)/a)^m + b^2*(1 + m)*x^2*(1 + (b*x)/a)^m - 
 a^2*(-1 + (1 + (b*x)/a)^m))*Log[c*x^n]))/(b^2*(1 + m)*(2 + m)*(1 + (b*x)/a)^m 
)
 

Integral number [170] \[ \int \frac {(a+b x)^m \log \left (c x^n\right )}{x} \, dx \]

[B]   time = 0.0419056 (sec), size = 89 ,normalized size = 5.24 \[ \frac {\left (1+\frac {a}{b x}\right )^{-m} (a+b x)^m \left (-n \, _3F_2\left (-m,-m,-m;1-m,1-m;-\frac {a}{b x}\right )+m \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,-\frac {a}{b x}\right ) \log \left (c x^n\right )\right )}{m^2} \]

[In]

Integrate[((a + b*x)^m*Log[c*x^n])/x,x]

output

((a + b*x)^m*(-(n*HypergeometricPFQ[{-m, -m, -m}, {1 - m, 1 - m}, -(a/(b*x))]) 
 + m*Hypergeometric2F1[-m, -m, 1 - m, -(a/(b*x))]*Log[c*x^n]))/(m^2*(1 + a/(b* 
x))^m)
 

Integral number [322] \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx \]

[B]   time = 0.703872 (sec), size = 108 ,normalized size = 4.32 \[ \frac {x (f x)^m \left (-b n \, _3F_2\left (1,\frac {1}{2}+\frac {m}{2},\frac {1}{2}+\frac {m}{2};\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};-\frac {e x^2}{d}\right )+(1+m) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {e x^2}{d}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{d (1+m)^2} \]

[In]

Integrate[((f*x)^m*(a + b*Log[c*x^n]))/(d + e*x^2),x]

output

(x*(f*x)^m*(-(b*n*HypergeometricPFQ[{1, 1/2 + m/2, 1/2 + m/2}, {3/2 + m/2, 3/2 
 + m/2}, -((e*x^2)/d)]) + (1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, - 
((e*x^2)/d)]*(a + b*Log[c*x^n])))/(d*(1 + m)^2)
 

Integral number [323] \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx \]

[B]   time = 0.136591 (sec), size = 108 ,normalized size = 4.32 \[ \frac {x (f x)^m \left (-b n \, _3F_2\left (2,\frac {1}{2}+\frac {m}{2},\frac {1}{2}+\frac {m}{2};\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};-\frac {e x^2}{d}\right )+(1+m) \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},\frac {3+m}{2},-\frac {e x^2}{d}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{d^2 (1+m)^2} \]

[In]

Integrate[((f*x)^m*(a + b*Log[c*x^n]))/(d + e*x^2)^2,x]

output

(x*(f*x)^m*(-(b*n*HypergeometricPFQ[{2, 1/2 + m/2, 1/2 + m/2}, {3/2 + m/2, 3/2 
 + m/2}, -((e*x^2)/d)]) + (1 + m)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, - 
((e*x^2)/d)]*(a + b*Log[c*x^n])))/(d^2*(1 + m)^2)
 

Integral number [406] \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx \]

[B]   time = 0.0916891 (sec), size = 87 ,normalized size = 3.78 \[ \frac {x^4 \left (-b n \, _3F_2\left (1,\frac {4}{r},\frac {4}{r};1+\frac {4}{r},1+\frac {4}{r};-\frac {e x^r}{d}\right )+4 \operatorname {Hypergeometric2F1}\left (1,\frac {4}{r},\frac {4+r}{r},-\frac {e x^r}{d}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{16 d} \]

[In]

Integrate[(x^3*(a + b*Log[c*x^n]))/(d + e*x^r),x]

output

(x^4*(-(b*n*HypergeometricPFQ[{1, 4/r, 4/r}, {1 + 4/r, 1 + 4/r}, -((e*x^r)/d)] 
) + 4*Hypergeometric2F1[1, 4/r, (4 + r)/r, -((e*x^r)/d)]*(a + b*Log[c*x^n])))/ 
(16*d)
 

Integral number [407] \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx \]

[B]   time = 0.0764756 (sec), size = 87 ,normalized size = 4.14 \[ \frac {x^2 \left (-b n \, _3F_2\left (1,\frac {2}{r},\frac {2}{r};1+\frac {2}{r},1+\frac {2}{r};-\frac {e x^r}{d}\right )+2 \operatorname {Hypergeometric2F1}\left (1,\frac {2}{r},\frac {2+r}{r},-\frac {e x^r}{d}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{4 d} \]

[In]

Integrate[(x*(a + b*Log[c*x^n]))/(d + e*x^r),x]

output

(x^2*(-(b*n*HypergeometricPFQ[{1, 2/r, 2/r}, {1 + 2/r, 1 + 2/r}, -((e*x^r)/d)] 
) + 2*Hypergeometric2F1[1, 2/r, (2 + r)/r, -((e*x^r)/d)]*(a + b*Log[c*x^n])))/ 
(4*d)
 

Integral number [409] \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^r\right )} \, dx \]

[B]   time = 0.0886791 (sec), size = 86 ,normalized size = 3.74 \[ -\frac {b n \, _3F_2\left (1,-\frac {2}{r},-\frac {2}{r};1-\frac {2}{r},1-\frac {2}{r};-\frac {e x^r}{d}\right )+2 \operatorname {Hypergeometric2F1}\left (1,-\frac {2}{r},\frac {-2+r}{r},-\frac {e x^r}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d x^2} \]

[In]

Integrate[(a + b*Log[c*x^n])/(x^3*(d + e*x^r)),x]

output

-1/4*(b*n*HypergeometricPFQ[{1, -2/r, -2/r}, {1 - 2/r, 1 - 2/r}, -((e*x^r)/d)] 
 + 2*Hypergeometric2F1[1, -2/r, (-2 + r)/r, -((e*x^r)/d)]*(a + b*Log[c*x^n]))/ 
(d*x^2)
 

Integral number [410] \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx \]

[B]   time = 0.0840796 (sec), size = 87 ,normalized size = 3.78 \[ \frac {x^3 \left (-b n \, _3F_2\left (1,\frac {3}{r},\frac {3}{r};1+\frac {3}{r},1+\frac {3}{r};-\frac {e x^r}{d}\right )+3 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{r},\frac {3+r}{r},-\frac {e x^r}{d}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{9 d} \]

[In]

Integrate[(x^2*(a + b*Log[c*x^n]))/(d + e*x^r),x]

output

(x^3*(-(b*n*HypergeometricPFQ[{1, 3/r, 3/r}, {1 + 3/r, 1 + 3/r}, -((e*x^r)/d)] 
) + 3*Hypergeometric2F1[1, 3/r, (3 + r)/r, -((e*x^r)/d)]*(a + b*Log[c*x^n])))/ 
(9*d)
 

Integral number [411] \[ \int \frac {a+b \log \left (c x^n\right )}{d+e x^r} \, dx \]

[B]   time = 0.0650704 (sec), size = 69 ,normalized size = 3.45 \[ \frac {x \left (-b n \, _3F_2\left (1,\frac {1}{r},\frac {1}{r};1+\frac {1}{r},1+\frac {1}{r};-\frac {e x^r}{d}\right )+\operatorname {Hypergeometric2F1}\left (1,\frac {1}{r},1+\frac {1}{r},-\frac {e x^r}{d}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{d} \]

[In]

Integrate[(a + b*Log[c*x^n])/(d + e*x^r),x]

output

(x*(-(b*n*HypergeometricPFQ[{1, r^(-1), r^(-1)}, {1 + r^(-1), 1 + r^(-1)}, -(( 
e*x^r)/d)]) + Hypergeometric2F1[1, r^(-1), 1 + r^(-1), -((e*x^r)/d)]*(a + b*Lo 
g[c*x^n])))/d
 

Integral number [412] \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^r\right )} \, dx \]

[B]   time = 0.0852945 (sec), size = 83 ,normalized size = 3.61 \[ -\frac {b n \, _3F_2\left (1,-\frac {1}{r},-\frac {1}{r};1-\frac {1}{r},1-\frac {1}{r};-\frac {e x^r}{d}\right )+\operatorname {Hypergeometric2F1}\left (1,-\frac {1}{r},\frac {-1+r}{r},-\frac {e x^r}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d x} \]

[In]

Integrate[(a + b*Log[c*x^n])/(x^2*(d + e*x^r)),x]

output

-((b*n*HypergeometricPFQ[{1, -r^(-1), -r^(-1)}, {1 - r^(-1), 1 - r^(-1)}, -((e 
*x^r)/d)] + Hypergeometric2F1[1, -r^(-1), (-1 + r)/r, -((e*x^r)/d)]*(a + b*Log 
[c*x^n]))/(d*x))
 

Integral number [413] \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx \]

[B]   time = 0.164696 (sec), size = 140 ,normalized size = 6.09 \[ \frac {x^4 \left (-b n (-4+r) \left (d+e x^r\right ) \, _3F_2\left (1,\frac {4}{r},\frac {4}{r};1+\frac {4}{r},1+\frac {4}{r};-\frac {e x^r}{d}\right )+16 d \left (a+b \log \left (c x^n\right )\right )+4 \left (d+e x^r\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {4}{r},\frac {4+r}{r},-\frac {e x^r}{d}\right ) \left (-b n+a (-4+r)+b (-4+r) \log \left (c x^n\right )\right )\right )}{16 d^2 r \left (d+e x^r\right )} \]

[In]

Integrate[(x^3*(a + b*Log[c*x^n]))/(d + e*x^r)^2,x]

output

(x^4*(-(b*n*(-4 + r)*(d + e*x^r)*HypergeometricPFQ[{1, 4/r, 4/r}, {1 + 4/r, 1 
+ 4/r}, -((e*x^r)/d)]) + 16*d*(a + b*Log[c*x^n]) + 4*(d + e*x^r)*Hypergeometri 
c2F1[1, 4/r, (4 + r)/r, -((e*x^r)/d)]*(-(b*n) + a*(-4 + r) + b*(-4 + r)*Log[c* 
x^n])))/(16*d^2*r*(d + e*x^r))
 

Integral number [414] \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx \]

[B]   time = 0.161746 (sec), size = 140 ,normalized size = 6.67 \[ \frac {x^2 \left (-b n (-2+r) \left (d+e x^r\right ) \, _3F_2\left (1,\frac {2}{r},\frac {2}{r};1+\frac {2}{r},1+\frac {2}{r};-\frac {e x^r}{d}\right )+4 d \left (a+b \log \left (c x^n\right )\right )+2 \left (d+e x^r\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {2}{r},\frac {2+r}{r},-\frac {e x^r}{d}\right ) \left (-b n+a (-2+r)+b (-2+r) \log \left (c x^n\right )\right )\right )}{4 d^2 r \left (d+e x^r\right )} \]

[In]

Integrate[(x*(a + b*Log[c*x^n]))/(d + e*x^r)^2,x]

output

(x^2*(-(b*n*(-2 + r)*(d + e*x^r)*HypergeometricPFQ[{1, 2/r, 2/r}, {1 + 2/r, 1 
+ 2/r}, -((e*x^r)/d)]) + 4*d*(a + b*Log[c*x^n]) + 2*(d + e*x^r)*Hypergeometric 
2F1[1, 2/r, (2 + r)/r, -((e*x^r)/d)]*(-(b*n) + a*(-2 + r) + b*(-2 + r)*Log[c*x 
^n])))/(4*d^2*r*(d + e*x^r))
 

Integral number [416] \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^r\right )^2} \, dx \]

[B]   time = 0.163581 (sec), size = 139 ,normalized size = 6.04 \[ -\frac {b n (2+r) \left (d+e x^r\right ) \, _3F_2\left (1,-\frac {2}{r},-\frac {2}{r};1-\frac {2}{r},1-\frac {2}{r};-\frac {e x^r}{d}\right )-4 d \left (a+b \log \left (c x^n\right )\right )+2 \left (d+e x^r\right ) \operatorname {Hypergeometric2F1}\left (1,-\frac {2}{r},\frac {-2+r}{r},-\frac {e x^r}{d}\right ) \left (-b n+a (2+r)+b (2+r) \log \left (c x^n\right )\right )}{4 d^2 r x^2 \left (d+e x^r\right )} \]

[In]

Integrate[(a + b*Log[c*x^n])/(x^3*(d + e*x^r)^2),x]

output

-1/4*(b*n*(2 + r)*(d + e*x^r)*HypergeometricPFQ[{1, -2/r, -2/r}, {1 - 2/r, 1 - 
 2/r}, -((e*x^r)/d)] - 4*d*(a + b*Log[c*x^n]) + 2*(d + e*x^r)*Hypergeometric2F 
1[1, -2/r, (-2 + r)/r, -((e*x^r)/d)]*(-(b*n) + a*(2 + r) + b*(2 + r)*Log[c*x^n 
]))/(d^2*r*x^2*(d + e*x^r))
 

Integral number [417] \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx \]

[B]   time = 0.156319 (sec), size = 140 ,normalized size = 6.09 \[ \frac {x^3 \left (-b n (-3+r) \left (d+e x^r\right ) \, _3F_2\left (1,\frac {3}{r},\frac {3}{r};1+\frac {3}{r},1+\frac {3}{r};-\frac {e x^r}{d}\right )+9 d \left (a+b \log \left (c x^n\right )\right )+3 \left (d+e x^r\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {3}{r},\frac {3+r}{r},-\frac {e x^r}{d}\right ) \left (-b n+a (-3+r)+b (-3+r) \log \left (c x^n\right )\right )\right )}{9 d^2 r \left (d+e x^r\right )} \]

[In]

Integrate[(x^2*(a + b*Log[c*x^n]))/(d + e*x^r)^2,x]

output

(x^3*(-(b*n*(-3 + r)*(d + e*x^r)*HypergeometricPFQ[{1, 3/r, 3/r}, {1 + 3/r, 1 
+ 3/r}, -((e*x^r)/d)]) + 9*d*(a + b*Log[c*x^n]) + 3*(d + e*x^r)*Hypergeometric 
2F1[1, 3/r, (3 + r)/r, -((e*x^r)/d)]*(-(b*n) + a*(-3 + r) + b*(-3 + r)*Log[c*x 
^n])))/(9*d^2*r*(d + e*x^r))
 

Integral number [418] \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^r\right )^2} \, dx \]

[B]   time = 1.70687 (sec), size = 161 ,normalized size = 8.05 \[ \frac {x \left (a d r \operatorname {Hypergeometric2F1}\left (2,\frac {1}{r},1+\frac {1}{r},-\frac {e x^r}{d}\right )+a e r x^r \operatorname {Hypergeometric2F1}\left (2,\frac {1}{r},1+\frac {1}{r},-\frac {e x^r}{d}\right )-b n (-1+r) \left (d+e x^r\right ) \, _3F_2\left (1,\frac {1}{r},\frac {1}{r};1+\frac {1}{r},1+\frac {1}{r};-\frac {e x^r}{d}\right )+b d \log \left (c x^n\right )-b \left (d+e x^r\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{r},1+\frac {1}{r},-\frac {e x^r}{d}\right ) \left (n-(-1+r) \log \left (c x^n\right )\right )\right )}{d^2 r \left (d+e x^r\right )} \]

[In]

Integrate[(a + b*Log[c*x^n])/(d + e*x^r)^2,x]

output

(x*(a*d*r*Hypergeometric2F1[2, r^(-1), 1 + r^(-1), -((e*x^r)/d)] + a*e*r*x^r*H 
ypergeometric2F1[2, r^(-1), 1 + r^(-1), -((e*x^r)/d)] - b*n*(-1 + r)*(d + e*x^ 
r)*HypergeometricPFQ[{1, r^(-1), r^(-1)}, {1 + r^(-1), 1 + r^(-1)}, -((e*x^r)/ 
d)] + b*d*Log[c*x^n] - b*(d + e*x^r)*Hypergeometric2F1[1, r^(-1), 1 + r^(-1), 
-((e*x^r)/d)]*(n - (-1 + r)*Log[c*x^n])))/(d^2*r*(d + e*x^r))
 

Integral number [419] \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^r\right )^2} \, dx \]

[B]   time = 0.138563 (sec), size = 135 ,normalized size = 5.87 \[ \frac {-b n (1+r) \left (d+e x^r\right ) \, _3F_2\left (1,-\frac {1}{r},-\frac {1}{r};1-\frac {1}{r},1-\frac {1}{r};-\frac {e x^r}{d}\right )+d \left (a+b \log \left (c x^n\right )\right )-\left (d+e x^r\right ) \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{r},\frac {-1+r}{r},-\frac {e x^r}{d}\right ) \left (a-b n+a r+b (1+r) \log \left (c x^n\right )\right )}{d^2 r x \left (d+e x^r\right )} \]

[In]

Integrate[(a + b*Log[c*x^n])/(x^2*(d + e*x^r)^2),x]

output

(-(b*n*(1 + r)*(d + e*x^r)*HypergeometricPFQ[{1, -r^(-1), -r^(-1)}, {1 - r^(-1 
), 1 - r^(-1)}, -((e*x^r)/d)]) + d*(a + b*Log[c*x^n]) - (d + e*x^r)*Hypergeome 
tric2F1[1, -r^(-1), (-1 + r)/r, -((e*x^r)/d)]*(a - b*n + a*r + b*(1 + r)*Log[c 
*x^n]))/(d^2*r*x*(d + e*x^r))
 

Integral number [444] \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx \]

[B]   time = 0.157935 (sec), size = 111 ,normalized size = 4.44 \[ \frac {x (f x)^m \left (-b n \, _3F_2\left (1,\frac {1}{r}+\frac {m}{r},\frac {1}{r}+\frac {m}{r};1+\frac {1}{r}+\frac {m}{r},1+\frac {1}{r}+\frac {m}{r};-\frac {e x^r}{d}\right )+(1+m) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{r},\frac {1+m+r}{r},-\frac {e x^r}{d}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{d (1+m)^2} \]

[In]

Integrate[((f*x)^m*(a + b*Log[c*x^n]))/(d + e*x^r),x]

output

(x*(f*x)^m*(-(b*n*HypergeometricPFQ[{1, r^(-1) + m/r, r^(-1) + m/r}, {1 + r^(- 
1) + m/r, 1 + r^(-1) + m/r}, -((e*x^r)/d)]) + (1 + m)*Hypergeometric2F1[1, (1 
+ m)/r, (1 + m + r)/r, -((e*x^r)/d)]*(a + b*Log[c*x^n])))/(d*(1 + m)^2)
 

Integral number [445] \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx \]

[B]   time = 0.296047 (sec), size = 177 ,normalized size = 7.08 \[ \frac {x (f x)^m \left (b n (1+m-r) \left (d+e x^r\right ) \, _3F_2\left (1,\frac {1}{r}+\frac {m}{r},\frac {1}{r}+\frac {m}{r};1+\frac {1}{r}+\frac {m}{r},1+\frac {1}{r}+\frac {m}{r};-\frac {e x^r}{d}\right )-(1+m) \left (-d (1+m) \left (a+b \log \left (c x^n\right )\right )+\left (d+e x^r\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{r},\frac {1+m+r}{r},-\frac {e x^r}{d}\right ) \left (b n+a (1+m-r)+b (1+m-r) \log \left (c x^n\right )\right )\right )\right )}{d^2 (1+m)^2 r \left (d+e x^r\right )} \]

[In]

Integrate[((f*x)^m*(a + b*Log[c*x^n]))/(d + e*x^r)^2,x]

output

(x*(f*x)^m*(b*n*(1 + m - r)*(d + e*x^r)*HypergeometricPFQ[{1, r^(-1) + m/r, r^ 
(-1) + m/r}, {1 + r^(-1) + m/r, 1 + r^(-1) + m/r}, -((e*x^r)/d)] - (1 + m)*(-( 
d*(1 + m)*(a + b*Log[c*x^n])) + (d + e*x^r)*Hypergeometric2F1[1, (1 + m)/r, (1 
 + m + r)/r, -((e*x^r)/d)]*(b*n + a*(1 + m - r) + b*(1 + m - r)*Log[c*x^n])))) 
/(d^2*(1 + m)^2*r*(d + e*x^r))
 

4.3 Test file Number [58]

4.3.1 Mathematica

Integral number [138] \[ \int (g x)^q \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx \]

[B]   time = 0.33242 (sec), size = 304 ,normalized size = 10.86 \[ \frac {x (g x)^q \left (-a k m+2 b k m n-a k m q-b k m n \, _3F_2\left (1,\frac {1}{m}+\frac {q}{m},\frac {1}{m}+\frac {q}{m};1+\frac {1}{m}+\frac {q}{m},1+\frac {1}{m}+\frac {q}{m};-\frac {f x^m}{e}\right )-b k m \log \left (c x^n\right )-b k m q \log \left (c x^n\right )+k m \operatorname {Hypergeometric2F1}\left (1,\frac {1+q}{m},\frac {1+m+q}{m},-\frac {f x^m}{e}\right ) \left (a-b n+a q+b (1+q) \log \left (c x^n\right )\right )+a \log \left (d \left (e+f x^m\right )^k\right )-b n \log \left (d \left (e+f x^m\right )^k\right )+2 a q \log \left (d \left (e+f x^m\right )^k\right )-b n q \log \left (d \left (e+f x^m\right )^k\right )+a q^2 \log \left (d \left (e+f x^m\right )^k\right )+b \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )+2 b q \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )+b q^2 \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )\right )}{(1+q)^3} \]

[In]

Integrate[(g*x)^q*(a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k],x]

output

(x*(g*x)^q*(-(a*k*m) + 2*b*k*m*n - a*k*m*q - b*k*m*n*HypergeometricPFQ[{1, m^( 
-1) + q/m, m^(-1) + q/m}, {1 + m^(-1) + q/m, 1 + m^(-1) + q/m}, -((f*x^m)/e)] 
- b*k*m*Log[c*x^n] - b*k*m*q*Log[c*x^n] + k*m*Hypergeometric2F1[1, (1 + q)/m, 
(1 + m + q)/m, -((f*x^m)/e)]*(a - b*n + a*q + b*(1 + q)*Log[c*x^n]) + a*Log[d* 
(e + f*x^m)^k] - b*n*Log[d*(e + f*x^m)^k] + 2*a*q*Log[d*(e + f*x^m)^k] - b*n*q 
*Log[d*(e + f*x^m)^k] + a*q^2*Log[d*(e + f*x^m)^k] + b*Log[c*x^n]*Log[d*(e + f 
*x^m)^k] + 2*b*q*Log[c*x^n]*Log[d*(e + f*x^m)^k] + b*q^2*Log[c*x^n]*Log[d*(e + 
 f*x^m)^k]))/(1 + q)^3
 

Integral number [144] \[ \int x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx \]

[B]   time = 0.152851 (sec), size = 292 ,normalized size = 11.23 \[ -\frac {x^3 \left (-6 b e k m n-2 b e k m^2 n+9 a f k m x^m \operatorname {Hypergeometric2F1}\left (1,\frac {3+m}{m},2+\frac {3}{m},-\frac {f x^m}{e}\right )+b e k m (3+m) n \, _3F_2\left (1,\frac {3}{m},\frac {3}{m};1+\frac {3}{m},1+\frac {3}{m};-\frac {f x^m}{e}\right )+b e k m (3+m) \operatorname {Hypergeometric2F1}\left (1,\frac {3}{m},\frac {3+m}{m},-\frac {f x^m}{e}\right ) \left (n-3 \log \left (c x^n\right )\right )+9 b e k m \log \left (c x^n\right )+3 b e k m^2 \log \left (c x^n\right )-27 a e \log \left (d \left (e+f x^m\right )^k\right )-9 a e m \log \left (d \left (e+f x^m\right )^k\right )+9 b e n \log \left (d \left (e+f x^m\right )^k\right )+3 b e m n \log \left (d \left (e+f x^m\right )^k\right )-27 b e \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )-9 b e m \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )\right )}{27 e (3+m)} \]

[In]

Integrate[x^2*(a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k],x]

output

-1/27*(x^3*(-6*b*e*k*m*n - 2*b*e*k*m^2*n + 9*a*f*k*m*x^m*Hypergeometric2F1[1, 
(3 + m)/m, 2 + 3/m, -((f*x^m)/e)] + b*e*k*m*(3 + m)*n*HypergeometricPFQ[{1, 3/ 
m, 3/m}, {1 + 3/m, 1 + 3/m}, -((f*x^m)/e)] + b*e*k*m*(3 + m)*Hypergeometric2F1 
[1, 3/m, (3 + m)/m, -((f*x^m)/e)]*(n - 3*Log[c*x^n]) + 9*b*e*k*m*Log[c*x^n] + 
3*b*e*k*m^2*Log[c*x^n] - 27*a*e*Log[d*(e + f*x^m)^k] - 9*a*e*m*Log[d*(e + f*x^ 
m)^k] + 9*b*e*n*Log[d*(e + f*x^m)^k] + 3*b*e*m*n*Log[d*(e + f*x^m)^k] - 27*b*e 
*Log[c*x^n]*Log[d*(e + f*x^m)^k] - 9*b*e*m*Log[c*x^n]*Log[d*(e + f*x^m)^k]))/( 
e*(3 + m))
 

Integral number [145] \[ \int x \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx \]

[B]   time = 0.136601 (sec), size = 292 ,normalized size = 12.17 \[ -\frac {x^2 \left (-4 b e k m n-2 b e k m^2 n+4 a f k m x^m \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{m},2+\frac {2}{m},-\frac {f x^m}{e}\right )+b e k m (2+m) n \, _3F_2\left (1,\frac {2}{m},\frac {2}{m};1+\frac {2}{m},1+\frac {2}{m};-\frac {f x^m}{e}\right )+b e k m (2+m) \operatorname {Hypergeometric2F1}\left (1,\frac {2}{m},\frac {2+m}{m},-\frac {f x^m}{e}\right ) \left (n-2 \log \left (c x^n\right )\right )+4 b e k m \log \left (c x^n\right )+2 b e k m^2 \log \left (c x^n\right )-8 a e \log \left (d \left (e+f x^m\right )^k\right )-4 a e m \log \left (d \left (e+f x^m\right )^k\right )+4 b e n \log \left (d \left (e+f x^m\right )^k\right )+2 b e m n \log \left (d \left (e+f x^m\right )^k\right )-8 b e \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )-4 b e m \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )\right )}{8 e (2+m)} \]

[In]

Integrate[x*(a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k],x]

output

-1/8*(x^2*(-4*b*e*k*m*n - 2*b*e*k*m^2*n + 4*a*f*k*m*x^m*Hypergeometric2F1[1, ( 
2 + m)/m, 2 + 2/m, -((f*x^m)/e)] + b*e*k*m*(2 + m)*n*HypergeometricPFQ[{1, 2/m 
, 2/m}, {1 + 2/m, 1 + 2/m}, -((f*x^m)/e)] + b*e*k*m*(2 + m)*Hypergeometric2F1[ 
1, 2/m, (2 + m)/m, -((f*x^m)/e)]*(n - 2*Log[c*x^n]) + 4*b*e*k*m*Log[c*x^n] + 2 
*b*e*k*m^2*Log[c*x^n] - 8*a*e*Log[d*(e + f*x^m)^k] - 4*a*e*m*Log[d*(e + f*x^m) 
^k] + 4*b*e*n*Log[d*(e + f*x^m)^k] + 2*b*e*m*n*Log[d*(e + f*x^m)^k] - 8*b*e*Lo 
g[c*x^n]*Log[d*(e + f*x^m)^k] - 4*b*e*m*Log[c*x^n]*Log[d*(e + f*x^m)^k]))/(e*( 
2 + m))
 

Integral number [146] \[ \int \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx \]

[B]   time = 0.152115 (sec), size = 165 ,normalized size = 7.17 \[ b k m n x-k m x \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )+x \left (b k m n-b k m n \, _3F_2\left (1,\frac {1}{m},\frac {1}{m};1+\frac {1}{m},1+\frac {1}{m};-\frac {f x^m}{e}\right )-b k m n \log (x)+k m \operatorname {Hypergeometric2F1}\left (1,\frac {1}{m},1+\frac {1}{m},-\frac {f x^m}{e}\right ) \left (a-b n+b \log \left (c x^n\right )\right )+a \log \left (d \left (e+f x^m\right )^k\right )-b n \log \left (d \left (e+f x^m\right )^k\right )+b \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )\right ) \]

[In]

Integrate[(a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k],x]

output

b*k*m*n*x - k*m*x*(a + b*(-(n*Log[x]) + Log[c*x^n])) + x*(b*k*m*n - b*k*m*n*Hy 
pergeometricPFQ[{1, m^(-1), m^(-1)}, {1 + m^(-1), 1 + m^(-1)}, -((f*x^m)/e)] - 
 b*k*m*n*Log[x] + k*m*Hypergeometric2F1[1, m^(-1), 1 + m^(-1), -((f*x^m)/e)]*( 
a - b*n + b*Log[c*x^n]) + a*Log[d*(e + f*x^m)^k] - b*n*Log[d*(e + f*x^m)^k] + 
b*Log[c*x^n]*Log[d*(e + f*x^m)^k])
 

Integral number [148] \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{x^2} \, dx \]

[B]   time = 0.125142 (sec), size = 282 ,normalized size = 10.85 \[ \frac {2 b e k m n-2 b e k m^2 n+a f k m x^m \operatorname {Hypergeometric2F1}\left (1,\frac {-1+m}{m},2-\frac {1}{m},-\frac {f x^m}{e}\right )+b e k (-1+m) m n \, _3F_2\left (1,-\frac {1}{m},-\frac {1}{m};1-\frac {1}{m},1-\frac {1}{m};-\frac {f x^m}{e}\right )+b e k m \log \left (c x^n\right )-b e k m^2 \log \left (c x^n\right )+b e k (-1+m) m \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{m},\frac {-1+m}{m},-\frac {f x^m}{e}\right ) \left (n+\log \left (c x^n\right )\right )+a e \log \left (d \left (e+f x^m\right )^k\right )-a e m \log \left (d \left (e+f x^m\right )^k\right )+b e n \log \left (d \left (e+f x^m\right )^k\right )-b e m n \log \left (d \left (e+f x^m\right )^k\right )+b e \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )-b e m \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )}{e (-1+m) x} \]

[In]

Integrate[((a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k])/x^2,x]

output

(2*b*e*k*m*n - 2*b*e*k*m^2*n + a*f*k*m*x^m*Hypergeometric2F1[1, (-1 + m)/m, 2 
- m^(-1), -((f*x^m)/e)] + b*e*k*(-1 + m)*m*n*HypergeometricPFQ[{1, -m^(-1), -m 
^(-1)}, {1 - m^(-1), 1 - m^(-1)}, -((f*x^m)/e)] + b*e*k*m*Log[c*x^n] - b*e*k*m 
^2*Log[c*x^n] + b*e*k*(-1 + m)*m*Hypergeometric2F1[1, -m^(-1), (-1 + m)/m, -(( 
f*x^m)/e)]*(n + Log[c*x^n]) + a*e*Log[d*(e + f*x^m)^k] - a*e*m*Log[d*(e + f*x^ 
m)^k] + b*e*n*Log[d*(e + f*x^m)^k] - b*e*m*n*Log[d*(e + f*x^m)^k] + b*e*Log[c* 
x^n]*Log[d*(e + f*x^m)^k] - b*e*m*Log[c*x^n]*Log[d*(e + f*x^m)^k])/(e*(-1 + m) 
*x)
 

Integral number [149] \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{x^3} \, dx \]

[B]   time = 0.128627 (sec), size = 292 ,normalized size = 11.23 \[ \frac {4 b e k m n-2 b e k m^2 n+4 a f k m x^m \operatorname {Hypergeometric2F1}\left (1,\frac {-2+m}{m},2-\frac {2}{m},-\frac {f x^m}{e}\right )+b e k (-2+m) m n \, _3F_2\left (1,-\frac {2}{m},-\frac {2}{m};1-\frac {2}{m},1-\frac {2}{m};-\frac {f x^m}{e}\right )+4 b e k m \log \left (c x^n\right )-2 b e k m^2 \log \left (c x^n\right )+b e k (-2+m) m \operatorname {Hypergeometric2F1}\left (1,-\frac {2}{m},\frac {-2+m}{m},-\frac {f x^m}{e}\right ) \left (n+2 \log \left (c x^n\right )\right )+8 a e \log \left (d \left (e+f x^m\right )^k\right )-4 a e m \log \left (d \left (e+f x^m\right )^k\right )+4 b e n \log \left (d \left (e+f x^m\right )^k\right )-2 b e m n \log \left (d \left (e+f x^m\right )^k\right )+8 b e \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )-4 b e m \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )}{8 e (-2+m) x^2} \]

[In]

Integrate[((a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k])/x^3,x]

output

(4*b*e*k*m*n - 2*b*e*k*m^2*n + 4*a*f*k*m*x^m*Hypergeometric2F1[1, (-2 + m)/m, 
2 - 2/m, -((f*x^m)/e)] + b*e*k*(-2 + m)*m*n*HypergeometricPFQ[{1, -2/m, -2/m}, 
 {1 - 2/m, 1 - 2/m}, -((f*x^m)/e)] + 4*b*e*k*m*Log[c*x^n] - 2*b*e*k*m^2*Log[c* 
x^n] + b*e*k*(-2 + m)*m*Hypergeometric2F1[1, -2/m, (-2 + m)/m, -((f*x^m)/e)]*( 
n + 2*Log[c*x^n]) + 8*a*e*Log[d*(e + f*x^m)^k] - 4*a*e*m*Log[d*(e + f*x^m)^k] 
+ 4*b*e*n*Log[d*(e + f*x^m)^k] - 2*b*e*m*n*Log[d*(e + f*x^m)^k] + 8*b*e*Log[c* 
x^n]*Log[d*(e + f*x^m)^k] - 4*b*e*m*Log[c*x^n]*Log[d*(e + f*x^m)^k])/(8*e*(-2 
+ m)*x^2)
 

Integral number [220] \[ \int -(d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx \]

[B]   time = 0.296332 (sec), size = 266 ,normalized size = 10.23 \[ -\frac {x (d x)^m \left (-a q-a m q+2 b n q-b n q \, _3F_2\left (1,\frac {1}{q}+\frac {m}{q},\frac {1}{q}+\frac {m}{q};1+\frac {1}{q}+\frac {m}{q},1+\frac {1}{q}+\frac {m}{q};e x^q\right )-b q \log \left (c x^n\right )-b m q \log \left (c x^n\right )+q \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{q},\frac {1+m+q}{q},e x^q\right ) \left (a+a m-b n+b (1+m) \log \left (c x^n\right )\right )+a \log \left (1-e x^q\right )+2 a m \log \left (1-e x^q\right )+a m^2 \log \left (1-e x^q\right )-b n \log \left (1-e x^q\right )-b m n \log \left (1-e x^q\right )+b \log \left (c x^n\right ) \log \left (1-e x^q\right )+2 b m \log \left (c x^n\right ) \log \left (1-e x^q\right )+b m^2 \log \left (c x^n\right ) \log \left (1-e x^q\right )\right )}{(1+m)^3} \]

[In]

Integrate[-((d*x)^m*(a + b*Log[c*x^n])*Log[1 - e*x^q]),x]

output

-((x*(d*x)^m*(-(a*q) - a*m*q + 2*b*n*q - b*n*q*HypergeometricPFQ[{1, q^(-1) + 
m/q, q^(-1) + m/q}, {1 + q^(-1) + m/q, 1 + q^(-1) + m/q}, e*x^q] - b*q*Log[c*x 
^n] - b*m*q*Log[c*x^n] + q*Hypergeometric2F1[1, (1 + m)/q, (1 + m + q)/q, e*x^ 
q]*(a + a*m - b*n + b*(1 + m)*Log[c*x^n]) + a*Log[1 - e*x^q] + 2*a*m*Log[1 - e 
*x^q] + a*m^2*Log[1 - e*x^q] - b*n*Log[1 - e*x^q] - b*m*n*Log[1 - e*x^q] + b*L 
og[c*x^n]*Log[1 - e*x^q] + 2*b*m*Log[c*x^n]*Log[1 - e*x^q] + b*m^2*Log[c*x^n]* 
Log[1 - e*x^q]))/(1 + m)^3)
 

4.3.2 Maple

Integral number [221] \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,e x^q\right ) \, dx \]

[B]   time = 0.089 (sec), size = 867 ,normalized size = 37.7 \[-\frac {\left (d x \right )^{m} x^{-m} \left (-e \right )^{-\frac {m}{q}-\frac {1}{q}} a \left (-\frac {q^{2} x^{1+m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (1-e \,x^{q}\right )}{\left (1+m \right )^{2}}-\frac {q \,x^{1+m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \operatorname {Li}_{2}\left (e \,x^{q}\right )}{1+m}-\frac {q^{2} x^{1+m +q} e \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \Phi \left (e \,x^{q}, 1, \frac {1+m +q}{q}\right )}{\left (1+m \right )^{2}}\right )}{q}-\frac {\left (d x \right )^{m} x^{-m} \left (-e \right )^{-\frac {m}{q}-\frac {1}{q}} b \ln \left (c \right ) \left (-\frac {q^{2} x^{1+m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (1-e \,x^{q}\right )}{\left (1+m \right )^{2}}-\frac {q \,x^{1+m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \operatorname {Li}_{2}\left (e \,x^{q}\right )}{1+m}-\frac {q^{2} x^{1+m +q} e \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \Phi \left (e \,x^{q}, 1, \frac {1+m +q}{q}\right )}{\left (1+m \right )^{2}}\right )}{q}+\left (\frac {\left (-e \right )^{-\frac {m}{q}-\frac {1}{q}} \ln \left (-e \right ) \left (d x \right )^{m} x^{-m} b n \left (-\frac {q^{2} x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (1-e \,x^{q}\right )}{\left (1+m \right )^{2}}-\frac {q \,x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \operatorname {Li}_{2}\left (e \,x^{q}\right )}{1+m}-\frac {q^{2} x^{q +m} e \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \Phi \left (e \,x^{q}, 1, \frac {1+m +q}{q}\right )}{\left (1+m \right )^{2}}\right )}{q^{2}}-\frac {\left (-e \right )^{-\frac {m}{q}-\frac {1}{q}} \left (d x \right )^{m} x^{-m} b n \left (-\frac {q^{2} x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (x \right ) \ln \left (1-e \,x^{q}\right )}{\left (1+m \right )^{2}}-\frac {q \,x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (-e \right ) \ln \left (1-e \,x^{q}\right )}{\left (1+m \right )^{2}}+\frac {2 q^{2} x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (1-e \,x^{q}\right )}{\left (1+m \right )^{3}}-\frac {q \,x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (x \right ) \operatorname {Li}_{2}\left (e \,x^{q}\right )}{1+m}-\frac {x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (-e \right ) \operatorname {Li}_{2}\left (e \,x^{q}\right )}{1+m}+\frac {q \,x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \operatorname {Li}_{2}\left (e \,x^{q}\right )}{\left (1+m \right )^{2}}-\frac {q^{2} x^{q +m} e \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (x \right ) \Phi \left (e \,x^{q}, 1, \frac {1+m +q}{q}\right )}{\left (1+m \right )^{2}}-\frac {q \,x^{q +m} e \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (-e \right ) \Phi \left (e \,x^{q}, 1, \frac {1+m +q}{q}\right )}{\left (1+m \right )^{2}}+\frac {2 q^{2} x^{q +m} e \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \Phi \left (e \,x^{q}, 1, \frac {1+m +q}{q}\right )}{\left (1+m \right )^{3}}+\frac {q \,x^{q +m} e \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \Phi \left (e \,x^{q}, 2, \frac {1+m +q}{q}\right )}{\left (1+m \right )^{2}}\right )}{q}\right ) x\]

[In]

int((d*x)^m*(a+b*ln(c*x^n))*polylog(2,e*x^q),x)

output

-(d*x)^m*x^(-m)*(-e)^(-m/q-1/q)*a/q*(-q^2*x^(1+m)*(-e)^(m/q+1/q)/(1+m)^2*ln(1- 
e*x^q)-q*x^(1+m)*(-e)^(m/q+1/q)/(1+m)*polylog(2,e*x^q)-q^2*x^(1+m+q)*e*(-e)^(m 
/q+1/q)/(1+m)^2*LerchPhi(e*x^q,1,(1+m+q)/q))-(d*x)^m*x^(-m)*(-e)^(-m/q-1/q)*b* 
ln(c)/q*(-q^2*x^(1+m)*(-e)^(m/q+1/q)/(1+m)^2*ln(1-e*x^q)-q*x^(1+m)*(-e)^(m/q+1 
/q)/(1+m)*polylog(2,e*x^q)-q^2*x^(1+m+q)*e*(-e)^(m/q+1/q)/(1+m)^2*LerchPhi(e*x 
^q,1,(1+m+q)/q))+((-e)^(-m/q-1/q)*ln(-e)/q^2*(d*x)^m*x^(-m)*b*n*(-q^2*x^m*(-e) 
^(m/q+1/q)/(1+m)^2*ln(1-e*x^q)-q*x^m*(-e)^(m/q+1/q)/(1+m)*polylog(2,e*x^q)-q^2 
*x^(q+m)*e*(-e)^(m/q+1/q)/(1+m)^2*LerchPhi(e*x^q,1,(1+m+q)/q))-(-e)^(-m/q-1/q) 
*(d*x)^m*x^(-m)*b*n/q*(-q^2*x^m*(-e)^(m/q+1/q)*ln(x)/(1+m)^2*ln(1-e*x^q)-q*x^m 
*(-e)^(m/q+1/q)*ln(-e)/(1+m)^2*ln(1-e*x^q)+2*q^2*x^m*(-e)^(m/q+1/q)/(1+m)^3*ln 
(1-e*x^q)-q*x^m*(-e)^(m/q+1/q)*ln(x)/(1+m)*polylog(2,e*x^q)-x^m*(-e)^(m/q+1/q) 
*ln(-e)/(1+m)*polylog(2,e*x^q)+q*x^m*(-e)^(m/q+1/q)/(1+m)^2*polylog(2,e*x^q)-q 
^2*x^(q+m)*e*(-e)^(m/q+1/q)*ln(x)/(1+m)^2*LerchPhi(e*x^q,1,(1+m+q)/q)-q*x^(q+m 
)*e*(-e)^(m/q+1/q)*ln(-e)/(1+m)^2*LerchPhi(e*x^q,1,(1+m+q)/q)+2*q^2*x^(q+m)*e* 
(-e)^(m/q+1/q)/(1+m)^3*LerchPhi(e*x^q,1,(1+m+q)/q)+q*x^(q+m)*e*(-e)^(m/q+1/q)/ 
(1+m)^2*LerchPhi(e*x^q,2,(1+m+q)/q)))*x
 

4.4 Test file Number [63]

4.4.1 Mathematica

Integral number [98] \[ \int x^2 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx \]

[B]   time = 3.12679 (sec), size = 909 ,normalized size = 50.5 \[ \frac {2 a p x \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2}{b}-\frac {2 a^{3/2} p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2}{b^{3/2}}+p x^3 \log \left (a+b x^2\right ) \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2+\frac {1}{3} x^3 \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2 \left (-2 p-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )+3 p^2 \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right ) \left (\frac {1}{3} x^3 \log ^2\left (a+b x^2\right )-\frac {4 \left (9 i a^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2+3 a^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-8+6 \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )+3 \log \left (a+b x^2\right )\right )+\sqrt {b} x \left (24 a-2 b x^2+\left (-9 a+3 b x^2\right ) \log \left (a+b x^2\right )\right )+9 i a^{3/2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {a}+\sqrt {b} x}{-i \sqrt {a}+\sqrt {b} x}\right )\right )}{27 b^{3/2}}\right )+\frac {p^3 \left (416 \sqrt {-a} a^{3/2} \sqrt {\frac {b x^2}{a+b x^2}} \sqrt {a+b x^2} \arcsin \left (\frac {\sqrt {a}}{\sqrt {a+b x^2}}\right )+\frac {2}{3} \sqrt {-a} b x^2 \left (624 a-16 b x^2+\left (-288 a+24 b x^2\right ) \log \left (a+b x^2\right )+18 \left (3 a-b x^2\right ) \log ^2\left (a+b x^2\right )+9 b x^2 \log ^3\left (a+b x^2\right )\right )+36 \sqrt {-a} a^{3/2} \sqrt {\frac {b x^2}{a+b x^2}} \left (8 \sqrt {a} \, _4F_3\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2},\frac {3}{2};\frac {a}{a+b x^2}\right )+\log \left (a+b x^2\right ) \left (4 \sqrt {a} \, _3F_2\left (\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};\frac {a}{a+b x^2}\right )+\sqrt {a+b x^2} \arcsin \left (\frac {\sqrt {a}}{\sqrt {a+b x^2}}\right ) \log \left (a+b x^2\right )\right )\right )-48 a^2 \left (4 \sqrt {b x^2} \text {arctanh}\left (\frac {\sqrt {b x^2}}{\sqrt {-a}}\right ) \left (\log \left (a+b x^2\right )-\log \left (1+\frac {b x^2}{a}\right )\right )-\sqrt {-a} \sqrt {-\frac {b x^2}{a}} \left (\log ^2\left (1+\frac {b x^2}{a}\right )-4 \log \left (1+\frac {b x^2}{a}\right ) \log \left (\frac {1}{2} \left (1+\sqrt {-\frac {b x^2}{a}}\right )\right )+2 \log ^2\left (\frac {1}{2} \left (1+\sqrt {-\frac {b x^2}{a}}\right )\right )-4 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {1}{2} \sqrt {-\frac {b x^2}{a}}\right )\right )\right )\right )}{18 \sqrt {-a} b^2 x} \]

[In]

Integrate[x^2*Log[c*(a + b*x^2)^p]^3,x]

output

(2*a*p*x*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2)/b - (2*a^(3/2)*p*ArcT 
an[(Sqrt[b]*x)/Sqrt[a]]*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2)/b^(3/2 
) + p*x^3*Log[a + b*x^2]*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2 + (x^3 
*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2*(-2*p - p*Log[a + b*x^2] + Log 
[c*(a + b*x^2)^p]))/3 + 3*p^2*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])*((x 
^3*Log[a + b*x^2]^2)/3 - (4*((9*I)*a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]]^2 + 3*a 
^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*(-8 + 6*Log[(2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b 
]*x)] + 3*Log[a + b*x^2]) + Sqrt[b]*x*(24*a - 2*b*x^2 + (-9*a + 3*b*x^2)*Log[a 
 + b*x^2]) + (9*I)*a^(3/2)*PolyLog[2, (I*Sqrt[a] + Sqrt[b]*x)/((-I)*Sqrt[a] + 
Sqrt[b]*x)]))/(27*b^(3/2))) + (p^3*(416*Sqrt[-a]*a^(3/2)*Sqrt[(b*x^2)/(a + b*x 
^2)]*Sqrt[a + b*x^2]*ArcSin[Sqrt[a]/Sqrt[a + b*x^2]] + (2*Sqrt[-a]*b*x^2*(624* 
a - 16*b*x^2 + (-288*a + 24*b*x^2)*Log[a + b*x^2] + 18*(3*a - b*x^2)*Log[a + b 
*x^2]^2 + 9*b*x^2*Log[a + b*x^2]^3))/3 + 36*Sqrt[-a]*a^(3/2)*Sqrt[(b*x^2)/(a + 
 b*x^2)]*(8*Sqrt[a]*HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3/2, 3/2}, a 
/(a + b*x^2)] + Log[a + b*x^2]*(4*Sqrt[a]*HypergeometricPFQ[{1/2, 1/2, 1/2}, { 
3/2, 3/2}, a/(a + b*x^2)] + Sqrt[a + b*x^2]*ArcSin[Sqrt[a]/Sqrt[a + b*x^2]]*Lo 
g[a + b*x^2])) - 48*a^2*(4*Sqrt[b*x^2]*ArcTanh[Sqrt[b*x^2]/Sqrt[-a]]*(Log[a + 
b*x^2] - Log[1 + (b*x^2)/a]) - Sqrt[-a]*Sqrt[-((b*x^2)/a)]*(Log[1 + (b*x^2)/a] 
^2 - 4*Log[1 + (b*x^2)/a]*Log[(1 + Sqrt[-((b*x^2)/a)])/2] + 2*Log[(1 + Sqrt[-( 
(b*x^2)/a)])/2]^2 - 4*PolyLog[2, 1/2 - Sqrt[-((b*x^2)/a)]/2]))))/(18*Sqrt[-...
 

Integral number [99] \[ \int \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx \]

[B]   time = 2.66678 (sec), size = 789 ,normalized size = 56.36 \[ \frac {6 \sqrt {a} p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2}{\sqrt {b}}+3 p x \log \left (a+b x^2\right ) \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2+x \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2 \left (-6 p-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )-\frac {3 p^2 \left (p \log \left (a+b x^2\right )-\log \left (c \left (a+b x^2\right )^p\right )\right ) \left (4 i \sqrt {a} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2+4 \sqrt {a} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-2+2 \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )+\log \left (a+b x^2\right )\right )+\sqrt {b} x \left (8-4 \log \left (a+b x^2\right )+\log ^2\left (a+b x^2\right )\right )+4 i \sqrt {a} \operatorname {PolyLog}\left (2,\frac {i \sqrt {a}+\sqrt {b} x}{-i \sqrt {a}+\sqrt {b} x}\right )\right )}{\sqrt {b}}+\frac {p^3 \left (-48 \sqrt {-a^2} \sqrt {\frac {b x^2}{a+b x^2}} \sqrt {a+b x^2} \arcsin \left (\frac {\sqrt {a}}{\sqrt {a+b x^2}}\right )+\sqrt {-a} b x^2 \left (-48+24 \log \left (a+b x^2\right )-6 \log ^2\left (a+b x^2\right )+\log ^3\left (a+b x^2\right )\right )-6 \sqrt {-a^2} \sqrt {\frac {b x^2}{a+b x^2}} \left (8 \sqrt {a} \, _4F_3\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2},\frac {3}{2};\frac {a}{a+b x^2}\right )+\log \left (a+b x^2\right ) \left (4 \sqrt {a} \, _3F_2\left (\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};\frac {a}{a+b x^2}\right )+\sqrt {a+b x^2} \arcsin \left (\frac {\sqrt {a}}{\sqrt {a+b x^2}}\right ) \log \left (a+b x^2\right )\right )\right )+24 a \sqrt {b x^2} \text {arctanh}\left (\frac {\sqrt {b x^2}}{\sqrt {-a}}\right ) \left (\log \left (a+b x^2\right )-\log \left (1+\frac {b x^2}{a}\right )\right )+6 (-a)^{3/2} \sqrt {-\frac {b x^2}{a}} \left (\log ^2\left (1+\frac {b x^2}{a}\right )-4 \log \left (1+\frac {b x^2}{a}\right ) \log \left (\frac {1}{2} \left (1+\sqrt {-\frac {b x^2}{a}}\right )\right )+2 \log ^2\left (\frac {1}{2} \left (1+\sqrt {-\frac {b x^2}{a}}\right )\right )-4 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {1}{2} \sqrt {-\frac {b x^2}{a}}\right )\right )\right )}{\sqrt {-a} b x} \]

[In]

Integrate[Log[c*(a + b*x^2)^p]^3,x]

output

(6*Sqrt[a]*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x 
^2)^p])^2)/Sqrt[b] + 3*p*x*Log[a + b*x^2]*(-(p*Log[a + b*x^2]) + Log[c*(a + b* 
x^2)^p])^2 + x*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2*(-6*p - p*Log[a 
+ b*x^2] + Log[c*(a + b*x^2)^p]) - (3*p^2*(p*Log[a + b*x^2] - Log[c*(a + b*x^2 
)^p])*((4*I)*Sqrt[a]*ArcTan[(Sqrt[b]*x)/Sqrt[a]]^2 + 4*Sqrt[a]*ArcTan[(Sqrt[b] 
*x)/Sqrt[a]]*(-2 + 2*Log[(2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)] + Log[a + b*x^2] 
) + Sqrt[b]*x*(8 - 4*Log[a + b*x^2] + Log[a + b*x^2]^2) + (4*I)*Sqrt[a]*PolyLo 
g[2, (I*Sqrt[a] + Sqrt[b]*x)/((-I)*Sqrt[a] + Sqrt[b]*x)]))/Sqrt[b] + (p^3*(-48 
*Sqrt[-a^2]*Sqrt[(b*x^2)/(a + b*x^2)]*Sqrt[a + b*x^2]*ArcSin[Sqrt[a]/Sqrt[a + 
b*x^2]] + Sqrt[-a]*b*x^2*(-48 + 24*Log[a + b*x^2] - 6*Log[a + b*x^2]^2 + Log[a 
 + b*x^2]^3) - 6*Sqrt[-a^2]*Sqrt[(b*x^2)/(a + b*x^2)]*(8*Sqrt[a]*Hypergeometri 
cPFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3/2, 3/2}, a/(a + b*x^2)] + Log[a + b*x^2]*(4 
*Sqrt[a]*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, a/(a + b*x^2)] + Sqrt[ 
a + b*x^2]*ArcSin[Sqrt[a]/Sqrt[a + b*x^2]]*Log[a + b*x^2])) + 24*a*Sqrt[b*x^2] 
*ArcTanh[Sqrt[b*x^2]/Sqrt[-a]]*(Log[a + b*x^2] - Log[1 + (b*x^2)/a]) + 6*(-a)^ 
(3/2)*Sqrt[-((b*x^2)/a)]*(Log[1 + (b*x^2)/a]^2 - 4*Log[1 + (b*x^2)/a]*Log[(1 + 
 Sqrt[-((b*x^2)/a)])/2] + 2*Log[(1 + Sqrt[-((b*x^2)/a)])/2]^2 - 4*PolyLog[2, 1 
/2 - Sqrt[-((b*x^2)/a)]/2])))/(Sqrt[-a]*b*x)
 

Integral number [100] \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx \]

[C]   time = 1.00436 (sec), size = 505 ,normalized size = 28.06 \[ \frac {p^3 \left (-96 \sqrt {a} \sqrt {1-\frac {a}{a+b x^2}} \, _4F_3\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2},\frac {3}{2};\frac {a}{a+b x^2}\right )-48 \sqrt {a} \sqrt {1-\frac {a}{a+b x^2}} \, _3F_2\left (\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};\frac {a}{a+b x^2}\right ) \log \left (a+b x^2\right )-2 \log ^2\left (a+b x^2\right ) \left (6 \sqrt {a+b x^2} \sqrt {1-\frac {a}{a+b x^2}} \arcsin \left (\frac {\sqrt {a}}{\sqrt {a+b x^2}}\right )+\sqrt {a} \log \left (a+b x^2\right )\right )\right )}{2 \sqrt {a} x}+\frac {6 \sqrt {b} p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2}{\sqrt {a}}-\frac {3 p \log \left (a+b x^2\right ) \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2}{x}-\frac {\left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^3}{x}+3 p^2 \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right ) \left (-\frac {\log ^2\left (a+b x^2\right )}{x}+\frac {4 \sqrt {b} \left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (i \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )+2 \log \left (\frac {2 i}{i-\frac {\sqrt {b} x}{\sqrt {a}}}\right )+\log \left (a+b x^2\right )\right )+i \operatorname {PolyLog}\left (2,\frac {i \sqrt {a}+\sqrt {b} x}{-i \sqrt {a}+\sqrt {b} x}\right )\right )}{\sqrt {a}}\right ) \]

[In]

Integrate[Log[c*(a + b*x^2)^p]^3/x^2,x]

output

(p^3*(-96*Sqrt[a]*Sqrt[1 - a/(a + b*x^2)]*HypergeometricPFQ[{1/2, 1/2, 1/2, 1/ 
2}, {3/2, 3/2, 3/2}, a/(a + b*x^2)] - 48*Sqrt[a]*Sqrt[1 - a/(a + b*x^2)]*Hyper 
geometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, a/(a + b*x^2)]*Log[a + b*x^2] - 2*Lo 
g[a + b*x^2]^2*(6*Sqrt[a + b*x^2]*Sqrt[1 - a/(a + b*x^2)]*ArcSin[Sqrt[a]/Sqrt[ 
a + b*x^2]] + Sqrt[a]*Log[a + b*x^2])))/(2*Sqrt[a]*x) + (6*Sqrt[b]*p*ArcTan[(S 
qrt[b]*x)/Sqrt[a]]*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2)/Sqrt[a] - ( 
3*p*Log[a + b*x^2]*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2)/x - (-(p*Lo 
g[a + b*x^2]) + Log[c*(a + b*x^2)^p])^3/x + 3*p^2*(-(p*Log[a + b*x^2]) + Log[c 
*(a + b*x^2)^p])*(-(Log[a + b*x^2]^2/x) + (4*Sqrt[b]*(ArcTan[(Sqrt[b]*x)/Sqrt[ 
a]]*(I*ArcTan[(Sqrt[b]*x)/Sqrt[a]] + 2*Log[(2*I)/(I - (Sqrt[b]*x)/Sqrt[a])] + 
Log[a + b*x^2]) + I*PolyLog[2, (I*Sqrt[a] + Sqrt[b]*x)/((-I)*Sqrt[a] + Sqrt[b] 
*x)]))/Sqrt[a])
 

Integral number [101] \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^4} \, dx \]

[B]   time = 2.19457 (sec), size = 851 ,normalized size = 47.28 \[ \frac {a^2 \left (p \log \left (a+b x^2\right )-\log \left (c \left (a+b x^2\right )^p\right )\right )^3-6 a b p x^2 \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2-6 \sqrt {a} b^{3/2} p x^3 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2-3 a^2 p \log \left (a+b x^2\right ) \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2+3 \sqrt {a} p^2 \left (p \log \left (a+b x^2\right )-\log \left (c \left (a+b x^2\right )^p\right )\right ) \left (a^{3/2} \log ^2\left (a+b x^2\right )+4 b x^2 \left (i \sqrt {b} x \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2+\sqrt {a} \log \left (a+b x^2\right )+\sqrt {b} x \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-2+2 \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )+\log \left (a+b x^2\right )\right )+i \sqrt {b} x \operatorname {PolyLog}\left (2,\frac {i \sqrt {a}+\sqrt {b} x}{-i \sqrt {a}+\sqrt {b} x}\right )\right )\right )+p^3 \left (48 a b x^2 \sqrt {\frac {b x^2}{a+b x^2}} \, _4F_3\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2},\frac {3}{2};\frac {a}{a+b x^2}\right )+24 \sqrt {-a} \left (b x^2\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {b x^2}}{\sqrt {-a}}\right ) \log \left (a+b x^2\right )+24 a b x^2 \sqrt {\frac {b x^2}{a+b x^2}} \, _3F_2\left (\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};\frac {a}{a+b x^2}\right ) \log \left (a+b x^2\right )-6 a b x^2 \log ^2\left (a+b x^2\right )+6 \sqrt {a} \left (\frac {b x^2}{a+b x^2}\right )^{3/2} \left (a+b x^2\right )^{3/2} \arcsin \left (\frac {\sqrt {a}}{\sqrt {a+b x^2}}\right ) \log ^2\left (a+b x^2\right )-a^2 \log ^3\left (a+b x^2\right )-24 \sqrt {-a} \left (b x^2\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {b x^2}}{\sqrt {-a}}\right ) \log \left (1+\frac {b x^2}{a}\right )-6 a^2 \left (-\frac {b x^2}{a}\right )^{3/2} \log ^2\left (1+\frac {b x^2}{a}\right )+24 a^2 \left (-\frac {b x^2}{a}\right )^{3/2} \log \left (1+\frac {b x^2}{a}\right ) \log \left (\frac {1}{2} \left (1+\sqrt {-\frac {b x^2}{a}}\right )\right )-12 a^2 \left (-\frac {b x^2}{a}\right )^{3/2} \log ^2\left (\frac {1}{2} \left (1+\sqrt {-\frac {b x^2}{a}}\right )\right )+24 a^2 \left (-\frac {b x^2}{a}\right )^{3/2} \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {1}{2} \sqrt {-\frac {b x^2}{a}}\right )\right )}{3 a^2 x^3} \]

[In]

Integrate[Log[c*(a + b*x^2)^p]^3/x^4,x]

output

(a^2*(p*Log[a + b*x^2] - Log[c*(a + b*x^2)^p])^3 - 6*a*b*p*x^2*(-(p*Log[a + b* 
x^2]) + Log[c*(a + b*x^2)^p])^2 - 6*Sqrt[a]*b^(3/2)*p*x^3*ArcTan[(Sqrt[b]*x)/S 
qrt[a]]*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2 - 3*a^2*p*Log[a + b*x^2 
]*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2 + 3*Sqrt[a]*p^2*(p*Log[a + b* 
x^2] - Log[c*(a + b*x^2)^p])*(a^(3/2)*Log[a + b*x^2]^2 + 4*b*x^2*(I*Sqrt[b]*x* 
ArcTan[(Sqrt[b]*x)/Sqrt[a]]^2 + Sqrt[a]*Log[a + b*x^2] + Sqrt[b]*x*ArcTan[(Sqr 
t[b]*x)/Sqrt[a]]*(-2 + 2*Log[(2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)] + Log[a + b* 
x^2]) + I*Sqrt[b]*x*PolyLog[2, (I*Sqrt[a] + Sqrt[b]*x)/((-I)*Sqrt[a] + Sqrt[b] 
*x)])) + p^3*(48*a*b*x^2*Sqrt[(b*x^2)/(a + b*x^2)]*HypergeometricPFQ[{1/2, 1/2 
, 1/2, 1/2}, {3/2, 3/2, 3/2}, a/(a + b*x^2)] + 24*Sqrt[-a]*(b*x^2)^(3/2)*ArcTa 
nh[Sqrt[b*x^2]/Sqrt[-a]]*Log[a + b*x^2] + 24*a*b*x^2*Sqrt[(b*x^2)/(a + b*x^2)] 
*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, a/(a + b*x^2)]*Log[a + b*x^2] 
- 6*a*b*x^2*Log[a + b*x^2]^2 + 6*Sqrt[a]*((b*x^2)/(a + b*x^2))^(3/2)*(a + b*x^ 
2)^(3/2)*ArcSin[Sqrt[a]/Sqrt[a + b*x^2]]*Log[a + b*x^2]^2 - a^2*Log[a + b*x^2] 
^3 - 24*Sqrt[-a]*(b*x^2)^(3/2)*ArcTanh[Sqrt[b*x^2]/Sqrt[-a]]*Log[1 + (b*x^2)/a 
] - 6*a^2*(-((b*x^2)/a))^(3/2)*Log[1 + (b*x^2)/a]^2 + 24*a^2*(-((b*x^2)/a))^(3 
/2)*Log[1 + (b*x^2)/a]*Log[(1 + Sqrt[-((b*x^2)/a)])/2] - 12*a^2*(-((b*x^2)/a)) 
^(3/2)*Log[(1 + Sqrt[-((b*x^2)/a)])/2]^2 + 24*a^2*(-((b*x^2)/a))^(3/2)*PolyLog 
[2, 1/2 - Sqrt[-((b*x^2)/a)]/2]))/(3*a^2*x^3)
 

Integral number [158] \[ \int (f x)^m \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx \]

[B]   time = 1.8218 (sec), size = 994 ,normalized size = 49.7 \[ \frac {(f x)^m \left ((1+m) p^3 x^2 \log ^3\left (d+e x^2\right )+\frac {6 p^3 \left (-\frac {e x^2}{d}\right )^{\frac {1-m}{2}} \left (-\left ((1+m) \left (d+e x^2\right ) \, _4F_3\left (1,1,1,\frac {1}{2}-\frac {m}{2};2,2,2;1+\frac {e x^2}{d}\right )\right )+(1+m) \left (d+e x^2\right ) \, _3F_2\left (1,1,\frac {1}{2}-\frac {m}{2};2,2;1+\frac {e x^2}{d}\right ) \log \left (d+e x^2\right )+d \left (-1+\left (-\frac {e x^2}{d}\right )^{\frac {1+m}{2}}\right ) \log ^2\left (d+e x^2\right )\right )}{e}+\frac {6 d (1+m) p^3 \left (\frac {e x^2}{d+e x^2}\right )^{\frac {1}{2}-\frac {m}{2}} \left (8 \, _4F_3\left (\frac {1}{2}-\frac {m}{2},\frac {1}{2}-\frac {m}{2},\frac {1}{2}-\frac {m}{2},\frac {1}{2}-\frac {m}{2};\frac {3}{2}-\frac {m}{2},\frac {3}{2}-\frac {m}{2},\frac {3}{2}-\frac {m}{2};\frac {d}{d+e x^2}\right )+(-1+m) \log \left (d+e x^2\right ) \left (-4 \, _3F_2\left (\frac {1}{2}-\frac {m}{2},\frac {1}{2}-\frac {m}{2},\frac {1}{2}-\frac {m}{2};\frac {3}{2}-\frac {m}{2},\frac {3}{2}-\frac {m}{2};\frac {d}{d+e x^2}\right )+(-1+m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {m}{2},\frac {1}{2}-\frac {m}{2},\frac {3}{2}-\frac {m}{2},\frac {d}{d+e x^2}\right ) \log \left (d+e x^2\right )\right )\right )}{e (-1+m)^3}-\frac {3 p^2 \left (-\frac {e x^2}{d}\right )^{\frac {1-m}{2}} \left (-\left ((1+m) \left (d+e x^2\right ) \, _4F_3\left (1,1,1,\frac {1}{2}-\frac {m}{2};2,2,2;1+\frac {e x^2}{d}\right )\right )+(1+m) \left (d+e x^2\right ) \, _3F_2\left (1,1,\frac {1}{2}-\frac {m}{2};2,2;1+\frac {e x^2}{d}\right ) \log \left (d+e x^2\right )+d \left (-1+\left (-\frac {e x^2}{d}\right )^{\frac {1+m}{2}}\right ) \log ^2\left (d+e x^2\right )\right ) \left (-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right )}{e}-\frac {3 m p^2 \left (-\frac {e x^2}{d}\right )^{\frac {1-m}{2}} \left (-\left ((1+m) \left (d+e x^2\right ) \, _4F_3\left (1,1,1,\frac {1}{2}-\frac {m}{2};2,2,2;1+\frac {e x^2}{d}\right )\right )+(1+m) \left (d+e x^2\right ) \, _3F_2\left (1,1,\frac {1}{2}-\frac {m}{2};2,2;1+\frac {e x^2}{d}\right ) \log \left (d+e x^2\right )+d \left (-1+\left (-\frac {e x^2}{d}\right )^{\frac {1+m}{2}}\right ) \log ^2\left (d+e x^2\right )\right ) \left (-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right )}{e}+\frac {3 p x^2 \left (-2 e x^2 \operatorname {Hypergeometric2F1}\left (1,\frac {3+m}{2},\frac {5+m}{2},-\frac {e x^2}{d}\right )+d (3+m) \log \left (d+e x^2\right )\right ) \left (-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right )^2}{d (3+m)}+\frac {3 m p x^2 \left (-2 e x^2 \operatorname {Hypergeometric2F1}\left (1,\frac {3+m}{2},\frac {5+m}{2},-\frac {e x^2}{d}\right )+d (3+m) \log \left (d+e x^2\right )\right ) \left (-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right )^2}{d (3+m)}+x^2 \left (-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right )^3+m x^2 \left (-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right )^3\right )}{(1+m)^2 x} \]

[In]

Integrate[(f*x)^m*Log[c*(d + e*x^2)^p]^3,x]

output

((f*x)^m*((1 + m)*p^3*x^2*Log[d + e*x^2]^3 + (6*p^3*(-((e*x^2)/d))^((1 - m)/2) 
*(-((1 + m)*(d + e*x^2)*HypergeometricPFQ[{1, 1, 1, 1/2 - m/2}, {2, 2, 2}, 1 + 
 (e*x^2)/d]) + (1 + m)*(d + e*x^2)*HypergeometricPFQ[{1, 1, 1/2 - m/2}, {2, 2} 
, 1 + (e*x^2)/d]*Log[d + e*x^2] + d*(-1 + (-((e*x^2)/d))^((1 + m)/2))*Log[d + 
e*x^2]^2))/e + (6*d*(1 + m)*p^3*((e*x^2)/(d + e*x^2))^(1/2 - m/2)*(8*Hypergeom 
etricPFQ[{1/2 - m/2, 1/2 - m/2, 1/2 - m/2, 1/2 - m/2}, {3/2 - m/2, 3/2 - m/2, 
3/2 - m/2}, d/(d + e*x^2)] + (-1 + m)*Log[d + e*x^2]*(-4*HypergeometricPFQ[{1/ 
2 - m/2, 1/2 - m/2, 1/2 - m/2}, {3/2 - m/2, 3/2 - m/2}, d/(d + e*x^2)] + (-1 + 
 m)*Hypergeometric2F1[1/2 - m/2, 1/2 - m/2, 3/2 - m/2, d/(d + e*x^2)]*Log[d + 
e*x^2])))/(e*(-1 + m)^3) - (3*p^2*(-((e*x^2)/d))^((1 - m)/2)*(-((1 + m)*(d + e 
*x^2)*HypergeometricPFQ[{1, 1, 1, 1/2 - m/2}, {2, 2, 2}, 1 + (e*x^2)/d]) + (1 
+ m)*(d + e*x^2)*HypergeometricPFQ[{1, 1, 1/2 - m/2}, {2, 2}, 1 + (e*x^2)/d]*L 
og[d + e*x^2] + d*(-1 + (-((e*x^2)/d))^((1 + m)/2))*Log[d + e*x^2]^2)*(-(p*Log 
[d + e*x^2]) + Log[c*(d + e*x^2)^p]))/e - (3*m*p^2*(-((e*x^2)/d))^((1 - m)/2)* 
(-((1 + m)*(d + e*x^2)*HypergeometricPFQ[{1, 1, 1, 1/2 - m/2}, {2, 2, 2}, 1 + 
(e*x^2)/d]) + (1 + m)*(d + e*x^2)*HypergeometricPFQ[{1, 1, 1/2 - m/2}, {2, 2}, 
 1 + (e*x^2)/d]*Log[d + e*x^2] + d*(-1 + (-((e*x^2)/d))^((1 + m)/2))*Log[d + e 
*x^2]^2)*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p]))/e + (3*p*x^2*(-2*e*x^2* 
Hypergeometric2F1[1, (3 + m)/2, (5 + m)/2, -((e*x^2)/d)] + d*(3 + m)*Log[d + e 
*x^2])*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2)/(d*(3 + m)) + (3*m*p...
 

Integral number [159] \[ \int (f x)^m \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx \]

[B]   time = 0.528109 (sec), size = 466 ,normalized size = 23.3 \[ \frac {(f x)^m \left (4 p^2 x \left (\frac {2 e x^2 \operatorname {Hypergeometric2F1}\left (1,\frac {3+m}{2},\frac {5+m}{2},-\frac {e x^2}{d}\right )}{d (3+m)}-\log \left (d+e x^2\right )\right )+(1+m) p^2 x \log ^2\left (d+e x^2\right )+\frac {4 d (1+m) p^2 \left (\frac {e x^2}{d+e x^2}\right )^{\frac {1}{2}-\frac {m}{2}} \left (-2 \, _3F_2\left (\frac {1}{2}-\frac {m}{2},\frac {1}{2}-\frac {m}{2},\frac {1}{2}-\frac {m}{2};\frac {3}{2}-\frac {m}{2},\frac {3}{2}-\frac {m}{2};\frac {d}{d+e x^2}\right )+(-1+m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {m}{2},\frac {1}{2}-\frac {m}{2},\frac {3}{2}-\frac {m}{2},\frac {d}{d+e x^2}\right ) \log \left (d+e x^2\right )\right )}{e (-1+m)^2 x}+\frac {2 p \left (2 e x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3+m}{2},\frac {5+m}{2},-\frac {e x^2}{d}\right )-d (3+m) x \log \left (d+e x^2\right )\right ) \left (p \log \left (d+e x^2\right )-\log \left (c \left (d+e x^2\right )^p\right )\right )}{d (3+m)}-\frac {2 m p \left (-2 e x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3+m}{2},\frac {5+m}{2},-\frac {e x^2}{d}\right )+d (3+m) x \log \left (d+e x^2\right )\right ) \left (p \log \left (d+e x^2\right )-\log \left (c \left (d+e x^2\right )^p\right )\right )}{d (3+m)}+x \left (-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right )^2+m x \left (-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right )^2\right )}{(1+m)^2} \]

[In]

Integrate[(f*x)^m*Log[c*(d + e*x^2)^p]^2,x]

output

((f*x)^m*(4*p^2*x*((2*e*x^2*Hypergeometric2F1[1, (3 + m)/2, (5 + m)/2, -((e*x^ 
2)/d)])/(d*(3 + m)) - Log[d + e*x^2]) + (1 + m)*p^2*x*Log[d + e*x^2]^2 + (4*d* 
(1 + m)*p^2*((e*x^2)/(d + e*x^2))^(1/2 - m/2)*(-2*HypergeometricPFQ[{1/2 - m/2 
, 1/2 - m/2, 1/2 - m/2}, {3/2 - m/2, 3/2 - m/2}, d/(d + e*x^2)] + (-1 + m)*Hyp 
ergeometric2F1[1/2 - m/2, 1/2 - m/2, 3/2 - m/2, d/(d + e*x^2)]*Log[d + e*x^2]) 
)/(e*(-1 + m)^2*x) + (2*p*(2*e*x^3*Hypergeometric2F1[1, (3 + m)/2, (5 + m)/2, 
-((e*x^2)/d)] - d*(3 + m)*x*Log[d + e*x^2])*(p*Log[d + e*x^2] - Log[c*(d + e*x 
^2)^p]))/(d*(3 + m)) - (2*m*p*(-2*e*x^3*Hypergeometric2F1[1, (3 + m)/2, (5 + m 
)/2, -((e*x^2)/d)] + d*(3 + m)*x*Log[d + e*x^2])*(p*Log[d + e*x^2] - Log[c*(d 
+ e*x^2)^p]))/(d*(3 + m)) + x*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2 + 
 m*x*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2))/(1 + m)^2
 

Integral number [277] \[ \int \left (f+g x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx \]

[B]   time = 9.0714 (sec), size = 1772 ,normalized size = 80.55 \[ \text {result too large to display} \]

[In]

Integrate[(f + g*x^2)*Log[c*(d + e*x^2)^p]^3,x]

output

(2*d*g*p*x*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2)/e + (6*Sqrt[d]*f*p* 
ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2)/Sq 
rt[e] - (2*d^(3/2)*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(-(p*Log[d + e*x^2]) + Log[ 
c*(d + e*x^2)^p])^2)/e^(3/2) + 3*f*p*x*Log[d + e*x^2]*(-(p*Log[d + e*x^2]) + L 
og[c*(d + e*x^2)^p])^2 + g*p*x^3*Log[d + e*x^2]*(-(p*Log[d + e*x^2]) + Log[c*( 
d + e*x^2)^p])^2 + f*x*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2*(-6*p - 
p*Log[d + e*x^2] + Log[c*(d + e*x^2)^p]) + (g*x^3*(-(p*Log[d + e*x^2]) + Log[c 
*(d + e*x^2)^p])^2*(-2*p - p*Log[d + e*x^2] + Log[c*(d + e*x^2)^p]))/3 + 3*f*p 
^2*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])*(x*Log[d + e*x^2]^2 - (4*((-I) 
*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2 + Sqrt[e]*x*(-2 + Log[d + e*x^2]) - Sqr 
t[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(-2 + 2*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]* 
x)] + Log[d + e*x^2]) - I*Sqrt[d]*PolyLog[2, (I*Sqrt[d] + Sqrt[e]*x)/((-I)*Sqr 
t[d] + Sqrt[e]*x)]))/Sqrt[e]) + 3*g*p^2*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^ 
2)^p])*((x^3*Log[d + e*x^2]^2)/3 - (4*((9*I)*d^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d 
]]^2 + 3*d^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(-8 + 6*Log[(2*Sqrt[d])/(Sqrt[d] 
+ I*Sqrt[e]*x)] + 3*Log[d + e*x^2]) + Sqrt[e]*x*(24*d - 2*e*x^2 + (-9*d + 3*e* 
x^2)*Log[d + e*x^2]) + (9*I)*d^(3/2)*PolyLog[2, (I*Sqrt[d] + Sqrt[e]*x)/((-I)* 
Sqrt[d] + Sqrt[e]*x)]))/(27*e^(3/2))) + (g*p^3*(416*Sqrt[-d]*d^(3/2)*Sqrt[d + 
e*x^2]*Sqrt[1 - d/(d + e*x^2)]*ArcSin[Sqrt[d]/Sqrt[d + e*x^2]] + 36*Sqrt[-d]*d 
^(3/2)*Sqrt[1 - d/(d + e*x^2)]*(8*Sqrt[d]*HypergeometricPFQ[{1/2, 1/2, 1/2,...
 

Integral number [298] \[ \int \left (f+g x^3\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx \]

[B]   time = 8.70025 (sec), size = 2385 ,normalized size = 99.38 \[ \text {Result too large to show} \]

[In]

Integrate[(f + g*x^3)^2*Log[c*(d + e*x^2)^p]^3,x]

output

(f*g*p^3*(d + e*x^2)*(-8*d*(-6 + 6*Log[d + e*x^2] - 3*Log[d + e*x^2]^2 + Log[d 
 + e*x^2]^3) + (d + e*x^2)*(-3 + 6*Log[d + e*x^2] - 6*Log[d + e*x^2]^2 + 4*Log 
[d + e*x^2]^3)))/(8*e^2) + 6*f*g*p^2*((x^4*Log[d + e*x^2]^2)/4 - e*((3*d*x^2)/ 
(4*e^2) - x^4/(8*e) - (3*d^2*Log[d + e*x^2])/(4*e^3) - (d*x^2*Log[d + e*x^2])/ 
(2*e^2) + (x^4*Log[d + e*x^2])/(4*e) + (d^2*Log[d + e*x^2]^2)/(4*e^3)))*(-(p*L 
og[d + e*x^2]) + Log[c*(d + e*x^2)^p]) + (3*d*f*g*p*x^2*(-(p*Log[d + e*x^2]) + 
 Log[c*(d + e*x^2)^p])^2)/(2*e) - (2*d^2*g^2*p*x^3*(-(p*Log[d + e*x^2]) + Log[ 
c*(d + e*x^2)^p])^2)/(7*e^2) + (6*d*g^2*p*x^5*(-(p*Log[d + e*x^2]) + Log[c*(d 
+ e*x^2)^p])^2)/(35*e) - (3*d^2*f*g*p*Log[d + e*x^2]*(-(p*Log[d + e*x^2]) + Lo 
g[c*(d + e*x^2)^p])^2)/(2*e^2) + (3*p*x*(14*f^2 + 7*f*g*x^3 + 2*g^2*x^6)*Log[d 
 + e*x^2]*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2)/14 + (f*g*x^4*(-(p*L 
og[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2*(-3*p + 2*(-(p*Log[d + e*x^2]) + Log[ 
c*(d + e*x^2)^p])))/4 + (g^2*x^7*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^ 
2*(-6*p + 7*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])))/49 + (x*(-(p*Log[d 
+ e*x^2]) + Log[c*(d + e*x^2)^p])^2*(-42*e^3*f^2*p + 6*d^3*g^2*p + 7*e^3*f^2*( 
-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])))/(7*e^3) - (6*ArcTan[(Sqrt[e]*x)/ 
Sqrt[d]]*(-7*d*e^3*f^2*p*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2 + d^4* 
g^2*p*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2))/(7*Sqrt[d]*e^(7/2)) + 3 
*f^2*p^2*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])*(x*Log[d + e*x^2]^2 - (4 
*((-I)*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2 + Sqrt[e]*x*(-2 + Log[d + e*x^...
 

Integral number [299] \[ \int \left (f+g x^3\right ) \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx \]

[B]   time = 1.29533 (sec), size = 1051 ,normalized size = 47.77 \[ \frac {1}{4} g x^4 \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac {6 \sqrt {d} f p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right )^2}{\sqrt {e}}+3 f p x \log \left (d+e x^2\right ) \left (-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right )^2+f x \left (-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right )^2 \left (-6 p-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right )-\frac {3}{4} g p \left (-\frac {7 d p^2 x^2}{2 e}+\frac {p^2 x^4}{4}+\frac {d^2 p^2 \log \left (d+e x^2\right )}{2 e^2}+\frac {3 d^2 p \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac {3 d p x^2 \log \left (c \left (d+e x^2\right )^p\right )}{e}-\frac {1}{2} p x^4 \log \left (c \left (d+e x^2\right )^p\right )-\frac {3 d^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}-\frac {d x^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}+\frac {1}{2} x^4 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {d^2 \log ^3\left (c \left (d+e x^2\right )^p\right )}{3 e^2 p}\right )+3 f p^2 \left (-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right ) \left (x \log ^2\left (d+e x^2\right )-\frac {4 \left (-i \sqrt {d} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2+\sqrt {e} x \left (-2+\log \left (d+e x^2\right )\right )-\sqrt {d} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (-2+2 \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )+\log \left (d+e x^2\right )\right )-i \sqrt {d} \operatorname {PolyLog}\left (2,\frac {i \sqrt {d}+\sqrt {e} x}{-i \sqrt {d}+\sqrt {e} x}\right )\right )}{\sqrt {e}}\right )+\frac {f p^3 \left (-48 \sqrt {-d^2} \sqrt {d+e x^2} \sqrt {1-\frac {d}{d+e x^2}} \arcsin \left (\frac {\sqrt {d}}{\sqrt {d+e x^2}}\right )-6 \sqrt {-d^2} \sqrt {1-\frac {d}{d+e x^2}} \left (8 \sqrt {d} \, _4F_3\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2},\frac {3}{2};\frac {d}{d+e x^2}\right )+4 \sqrt {d} \, _3F_2\left (\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};\frac {d}{d+e x^2}\right ) \log \left (d+e x^2\right )+\sqrt {d+e x^2} \arcsin \left (\frac {\sqrt {d}}{\sqrt {d+e x^2}}\right ) \log ^2\left (d+e x^2\right )\right )+\sqrt {-d} e x^2 \left (-48+24 \log \left (d+e x^2\right )-6 \log ^2\left (d+e x^2\right )+\log ^3\left (d+e x^2\right )\right )+24 d \sqrt {e x^2} \text {arctanh}\left (\frac {\sqrt {e x^2}}{\sqrt {-d}}\right ) \left (\log \left (d+e x^2\right )-\log \left (\frac {d+e x^2}{d}\right )\right )+6 (-d)^{3/2} \sqrt {1-\frac {d+e x^2}{d}} \left (\log ^2\left (\frac {d+e x^2}{d}\right )-4 \log \left (\frac {d+e x^2}{d}\right ) \log \left (\frac {1}{2} \left (1+\sqrt {1-\frac {d+e x^2}{d}}\right )\right )+2 \log ^2\left (\frac {1}{2} \left (1+\sqrt {1-\frac {d+e x^2}{d}}\right )\right )-4 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {1}{2} \sqrt {1-\frac {d+e x^2}{d}}\right )\right )\right )}{\sqrt {-d} e x} \]

[In]

Integrate[(f + g*x^3)*Log[c*(d + e*x^2)^p]^3,x]

output

(g*x^4*Log[c*(d + e*x^2)^p]^3)/4 + (6*Sqrt[d]*f*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]* 
(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2)/Sqrt[e] + 3*f*p*x*Log[d + e*x^ 
2]*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2 + f*x*(-(p*Log[d + e*x^2]) + 
 Log[c*(d + e*x^2)^p])^2*(-6*p - p*Log[d + e*x^2] + Log[c*(d + e*x^2)^p]) - (3 
*g*p*((-7*d*p^2*x^2)/(2*e) + (p^2*x^4)/4 + (d^2*p^2*Log[d + e*x^2])/(2*e^2) + 
(3*d^2*p*Log[c*(d + e*x^2)^p])/e^2 + (3*d*p*x^2*Log[c*(d + e*x^2)^p])/e - (p*x 
^4*Log[c*(d + e*x^2)^p])/2 - (3*d^2*Log[c*(d + e*x^2)^p]^2)/(2*e^2) - (d*x^2*L 
og[c*(d + e*x^2)^p]^2)/e + (x^4*Log[c*(d + e*x^2)^p]^2)/2 + (d^2*Log[c*(d + e* 
x^2)^p]^3)/(3*e^2*p)))/4 + 3*f*p^2*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p] 
)*(x*Log[d + e*x^2]^2 - (4*((-I)*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2 + Sqrt[ 
e]*x*(-2 + Log[d + e*x^2]) - Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(-2 + 2*Log[( 
2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)] + Log[d + e*x^2]) - I*Sqrt[d]*PolyLog[2, ( 
I*Sqrt[d] + Sqrt[e]*x)/((-I)*Sqrt[d] + Sqrt[e]*x)]))/Sqrt[e]) + (f*p^3*(-48*Sq 
rt[-d^2]*Sqrt[d + e*x^2]*Sqrt[1 - d/(d + e*x^2)]*ArcSin[Sqrt[d]/Sqrt[d + e*x^2 
]] - 6*Sqrt[-d^2]*Sqrt[1 - d/(d + e*x^2)]*(8*Sqrt[d]*HypergeometricPFQ[{1/2, 1 
/2, 1/2, 1/2}, {3/2, 3/2, 3/2}, d/(d + e*x^2)] + 4*Sqrt[d]*HypergeometricPFQ[{ 
1/2, 1/2, 1/2}, {3/2, 3/2}, d/(d + e*x^2)]*Log[d + e*x^2] + Sqrt[d + e*x^2]*Ar 
cSin[Sqrt[d]/Sqrt[d + e*x^2]]*Log[d + e*x^2]^2) + Sqrt[-d]*e*x^2*(-48 + 24*Log 
[d + e*x^2] - 6*Log[d + e*x^2]^2 + Log[d + e*x^2]^3) + 24*d*Sqrt[e*x^2]*ArcTan 
h[Sqrt[e*x^2]/Sqrt[-d]]*(Log[d + e*x^2] - Log[(d + e*x^2)/d]) + 6*(-d)^(3/2...
 

Integral number [485] \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx \]

[A]   time = 7.79559 (sec), size = 1552 ,normalized size = 64.67 \[ \text {result too large to display} \]

[In]

Integrate[x^2*(a + b*Log[c*(d + e*x^(2/3))^n])^3,x]

output

(-2*b*d^4*n*x^(1/3)*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^ 
2)/e^4 + (2*b*d^3*n*x*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n] 
)^2)/(3*e^3) - (2*b*d^2*n*x^(5/3)*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e 
*x^(2/3))^n])^2)/(5*e^2) + (2*b*d*n*x^(7/3)*(a - b*n*Log[d + e*x^(2/3)] + b*Lo 
g[c*(d + e*x^(2/3))^n])^2)/(7*e) + (2*b*d^(9/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqr 
t[d]]*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2)/e^(9/2) + b 
*n*x^3*Log[d + e*x^(2/3)]*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3) 
)^n])^2 + (x^3*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2*(3* 
a - 2*b*n - 3*b*n*Log[d + e*x^(2/3)] + 3*b*Log[c*(d + e*x^(2/3))^n]))/9 - (b^3 
*n^3*(1094783760*d^(9/2)*Sqrt[d + e*x^(2/3)]*Sqrt[(e*x^(2/3))/(d + e*x^(2/3))] 
*ArcSin[Sqrt[d]/Sqrt[d + e*x^(2/3)]] - e*x^(2/3)*(-16*(68423985*d^4 - 4186770* 
d^3*e*x^(2/3) + 871542*d^2*e^2*x^(4/3) - 217125*d*e^3*x^2 + 42875*e^4*x^(8/3)) 
 + 2520*(177345*d^4 - 26040*d^3*e*x^(2/3) + 9009*d^2*e^2*x^(4/3) - 3600*d*e^3* 
x^2 + 1225*e^4*x^(8/3))*Log[d + e*x^(2/3)] - 198450*(315*d^4 - 105*d^3*e*x^(2/ 
3) + 63*d^2*e^2*x^(4/3) - 45*d*e^3*x^2 + 35*e^4*x^(8/3))*Log[d + e*x^(2/3)]^2 
+ 10418625*e^4*x^(8/3)*Log[d + e*x^(2/3)]^3) + 62511750*d^(9/2)*Sqrt[(e*x^(2/3 
))/(d + e*x^(2/3))]*(8*Sqrt[d]*HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3 
/2, 3/2}, d/(d + e*x^(2/3))] + Log[d + e*x^(2/3)]*(4*Sqrt[d]*HypergeometricPFQ 
[{1/2, 1/2, 1/2}, {3/2, 3/2}, d/(d + e*x^(2/3))] + Sqrt[d + e*x^(2/3)]*ArcSin[ 
Sqrt[d]/Sqrt[d + e*x^(2/3)]]*Log[d + e*x^(2/3)])) + 111727350*(-d)^(9/2)*(4...
 

Integral number [486] \[ \int \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx \]

[B]   time = 5.89109 (sec), size = 1299 ,normalized size = 64.95 \[ \text {result too large to display} \]

[In]

Integrate[(a + b*Log[c*(d + e*x^(2/3))^n])^3,x]

output

(6*b*d*n*x^(1/3)*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2)/ 
e - (6*b*d^(3/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(a - b*n*Log[d + e*x^(2/3 
)] + b*Log[c*(d + e*x^(2/3))^n])^2)/e^(3/2) + 3*b*n*x*Log[d + e*x^(2/3)]*(a - 
b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2 + x*(a - b*n*Log[d + e* 
x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2*(a - 2*b*n - b*n*Log[d + e*x^(2/3)] + 
 b*Log[c*(d + e*x^(2/3))^n]) + (b^2*n^2*x^(1/3)*(a - b*n*Log[d + e*x^(2/3)] + 
b*Log[c*(d + e*x^(2/3))^n])*((-96*d^(3/2)*ArcSin[Sqrt[d]/Sqrt[d + e*x^(2/3)]]) 
/(Sqrt[d + e*x^(2/3)]*Sqrt[(e*x^(2/3))/(d + e*x^(2/3))]) - d*(104 - 48*Log[d + 
 e*x^(2/3)] + 9*Log[d + e*x^(2/3)]^2) + (d + e*x^(2/3))*(8 - 12*Log[d + e*x^(2 
/3)] + 9*Log[d + e*x^(2/3)]^2) + (36*(-d)^(3/2)*ArcTanh[Sqrt[e*x^(2/3)]/Sqrt[- 
d]]*(Log[d + e*x^(2/3)] - Log[1 + (e*x^(2/3))/d]))/Sqrt[e*x^(2/3)] + (9*d*(2*L 
og[(1 + Sqrt[-((e*x^(2/3))/d)])/2]^2 - 4*Log[(1 + Sqrt[-((e*x^(2/3))/d)])/2]*L 
og[1 + (e*x^(2/3))/d] + Log[1 + (e*x^(2/3))/d]^2 - 4*PolyLog[2, 1/2 - Sqrt[-(( 
e*x^(2/3))/d)]/2]))/Sqrt[-((e*x^(2/3))/d)]))/(3*e) + (b^3*n^3*(624*d*e*x^(2/3) 
 - 16*e^2*x^(4/3) + 624*d^(3/2)*Sqrt[d + e*x^(2/3)]*Sqrt[(e*x^(2/3))/(d + e*x^ 
(2/3))]*ArcSin[Sqrt[d]/Sqrt[d + e*x^(2/3)]] + 432*d^2*Sqrt[(e*x^(2/3))/(d + e* 
x^(2/3))]*HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3/2, 3/2}, d/(d + e*x^ 
(2/3))] + 144*d^2*Sqrt[-((e*x^(2/3))/d)]*Log[(1 + Sqrt[-((e*x^(2/3))/d)])/2]^2 
 - 288*d*e*x^(2/3)*Log[d + e*x^(2/3)] + 24*e^2*x^(4/3)*Log[d + e*x^(2/3)] + 28 
8*Sqrt[-d]*d*Sqrt[e*x^(2/3)]*ArcTanh[Sqrt[e*x^(2/3)]/Sqrt[-d]]*Log[d + e*x^...
 

Integral number [487] \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^2} \, dx \]

[B]   time = 6.62541 (sec), size = 1158 ,normalized size = 48.25 \[ \text {result too large to display} \]

[In]

Integrate[(a + b*Log[c*(d + e*x^(2/3))^n])^3/x^2,x]

output

(-6*b*e*n*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2)/(d*x^(1 
/3)) - (6*b*e^(3/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(a - b*n*Log[d + e*x^( 
2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2)/d^(3/2) - (3*b*n*Log[d + e*x^(2/3)]*(a 
- b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2)/x - (a - b*n*Log[d + 
 e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^3/x + (3*b^2*e*n^2*(a - b*n*Log[d + 
e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])*((-16*Sqrt[d + e*x^(2/3)]*Sqrt[(e*x^( 
2/3))/(d + e*x^(2/3))]*ArcSin[Sqrt[d]/Sqrt[d + e*x^(2/3)]])/d^(3/2) - (8*Log[d 
 + e*x^(2/3)])/d - (2*Log[d + e*x^(2/3)]^2)/(e*x^(2/3)) - (8*Sqrt[e*x^(2/3)]*A 
rcTanh[Sqrt[e*x^(2/3)]/Sqrt[-d]]*(Log[d + e*x^(2/3)] - Log[1 + (e*x^(2/3))/d]) 
)/(-d)^(3/2) - (2*Sqrt[-((e*x^(2/3))/d)]*(2*Log[(1 + Sqrt[-((e*x^(2/3))/d)])/2 
]^2 - 4*Log[(1 + Sqrt[-((e*x^(2/3))/d)])/2]*Log[1 + (e*x^(2/3))/d] + Log[1 + ( 
e*x^(2/3))/d]^2 - 4*PolyLog[2, 1/2 - Sqrt[-((e*x^(2/3))/d)]/2]))/d))/(2*x^(1/3 
)) + (b^3*n^3*(48*Sqrt[-d^2]*e*Sqrt[(e*x^(2/3))/(d + e*x^(2/3))]*x^(2/3)*Hyper 
geometricPFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3/2, 3/2}, d/(d + e*x^(2/3))] - 12*d* 
Sqrt[-d^2]*(-((e*x^(2/3))/d))^(3/2)*Log[(1 + Sqrt[-((e*x^(2/3))/d)])/2]^2 - 24 
*Sqrt[d]*(e*x^(2/3))^(3/2)*ArcTanh[Sqrt[e*x^(2/3)]/Sqrt[-d]]*Log[d + e*x^(2/3) 
] + 24*Sqrt[-d^2]*e*Sqrt[(e*x^(2/3))/(d + e*x^(2/3))]*x^(2/3)*HypergeometricPF 
Q[{1/2, 1/2, 1/2}, {3/2, 3/2}, d/(d + e*x^(2/3))]*Log[d + e*x^(2/3)] - 6*Sqrt[ 
-d^2]*e*x^(2/3)*Log[d + e*x^(2/3)]^2 + 6*Sqrt[-d]*(d + e*x^(2/3))^(3/2)*((e*x^ 
(2/3))/(d + e*x^(2/3)))^(3/2)*ArcSin[Sqrt[d]/Sqrt[d + e*x^(2/3)]]*Log[d + e...
 

Integral number [488] \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^4} \, dx \]

[B]   time = 7.58333 (sec), size = 1385 ,normalized size = 57.71 \[ \text {result too large to display} \]

[In]

Integrate[(a + b*Log[c*(d + e*x^(2/3))^n])^3/x^4,x]

output

((-60*b*e*n*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2)/(d*x^ 
(7/3)) + (84*b*e^2*n*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n]) 
^2)/(d^2*x^(5/3)) - (140*b*e^3*n*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e* 
x^(2/3))^n])^2)/(d^3*x) + (420*b*e^4*n*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*( 
d + e*x^(2/3))^n])^2)/(d^4*x^(1/3)) + (420*b*e^(9/2)*n*ArcTan[(Sqrt[e]*x^(1/3) 
)/Sqrt[d]]*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2)/d^(9/2 
) - (210*b*n*Log[d + e*x^(2/3)]*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x 
^(2/3))^n])^2)/x^3 - (70*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3)) 
^n])^3)/x^3 - (2*b^3*n^3*(1376*e^3*(d + e*x^(2/3))^(3/2)*((e*x^(2/3))/(d + e*x 
^(2/3)))^(3/2)*x^2*ArcSin[Sqrt[d]/Sqrt[d + e*x^(2/3)]] + Sqrt[d]*(16*e^3*(d - 
15*e*x^(2/3))*x^2 + 8*(3*d^2*e^2*x^(4/3) - 12*d*e^3*x^2 + 71*e^4*x^(8/3))*Log[ 
d + e*x^(2/3)] + (30*d^3*e*x^(2/3) - 42*d^2*e^2*x^(4/3) + 70*d*e^3*x^2 - 210*e 
^4*x^(8/3))*Log[d + e*x^(2/3)]^2 + 35*d^4*Log[d + e*x^(2/3)]^3) + 210*e^4*Sqrt 
[(e*x^(2/3))/(d + e*x^(2/3))]*x^(8/3)*(8*Sqrt[d]*HypergeometricPFQ[{1/2, 1/2, 
1/2, 1/2}, {3/2, 3/2, 3/2}, d/(d + e*x^(2/3))] + Log[d + e*x^(2/3)]*(4*Sqrt[d] 
*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, d/(d + e*x^(2/3))] + Sqrt[d + 
e*x^(2/3)]*ArcSin[Sqrt[d]/Sqrt[d + e*x^(2/3)]]*Log[d + e*x^(2/3)])) + (352*d^( 
3/2)*e^4*x^(8/3)*(4*Sqrt[e*x^(2/3)]*ArcTanh[Sqrt[e*x^(2/3)]/Sqrt[-d]]*(Log[d + 
 e*x^(2/3)] - Log[1 + (e*x^(2/3))/d]) - Sqrt[-d]*Sqrt[-((e*x^(2/3))/d)]*(2*Log 
[(1 + Sqrt[-((e*x^(2/3))/d)])/2]^2 - 4*Log[(1 + Sqrt[-((e*x^(2/3))/d)])/2]*...
 

Integral number [528] \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx \]

[B]   time = 23.0946 (sec), size = 5975 ,normalized size = 248.96 \[ \text {Result too large to show} \]

[In]

Integrate[x^2*(a + b*Log[c*(d + e/x^(2/3))^n])^3,x]

output

Result too large to show
 

Integral number [530] \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{x^2} \, dx \]

[B]   time = 13.157 (sec), size = 5504 ,normalized size = 229.33 \[ \text {Result too large to show} \]

[In]

Integrate[(a + b*Log[c*(d + e/x^(2/3))^n])^3/x^2,x]

output

Result too large to show
 

Integral number [531] \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{x^4} \, dx \]

[B]   time = 21.2985 (sec), size = 6328 ,normalized size = 263.67 \[ \text {Result too large to show} \]

[In]

Integrate[(a + b*Log[c*(d + e/x^(2/3))^n])^3/x^4,x]

output

Result too large to show
 

4.5 Test file Number [79]

4.5.1 Mathematica

Integral number [399] \[ \int \frac {\cos ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \]

[C]   time = 0.895414 (sec), size = 394 ,normalized size = 17.13 \[ \frac {-i \text {RootSum}\left [-i b+3 i b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 i b \text {$\#$1}^4+i b \text {$\#$1}^6\& ,\frac {2 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+4 i a \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}+2 a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}+12 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-6 i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-4 i a \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^3-2 a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3+2 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4}{b \text {$\#$1}-4 i a \text {$\#$1}^2-2 b \text {$\#$1}^3+b \text {$\#$1}^5}\& \right ]+\frac {24 \cos (c+d x) (a+b \sin (c+d x))}{4 a+3 b \sin (c+d x)-b \sin (3 (c+d x))}}{18 a b d} \]

[In]

Integrate[Cos[c + d*x]^4/(a + b*Sin[c + d*x]^3)^2,x]

output

((-I)*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , ( 
2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - I*b*Log[1 - 2*Cos[c + d*x]*#1 + 
 #1^2] + (4*I)*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + 2*a*Log[1 - 2*C 
os[c + d*x]*#1 + #1^2]*#1 + 12*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 
 - (6*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (4*I)*a*ArcTan[Sin[c + d*x 
]/(Cos[c + d*x] - #1)]*#1^3 - 2*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3 + 2*b 
*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - I*b*Log[1 - 2*Cos[c + d*x]*#1 
 + #1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^5) & ] + (24*Cos[c + d*x 
]*(a + b*Sin[c + d*x]))/(4*a + 3*b*Sin[c + d*x] - b*Sin[3*(c + d*x)]))/(18*a*b 
*d)
 

Integral number [400] \[ \int \frac {\cos ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \]

[C]   time = 0.765131 (sec), size = 273 ,normalized size = 11.87 \[ \frac {-i \text {RootSum}\left [-i b+3 i b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 i b \text {$\#$1}^4+i b \text {$\#$1}^6\& ,\frac {2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+12 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-6 i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4}{b \text {$\#$1}-4 i a \text {$\#$1}^2-2 b \text {$\#$1}^3+b \text {$\#$1}^5}\& \right ]+\frac {12 \sin (2 (c+d x))}{4 a+3 b \sin (c+d x)-b \sin (3 (c+d x))}}{18 a d} \]

[In]

Integrate[Cos[c + d*x]^2/(a + b*Sin[c + d*x]^3)^2,x]

output

((-I)*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , ( 
2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^ 
2] + 12*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - (6*I)*Log[1 - 2*Cos[c 
+ d*x]*#1 + #1^2]*#1^2 + 2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - I*L 
og[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^ 
5) & ] + (12*Sin[2*(c + d*x)])/(4*a + 3*b*Sin[c + d*x] - b*Sin[3*(c + d*x)]))/ 
(18*a*d)
 

Integral number [401] \[ \int \frac {1}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \]

[C]   time = 0.608929 (sec), size = 502 ,normalized size = 35.86 \[ \frac {\frac {i \text {RootSum}\left [-i b+3 i b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 i b \text {$\#$1}^4+i b \text {$\#$1}^6\& ,\frac {2 b^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+4 i a b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}+2 a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}-24 a^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+12 b^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+12 i a^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-6 i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-4 i a b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^3-2 a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3+2 b^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4}{b \text {$\#$1}-4 i a \text {$\#$1}^2-2 b \text {$\#$1}^3+b \text {$\#$1}^5}\& \right ]}{a^2-b^2}-\frac {12 b \cos (c+d x) (-3 a+a \cos (2 (c+d x))+2 b \sin (c+d x))}{(a-b) (a+b) (4 a+3 b \sin (c+d x)-b \sin (3 (c+d x)))}}{18 a d} \]

[In]

Integrate[(a + b*Sin[c + d*x]^3)^(-2),x]

output

((I*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (2* 
b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - I*b^2*Log[1 - 2*Cos[c + d*x]*#1 
 + #1^2] + (4*I)*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + 2*a*b*Log[1 
 - 2*Cos[c + d*x]*#1 + #1^2]*#1 - 24*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - # 
1)]*#1^2 + 12*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (12*I)*a^2*L 
og[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (6*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1 + 
#1^2]*#1^2 - (4*I)*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - 2*a*b*L 
og[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3 + 2*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x 
] - #1)]*#1^4 - I*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a* 
#1^2 - 2*b*#1^3 + b*#1^5) & ])/(a^2 - b^2) - (12*b*Cos[c + d*x]*(-3*a + a*Cos[ 
2*(c + d*x)] + 2*b*Sin[c + d*x]))/((a - b)*(a + b)*(4*a + 3*b*Sin[c + d*x] - b 
*Sin[3*(c + d*x)])))/(18*a*d)
 

Integral number [402] \[ \int \frac {\sec ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \]

[C]   time = 2.21428 (sec), size = 845 ,normalized size = 36.74 \[ \frac {-\frac {i b \text {RootSum}\left [-i b+3 i b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 i b \text {$\#$1}^4+i b \text {$\#$1}^6\& ,\frac {16 a^2 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+2 b^3 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-8 i a^2 b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )-i b^3 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+20 i a^3 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}+16 i a b^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}+10 a^3 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}+8 a b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}-120 a^2 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+12 b^3 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+60 i a^2 b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-6 i b^3 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-20 i a^3 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^3-16 i a b^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^3-10 a^3 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3-8 a b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3+16 a^2 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4+2 b^3 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-8 i a^2 b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4-i b^3 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4}{b \text {$\#$1}-4 i a \text {$\#$1}^2-2 b \text {$\#$1}^3+b \text {$\#$1}^5}\& \right ]}{a \left (a^2-b^2\right )^2}+\frac {18 \sin \left (\frac {1}{2} (c+d x)\right )}{(a+b)^2 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {18 \sin \left (\frac {1}{2} (c+d x)\right )}{(a-b)^2 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {12 b \cos (c+d x) \left (-2 a^3-7 a b^2+3 a b^2 \cos (2 (c+d x))+2 b \left (2 a^2+b^2\right ) \sin (c+d x)\right )}{a (a-b)^2 (a+b)^2 (4 a+3 b \sin (c+d x)-b \sin (3 (c+d x)))}}{18 d} \]

[In]

Integrate[Sec[c + d*x]^2/(a + b*Sin[c + d*x]^3)^2,x]

output

(((-I)*b*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & 
, (16*a^2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + 2*b^3*ArcTan[Sin[c + d* 
x]/(Cos[c + d*x] - #1)] - (8*I)*a^2*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - I*b^ 
3*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + (20*I)*a^3*ArcTan[Sin[c + d*x]/(Cos[c + 
d*x] - #1)]*#1 + (16*I)*a*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + 10 
*a^3*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 + 8*a*b^2*Log[1 - 2*Cos[c + d*x]*#1 
+ #1^2]*#1 - 120*a^2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 12*b^3* 
ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (60*I)*a^2*b*Log[1 - 2*Cos[c + 
 d*x]*#1 + #1^2]*#1^2 - (6*I)*b^3*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (20 
*I)*a^3*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - (16*I)*a*b^2*ArcTan[Si 
n[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - 10*a^3*Log[1 - 2*Cos[c + d*x]*#1 + #1^2 
]*#1^3 - 8*a*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3 + 16*a^2*b*ArcTan[Sin[ 
c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + 2*b^3*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - 
 #1)]*#1^4 - (8*I)*a^2*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - I*b^3*Log[1 
- 2*Cos[c + d*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^5) & 
])/(a*(a^2 - b^2)^2) + (18*Sin[(c + d*x)/2])/((a + b)^2*(Cos[(c + d*x)/2] - Si 
n[(c + d*x)/2])) + (18*Sin[(c + d*x)/2])/((a - b)^2*(Cos[(c + d*x)/2] + Sin[(c 
 + d*x)/2])) + (12*b*Cos[c + d*x]*(-2*a^3 - 7*a*b^2 + 3*a*b^2*Cos[2*(c + d*x)] 
 + 2*b*(2*a^2 + b^2)*Sin[c + d*x]))/(a*(a - b)^2*(a + b)^2*(4*a + 3*b*Sin[c + 
d*x] - b*Sin[3*(c + d*x)])))/(18*d)
 

Integral number [403] \[ \int \frac {\sec ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \]

[C]   time = 2.2548 (sec), size = 1158 ,normalized size = 50.35 \[ \text {result too large to display} \]

[In]

Integrate[Sec[c + d*x]^4/(a + b*Sin[c + d*x]^3)^2,x]

output

((4*I)*b^2*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 
& , (14*a^4*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + 74*a^2*b^2*ArcTan[Sin[c 
 + d*x]/(Cos[c + d*x] - #1)] + 2*b^4*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] 
- (7*I)*a^4*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - (37*I)*a^2*b^2*Log[1 - 2*Cos[c 
 + d*x]*#1 + #1^2] - I*b^4*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + (144*I)*a^3*b*A 
rcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + (36*I)*a*b^3*ArcTan[Sin[c + d*x]/ 
(Cos[c + d*x] - #1)]*#1 + 72*a^3*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 + 18*a 
*b^3*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 - 180*a^4*ArcTan[Sin[c + d*x]/(Cos[c 
 + d*x] - #1)]*#1^2 - 372*a^2*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^ 
2 + 12*b^4*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (90*I)*a^4*Log[1 - 
2*Cos[c + d*x]*#1 + #1^2]*#1^2 + (186*I)*a^2*b^2*Log[1 - 2*Cos[c + d*x]*#1 + # 
1^2]*#1^2 - (6*I)*b^4*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (144*I)*a^3*b*A 
rcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - (36*I)*a*b^3*ArcTan[Sin[c + d*x 
]/(Cos[c + d*x] - #1)]*#1^3 - 72*a^3*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3 
- 18*a*b^3*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3 + 14*a^4*ArcTan[Sin[c + d*x] 
/(Cos[c + d*x] - #1)]*#1^4 + 74*a^2*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1 
)]*#1^4 + 2*b^4*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - (7*I)*a^4*Log[ 
1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - (37*I)*a^2*b^2*Log[1 - 2*Cos[c + d*x]*#1 
+ #1^2]*#1^4 - I*b^4*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a*# 
1^2 - 2*b*#1^3 + b*#1^5) & ] + (3*Sec[c + d*x]^3*(48*a^5*b + 568*a^3*b^3 + ...
 

4.5.2 Fricas

Integral number [399] \[ \int \frac {\cos ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \]

[C]   time = 9.39875 (sec), size = 9984 ,normalized size = 434.09 \[ \text {Too large to display} \]

[In]

integrate(cos(d*x+c)^4/(a+b*sin(d*x+c)^3)^2,x, algorithm=""fricas"")

output

-1/36*(sqrt(2)*sqrt(1/2)*(a^2*b*d - (a*b^2*d*cos(d*x + c)^2 - a*b^2*d)*sin(d*x 
 + c))*sqrt((((1/2)^(1/3)*(I*sqrt(3) + 1)*(27/(a^6*b^6*d^6) - (a^2 - 4*b^2)^3/ 
(a^10*b^8*d^6) - (a^6 + 15*a^4*b^2 + 48*a^2*b^4 - 64*b^6)/(a^10*b^8*d^6))^(1/3 
) + 6/(a^2*b^2*d^2))*a^2*b^2*d^2 + 3*sqrt(1/3)*a^2*b^2*d^2*sqrt(-(((1/2)^(1/3) 
*(I*sqrt(3) + 1)*(27/(a^6*b^6*d^6) - (a^2 - 4*b^2)^3/(a^10*b^8*d^6) - (a^6 + 1 
5*a^4*b^2 + 48*a^2*b^4 - 64*b^6)/(a^10*b^8*d^6))^(1/3) + 6/(a^2*b^2*d^2))^2*a^ 
4*b^4*d^4 - 12*((1/2)^(1/3)*(I*sqrt(3) + 1)*(27/(a^6*b^6*d^6) - (a^2 - 4*b^2)^ 
3/(a^10*b^8*d^6) - (a^6 + 15*a^4*b^2 + 48*a^2*b^4 - 64*b^6)/(a^10*b^8*d^6))^(1 
/3) + 6/(a^2*b^2*d^2))*a^2*b^2*d^2 + 36)/(a^4*b^4*d^4)) - 18)/(a^2*b^2*d^2))*l 
og(1/4*(a^9*b^5 + 8*a^7*b^7)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(27/(a^6*b^6*d^6) - 
(a^2 - 4*b^2)^3/(a^10*b^8*d^6) - (a^6 + 15*a^4*b^2 + 48*a^2*b^4 - 64*b^6)/(a^1 
0*b^8*d^6))^(1/3) + 6/(a^2*b^2*d^2))^2*d^4*sin(d*x + c) - 2*a^6 + 96*a^2*b^4 - 
 256*b^6 - 3*(a^7*b^3 + 8*a^5*b^5)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(27/(a^6*b^6*d 
^6) - (a^2 - 4*b^2)^3/(a^10*b^8*d^6) - (a^6 + 15*a^4*b^2 + 48*a^2*b^4 - 64*b^6 
)/(a^10*b^8*d^6))^(1/3) + 6/(a^2*b^2*d^2))*d^2*sin(d*x + c) - 1/4*sqrt(2)*sqrt 
(1/2)*(((1/2)^(1/3)*(I*sqrt(3) + 1)*(27/(a^6*b^6*d^6) - (a^2 - 4*b^2)^3/(a^10* 
b^8*d^6) - (a^6 + 15*a^4*b^2 + 48*a^2*b^4 - 64*b^6)/(a^10*b^8*d^6))^(1/3) + 6/ 
(a^2*b^2*d^2))^2*a^9*b^7*d^5*cos(d*x + c) - 16*(a^7*b^5 - a^5*b^7)*((1/2)^(1/3 
)*(I*sqrt(3) + 1)*(27/(a^6*b^6*d^6) - (a^2 - 4*b^2)^3/(a^10*b^8*d^6) - (a^6 + 
15*a^4*b^2 + 48*a^2*b^4 - 64*b^6)/(a^10*b^8*d^6))^(1/3) + 6/(a^2*b^2*d^2))*...
 

Integral number [400] \[ \int \frac {\cos ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \]

[C]   time = 3.24086 (sec), size = 36403 ,normalized size = 1582.74 \[ \text {Too large to display} \]

[In]

integrate(cos(d*x+c)^2/(a+b*sin(d*x+c)^3)^2,x, algorithm=""fricas"")

output

1/324*(3*sqrt(2/3)*sqrt(1/6)*(a^2*d - (a*b*d*cos(d*x + c)^2 - a*b*d)*sin(d*x + 
 c))*sqrt(-((a^4 - a^2*b^2)*((-I*sqrt(3) + 1)*(3/(a^6*b^2*d^4 - a^4*b^4*d^4) - 
 1/(a^4*d^2 - a^2*b^2*d^2)^2)/(-1/1062882*(a^4 - 16*a^2*b^2 + 64*b^4)/(a^12*b^ 
4*d^6 - a^10*b^6*d^6) + 1/118098/((a^6*b^2*d^4 - a^4*b^4*d^4)*(a^4*d^2 - a^2*b 
^2*d^2)) - 1/531441/(a^4*d^2 - a^2*b^2*d^2)^3 + 1/1062882*(a^6 + 28*a^4*b^2 - 
80*a^2*b^4 + 64*b^6)/((a^2 - b^2)^2*a^10*b^4*d^6))^(1/3) - 6561*(I*sqrt(3) + 1 
)*(-1/1062882*(a^4 - 16*a^2*b^2 + 64*b^4)/(a^12*b^4*d^6 - a^10*b^6*d^6) + 1/11 
8098/((a^6*b^2*d^4 - a^4*b^4*d^4)*(a^4*d^2 - a^2*b^2*d^2)) - 1/531441/(a^4*d^2 
 - a^2*b^2*d^2)^3 + 1/1062882*(a^6 + 28*a^4*b^2 - 80*a^2*b^4 + 64*b^6)/((a^2 - 
 b^2)^2*a^10*b^4*d^6))^(1/3) - 162/(a^4*d^2 - a^2*b^2*d^2))*d^2 + 3*sqrt(1/3)* 
(a^4 - a^2*b^2)*d^2*sqrt(-((a^8*b^2 - 2*a^6*b^4 + a^4*b^6)*((-I*sqrt(3) + 1)*( 
3/(a^6*b^2*d^4 - a^4*b^4*d^4) - 1/(a^4*d^2 - a^2*b^2*d^2)^2)/(-1/1062882*(a^4 
- 16*a^2*b^2 + 64*b^4)/(a^12*b^4*d^6 - a^10*b^6*d^6) + 1/118098/((a^6*b^2*d^4 
- a^4*b^4*d^4)*(a^4*d^2 - a^2*b^2*d^2)) - 1/531441/(a^4*d^2 - a^2*b^2*d^2)^3 + 
 1/1062882*(a^6 + 28*a^4*b^2 - 80*a^2*b^4 + 64*b^6)/((a^2 - b^2)^2*a^10*b^4*d^ 
6))^(1/3) - 6561*(I*sqrt(3) + 1)*(-1/1062882*(a^4 - 16*a^2*b^2 + 64*b^4)/(a^12 
*b^4*d^6 - a^10*b^6*d^6) + 1/118098/((a^6*b^2*d^4 - a^4*b^4*d^4)*(a^4*d^2 - a^ 
2*b^2*d^2)) - 1/531441/(a^4*d^2 - a^2*b^2*d^2)^3 + 1/1062882*(a^6 + 28*a^4*b^2 
 - 80*a^2*b^4 + 64*b^6)/((a^2 - b^2)^2*a^10*b^4*d^6))^(1/3) - 162/(a^4*d^2 - a 
^2*b^2*d^2))^2*d^4 + 324*(a^4*b^2 - a^2*b^4)*((-I*sqrt(3) + 1)*(3/(a^6*b^2*...
 

Integral number [401] \[ \int \frac {1}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \]

[C]   time = 8.47099 (sec), size = 70185 ,normalized size = 5013.21 \[ \text {Too large to display} \]

[In]

integrate(1/(a+b*sin(d*x+c)^3)^2,x, algorithm=""fricas"")

output

-1/108*(36*a*b*cos(d*x + c)^3 + 36*b^2*cos(d*x + c)*sin(d*x + c) - sqrt(2/3)*s 
qrt(1/2)*((a^4 - a^2*b^2)*d - ((a^3*b - a*b^3)*d*cos(d*x + c)^2 - (a^3*b - a*b 
^3)*d)*sin(d*x + c))*sqrt(-(1458*a^4 + 486*a^2*b^2 - 486*b^4 - (a^8 - 3*a^6*b^ 
2 + 3*a^4*b^4 - a^2*b^6)*((-I*sqrt(3) + 1)*(3*(3*a^4 + a^2*b^2 - b^4)^2/(a^8*d 
^2 - 3*a^6*b^2*d^2 + 3*a^4*b^4*d^2 - a^2*b^6*d^2)^2 - (27*a^2 - 11*b^2)/(a^10* 
d^4 - 3*a^8*b^2*d^4 + 3*a^6*b^4*d^4 - a^4*b^6*d^4))/(-1/1062882*(729*a^4 - 432 
*a^2*b^2 + 64*b^4)/(a^16*d^6 - 3*a^14*b^2*d^6 + 3*a^12*b^4*d^6 - a^10*b^6*d^6) 
 - 1/19683*(3*a^4 + a^2*b^2 - b^4)^3/(a^8*d^2 - 3*a^6*b^2*d^2 + 3*a^4*b^4*d^2 
- a^2*b^6*d^2)^3 + 1/39366*(3*a^4 + a^2*b^2 - b^4)*(27*a^2 - 11*b^2)/((a^10*d^ 
4 - 3*a^8*b^2*d^4 + 3*a^6*b^4*d^4 - a^4*b^6*d^4)*(a^8*d^2 - 3*a^6*b^2*d^2 + 3* 
a^4*b^4*d^2 - a^2*b^6*d^2)) + 1/1062882*(3375*a^8 - 4573*a^6*b^2 + 2460*a^4*b^ 
4 - 624*a^2*b^6 + 64*b^8)*b^2/((a^2 - b^2)^6*a^10*d^6))^(1/3) + 2187*(I*sqrt(3 
) + 1)*(-1/1062882*(729*a^4 - 432*a^2*b^2 + 64*b^4)/(a^16*d^6 - 3*a^14*b^2*d^6 
 + 3*a^12*b^4*d^6 - a^10*b^6*d^6) - 1/19683*(3*a^4 + a^2*b^2 - b^4)^3/(a^8*d^2 
 - 3*a^6*b^2*d^2 + 3*a^4*b^4*d^2 - a^2*b^6*d^2)^3 + 1/39366*(3*a^4 + a^2*b^2 - 
 b^4)*(27*a^2 - 11*b^2)/((a^10*d^4 - 3*a^8*b^2*d^4 + 3*a^6*b^4*d^4 - a^4*b^6*d 
^4)*(a^8*d^2 - 3*a^6*b^2*d^2 + 3*a^4*b^4*d^2 - a^2*b^6*d^2)) + 1/1062882*(3375 
*a^8 - 4573*a^6*b^2 + 2460*a^4*b^4 - 624*a^2*b^6 + 64*b^8)*b^2/((a^2 - b^2)^6* 
a^10*d^6))^(1/3) + 162*(3*a^4 + a^2*b^2 - b^4)/(a^8*d^2 - 3*a^6*b^2*d^2 + 3*a^ 
4*b^4*d^2 - a^2*b^6*d^2))*d^2 + 3*sqrt(1/3)*(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - ...
 

Integral number [402] \[ \int \frac {\sec ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \]

[C]   time = 43.884 (sec), size = 102913 ,normalized size = 4474.48 \[ \text {Too large to display} \]

[In]

integrate(sec(d*x+c)^2/(a+b*sin(d*x+c)^3)^2,x, algorithm=""fricas"")

output

1/108*(108*(a^3*b + 2*a*b^3)*cos(d*x + c)^4 - 108*a^3*b + 108*a*b^3 - sqrt(2)* 
sqrt(1/2)*((a^6 - 2*a^4*b^2 + a^2*b^4)*d*cos(d*x + c) - ((a^5*b - 2*a^3*b^3 + 
a*b^5)*d*cos(d*x + c)^3 - (a^5*b - 2*a^3*b^3 + a*b^5)*d*cos(d*x + c))*sin(d*x 
+ c))*sqrt(-(5670*a^6*b^2 + 31590*a^4*b^4 + 2916*a^2*b^6 - 810*b^8 - (a^12 - 5 
*a^10*b^2 + 10*a^8*b^4 - 10*a^6*b^6 + 5*a^4*b^8 - a^2*b^10)*((-I*sqrt(3) + 1)* 
((35*a^6*b^2 + 195*a^4*b^4 + 18*a^2*b^6 - 5*b^8)^2/(a^12*d^2 - 5*a^10*b^2*d^2 
+ 10*a^8*b^4*d^2 - 10*a^6*b^6*d^2 + 5*a^4*b^8*d^2 - a^2*b^10*d^2)^2 - 45*(10*a 
^2*b^4 - b^6)/(a^14*d^4 - 5*a^12*b^2*d^4 + 10*a^10*b^4*d^4 - 10*a^8*b^6*d^4 + 
5*a^6*b^8*d^4 - a^4*b^10*d^4))/(-1/19683*(35*a^6*b^2 + 195*a^4*b^4 + 18*a^2*b^ 
6 - 5*b^8)^3/(a^12*d^2 - 5*a^10*b^2*d^2 + 10*a^8*b^4*d^2 - 10*a^6*b^6*d^2 + 5* 
a^4*b^8*d^2 - a^2*b^10*d^2)^3 - 1/1062882*(15625*a^4*b^4 - 2000*a^2*b^6 + 64*b 
^8)/(a^20*d^6 - 5*a^18*b^2*d^6 + 10*a^16*b^4*d^6 - 10*a^14*b^6*d^6 + 5*a^12*b^ 
8*d^6 - a^10*b^10*d^6) + 5/1458*(35*a^6*b^2 + 195*a^4*b^4 + 18*a^2*b^6 - 5*b^8 
)*(10*a^2*b^4 - b^6)/((a^14*d^4 - 5*a^12*b^2*d^4 + 10*a^10*b^4*d^4 - 10*a^8*b^ 
6*d^4 + 5*a^6*b^8*d^4 - a^4*b^10*d^4)*(a^12*d^2 - 5*a^10*b^2*d^2 + 10*a^8*b^4* 
d^2 - 10*a^6*b^6*d^2 + 5*a^4*b^8*d^2 - a^2*b^10*d^2)) - 1/1062882*(15625*a^14 
+ 959375*a^12*b^2 + 24861*a^10*b^4 - 1094705*a^8*b^6 + 307475*a^6*b^8 - 37740* 
a^4*b^10 + 2320*a^2*b^12 - 64*b^14)*b^4/((a^2 - b^2)^10*a^10*d^6))^(1/3) + 729 
*(I*sqrt(3) + 1)*(-1/19683*(35*a^6*b^2 + 195*a^4*b^4 + 18*a^2*b^6 - 5*b^8)^3/( 
a^12*d^2 - 5*a^10*b^2*d^2 + 10*a^8*b^4*d^2 - 10*a^6*b^6*d^2 + 5*a^4*b^8*d^2...
 

Integral number [403] \[ \int \frac {\sec ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \]

[C]   time = 140.116 (sec), size = 133123 ,normalized size = 5787.96 \[ \text {Too large to display} \]

[In]

integrate(sec(d*x+c)^4/(a+b*sin(d*x+c)^3)^2,x, algorithm=""fricas"")

output

1/108*(36*(2*a^5*b - 30*a^3*b^3 - 17*a*b^5)*cos(d*x + c)^6 - 36*a^5*b + 72*a^3 
*b^3 - 36*a*b^5 - 108*(a^5*b - 21*a^3*b^3 - 10*a*b^5)*cos(d*x + c)^4 + sqrt(2/ 
3)*sqrt(1/6)*((a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6)*d*cos(d*x + c)^3 - ((a^7 
*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*d*cos(d*x + c)^5 - (a^7*b - 3*a^5*b^3 + 3* 
a^3*b^5 - a*b^7)*d*cos(d*x + c)^3)*sin(d*x + c))*sqrt(-(573480*a^8*b^4 + 42933 
24*a^6*b^6 + 3847662*a^4*b^8 + 159894*a^2*b^10 - 17010*b^12 - (a^16 - 7*a^14*b 
^2 + 21*a^12*b^4 - 35*a^10*b^6 + 35*a^8*b^8 - 21*a^6*b^10 + 7*a^4*b^12 - a^2*b 
^14)*((-I*sqrt(3) + 1)*((1180*a^8*b^4 + 8834*a^6*b^6 + 7917*a^4*b^8 + 329*a^2* 
b^10 - 35*b^12)^2/(a^16*d^2 - 7*a^14*b^2*d^2 + 21*a^12*b^4*d^2 - 35*a^10*b^6*d 
^2 + 35*a^8*b^8*d^2 - 21*a^6*b^10*d^2 + 7*a^4*b^12*d^2 - a^2*b^14*d^2)^2 + 15* 
(1029*a^4*b^6 - 3173*a^2*b^8 + 119*b^10)/(a^18*d^4 - 7*a^16*b^2*d^4 + 21*a^14* 
b^4*d^4 - 35*a^12*b^6*d^4 + 35*a^10*b^8*d^4 - 21*a^8*b^10*d^4 + 7*a^6*b^12*d^4 
 - a^4*b^14*d^4))/(-1/531441*(1180*a^8*b^4 + 8834*a^6*b^6 + 7917*a^4*b^8 + 329 
*a^2*b^10 - 35*b^12)^3/(a^16*d^2 - 7*a^14*b^2*d^2 + 21*a^12*b^4*d^2 - 35*a^10* 
b^6*d^2 + 35*a^8*b^8*d^2 - 21*a^6*b^10*d^2 + 7*a^4*b^12*d^2 - a^2*b^14*d^2)^3 
- 1/1062882*(117649*a^4*b^8 - 5488*a^2*b^10 + 64*b^12)/(a^24*d^6 - 7*a^22*b^2* 
d^6 + 21*a^20*b^4*d^6 - 35*a^18*b^6*d^6 + 35*a^16*b^8*d^6 - 21*a^14*b^10*d^6 + 
 7*a^12*b^12*d^6 - a^10*b^14*d^6) - 5/118098*(1180*a^8*b^4 + 8834*a^6*b^6 + 79 
17*a^4*b^8 + 329*a^2*b^10 - 35*b^12)*(1029*a^4*b^6 - 3173*a^2*b^8 + 119*b^10)/ 
((a^18*d^4 - 7*a^16*b^2*d^4 + 21*a^14*b^4*d^4 - 35*a^12*b^6*d^4 + 35*a^10*b...
 

4.5.3 Mupad

Integral number [399] \[ \int \frac {\cos ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \]

[B]   time = 15.0511 (sec), size = -1 ,normalized size = -0.04 \[ \text {Too large to display} \]

[In]

int(cos(c + d*x)^4/(a + b*sin(c + d*x)^3)^2,x)

output

2/(3*d*(a*b + 8*b^2*tan(c/2 + (d*x)/2)^3 + 3*a*b*tan(c/2 + (d*x)/2)^2 + 3*a*b* 
tan(c/2 + (d*x)/2)^4 + a*b*tan(c/2 + (d*x)/2)^6)) + symsum(log((638976*a^2*b^4 
 - 655360*b^6 - 8192*a^6 + 24576*a^4*b^2 - 2949120*root(531441*a^10*b^8*d^6 + 
59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, 
 d, k)*a^3*b^5 + 2138112*root(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a 
^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, d, k)*a^5*b^3 - 9437184*r 
oot(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 
15*a^4*b^2 + a^6 - 64*b^6, d, k)*b^8*tan(c/2 + (d*x)/2) - 786432*a*b^5*tan(c/2 
 + (d*x)/2) + 98304*a^5*b*tan(c/2 + (d*x)/2) - 21233664*root(531441*a^10*b^8*d 
^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64 
*b^6, d, k)^2*a^2*b^8 + 18579456*root(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 
+ 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, d, k)^2*a^4*b^6 + 
 2654208*root(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48* 
a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, d, k)^2*a^6*b^4 - 167215104*root(531441*a 
^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + 
 a^6 - 64*b^6, d, k)^3*a^5*b^7 + 113467392*root(531441*a^10*b^8*d^6 + 59049*a^ 
8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, d, k)^3 
*a^7*b^5 - 107495424*root(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b 
^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, d, k)^4*a^6*b^8 + 107495424*r 
oot(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^4...
 

Integral number [400] \[ \int \frac {\cos ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \]

[B]   time = 15.2435 (sec), size = -1 ,normalized size = -0.04 \[ \text {Too large to display} \]

[In]

int(cos(c + d*x)^2/(a + b*sin(c + d*x)^3)^2,x)

output

symsum(log(-((131072*b^2)/243 - (16384*a^2)/243 + (8192*root(531441*a^12*b^4*d 
^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + 
a^4 + 64*b^4, d, k)*a^4*tan(c/2 + (d*x)/2))/27 + (1048576*root(531441*a^12*b^4 
*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 
+ a^4 + 64*b^4, d, k)*b^4*tan(c/2 + (d*x)/2))/27 + (262144*root(531441*a^12*b^ 
4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 
 + a^4 + 64*b^4, d, k)^2*a^2*b^4)/3 - (131072*root(531441*a^12*b^4*d^6 - 53144 
1*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b 
^4, d, k)^2*a^4*b^2)/3 - 98304*root(531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 
+ 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d, k)^3*a^5 
*b^3 + 442368*root(531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d 
^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d, k)^4*a^6*b^4 + 221184*roo 
t(531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2* 
d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d, k)^4*a^8*b^2 + 7962624*root(531441*a^12*b^ 
4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 
 + a^4 + 64*b^4, d, k)^5*a^7*b^5 - 5971968*root(531441*a^12*b^4*d^6 - 531441*a 
^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, 
 d, k)^5*a^9*b^3 + (131072*root(531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19 
683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d, k)*a*b^3)/27 
 - (65536*root(531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^...
 

Integral number [401] \[ \int \frac {1}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \]

[B]   time = 17.0233 (sec), size = -1 ,normalized size = -0.07 \[ \text {Too large to display} \]

[In]

int(1/(a + b*sin(c + d*x)^3)^2,x)

output

symsum(log(- (8192*(80*b^6 - 270*a^2*b^4))/(243*(a^7 + a^3*b^4 - 2*a^5*b^2)) - 
 root(1594323*a^14*b^2*d^6 - 1594323*a^12*b^4*d^6 + 531441*a^10*b^6*d^6 - 5314 
41*a^16*d^6 - 59049*a^10*b^2*d^4 + 59049*a^8*b^4*d^4 - 177147*a^12*d^4 + 8019* 
a^6*b^2*d^2 - 19683*a^8*d^2 + 432*a^2*b^2 - 64*b^4 - 729*a^4, d, k)*((8192*(14 
4*a*b^7 + 648*a^3*b^5 - 2187*a^5*b^3))/(243*(a^7 + a^3*b^4 - 2*a^5*b^2)) - roo 
t(1594323*a^14*b^2*d^6 - 1594323*a^12*b^4*d^6 + 531441*a^10*b^6*d^6 - 531441*a 
^16*d^6 - 59049*a^10*b^2*d^4 + 59049*a^8*b^4*d^4 - 177147*a^12*d^4 + 8019*a^6* 
b^2*d^2 - 19683*a^8*d^2 + 432*a^2*b^2 - 64*b^4 - 729*a^4, d, k)*(root(1594323* 
a^14*b^2*d^6 - 1594323*a^12*b^4*d^6 + 531441*a^10*b^6*d^6 - 531441*a^16*d^6 - 
59049*a^10*b^2*d^4 + 59049*a^8*b^4*d^4 - 177147*a^12*d^4 + 8019*a^6*b^2*d^2 - 
19683*a^8*d^2 + 432*a^2*b^2 - 64*b^4 - 729*a^4, d, k)*((8192*(26973*a^7*b^5 - 
20412*a^5*b^7 + 39366*a^9*b^3))/(243*(a^7 + a^3*b^4 - 2*a^5*b^2)) - root(15943 
23*a^14*b^2*d^6 - 1594323*a^12*b^4*d^6 + 531441*a^10*b^6*d^6 - 531441*a^16*d^6 
 - 59049*a^10*b^2*d^4 + 59049*a^8*b^4*d^4 - 177147*a^12*d^4 + 8019*a^6*b^2*d^2 
 - 19683*a^8*d^2 + 432*a^2*b^2 - 64*b^4 - 729*a^4, d, k)*(root(1594323*a^14*b^ 
2*d^6 - 1594323*a^12*b^4*d^6 + 531441*a^10*b^6*d^6 - 531441*a^16*d^6 - 59049*a 
^10*b^2*d^4 + 59049*a^8*b^4*d^4 - 177147*a^12*d^4 + 8019*a^6*b^2*d^2 - 19683*a 
^8*d^2 + 432*a^2*b^2 - 64*b^4 - 729*a^4, d, k)*((8192*(236196*a^7*b^9 - 649539 
*a^9*b^7 + 590490*a^11*b^5 - 177147*a^13*b^3))/(243*(a^7 + a^3*b^4 - 2*a^5*b^2 
)) + (8192*tan(c/2 + (d*x)/2)*(6561*a^8*b^8 - 13122*a^10*b^6 + 6561*a^12*b^...
 

Integral number [402] \[ \int \frac {\sec ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \]

[B]   time = 19.147 (sec), size = -1 ,normalized size = -0.04 \[ \text {Too large to display} \]

[In]

int(1/(cos(c + d*x)^2*(a + b*sin(c + d*x)^3)^2),x)

output

symsum(log(5479612416*a^8*b^36 - 180486144*a^6*b^38 - root(5314410*a^16*b^4*d^ 
6 - 5314410*a^14*b^6*d^6 - 2657205*a^18*b^2*d^6 + 2657205*a^12*b^8*d^6 - 53144 
1*a^10*b^10*d^6 + 531441*a^20*d^6 + 11514555*a^12*b^4*d^4 + 2066715*a^14*b^2*d 
^4 + 1062882*a^10*b^6*d^4 - 295245*a^8*b^8*d^4 + 984150*a^8*b^4*d^2 - 98415*a^ 
6*b^6*d^2 + 15625*a^4*b^4 - 2000*a^2*b^6 + 64*b^8, d, k)*(tan(c/2 + (d*x)/2)*( 
764411904*a^6*b^40 - 27805483008*a^8*b^38 + 437297356800*a^10*b^36 - 367246172 
1600*a^12*b^34 + 19250011791360*a^14*b^32 - 69150635753472*a^16*b^30 + 1801658 
72001024*a^18*b^28 - 352655758540800*a^20*b^26 + 529923028377600*a^22*b^24 - 6 
18699706859520*a^24*b^22 + 563713761042432*a^26*b^20 - 399760062234624*a^28*b^ 
18 + 218398602240000*a^30*b^16 - 90108039168000*a^32*b^14 + 27130620764160*a^3 
4*b^12 - 5617221156864*a^36*b^10 + 713536708608*a^38*b^8 - 41803776000*a^40*b^ 
6) - root(5314410*a^16*b^4*d^6 - 5314410*a^14*b^6*d^6 - 2657205*a^18*b^2*d^6 + 
 2657205*a^12*b^8*d^6 - 531441*a^10*b^10*d^6 + 531441*a^20*d^6 + 11514555*a^12 
*b^4*d^4 + 2066715*a^14*b^2*d^4 + 1062882*a^10*b^6*d^4 - 295245*a^8*b^8*d^4 + 
984150*a^8*b^4*d^2 - 98415*a^6*b^6*d^2 + 15625*a^4*b^4 - 2000*a^2*b^6 + 64*b^8 
, d, k)*(root(5314410*a^16*b^4*d^6 - 5314410*a^14*b^6*d^6 - 2657205*a^18*b^2*d 
^6 + 2657205*a^12*b^8*d^6 - 531441*a^10*b^10*d^6 + 531441*a^20*d^6 + 11514555* 
a^12*b^4*d^4 + 2066715*a^14*b^2*d^4 + 1062882*a^10*b^6*d^4 - 295245*a^8*b^8*d^ 
4 + 984150*a^8*b^4*d^2 - 98415*a^6*b^6*d^2 + 15625*a^4*b^4 - 2000*a^2*b^6 + 64 
*b^8, d, k)*(tan(c/2 + (d*x)/2)*(157695787008*a^12*b^38 - 4039140556800*a^1...
 

Integral number [403] \[ \int \frac {\sec ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \]

[B]   time = 23.8272 (sec), size = -1 ,normalized size = -0.04 \[ \text {Too large to display} \]

[In]

int(1/(cos(c + d*x)^4*(a + b*sin(c + d*x)^3)^2),x)

output

symsum(log(26838024192*a^8*b^54 - tan(c/2 + (d*x)/2)*(7962624000*a^7*b^55 - 50 
8612608000*a^9*b^53 + 8841498624000*a^11*b^51 - 82283765760000*a^13*b^49 + 501 
714984960000*a^15*b^47 - 2205295497216000*a^17*b^45 + 7379181637632000*a^19*b^ 
43 - 19451488075776000*a^21*b^41 + 41318016122880000*a^23*b^39 - 7181143216128 
0000*a^25*b^37 + 103155513237504000*a^27*b^35 - 123224906907648000*a^29*b^33 + 
 122756816093184000*a^31*b^31 - 101967282708480000*a^33*b^29 + 703968720076800 
00*a^35*b^27 - 40129785593856000*a^37*b^25 + 18687625592832000*a^39*b^23 - 699 
4754113536000*a^41*b^21 + 2053854351360000*a^43*b^19 - 455730831360000*a^45*b^ 
17 + 71860690944000*a^47*b^15 - 7177310208000*a^49*b^13 + 341397504000*a^51*b^ 
11) - 392822784*a^6*b^56 - root(18600435*a^18*b^6*d^6 - 18600435*a^16*b^8*d^6 
- 11160261*a^20*b^4*d^6 + 11160261*a^14*b^10*d^6 + 3720087*a^22*b^2*d^6 - 3720 
087*a^12*b^12*d^6 + 531441*a^10*b^14*d^6 - 531441*a^24*d^6 - 173879622*a^14*b^ 
6*d^4 - 155830311*a^12*b^8*d^4 - 23225940*a^16*b^4*d^4 - 6475707*a^10*b^10*d^4 
 + 688905*a^8*b^12*d^4 - 11565585*a^8*b^8*d^2 + 3750705*a^10*b^6*d^2 + 433755* 
a^6*b^10*d^2 - 117649*a^4*b^8 + 5488*a^2*b^10 - 64*b^12, d, k)*(tan(c/2 + (d*x 
)/2)*(764411904*a^6*b^58 - 61439606784*a^8*b^56 + 2110475575296*a^10*b^54 - 33 
643637121024*a^12*b^52 + 319697763065856*a^14*b^50 - 2067381036048384*a^16*b^4 
8 + 9810082122817536*a^18*b^46 - 35797302942326784*a^20*b^44 + 103613766013034 
496*a^22*b^42 - 243004699498881024*a^24*b^40 + 468678655511248896*a^26*b^38 - 
750973819695611904*a^28*b^36 + 1006348379003928576*a^30*b^34 - 113202827820...
 

4.6 Test file Number [151]

4.6.1 Mathematica

Integral number [65] \[ \int \frac {\arctan (a+b x)}{\sqrt [3]{1+a^2+2 a b x+b^2 x^2}} \, dx \]

[B]   time = 0.31135 (sec), size = 163 ,normalized size = 5.82 \[ \frac {6 \operatorname {Gamma}\left (\frac {11}{6}\right ) \operatorname {Gamma}\left (\frac {7}{3}\right ) \left (15+10 (a+b x) \arctan (a+b x)+\frac {4 (a+b x) \arctan (a+b x) \operatorname {Hypergeometric2F1}\left (1,\frac {4}{3},\frac {11}{6},\frac {1}{1+(a+b x)^2}\right )}{1+(a+b x)^2}\right )+\frac {5 \sqrt [3]{2} \sqrt {\pi } \operatorname {Gamma}\left (\frac {5}{3}\right ) \, _3F_2\left (1,\frac {4}{3},\frac {4}{3};\frac {11}{6},\frac {7}{3};\frac {1}{1+(a+b x)^2}\right )}{1+(a+b x)^2}}{20 b \sqrt [3]{1+a^2+2 a b x+b^2 x^2} \operatorname {Gamma}\left (\frac {11}{6}\right ) \operatorname {Gamma}\left (\frac {7}{3}\right )} \]

[In]

Integrate[ArcTan[a + b*x]/(1 + a^2 + 2*a*b*x + b^2*x^2)^(1/3),x]

output

(6*Gamma[11/6]*Gamma[7/3]*(15 + 10*(a + b*x)*ArcTan[a + b*x] + (4*(a + b*x)*Ar 
cTan[a + b*x]*Hypergeometric2F1[1, 4/3, 11/6, (1 + (a + b*x)^2)^(-1)])/(1 + (a 
 + b*x)^2)) + (5*2^(1/3)*Sqrt[Pi]*Gamma[5/3]*HypergeometricPFQ[{1, 4/3, 4/3}, 
{11/6, 7/3}, (1 + (a + b*x)^2)^(-1)])/(1 + (a + b*x)^2))/(20*b*(1 + a^2 + 2*a* 
b*x + b^2*x^2)^(1/3)*Gamma[11/6]*Gamma[7/3])
 

Integral number [66] \[ \int \frac {\arctan (a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx \]

[B]   time = 0.151768 (sec), size = 165 ,normalized size = 5. \[ \frac {6 \operatorname {Gamma}\left (\frac {11}{6}\right ) \operatorname {Gamma}\left (\frac {7}{3}\right ) \left (15+10 (a+b x) \arctan (a+b x)+\frac {4 (a+b x) \arctan (a+b x) \operatorname {Hypergeometric2F1}\left (1,\frac {4}{3},\frac {11}{6},\frac {1}{1+(a+b x)^2}\right )}{1+(a+b x)^2}\right )+\frac {5 \sqrt [3]{2} \sqrt {\pi } \operatorname {Gamma}\left (\frac {5}{3}\right ) \, _3F_2\left (1,\frac {4}{3},\frac {4}{3};\frac {11}{6},\frac {7}{3};\frac {1}{1+(a+b x)^2}\right )}{1+(a+b x)^2}}{20 b \sqrt [3]{c \left (1+a^2+2 a b x+b^2 x^2\right )} \operatorname {Gamma}\left (\frac {11}{6}\right ) \operatorname {Gamma}\left (\frac {7}{3}\right )} \]

[In]

Integrate[ArcTan[a + b*x]/((1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2)^(1/3),x]

output

(6*Gamma[11/6]*Gamma[7/3]*(15 + 10*(a + b*x)*ArcTan[a + b*x] + (4*(a + b*x)*Ar 
cTan[a + b*x]*Hypergeometric2F1[1, 4/3, 11/6, (1 + (a + b*x)^2)^(-1)])/(1 + (a 
 + b*x)^2)) + (5*2^(1/3)*Sqrt[Pi]*Gamma[5/3]*HypergeometricPFQ[{1, 4/3, 4/3}, 
{11/6, 7/3}, (1 + (a + b*x)^2)^(-1)])/(1 + (a + b*x)^2))/(20*b*(c*(1 + a^2 + 2 
*a*b*x + b^2*x^2))^(1/3)*Gamma[11/6]*Gamma[7/3])
 

Integral number [69] \[ \int \frac {(a+b x)^2 \arctan (a+b x)}{\sqrt [3]{1+a^2+2 a b x+b^2 x^2}} \, dx \]

[B]   time = 4.75302 (sec), size = 181 ,normalized size = 5.17 \[ -\frac {3 \left (1+(a+b x)^2\right )^{2/3} \left (\frac {5 \sqrt [3]{2} \sqrt {\pi } \operatorname {Gamma}\left (\frac {5}{3}\right ) \, _3F_2\left (1,\frac {4}{3},\frac {4}{3};\frac {11}{6},\frac {7}{3};\frac {1}{1+(a+b x)^2}\right )}{\left (1+(a+b x)^2\right )^2}+\operatorname {Gamma}\left (\frac {11}{6}\right ) \operatorname {Gamma}\left (\frac {7}{3}\right ) \left (15+\frac {90}{1+(a+b x)^2}+\frac {24 (a+b x) \arctan (a+b x) \operatorname {Hypergeometric2F1}\left (1,\frac {4}{3},\frac {11}{6},\frac {1}{1+(a+b x)^2}\right )}{\left (1+(a+b x)^2\right )^2}+5 \arctan (a+b x) (-4 (a+b x)+6 \sin (2 \arctan (a+b x)))\right )\right )}{140 b \operatorname {Gamma}\left (\frac {11}{6}\right ) \operatorname {Gamma}\left (\frac {7}{3}\right )} \]

[In]

Integrate[((a + b*x)^2*ArcTan[a + b*x])/(1 + a^2 + 2*a*b*x + b^2*x^2)^(1/3),x]

output

(-3*(1 + (a + b*x)^2)^(2/3)*((5*2^(1/3)*Sqrt[Pi]*Gamma[5/3]*HypergeometricPFQ[ 
{1, 4/3, 4/3}, {11/6, 7/3}, (1 + (a + b*x)^2)^(-1)])/(1 + (a + b*x)^2)^2 + Gam 
ma[11/6]*Gamma[7/3]*(15 + 90/(1 + (a + b*x)^2) + (24*(a + b*x)*ArcTan[a + b*x] 
*Hypergeometric2F1[1, 4/3, 11/6, (1 + (a + b*x)^2)^(-1)])/(1 + (a + b*x)^2)^2 
+ 5*ArcTan[a + b*x]*(-4*(a + b*x) + 6*Sin[2*ArcTan[a + b*x]]))))/(140*b*Gamma[ 
11/6]*Gamma[7/3])
 

Integral number [70] \[ \int \frac {(a+b x)^2 \arctan (a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx \]

[B]   time = 0.727785 (sec), size = 225 ,normalized size = 5.62 \[ -\frac {3 \sqrt [3]{1+a^2+2 a b x+b^2 x^2} \left (1+(a+b x)^2\right )^{2/3} \left (\frac {5 \sqrt [3]{2} \sqrt {\pi } \operatorname {Gamma}\left (\frac {5}{3}\right ) \, _3F_2\left (1,\frac {4}{3},\frac {4}{3};\frac {11}{6},\frac {7}{3};\frac {1}{1+(a+b x)^2}\right )}{\left (1+(a+b x)^2\right )^2}+\operatorname {Gamma}\left (\frac {11}{6}\right ) \operatorname {Gamma}\left (\frac {7}{3}\right ) \left (15+\frac {90}{1+(a+b x)^2}+\frac {24 (a+b x) \arctan (a+b x) \operatorname {Hypergeometric2F1}\left (1,\frac {4}{3},\frac {11}{6},\frac {1}{1+(a+b x)^2}\right )}{\left (1+(a+b x)^2\right )^2}+5 \arctan (a+b x) (-4 (a+b x)+6 \sin (2 \arctan (a+b x)))\right )\right )}{140 b \sqrt [3]{c \left (1+a^2+2 a b x+b^2 x^2\right )} \operatorname {Gamma}\left (\frac {11}{6}\right ) \operatorname {Gamma}\left (\frac {7}{3}\right )} \]

[In]

Integrate[((a + b*x)^2*ArcTan[a + b*x])/((1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2)^(1/3),x]

output

(-3*(1 + a^2 + 2*a*b*x + b^2*x^2)^(1/3)*(1 + (a + b*x)^2)^(2/3)*((5*2^(1/3)*Sq 
rt[Pi]*Gamma[5/3]*HypergeometricPFQ[{1, 4/3, 4/3}, {11/6, 7/3}, (1 + (a + b*x) 
^2)^(-1)])/(1 + (a + b*x)^2)^2 + Gamma[11/6]*Gamma[7/3]*(15 + 90/(1 + (a + b*x 
)^2) + (24*(a + b*x)*ArcTan[a + b*x]*Hypergeometric2F1[1, 4/3, 11/6, (1 + (a + 
 b*x)^2)^(-1)])/(1 + (a + b*x)^2)^2 + 5*ArcTan[a + b*x]*(-4*(a + b*x) + 6*Sin[ 
2*ArcTan[a + b*x]]))))/(140*b*(c*(1 + a^2 + 2*a*b*x + b^2*x^2))^(1/3)*Gamma[11 
/6]*Gamma[7/3])
 

4.7 Test file Number [154]

4.7.1 Mathematica

Integral number [116] \[ \int \frac {\cot ^{-1}(a+b x)}{\sqrt [3]{1+a^2+2 a b x+b^2 x^2}} \, dx \]

[B]   time = 0.30209 (sec), size = 177 ,normalized size = 6.32 \[ \frac {6 \operatorname {Gamma}\left (\frac {11}{6}\right ) \operatorname {Gamma}\left (\frac {7}{3}\right ) \left (5 \left (1+a^2+2 a b x+b^2 x^2\right ) \left (-3+2 (a+b x) \cot ^{-1}(a+b x)\right )+4 (a+b x) \cot ^{-1}(a+b x) \operatorname {Hypergeometric2F1}\left (1,\frac {4}{3},\frac {11}{6},\frac {1}{1+a^2+2 a b x+b^2 x^2}\right )\right )-5 \sqrt [3]{2} \sqrt {\pi } \operatorname {Gamma}\left (\frac {5}{3}\right ) \, _3F_2\left (1,\frac {4}{3},\frac {4}{3};\frac {11}{6},\frac {7}{3};\frac {1}{1+a^2+2 a b x+b^2 x^2}\right )}{20 b \left (1+a^2+2 a b x+b^2 x^2\right )^{4/3} \operatorname {Gamma}\left (\frac {11}{6}\right ) \operatorname {Gamma}\left (\frac {7}{3}\right )} \]

[In]

Integrate[ArcCot[a + b*x]/(1 + a^2 + 2*a*b*x + b^2*x^2)^(1/3),x]

output

(6*Gamma[11/6]*Gamma[7/3]*(5*(1 + a^2 + 2*a*b*x + b^2*x^2)*(-3 + 2*(a + b*x)*A 
rcCot[a + b*x]) + 4*(a + b*x)*ArcCot[a + b*x]*Hypergeometric2F1[1, 4/3, 11/6, 
(1 + a^2 + 2*a*b*x + b^2*x^2)^(-1)]) - 5*2^(1/3)*Sqrt[Pi]*Gamma[5/3]*Hypergeom 
etricPFQ[{1, 4/3, 4/3}, {11/6, 7/3}, (1 + a^2 + 2*a*b*x + b^2*x^2)^(-1)])/(20* 
b*(1 + a^2 + 2*a*b*x + b^2*x^2)^(4/3)*Gamma[11/6]*Gamma[7/3])
 

Integral number [117] \[ \int \frac {\cot ^{-1}(a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx \]

[B]   time = 0.141477 (sec), size = 180 ,normalized size = 5.45 \[ \frac {c \left (6 \operatorname {Gamma}\left (\frac {11}{6}\right ) \operatorname {Gamma}\left (\frac {7}{3}\right ) \left (5 \left (1+a^2+2 a b x+b^2 x^2\right ) \left (-3+2 (a+b x) \cot ^{-1}(a+b x)\right )+4 (a+b x) \cot ^{-1}(a+b x) \operatorname {Hypergeometric2F1}\left (1,\frac {4}{3},\frac {11}{6},\frac {1}{1+a^2+2 a b x+b^2 x^2}\right )\right )-5 \sqrt [3]{2} \sqrt {\pi } \operatorname {Gamma}\left (\frac {5}{3}\right ) \, _3F_2\left (1,\frac {4}{3},\frac {4}{3};\frac {11}{6},\frac {7}{3};\frac {1}{1+a^2+2 a b x+b^2 x^2}\right )\right )}{20 b \left (c \left (1+a^2+2 a b x+b^2 x^2\right )\right )^{4/3} \operatorname {Gamma}\left (\frac {11}{6}\right ) \operatorname {Gamma}\left (\frac {7}{3}\right )} \]

[In]

Integrate[ArcCot[a + b*x]/((1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2)^(1/3),x]

output

(c*(6*Gamma[11/6]*Gamma[7/3]*(5*(1 + a^2 + 2*a*b*x + b^2*x^2)*(-3 + 2*(a + b*x 
)*ArcCot[a + b*x]) + 4*(a + b*x)*ArcCot[a + b*x]*Hypergeometric2F1[1, 4/3, 11/ 
6, (1 + a^2 + 2*a*b*x + b^2*x^2)^(-1)]) - 5*2^(1/3)*Sqrt[Pi]*Gamma[5/3]*Hyperg 
eometricPFQ[{1, 4/3, 4/3}, {11/6, 7/3}, (1 + a^2 + 2*a*b*x + b^2*x^2)^(-1)]))/ 
(20*b*(c*(1 + a^2 + 2*a*b*x + b^2*x^2))^(4/3)*Gamma[11/6]*Gamma[7/3])
 

Integral number [120] \[ \int \frac {(a+b x)^2 \cot ^{-1}(a+b x)}{\sqrt [3]{1+a^2+2 a b x+b^2 x^2}} \, dx \]

[B]   time = 0.740671 (sec), size = 198 ,normalized size = 5.66 \[ \frac {3 \left (\operatorname {Gamma}\left (\frac {11}{6}\right ) \operatorname {Gamma}\left (\frac {7}{3}\right ) \left (5 \left (1+(a+b x)^2\right ) \left (3 \left (7+(a+b x)^2\right )+4 (a+b x) \left (-2+(a+b x)^2\right ) \cot ^{-1}(a+b x)\right )-24 (a+b x) \cot ^{-1}(a+b x) \operatorname {Hypergeometric2F1}\left (1,\frac {4}{3},\frac {11}{6},\frac {1}{1+a^2+2 a b x+b^2 x^2}\right )\right )+5 \sqrt [3]{2} \sqrt {\pi } \operatorname {Gamma}\left (\frac {5}{3}\right ) \, _3F_2\left (1,\frac {4}{3},\frac {4}{3};\frac {11}{6},\frac {7}{3};\frac {1}{1+a^2+2 a b x+b^2 x^2}\right )\right )}{140 b \sqrt [3]{1+a^2+2 a b x+b^2 x^2} \left (1+(a+b x)^2\right ) \operatorname {Gamma}\left (\frac {11}{6}\right ) \operatorname {Gamma}\left (\frac {7}{3}\right )} \]

[In]

Integrate[((a + b*x)^2*ArcCot[a + b*x])/(1 + a^2 + 2*a*b*x + b^2*x^2)^(1/3),x]

output

(3*(Gamma[11/6]*Gamma[7/3]*(5*(1 + (a + b*x)^2)*(3*(7 + (a + b*x)^2) + 4*(a + 
b*x)*(-2 + (a + b*x)^2)*ArcCot[a + b*x]) - 24*(a + b*x)*ArcCot[a + b*x]*Hyperg 
eometric2F1[1, 4/3, 11/6, (1 + a^2 + 2*a*b*x + b^2*x^2)^(-1)]) + 5*2^(1/3)*Sqr 
t[Pi]*Gamma[5/3]*HypergeometricPFQ[{1, 4/3, 4/3}, {11/6, 7/3}, (1 + a^2 + 2*a* 
b*x + b^2*x^2)^(-1)]))/(140*b*(1 + a^2 + 2*a*b*x + b^2*x^2)^(1/3)*(1 + (a + b* 
x)^2)*Gamma[11/6]*Gamma[7/3])
 

Integral number [121] \[ \int \frac {(a+b x)^2 \cot ^{-1}(a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx \]

[B]   time = 0.321534 (sec), size = 200 ,normalized size = 5. \[ \frac {3 \left (\operatorname {Gamma}\left (\frac {11}{6}\right ) \operatorname {Gamma}\left (\frac {7}{3}\right ) \left (5 \left (1+(a+b x)^2\right ) \left (3 \left (7+(a+b x)^2\right )+4 (a+b x) \left (-2+(a+b x)^2\right ) \cot ^{-1}(a+b x)\right )-24 (a+b x) \cot ^{-1}(a+b x) \operatorname {Hypergeometric2F1}\left (1,\frac {4}{3},\frac {11}{6},\frac {1}{1+a^2+2 a b x+b^2 x^2}\right )\right )+5 \sqrt [3]{2} \sqrt {\pi } \operatorname {Gamma}\left (\frac {5}{3}\right ) \, _3F_2\left (1,\frac {4}{3},\frac {4}{3};\frac {11}{6},\frac {7}{3};\frac {1}{1+a^2+2 a b x+b^2 x^2}\right )\right )}{140 b \sqrt [3]{c \left (1+a^2+2 a b x+b^2 x^2\right )} \left (1+(a+b x)^2\right ) \operatorname {Gamma}\left (\frac {11}{6}\right ) \operatorname {Gamma}\left (\frac {7}{3}\right )} \]

[In]

Integrate[((a + b*x)^2*ArcCot[a + b*x])/((1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2)^(1/3),x]

output

(3*(Gamma[11/6]*Gamma[7/3]*(5*(1 + (a + b*x)^2)*(3*(7 + (a + b*x)^2) + 4*(a + 
b*x)*(-2 + (a + b*x)^2)*ArcCot[a + b*x]) - 24*(a + b*x)*ArcCot[a + b*x]*Hyperg 
eometric2F1[1, 4/3, 11/6, (1 + a^2 + 2*a*b*x + b^2*x^2)^(-1)]) + 5*2^(1/3)*Sqr 
t[Pi]*Gamma[5/3]*HypergeometricPFQ[{1, 4/3, 4/3}, {11/6, 7/3}, (1 + a^2 + 2*a* 
b*x + b^2*x^2)^(-1)]))/(140*b*(c*(1 + a^2 + 2*a*b*x + b^2*x^2))^(1/3)*(1 + (a 
+ b*x)^2)*Gamma[11/6]*Gamma[7/3])
 

4.8 Test file Number [173]

4.8.1 Mathematica

Integral number [74] \[ \int \frac {\sinh ^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx \]

[B]   time = 1.06392 (sec), size = 826 ,normalized size = 35.91 \[ \frac {-9 a \left (a^2+3 b^2\right ) \cosh (c+d x)+a^3 \cosh (3 (c+d x))-a b^2 \cosh (3 (c+d x))-2 a b \text {RootSum}\left [a-b+3 a \text {$\#$1}^2+3 b \text {$\#$1}^2+3 a \text {$\#$1}^4-3 b \text {$\#$1}^4+a \text {$\#$1}^6+b \text {$\#$1}^6\& ,\frac {3 a^2 c+3 a b c+3 b^2 c+3 a^2 d x+3 a b d x+3 b^2 d x+6 a^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+6 a b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+6 b^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+2 a^2 c \text {$\#$1}^2-2 b^2 c \text {$\#$1}^2+2 a^2 d x \text {$\#$1}^2-2 b^2 d x \text {$\#$1}^2+4 a^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2-4 b^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2+3 a^2 c \text {$\#$1}^4-3 a b c \text {$\#$1}^4+3 b^2 c \text {$\#$1}^4+3 a^2 d x \text {$\#$1}^4-3 a b d x \text {$\#$1}^4+3 b^2 d x \text {$\#$1}^4+6 a^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4-6 a b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4+6 b^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}+b \text {$\#$1}+2 a \text {$\#$1}^3-2 b \text {$\#$1}^3+a \text {$\#$1}^5+b \text {$\#$1}^5}\& \right ]+27 a^2 b \sinh (c+d x)+9 b^3 \sinh (c+d x)-a^2 b \sinh (3 (c+d x))+b^3 \sinh (3 (c+d x))}{12 (a-b)^2 (a+b)^2 d} \]

[In]

Integrate[Sinh[c + d*x]^3/(a + b*Tanh[c + d*x]^3),x]

output

(-9*a*(a^2 + 3*b^2)*Cosh[c + d*x] + a^3*Cosh[3*(c + d*x)] - a*b^2*Cosh[3*(c + 
d*x)] - 2*a*b*RootSum[a - b + 3*a*#1^2 + 3*b*#1^2 + 3*a*#1^4 - 3*b*#1^4 + a*#1 
^6 + b*#1^6 & , (3*a^2*c + 3*a*b*c + 3*b^2*c + 3*a^2*d*x + 3*a*b*d*x + 3*b^2*d 
*x + 6*a^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - 
 Sinh[(c + d*x)/2]*#1] + 6*a*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Co 
sh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 6*b^2*Log[-Cosh[(c + d*x)/2] - Si 
nh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 2*a^2*c*#1^2 
- 2*b^2*c*#1^2 + 2*a^2*d*x*#1^2 - 2*b^2*d*x*#1^2 + 4*a^2*Log[-Cosh[(c + d*x)/2 
] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 - 4* 
b^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[( 
c + d*x)/2]*#1]*#1^2 + 3*a^2*c*#1^4 - 3*a*b*c*#1^4 + 3*b^2*c*#1^4 + 3*a^2*d*x* 
#1^4 - 3*a*b*d*x*#1^4 + 3*b^2*d*x*#1^4 + 6*a^2*Log[-Cosh[(c + d*x)/2] - Sinh[( 
c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 - 6*a*b*Log[-C 
osh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2 
]*#1]*#1^4 + 6*b^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x) 
/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4)/(a*#1 + b*#1 + 2*a*#1^3 - 2*b*#1^3 + a*#1 
^5 + b*#1^5) & ] + 27*a^2*b*Sinh[c + d*x] + 9*b^3*Sinh[c + d*x] - a^2*b*Sinh[3 
*(c + d*x)] + b^3*Sinh[3*(c + d*x)])/(12*(a - b)^2*(a + b)^2*d)
 

Integral number [76] \[ \int \frac {\sinh (c+d x)}{a+b \tanh ^3(c+d x)} \, dx \]

[B]   time = 0.718557 (sec), size = 409 ,normalized size = 19.48 \[ \frac {6 a \cosh (c+d x)+b \text {RootSum}\left [a-b+3 a \text {$\#$1}^2+3 b \text {$\#$1}^2+3 a \text {$\#$1}^4-3 b \text {$\#$1}^4+a \text {$\#$1}^6+b \text {$\#$1}^6\& ,\frac {2 a c+b c+2 a d x+b d x+4 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+2 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+2 a c \text {$\#$1}^4-b c \text {$\#$1}^4+2 a d x \text {$\#$1}^4-b d x \text {$\#$1}^4+4 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4-2 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}+b \text {$\#$1}+2 a \text {$\#$1}^3-2 b \text {$\#$1}^3+a \text {$\#$1}^5+b \text {$\#$1}^5}\& \right ]-6 b \sinh (c+d x)}{6 (a-b) (a+b) d} \]

[In]

Integrate[Sinh[c + d*x]/(a + b*Tanh[c + d*x]^3),x]

output

(6*a*Cosh[c + d*x] + b*RootSum[a - b + 3*a*#1^2 + 3*b*#1^2 + 3*a*#1^4 - 3*b*#1 
^4 + a*#1^6 + b*#1^6 & , (2*a*c + b*c + 2*a*d*x + b*d*x + 4*a*Log[-Cosh[(c + d 
*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 2* 
b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c 
+ d*x)/2]*#1] + 2*a*c*#1^4 - b*c*#1^4 + 2*a*d*x*#1^4 - b*d*x*#1^4 + 4*a*Log[-C 
osh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2 
]*#1]*#1^4 - 2*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2 
]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4)/(a*#1 + b*#1 + 2*a*#1^3 - 2*b*#1^3 + a*#1^5 
 + b*#1^5) & ] - 6*b*Sinh[c + d*x])/(6*(a - b)*(a + b)*d)
 

Integral number [77] \[ \int \frac {\text {csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx \]

[B]   time = 0.62824 (sec), size = 331 ,normalized size = 15.76 \[ -\frac {6 \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )-6 \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )+b \text {RootSum}\left [a-b+3 a \text {$\#$1}^2+3 b \text {$\#$1}^2+3 a \text {$\#$1}^4-3 b \text {$\#$1}^4+a \text {$\#$1}^6+b \text {$\#$1}^6\& ,\frac {c+d x+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )-2 c \text {$\#$1}^2-2 d x \text {$\#$1}^2-4 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2+c \text {$\#$1}^4+d x \text {$\#$1}^4+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}+b \text {$\#$1}+2 a \text {$\#$1}^3-2 b \text {$\#$1}^3+a \text {$\#$1}^5+b \text {$\#$1}^5}\& \right ]}{6 a d} \]

[In]

Integrate[Csch[c + d*x]/(a + b*Tanh[c + d*x]^3),x]

output

-1/6*(6*Log[Cosh[(c + d*x)/2]] - 6*Log[Sinh[(c + d*x)/2]] + b*RootSum[a - b + 
3*a*#1^2 + 3*b*#1^2 + 3*a*#1^4 - 3*b*#1^4 + a*#1^6 + b*#1^6 & , (c + d*x + 2*L 
og[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d 
*x)/2]*#1] - 2*c*#1^2 - 2*d*x*#1^2 - 4*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x) 
/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 + c*#1^4 + d*x*#1^4 + 
2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c 
+ d*x)/2]*#1]*#1^4)/(a*#1 + b*#1 + 2*a*#1^3 - 2*b*#1^3 + a*#1^5 + b*#1^5) & ]) 
/(a*d)
 

Integral number [79] \[ \int \frac {\text {csch}^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx \]

[B]   time = 0.918625 (sec), size = 214 ,normalized size = 9.3 \[ -\frac {16 b \text {RootSum}\left [a-b+3 a \text {$\#$1}^2+3 b \text {$\#$1}^2+3 a \text {$\#$1}^4-3 b \text {$\#$1}^4+a \text {$\#$1}^6+b \text {$\#$1}^6\& ,\frac {c \text {$\#$1}+d x \text {$\#$1}+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}}{a+b+2 a \text {$\#$1}^2-2 b \text {$\#$1}^2+a \text {$\#$1}^4+b \text {$\#$1}^4}\& \right ]+3 \left (\text {csch}^2\left (\frac {1}{2} (c+d x)\right )-4 \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )+4 \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )+\text {sech}^2\left (\frac {1}{2} (c+d x)\right )\right )}{24 a d} \]

[In]

Integrate[Csch[c + d*x]^3/(a + b*Tanh[c + d*x]^3),x]

output

-1/24*(16*b*RootSum[a - b + 3*a*#1^2 + 3*b*#1^2 + 3*a*#1^4 - 3*b*#1^4 + a*#1^6 
 + b*#1^6 & , (c*#1 + d*x*#1 + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + 
Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1)/(a + b + 2*a*#1^2 - 2*b*#1^2 
+ a*#1^4 + b*#1^4) & ] + 3*(Csch[(c + d*x)/2]^2 - 4*Log[Cosh[(c + d*x)/2]] + 4 
*Log[Sinh[(c + d*x)/2]] + Sech[(c + d*x)/2]^2))/(a*d)
 

4.8.2 Fricas

Integral number [74] \[ \int \frac {\sinh ^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx \]

[C]   time = 4.16929 (sec), size = 62017 ,normalized size = 2696.39 \[ \text {Too large to display} \]

[In]

integrate(sinh(d*x+c)^3/(a+b*tanh(d*x+c)^3),x, algorithm=""fricas"")

output

1/24*((a^3 - a^2*b - a*b^2 + b^3)*cosh(d*x + c)^6 + 6*(a^3 - a^2*b - a*b^2 + b 
^3)*cosh(d*x + c)*sinh(d*x + c)^5 + (a^3 - a^2*b - a*b^2 + b^3)*sinh(d*x + c)^ 
6 - 9*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cosh(d*x + c)^4 - 3*(3*a^3 - 9*a^2*b + 9 
*a*b^2 - 3*b^3 - 5*(a^3 - a^2*b - a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^ 
4 + 4*(5*(a^3 - a^2*b - a*b^2 + b^3)*cosh(d*x + c)^3 - 9*(a^3 - 3*a^2*b + 3*a* 
b^2 - b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - 4*sqrt(2/3)*sqrt(1/6)*((a^4 - 2*a^ 
2*b^2 + b^4)*d*cosh(d*x + c)^3 + 3*(a^4 - 2*a^2*b^2 + b^4)*d*cosh(d*x + c)^2*s 
inh(d*x + c) + 3*(a^4 - 2*a^2*b^2 + b^4)*d*cosh(d*x + c)*sinh(d*x + c)^2 + (a^ 
4 - 2*a^2*b^2 + b^4)*d*sinh(d*x + c)^3)*sqrt(-(810*a^6*b^2 + 2754*a^4*b^4 + 81 
0*a^2*b^6 - (a^10 - 5*a^8*b^2 + 10*a^6*b^4 - 10*a^4*b^6 + 5*a^2*b^8 - b^10)*(( 
5*a^2*b^2/(a^8*d^4 - 4*a^6*b^2*d^4 + 6*a^4*b^4*d^4 - 4*a^2*b^6*d^4 + b^8*d^4) 
+ 9*(5*a^6*b^2 + 17*a^4*b^4 + 5*a^2*b^6)^2/(a^10*d^2 - 5*a^8*b^2*d^2 + 10*a^6* 
b^4*d^2 - 10*a^4*b^6*d^2 + 5*a^2*b^8*d^2 - b^10*d^2)^2)*(-I*sqrt(3) + 1)/(-1/1 
458*a^2*b^2/(a^10*d^6 - 5*a^8*b^2*d^6 + 10*a^6*b^4*d^6 - 10*a^4*b^6*d^6 + 5*a^ 
2*b^8*d^6 - b^10*d^6) - 5/162*(5*a^6*b^2 + 17*a^4*b^4 + 5*a^2*b^6)*a^2*b^2/((a 
^10*d^2 - 5*a^8*b^2*d^2 + 10*a^6*b^4*d^2 - 10*a^4*b^6*d^2 + 5*a^2*b^8*d^2 - b^ 
10*d^2)*(a^8*d^4 - 4*a^6*b^2*d^4 + 6*a^4*b^4*d^4 - 4*a^2*b^6*d^4 + b^8*d^4)) - 
 1/27*(5*a^6*b^2 + 17*a^4*b^4 + 5*a^2*b^6)^3/(a^10*d^2 - 5*a^8*b^2*d^2 + 10*a^ 
6*b^4*d^2 - 10*a^4*b^6*d^2 + 5*a^2*b^8*d^2 - b^10*d^2)^3 + 1/1458*(a^10 - 30*a 
^8*b^2 - 700*a^6*b^4 - 700*a^4*b^6 - 30*a^2*b^8 + b^10)*a^2*b^2/((a^2 - b^2...
 

Integral number [76] \[ \int \frac {\sinh (c+d x)}{a+b \tanh ^3(c+d x)} \, dx \]

[C]   time = 1.73806 (sec), size = 40923 ,normalized size = 1948.71 \[ \text {Too large to display} \]

[In]

integrate(sinh(d*x+c)/(a+b*tanh(d*x+c)^3),x, algorithm=""fricas"")

output

-1/6*(sqrt(2/3)*sqrt(1/2)*((a^2 - b^2)*d*cosh(d*x + c) + (a^2 - b^2)*d*sinh(d* 
x + c))*sqrt(-(108*a^2*b^2 + 54*b^4 - (a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*((b^ 
2/(a^6*d^4 - 3*a^4*b^2*d^4 + 3*a^2*b^4*d^4 - b^6*d^4) + 3*(2*a^2*b^2 + b^4)^2/ 
(a^6*d^2 - 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2)^2)*(-I*sqrt(3) + 1)/(-1/14 
58*b^2/(a^8*d^6 - 3*a^6*b^2*d^6 + 3*a^4*b^4*d^6 - a^2*b^6*d^6) - 1/54*(2*a^2*b 
^2 + b^4)*b^2/((a^6*d^4 - 3*a^4*b^2*d^4 + 3*a^2*b^4*d^4 - b^6*d^4)*(a^6*d^2 - 
3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^6*d^2 
- 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2)^3 - 1/1458*(a^6 - 3*a^4*b^2 - 24*a^ 
2*b^4 - b^6)*b^2/((a^2 - b^2)^6*a^2*d^6))^(1/3) + 27*(-1/1458*b^2/(a^8*d^6 - 3 
*a^6*b^2*d^6 + 3*a^4*b^4*d^6 - a^2*b^6*d^6) - 1/54*(2*a^2*b^2 + b^4)*b^2/((a^6 
*d^4 - 3*a^4*b^2*d^4 + 3*a^2*b^4*d^4 - b^6*d^4)*(a^6*d^2 - 3*a^4*b^2*d^2 + 3*a 
^2*b^4*d^2 - b^6*d^2)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^6*d^2 - 3*a^4*b^2*d^2 + 3 
*a^2*b^4*d^2 - b^6*d^2)^3 - 1/1458*(a^6 - 3*a^4*b^2 - 24*a^2*b^4 - b^6)*b^2/(( 
a^2 - b^2)^6*a^2*d^6))^(1/3)*(I*sqrt(3) + 1) + 18*(2*a^2*b^2 + b^4)/(a^6*d^2 - 
 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2))*d^2 + 3*sqrt(1/3)*(a^6 - 3*a^4*b^2 
+ 3*a^2*b^4 - b^6)*d^2*sqrt((432*a^6*b^2 + 2592*a^4*b^4 + 5184*a^2*b^6 + 540*b 
^8 - (a^12 - 6*a^10*b^2 + 15*a^8*b^4 - 20*a^6*b^6 + 15*a^4*b^8 - 6*a^2*b^10 + 
b^12)*((b^2/(a^6*d^4 - 3*a^4*b^2*d^4 + 3*a^2*b^4*d^4 - b^6*d^4) + 3*(2*a^2*b^2 
 + b^4)^2/(a^6*d^2 - 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2)^2)*(-I*sqrt(3) + 
 1)/(-1/1458*b^2/(a^8*d^6 - 3*a^6*b^2*d^6 + 3*a^4*b^4*d^6 - a^2*b^6*d^6) - ...
 

Integral number [77] \[ \int \frac {\text {csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx \]

[C]   time = 1.72563 (sec), size = 20085 ,normalized size = 956.43 \[ \text {Too large to display} \]

[In]

integrate(csch(d*x+c)/(a+b*tanh(d*x+c)^3),x, algorithm=""fricas"")

output

-1/6*(sqrt(2/3)*sqrt(1/6)*a*d*sqrt(((a^4 - a^2*b^2)*((b^4/(a^4*d^2 - a^2*b^2*d 
^2)^2 + b^2/(a^6*d^4 - a^4*b^2*d^4))*(-I*sqrt(3) + 1)/(-1/729*b^6/(a^4*d^2 - a 
^2*b^2*d^2)^3 - 1/486*b^4/((a^6*d^4 - a^4*b^2*d^4)*(a^4*d^2 - a^2*b^2*d^2)) - 
1/1458*b^2/(a^8*d^6 - a^6*b^2*d^6) + 1/1458*b^2/((a^2 - b^2)^2*a^4*d^6))^(1/3) 
 + 81*(-1/729*b^6/(a^4*d^2 - a^2*b^2*d^2)^3 - 1/486*b^4/((a^6*d^4 - a^4*b^2*d^ 
4)*(a^4*d^2 - a^2*b^2*d^2)) - 1/1458*b^2/(a^8*d^6 - a^6*b^2*d^6) + 1/1458*b^2/ 
((a^2 - b^2)^2*a^4*d^6))^(1/3)*(I*sqrt(3) + 1) + 18*b^2/(a^4*d^2 - a^2*b^2*d^2 
))*d^2 + 3*sqrt(1/3)*(a^4 - a^2*b^2)*d^2*sqrt(-((a^8 - 2*a^6*b^2 + a^4*b^4)*(( 
b^4/(a^4*d^2 - a^2*b^2*d^2)^2 + b^2/(a^6*d^4 - a^4*b^2*d^4))*(-I*sqrt(3) + 1)/ 
(-1/729*b^6/(a^4*d^2 - a^2*b^2*d^2)^3 - 1/486*b^4/((a^6*d^4 - a^4*b^2*d^4)*(a^ 
4*d^2 - a^2*b^2*d^2)) - 1/1458*b^2/(a^8*d^6 - a^6*b^2*d^6) + 1/1458*b^2/((a^2 
- b^2)^2*a^4*d^6))^(1/3) + 81*(-1/729*b^6/(a^4*d^2 - a^2*b^2*d^2)^3 - 1/486*b^ 
4/((a^6*d^4 - a^4*b^2*d^4)*(a^4*d^2 - a^2*b^2*d^2)) - 1/1458*b^2/(a^8*d^6 - a^ 
6*b^2*d^6) + 1/1458*b^2/((a^2 - b^2)^2*a^4*d^6))^(1/3)*(I*sqrt(3) + 1) + 18*b^ 
2/(a^4*d^2 - a^2*b^2*d^2))^2*d^4 - 1296*a^2*b^2 + 324*b^4 - 36*(a^4*b^2 - a^2* 
b^4)*((b^4/(a^4*d^2 - a^2*b^2*d^2)^2 + b^2/(a^6*d^4 - a^4*b^2*d^4))*(-I*sqrt(3 
) + 1)/(-1/729*b^6/(a^4*d^2 - a^2*b^2*d^2)^3 - 1/486*b^4/((a^6*d^4 - a^4*b^2*d 
^4)*(a^4*d^2 - a^2*b^2*d^2)) - 1/1458*b^2/(a^8*d^6 - a^6*b^2*d^6) + 1/1458*b^2 
/((a^2 - b^2)^2*a^4*d^6))^(1/3) + 81*(-1/729*b^6/(a^4*d^2 - a^2*b^2*d^2)^3 - 1 
/486*b^4/((a^6*d^4 - a^4*b^2*d^4)*(a^4*d^2 - a^2*b^2*d^2)) - 1/1458*b^2/(a^...
 

Integral number [79] \[ \int \frac {\text {csch}^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx \]

[C]   time = 2.9369 (sec), size = 6846 ,normalized size = 297.65 \[ \text {Too large to display} \]

[In]

integrate(csch(d*x+c)^3/(a+b*tanh(d*x+c)^3),x, algorithm=""fricas"")

output

-1/6*(6*cosh(d*x + c)^3 + 18*cosh(d*x + c)*sinh(d*x + c)^2 + 6*sinh(d*x + c)^3 
 + (a*d*cosh(d*x + c)^4 + 4*a*d*cosh(d*x + c)*sinh(d*x + c)^3 + a*d*sinh(d*x + 
 c)^4 - 2*a*d*cosh(d*x + c)^2 + 2*(3*a*d*cosh(d*x + c)^2 - a*d)*sinh(d*x + c)^ 
2 + a*d + 4*(a*d*cosh(d*x + c)^3 - a*d*cosh(d*x + c))*sinh(d*x + c))*sqrt((1/2 
)^(1/3)*(I*sqrt(3) + 1)*((a^2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^ 
6))^(1/3) + 3*sqrt(1/3)*sqrt(-(((1/2)^(1/3)*(I*sqrt(3) + 1)*((a^2 + b^2)*b^2/( 
a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3) - 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) 
+ 1)/(a^6*d^4*((a^2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3)) 
)^2*a^6*d^4 + 16*b^2)/(a^6*d^4)) - 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/(a^6*d^4 
*((a^2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3)))*log(4*(a^3* 
b + a^2*b^2 + a*b^3 + b^4)*cosh(d*x + c) + 4*(a^3*b + a^2*b^2 + a*b^3 + b^4)*s 
inh(d*x + c) + 1/2*(((1/2)^(1/3)*(I*sqrt(3) + 1)*((a^2 + b^2)*b^2/(a^10*d^6) - 
 (a^2*b^2 - b^4)/(a^10*d^6))^(1/3) - 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/(a^6*d 
^4*((a^2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3)))^2*a^9*d^5 
 + (a^7 - a^5*b^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((a^2 + b^2)*b^2/(a^10*d^6) - 
(a^2*b^2 - b^4)/(a^10*d^6))^(1/3) - 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/(a^6*d^ 
4*((a^2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3)))*d^3 + 4*(a 
^4*b + 2*a^3*b^2 + a^2*b^3)*d - 3*sqrt(1/3)*(((1/2)^(1/3)*(I*sqrt(3) + 1)*((a^ 
2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3) - 2*(1/2)^(2/3)*b^ 
2*(-I*sqrt(3) + 1)/(a^6*d^4*((a^2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(...
 

4.8.3 Mupad

Integral number [76] \[ \int \frac {\sinh (c+d x)}{a+b \tanh ^3(c+d x)} \, dx \]

[B]   time = 83.4069 (sec), size = -1 ,normalized size = -0.05 \[ \text {Too large to display} \]

[In]

int(sinh(c + d*x)/(a + b*tanh(c + d*x)^3),x)

output

exp(- c - d*x)/(2*(a*d - b*d)) + symsum(log((81920*a^2*b^5*exp(d*x)*exp(root(2 
187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6 
*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, 
z, k)) + 221184*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6 
*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a 
^2*b^2*d^2*z^2 - b^2, z, k)^3*a^2*b^8*d^3 - 3538944*root(2187*a^6*b^2*d^6*z^6 
- 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2* 
d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)^3*a^3*b^7*d^3 
+ 1990656*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z 
^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2 
*d^2*z^2 - b^2, z, k)^3*a^4*b^6*d^3 + 3538944*root(2187*a^6*b^2*d^6*z^6 - 2187 
*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^ 
4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)^3*a^5*b^5*d^3 - 2211 
840*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 7 
29*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z 
^2 - b^2, z, k)^3*a^6*b^4*d^3 + 7962624*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b 
^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 72 
9*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)^5*a^3*b^9*d^5 + 15925248*r 
oot(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^ 
8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2...
 

Integral number [77] \[ \int \frac {\text {csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx \]

[B]   time = 15.2722 (sec), size = -1 ,normalized size = -0.05 \[ \text {Too large to display} \]

[In]

int(1/(sinh(c + d*x)*(a + b*tanh(c + d*x)^3)),x)

output

symsum(log(-(1409286144*b^6*exp(d*x)*exp(root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^ 
6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)) + 134217728*roo 
t(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2 
*z^2 - b^2, z, k)*b^7*d + 1879048192*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^ 
6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)*a*b^6*d - 2818572288 
*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2 
*d^2*z^2 - b^2, z, k)^3*a^2*b^7*d^3 - 40869298176*root(729*a^6*b^2*d^6*z^6 - 7 
29*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^3*a^3*b 
^6*d^3 + 28185722880*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2* 
d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^3*a^4*b^5*d^3 + 15502147584*root(729 
*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 
- b^2, z, k)^3*a^5*b^4*d^3 + 18119393280*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^ 
6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^5*a^4*b^7*d^5 + 
235552112640*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 
+ 27*a^2*b^2*d^2*z^2 - b^2, z, k)^5*a^5*b^6*d^5 + 14495514624*root(729*a^6*b^2 
*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z 
, k)^5*a^6*b^5*d^5 - 219244658688*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 
 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^5*a^7*b^4*d^5 - 4892236 
1856*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2 
*b^2*d^2*z^2 - b^2, z, k)^5*a^8*b^3*d^5 - 32614907904*root(729*a^6*b^2*d^6*...
 

Integral number [79] \[ \int \frac {\text {csch}^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx \]

[B]   time = 27.7824 (sec), size = -1 ,normalized size = -0.04 \[ \text {Too large to display} \]

[In]

int(1/(sinh(c + d*x)^3*(a + b*tanh(c + d*x)^3)),x)

output

exp(c + d*x)/(a*d - a*d*exp(2*c + 2*d*x)) - (2*exp(c + d*x))/(a*d - 2*a*d*exp( 
2*c + 2*d*x) + a*d*exp(4*c + 4*d*x)) + symsum(log((570425344*a^4*b^6*exp(d*x)* 
exp(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) - 33554 
432*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)*a*b^10*d 
 - 553648128*a^2*b^8*exp(d*x)*exp(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + 
 a^2*b^2 - b^4, z, k)) - 167772160*a^3*b^7*exp(d*x)*exp(root(729*a^10*d^6*z^6 
+ 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) - 16777216*b^10*exp(d*x)*exp(root 
(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) + 192937984*a^5 
*b^5*exp(d*x)*exp(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, 
z, k)) + 2617245696*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4 
, z, k)^3*a^5*b^8*d^3 - 150994944*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + 
 a^2*b^2 - b^4, z, k)^3*a^6*b^7*d^3 - 1384120320*root(729*a^10*d^6*z^6 + 27*a^ 
4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^3*a^7*b^6*d^3 + 2415919104*root(729*a^10* 
d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^3*a^8*b^5*d^3 - 3498049536 
*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^3*a^9*b^4*d 
^3 + 5435817984*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, 
 k)^5*a^8*b^7*d^5 + 679477248*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2 
*b^2 - b^4, z, k)^5*a^9*b^6*d^5 - 70665633792*root(729*a^10*d^6*z^6 + 27*a^4*b 
^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^5*a^10*b^5*d^5 + 52319748096*root(729*a^10*d 
^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^5*a^11*b^4*d^5 + 1223059...
 

4.9 Test file Number [206]

4.9.1 Mathematica

Integral number [115] \[ \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^2} \, dx \]

[A]   time = 0.0255593 (sec), size = 13 ,normalized size = 1.08 \[ -\frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x} \]

[In]

Integrate[(Cos[b*x]*CosIntegral[b*x])/x^2,x]

output

-((Cos[b*x]*CosIntegral[b*x])/x)
 

4.9.2 Fricas

Integral number [16] \[ \int \frac {\text {Si}(b x)^2}{x^3} \, dx \]

[C]   time = 0.255927 (sec), size = 74 ,normalized size = 7.4 \[ \frac {4 \, b^{2} x^{2} \operatorname {Ci}\left (2 \, b x\right ) - 2 \, b x \cos \left (b x\right ) \operatorname {Si}\left (b x\right ) - {\left (b^{2} x^{2} + 2\right )} \operatorname {Si}\left (b x\right )^{2} + \cos \left (b x\right )^{2} - 2 \, {\left (2 \, b x \cos \left (b x\right ) + \operatorname {Si}\left (b x\right )\right )} \sin \left (b x\right ) - 1}{4 \, x^{2}} \]

[In]

integrate(sin_integral(b*x)^2/x^3,x, algorithm=""fricas"")

output

1/4*(4*b^2*x^2*cos_integral(2*b*x) - 2*b*x*cos(b*x)*sin_integral(b*x) - (b^2*x 
^2 + 2)*sin_integral(b*x)^2 + cos(b*x)^2 - 2*(2*b*x*cos(b*x) + sin_integral(b* 
x))*sin(b*x) - 1)/x^2