3 Listing of integrals solved by CAS which has no known antiderivatives

 3.1 Test file Number [5] 0_Independent_test_suites/Hearn_Problems
 3.2 Test file Number [53] 3_Logarithms/3.1_u-a+b_log-c_x^n-^p
 3.3 Test file Number [55] 3_Logarithms/3.3_u-a+b_log-c-d+e_x^m-^n-^p
 3.4 Test file Number [72] 4_Trig_functions/4.1a_Sine/4.1.7(dtrig)^m(a+b(csin)^n)^p
 3.5 Test file Number [112] 4_Trig_functions/4.3a_Secant/4.3.1.3(dsin)^n(a+bsec)^m
 3.6 Test file Number [142] 5_Inverse_trig_functions/5.2a_Inverse_tangent/5.2.2u(a+barctan(c+dx))^p
 3.7 Test file Number [146] 5_Inverse_trig_functions/5.2b_Inverse_cotangent/5.2.1Inversecotangentfunctions
 3.8 Test file Number [165] 6_Hyperbolic_functions/6.2a_Hyperbolic_tangent/6.2.7(dhyper)^m(a+b(ctanh)^n)^p

3.1 Test file Number [5] 0_Independent_test_suites/Hearn_Problems

3.1.1 Mathematica

Integral number [281] \[ \int \left (\sqrt{9-4 \sqrt{2}} x-\sqrt{2} \sqrt{1+4 x+2 x^2+x^4}\right ) \, dx \]

[B]   time = 6.07292 (sec), size = 3168 ,normalized size = 66. \[ \text{Result too large to show} \]

[In]  Integrate[Sqrt[9 - 4*Sqrt[2]]*x - Sqrt[2]*Sqrt[1 + 4*x + 2*x^2 + x^4],x]

[Out]

(Sqrt[9 - 4*Sqrt[2]]*x^2)/2 - (Sqrt[2]*x*Sqrt[1 + 4*x + 2*x^2 + x^4])/3 - (2*Sqrt[2]*((6*(x - Root[1 + 3*#1 -
#1^2 + #1^3 & , 1, 0])^2*(-(EllipticF[ArcSin[Sqrt[-(((1 + x)*(Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 +
 3*#1 - #1^2 + #1^3 & , 3, 0]))/((x - Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0])*(1 + Root[1 + 3*#1 - #1^2 + #1^3
& , 3, 0])))]], ((Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3*#1 - #1^2 + #1^3 & , 2, 0])*(1 + Root[1 +
 3*#1 - #1^2 + #1^3 & , 3, 0]))/((1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 2, 0])*(Root[1 + 3*#1 - #1^2 + #1^3 & ,
1, 0] - Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))]*Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0]) + EllipticPi[(1 + Root
[1 + 3*#1 - #1^2 + #1^3 & , 3, 0])/(-Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] + Root[1 + 3*#1 - #1^2 + #1^3 & , 3
, 0]), ArcSin[Sqrt[-(((1 + x)*(Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))
/((x - Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0])*(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0])))]], ((Root[1 + 3*#1
 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3*#1 - #1^2 + #1^3 & , 2, 0])*(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))
/((1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 2, 0])*(Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3*#1 - #1^2 +
#1^3 & , 3, 0]))]*(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0]))*Sqrt[(x - Root[1 + 3*#1 - #1^2 + #1^3 & , 2, 0]
)/((x - Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0])*(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 2, 0]))]*(-1 - Root[1 + 3*
#1 - #1^2 + #1^3 & , 3, 0])*Sqrt[(x - Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0])/((x - Root[1 + 3*#1 - #1^2 + #1^3
 & , 1, 0])*(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))]*Sqrt[-(((1 + x)*(Root[1 + 3*#1 - #1^2 + #1^3 & , 1,
0] - Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))/((x - Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0])*(1 + Root[1 + 3*#1 -
 #1^2 + #1^3 & , 3, 0])))])/(Sqrt[1 + 4*x + 2*x^2 + x^4]*(Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3*#
1 - #1^2 + #1^3 & , 3, 0])) + (2*EllipticF[ArcSin[Sqrt[((1 + x)*(-Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] + Root
[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))/((x - Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0])*(1 + Root[1 + 3*#1 - #1^2 + #
1^3 & , 3, 0]))]], ((Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3*#1 - #1^2 + #1^3 & , 2, 0])*(-1 - Root
[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))/((-1 - Root[1 + 3*#1 - #1^2 + #1^3 & , 2, 0])*(Root[1 + 3*#1 - #1^2 + #1^3
 & , 1, 0] - Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))]*(x - Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0])^2*Sqrt[(x -
Root[1 + 3*#1 - #1^2 + #1^3 & , 2, 0])/((x - Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0])*(1 + Root[1 + 3*#1 - #1^2
+ #1^3 & , 2, 0]))]*(-1 - Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0])*Sqrt[(x - Root[1 + 3*#1 - #1^2 + #1^3 & , 3,
0])/((x - Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0])*(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))]*Sqrt[((1 + x)*(
-Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] + Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))/((x - Root[1 + 3*#1 - #1^2 +
#1^3 & , 1, 0])*(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))])/(Sqrt[1 + 4*x + 2*x^2 + x^4]*(-Root[1 + 3*#1 -
#1^2 + #1^3 & , 1, 0] + Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0])) + ((1 + x)*(x - Root[1 + 3*#1 - #1^2 + #1^3 &
, 2, 0])*(x - Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]) + (x - Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0])^2*(1 + Root
[1 + 3*#1 - #1^2 + #1^3 & , 1, 0])*Sqrt[(x - Root[1 + 3*#1 - #1^2 + #1^3 & , 2, 0])/((x - Root[1 + 3*#1 - #1^2
 + #1^3 & , 1, 0])*(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 2, 0]))]*Sqrt[(x - Root[1 + 3*#1 - #1^2 + #1^3 & , 3,
0])/((x - Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0])*(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))]*Sqrt[-(((1 + x)
*(Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))/((x - Root[1 + 3*#1 - #1^2 +
 #1^3 & , 1, 0])*(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0])))]*(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0])*((
EllipticE[ArcSin[Sqrt[-(((1 + x)*(Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0
]))/((x - Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0])*(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0])))]], ((Root[1 + 3
*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3*#1 - #1^2 + #1^3 & , 2, 0])*(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0
]))/((1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 2, 0])*(Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3*#1 - #1^2
 + #1^3 & , 3, 0]))]*(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 2, 0]))/(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0])
- (EllipticPi[(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0])/(-Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] + Root[1 + 3
*#1 - #1^2 + #1^3 & , 3, 0]), ArcSin[Sqrt[-(((1 + x)*(Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3*#1 -
#1^2 + #1^3 & , 3, 0]))/((x - Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0])*(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0
])))]], ((Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3*#1 - #1^2 + #1^3 & , 2, 0])*(1 + Root[1 + 3*#1 -
#1^2 + #1^3 & , 3, 0]))/((1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 2, 0])*(Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] -
Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))]*(1 - Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3*#1 - #1^2 + #
1^3 & , 2, 0] - Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))/(-Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] + Root[1 + 3*#
1 - #1^2 + #1^3 & , 3, 0]) + (EllipticF[ArcSin[Sqrt[-(((1 + x)*(Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1
 + 3*#1 - #1^2 + #1^3 & , 3, 0]))/((x - Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0])*(1 + Root[1 + 3*#1 - #1^2 + #1^
3 & , 3, 0])))]], ((Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3*#1 - #1^2 + #1^3 & , 2, 0])*(1 + Root[1
 + 3*#1 - #1^2 + #1^3 & , 3, 0]))/((1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 2, 0])*(Root[1 + 3*#1 - #1^2 + #1^3 &
, 1, 0] - Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))]*(Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] + Root[1 + 3*#1 - #1
^2 + #1^3 & , 1, 0]*(-Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]) - Root[1
+ 3*#1 - #1^2 + #1^3 & , 3, 0]))/((1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0])*(-Root[1 + 3*#1 - #1^2 + #1^3 &
, 1, 0] + Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))))/Sqrt[1 + 4*x + 2*x^2 + x^4]))/3

3.1.2 Maple

Integral number [281] \[ \int \left (\sqrt{9-4 \sqrt{2}} x-\sqrt{2} \sqrt{1+4 x+2 x^2+x^4}\right ) \, dx \]

[A]   time = 0.504 (sec), size = 4640 ,normalized size = 96.67 \[ \text{output too large to display} \]

[In]  int(-2^(1/2)*(x^4+2*x^2+4*x+1)^(1/2)+x*(-1+2*2^(1/2)),x)

[Out]

1/2*x^2*(-1+2*2^(1/2))-2^(1/2)*(1/3*x*(x^4+2*x^2+4*x+1)^(1/2)+4/3*(-4/3-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*33
^(1/2))^(1/3)+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*((1/2*(26+6*33^(1/2))^(1/3
)-4/(26+6*33^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*(1+x)/(1/6*(26
+6*33^(1/2))^(1/3)-4/3/(26+6*33^(1/2))^(1/3)+4/3-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))
^(1/3)))/(x+1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)-1/3))^(1/2)*(x+1/3*(26+6*33^(1/2))^(1/3)-8/3/(
26+6*33^(1/2))^(1/3)-1/3)^2*((-1/3*(26+6*33^(1/2))^(1/3)+8/3/(26+6*33^(1/2))^(1/3)+4/3)*(x-1/6*(26+6*33^(1/2))
^(1/3)+4/3/(26+6*33^(1/2))^(1/3)-1/3-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(1/
6*(26+6*33^(1/2))^(1/3)-4/3/(26+6*33^(1/2))^(1/3)+4/3+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(
1/2))^(1/3)))/(x+1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)-1/3))^(1/2)*((-1/3*(26+6*33^(1/2))^(1/3)+
8/3/(26+6*33^(1/2))^(1/3)+4/3)*(x-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)-1/3+1/2*I*3^(1/2)*(-1/3*
(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(1/6*(26+6*33^(1/2))^(1/3)-4/3/(26+6*33^(1/2))^(1/3)+4/3-1/2
*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(x+1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(
1/2))^(1/3)-1/3))^(1/2)/(1/2*(26+6*33^(1/2))^(1/3)-4/(26+6*33^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))
^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(-1/3*(26+6*33^(1/2))^(1/3)+8/3/(26+6*33^(1/2))^(1/3)+4/3)/((1+x)*(x+1/3*(2
6+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)-1/3)*(x-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)-1/3-
1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*(x-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*3
3^(1/2))^(1/3)-1/3+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3))))^(1/2)*EllipticF(((1/
2*(26+6*33^(1/2))^(1/3)-4/(26+6*33^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^
(1/3)))*(1+x)/(1/6*(26+6*33^(1/2))^(1/3)-4/3/(26+6*33^(1/2))^(1/3)+4/3-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/
3)-8/3/(26+6*33^(1/2))^(1/3)))/(x+1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)-1/3))^(1/2),((-1/2*(26+6
*33^(1/2))^(1/3)+4/(26+6*33^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))
*(-4/3-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6
*33^(1/2))^(1/3)))/(-4/3-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2
))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(-1/2*(26+6*33^(1/2))^(1/3)+4/(26+6*33^(1/2))^(1/3)+1/2*I*3^(1/2)*(-1/3*(
26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3))))^(1/2))+4*(-4/3-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2))
^(1/3)+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*((1/2*(26+6*33^(1/2))^(1/3)-4/(26
+6*33^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*(1+x)/(1/6*(26+6*33^(
1/2))^(1/3)-4/3/(26+6*33^(1/2))^(1/3)+4/3-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3))
)/(x+1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)-1/3))^(1/2)*(x+1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33
^(1/2))^(1/3)-1/3)^2*((-1/3*(26+6*33^(1/2))^(1/3)+8/3/(26+6*33^(1/2))^(1/3)+4/3)*(x-1/6*(26+6*33^(1/2))^(1/3)+
4/3/(26+6*33^(1/2))^(1/3)-1/3-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(1/6*(26+6
*33^(1/2))^(1/3)-4/3/(26+6*33^(1/2))^(1/3)+4/3+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(
1/3)))/(x+1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)-1/3))^(1/2)*((-1/3*(26+6*33^(1/2))^(1/3)+8/3/(26
+6*33^(1/2))^(1/3)+4/3)*(x-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)-1/3+1/2*I*3^(1/2)*(-1/3*(26+6*3
3^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(1/6*(26+6*33^(1/2))^(1/3)-4/3/(26+6*33^(1/2))^(1/3)+4/3-1/2*I*3^(1
/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(x+1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(
1/3)-1/3))^(1/2)/(1/2*(26+6*33^(1/2))^(1/3)-4/(26+6*33^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-
8/3/(26+6*33^(1/2))^(1/3)))/(-1/3*(26+6*33^(1/2))^(1/3)+8/3/(26+6*33^(1/2))^(1/3)+4/3)/((1+x)*(x+1/3*(26+6*33^
(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)-1/3)*(x-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)-1/3-1/2*I*3
^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*(x-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2)
)^(1/3)-1/3+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3))))^(1/2)*((-1/3*(26+6*33^(1/2)
)^(1/3)+8/3/(26+6*33^(1/2))^(1/3)+1/3)*EllipticF(((1/2*(26+6*33^(1/2))^(1/3)-4/(26+6*33^(1/2))^(1/3)-1/2*I*3^(
1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*(1+x)/(1/6*(26+6*33^(1/2))^(1/3)-4/3/(26+6*33^(1/
2))^(1/3)+4/3-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(x+1/3*(26+6*33^(1/2))^(1/
3)-8/3/(26+6*33^(1/2))^(1/3)-1/3))^(1/2),((-1/2*(26+6*33^(1/2))^(1/3)+4/(26+6*33^(1/2))^(1/3)-1/2*I*3^(1/2)*(-
1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*(-4/3-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3
)+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(-4/3-1/6*(26+6*33^(1/2))^(1/3)+4/3/(2
6+6*33^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(-1/2*(26+6*33^(1/2)
)^(1/3)+4/(26+6*33^(1/2))^(1/3)+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3))))^(1/2))+
(-4/3+1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3))*EllipticPi(((1/2*(26+6*33^(1/2))^(1/3)-4/(26+6*33^(
1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*(1+x)/(1/6*(26+6*33^(1/2))^(
1/3)-4/3/(26+6*33^(1/2))^(1/3)+4/3-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(x+1/
3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)-1/3))^(1/2),(1/6*(26+6*33^(1/2))^(1/3)-4/3/(26+6*33^(1/2))^(
1/3)+4/3-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(1/2*(26+6*33^(1/2))^(1/3)-4/(2
6+6*33^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3))),((-1/2*(26+6*33^(1/2
))^(1/3)+4/(26+6*33^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*(-4/3-1
/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2
))^(1/3)))/(-4/3-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)
-8/3/(26+6*33^(1/2))^(1/3)))/(-1/2*(26+6*33^(1/2))^(1/3)+4/(26+6*33^(1/2))^(1/3)+1/2*I*3^(1/2)*(-1/3*(26+6*33^
(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3))))^(1/2)))+2/3*((1+x)*(x-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2))^
(1/3)-1/3-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*(x-1/6*(26+6*33^(1/2))^(1/3)+4
/3/(26+6*33^(1/2))^(1/3)-1/3+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))+(-4/3-1/6*(
26+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(
1/3)))*((1/2*(26+6*33^(1/2))^(1/3)-4/(26+6*33^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6
*33^(1/2))^(1/3)))*(1+x)/(1/6*(26+6*33^(1/2))^(1/3)-4/3/(26+6*33^(1/2))^(1/3)+4/3-1/2*I*3^(1/2)*(-1/3*(26+6*33
^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(x+1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)-1/3))^(1/2)*(
x+1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)-1/3)^2*((-1/3*(26+6*33^(1/2))^(1/3)+8/3/(26+6*33^(1/2))^
(1/3)+4/3)*(x-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)-1/3-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3
)-8/3/(26+6*33^(1/2))^(1/3)))/(1/6*(26+6*33^(1/2))^(1/3)-4/3/(26+6*33^(1/2))^(1/3)+4/3+1/2*I*3^(1/2)*(-1/3*(26
+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(x+1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)-1/3))^(1
/2)*((-1/3*(26+6*33^(1/2))^(1/3)+8/3/(26+6*33^(1/2))^(1/3)+4/3)*(x-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2
))^(1/3)-1/3+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(1/6*(26+6*33^(1/2))^(1/3)-
4/3/(26+6*33^(1/2))^(1/3)+4/3-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(x+1/3*(26
+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)-1/3))^(1/2)*((1/2*(26+6*33^(1/2))^(1/3)-4/(26+6*33^(1/2))^(1/3)-1
/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3))+(1/6*(26+6*33^(1/2))^(1/3)-4/3/(26+6*33^(1
/2))^(1/3)+1/3-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*(-1/3*(26+6*33^(1/2))^(1/
3)+8/3/(26+6*33^(1/2))^(1/3)+1/3)+(-1/3*(26+6*33^(1/2))^(1/3)+8/3/(26+6*33^(1/2))^(1/3)+1/3)^2)/(1/2*(26+6*33^
(1/2))^(1/3)-4/(26+6*33^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(-1
/3*(26+6*33^(1/2))^(1/3)+8/3/(26+6*33^(1/2))^(1/3)+4/3)*EllipticF(((1/2*(26+6*33^(1/2))^(1/3)-4/(26+6*33^(1/2)
)^(1/3)-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*(1+x)/(1/6*(26+6*33^(1/2))^(1/3)
-4/3/(26+6*33^(1/2))^(1/3)+4/3-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(x+1/3*(2
6+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)-1/3))^(1/2),((-1/2*(26+6*33^(1/2))^(1/3)+4/(26+6*33^(1/2))^(1/3)
-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*(-4/3-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26
+6*33^(1/2))^(1/3)+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(-4/3-1/6*(26+6*33^(1
/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(-1
/2*(26+6*33^(1/2))^(1/3)+4/(26+6*33^(1/2))^(1/3)+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))
^(1/3))))^(1/2))+(-4/3-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))
^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*EllipticE(((1/2*(26+6*33^(1/2))^(1/3)-4/(26+6*33^(1/2))^(1/3)-1/2*I*3^(1/2)
*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*(1+x)/(1/6*(26+6*33^(1/2))^(1/3)-4/3/(26+6*33^(1/2))^
(1/3)+4/3-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(x+1/3*(26+6*33^(1/2))^(1/3)-8
/3/(26+6*33^(1/2))^(1/3)-1/3))^(1/2),((-1/2*(26+6*33^(1/2))^(1/3)+4/(26+6*33^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*
(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*(-4/3-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)+1/
2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(-4/3-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*
33^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(-1/2*(26+6*33^(1/2))^(1
/3)+4/(26+6*33^(1/2))^(1/3)+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3))))^(1/2))/(-1/
3*(26+6*33^(1/2))^(1/3)+8/3/(26+6*33^(1/2))^(1/3)+4/3)))/((1+x)*(x+1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2
))^(1/3)-1/3)*(x-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)-1/3-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(
1/3)-8/3/(26+6*33^(1/2))^(1/3)))*(x-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)-1/3+1/2*I*3^(1/2)*(-1/
3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3))))^(1/2))

3.1.3 Maxima

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

[A]   time = 0.132088 (sec), size = 324 ,normalized size = 23.14 \[ -\frac{{\left (x e^{\left (\frac{4 \, k \cos \left (2 \, x\right ) \cos \left (x\right )}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1} + \frac{4 \, k \sin \left (2 \, x\right ) \sin \left (x\right )}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}\right )} + x e^{\left (\frac{4 \, k \cos \left (x\right )}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}\right )}\right )} e^{\left (-\frac{2 \, k \cos \left (2 \, x\right ) \cos \left (x\right )}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1} - \frac{2 \, k \sin \left (2 \, x\right ) \sin \left (x\right )}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1} - \frac{2 \, k \cos \left (x\right )}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}\right )} \sin \left (\frac{2 \,{\left (k \cos \left (x\right ) \sin \left (2 \, x\right ) - k \cos \left (2 \, x\right ) \sin \left (x\right ) + k \sin \left (x\right )\right )}}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}\right )}{2 \, k} \]

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

[Out]

-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

3.2 Test file Number [53] 3_Logarithms/3.1_u-a+b_log-c_x^n-^p

3.2.1 Mathematica

Integral number [115] \[ \int \frac{(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x} \, dx \]

[B]   time = 0.0844013 (sec), size = 72 ,normalized size = 2.77 \[ \frac{x (f x)^m \left ((m+1) \text{Hypergeometric2F1}\left (1,m+1,m+2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \text{HypergeometricPFQ}\left (\{1,m+1,m+1\},\{m+2,m+2\},-\frac{e x}{d}\right )\right )}{d (m+1)^2} \]

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

[Out]

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

Integral number [116] \[ \int \frac{(f x)^m \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx \]

[B]   time = 0.0908573 (sec), size = 72 ,normalized size = 2.77 \[ \frac{x (f x)^m \left ((m+1) \text{Hypergeometric2F1}\left (2,m+1,m+2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \text{HypergeometricPFQ}\left (\{2,m+1,m+1\},\{m+2,m+2\},-\frac{e x}{d}\right )\right )}{d^2 (m+1)^2} \]

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

[Out]

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

Integral number [117] \[ \int x (a+b x)^m \log \left (c x^n\right ) \, dx \]

[B]   time = 0.346659 (sec), size = 173 ,normalized size = 9.61 \[ \frac{(a+b x)^m \left (\frac{b x}{a}+1\right )^{-m} \left (a b (m+2) n x \text{HypergeometricPFQ}\left (\{1,1,-m-1\},\{2,2\},-\frac{b x}{a}\right )+\left (-a^2 \left (\left (\frac{b x}{a}+1\right )^m-1\right )+b^2 (m+1) x^2 \left (\frac{b x}{a}+1\right )^m+a b m x \left (\frac{b x}{a}+1\right )^m\right ) \log \left (c x^n\right )-n \left (a^2 \left (\left (\frac{b x}{a}+1\right )^m-1\right )+b^2 x^2 \left (\frac{b x}{a}+1\right )^m+2 a b x \left (\frac{b x}{a}+1\right )^m\right )\right )}{b^2 (m+1) (m+2)} \]

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

[Out]

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

[B]   time = 0.0631966 (sec), size = 89 ,normalized size = 4.45 \[ \frac{\left (\frac{a}{b x}+1\right )^{-m} (a+b x)^m \left (m \log \left (c x^n\right ) \text{Hypergeometric2F1}\left (-m,-m,1-m,-\frac{a}{b x}\right )-n \text{HypergeometricPFQ}\left (\{-m,-m,-m\},\{1-m,1-m\},-\frac{a}{b x}\right )\right )}{m^2} \]

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

[Out]

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

[B]   time = 0.10691 (sec), size = 108 ,normalized size = 3.86 \[ \frac{x (f x)^m \left ((m+1) \text{Hypergeometric2F1}\left (1,\frac{m+1}{2},\frac{m+3}{2},-\frac{e x^2}{d}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \text{HypergeometricPFQ}\left (\left \{1,\frac{m}{2}+\frac{1}{2},\frac{m}{2}+\frac{1}{2}\right \},\left \{\frac{m}{2}+\frac{3}{2},\frac{m}{2}+\frac{3}{2}\right \},-\frac{e x^2}{d}\right )\right )}{d (m+1)^2} \]

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

[Out]

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

[B]   time = 0.112349 (sec), size = 108 ,normalized size = 3.86 \[ \frac{x (f x)^m \left ((m+1) \text{Hypergeometric2F1}\left (2,\frac{m+1}{2},\frac{m+3}{2},-\frac{e x^2}{d}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \text{HypergeometricPFQ}\left (\left \{2,\frac{m}{2}+\frac{1}{2},\frac{m}{2}+\frac{1}{2}\right \},\left \{\frac{m}{2}+\frac{3}{2},\frac{m}{2}+\frac{3}{2}\right \},-\frac{e x^2}{d}\right )\right )}{d^2 (m+1)^2} \]

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

[Out]

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

[B]   time = 0.0908185 (sec), size = 87 ,normalized size = 3.35 \[ \frac{x^4 \left (4 \text{Hypergeometric2F1}\left (1,\frac{4}{r},\frac{r+4}{r},-\frac{e x^r}{d}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \text{HypergeometricPFQ}\left (\left \{1,\frac{4}{r},\frac{4}{r}\right \},\left \{\frac{4}{r}+1,\frac{4}{r}+1\right \},-\frac{e x^r}{d}\right )\right )}{16 d} \]

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

[Out]

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

[B]   time = 0.0846185 (sec), size = 87 ,normalized size = 3.62 \[ \frac{x^2 \left (2 \text{Hypergeometric2F1}\left (1,\frac{2}{r},\frac{r+2}{r},-\frac{e x^r}{d}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \text{HypergeometricPFQ}\left (\left \{1,\frac{2}{r},\frac{2}{r}\right \},\left \{\frac{2}{r}+1,\frac{2}{r}+1\right \},-\frac{e x^r}{d}\right )\right )}{4 d} \]

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

[Out]

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

[B]   time = 0.0839667 (sec), size = 86 ,normalized size = 3.31 \[ -\frac{b n \text{HypergeometricPFQ}\left (\left \{1,-\frac{2}{r},-\frac{2}{r}\right \},\left \{1-\frac{2}{r},1-\frac{2}{r}\right \},-\frac{e x^r}{d}\right )+2 \text{Hypergeometric2F1}\left (1,-\frac{2}{r},\frac{r-2}{r},-\frac{e x^r}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d x^2} \]

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

[Out]

-(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*x^2)

Integral number [358] \[ \int \frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx \]

[B]   time = 0.0875054 (sec), size = 87 ,normalized size = 3.35 \[ \frac{x^3 \left (3 \text{Hypergeometric2F1}\left (1,\frac{3}{r},\frac{r+3}{r},-\frac{e x^r}{d}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \text{HypergeometricPFQ}\left (\left \{1,\frac{3}{r},\frac{3}{r}\right \},\left \{\frac{3}{r}+1,\frac{3}{r}+1\right \},-\frac{e x^r}{d}\right )\right )}{9 d} \]

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

[Out]

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

[B]   time = 0.064203 (sec), size = 69 ,normalized size = 3. \[ \frac{x \left (\text{Hypergeometric2F1}\left (1,\frac{1}{r},\frac{1}{r}+1,-\frac{e x^r}{d}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \text{HypergeometricPFQ}\left (\left \{1,\frac{1}{r},\frac{1}{r}\right \},\left \{\frac{1}{r}+1,\frac{1}{r}+1\right \},-\frac{e x^r}{d}\right )\right )}{d} \]

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

[Out]

(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*Log[c*x^n])))/d

Integral number [360] \[ \int \frac{a+b \log \left (c x^n\right )}{x^2 \left (d+e x^r\right )} \, dx \]

[B]   time = 0.0769575 (sec), size = 83 ,normalized size = 3.19 \[ -\frac{b n \text{HypergeometricPFQ}\left (\left \{1,-\frac{1}{r},-\frac{1}{r}\right \},\left \{1-\frac{1}{r},1-\frac{1}{r}\right \},-\frac{e x^r}{d}\right )+\text{Hypergeometric2F1}\left (1,-\frac{1}{r},\frac{r-1}{r},-\frac{e x^r}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d x} \]

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

[Out]

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

[B]   time = 0.290929 (sec), size = 140 ,normalized size = 5.38 \[ \frac{x^4 \left (-b n (r-4) \left (d+e x^r\right ) \text{HypergeometricPFQ}\left (\left \{1,\frac{4}{r},\frac{4}{r}\right \},\left \{\frac{4}{r}+1,\frac{4}{r}+1\right \},-\frac{e x^r}{d}\right )+4 \left (d+e x^r\right ) \text{Hypergeometric2F1}\left (1,\frac{4}{r},\frac{r+4}{r},-\frac{e x^r}{d}\right ) \left (a (r-4)+b (r-4) \log \left (c x^n\right )-b n\right )+16 d \left (a+b \log \left (c x^n\right )\right )\right )}{16 d^2 r \left (d+e x^r\right )} \]

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

[Out]

(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)*Hypergeometric2F1[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 [362] \[ \int \frac{x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx \]

[B]   time = 0.282185 (sec), size = 140 ,normalized size = 5.83 \[ \frac{x^2 \left (-b n (r-2) \left (d+e x^r\right ) \text{HypergeometricPFQ}\left (\left \{1,\frac{2}{r},\frac{2}{r}\right \},\left \{\frac{2}{r}+1,\frac{2}{r}+1\right \},-\frac{e x^r}{d}\right )+2 \left (d+e x^r\right ) \text{Hypergeometric2F1}\left (1,\frac{2}{r},\frac{r+2}{r},-\frac{e x^r}{d}\right ) \left (a (r-2)+b (r-2) \log \left (c x^n\right )-b n\right )+4 d \left (a+b \log \left (c x^n\right )\right )\right )}{4 d^2 r \left (d+e x^r\right )} \]

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

[Out]

(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)*Hypergeometric2F1[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 [364] \[ \int \frac{a+b \log \left (c x^n\right )}{x^3 \left (d+e x^r\right )^2} \, dx \]

[B]   time = 0.282261 (sec), size = 139 ,normalized size = 5.35 \[ -\frac{b n (r+2) \left (d+e x^r\right ) \text{HypergeometricPFQ}\left (\left \{1,-\frac{2}{r},-\frac{2}{r}\right \},\left \{1-\frac{2}{r},1-\frac{2}{r}\right \},-\frac{e x^r}{d}\right )+2 \left (d+e x^r\right ) \text{Hypergeometric2F1}\left (1,-\frac{2}{r},\frac{r-2}{r},-\frac{e x^r}{d}\right ) \left (a (r+2)+b (r+2) \log \left (c x^n\right )-b n\right )-4 d \left (a+b \log \left (c x^n\right )\right )}{4 d^2 r x^2 \left (d+e x^r\right )} \]

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

[Out]

-(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*Lo
g[c*x^n]) + 2*(d + e*x^r)*Hypergeometric2F1[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*x^2*(d + e*x^r))

Integral number [365] \[ \int \frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx \]

[B]   time = 0.288479 (sec), size = 140 ,normalized size = 5.38 \[ \frac{x^3 \left (-b n (r-3) \left (d+e x^r\right ) \text{HypergeometricPFQ}\left (\left \{1,\frac{3}{r},\frac{3}{r}\right \},\left \{\frac{3}{r}+1,\frac{3}{r}+1\right \},-\frac{e x^r}{d}\right )+3 \left (d+e x^r\right ) \text{Hypergeometric2F1}\left (1,\frac{3}{r},\frac{r+3}{r},-\frac{e x^r}{d}\right ) \left (a (r-3)+b (r-3) \log \left (c x^n\right )-b n\right )+9 d \left (a+b \log \left (c x^n\right )\right )\right )}{9 d^2 r \left (d+e x^r\right )} \]

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

[Out]

(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)*Hypergeometric2F1[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 [366] \[ \int \frac{a+b \log \left (c x^n\right )}{\left (d+e x^r\right )^2} \, dx \]

[B]   time = 3.6981 (sec), size = 121 ,normalized size = 5.26 \[ \frac{x \left (-b n (r-1) \left (d+e x^r\right ) \text{HypergeometricPFQ}\left (\left \{1,\frac{1}{r},\frac{1}{r}\right \},\left \{\frac{1}{r}+1,\frac{1}{r}+1\right \},-\frac{e x^r}{d}\right )+\left (d+e x^r\right ) \text{Hypergeometric2F1}\left (1,\frac{1}{r},\frac{1}{r}+1,-\frac{e x^r}{d}\right ) \left (a (r-1)+b (r-1) \log \left (c x^n\right )-b n\right )+d \left (a+b \log \left (c x^n\right )\right )\right )}{d^2 r \left (d+e x^r\right )} \]

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

[Out]

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

Integral number [367] \[ \int \frac{a+b \log \left (c x^n\right )}{x^2 \left (d+e x^r\right )^2} \, dx \]

[B]   time = 0.225633 (sec), size = 135 ,normalized size = 5.19 \[ \frac{-b n (r+1) \left (d+e x^r\right ) \text{HypergeometricPFQ}\left (\left \{1,-\frac{1}{r},-\frac{1}{r}\right \},\left \{1-\frac{1}{r},1-\frac{1}{r}\right \},-\frac{e x^r}{d}\right )-\left (d+e x^r\right ) \text{Hypergeometric2F1}\left (1,-\frac{1}{r},\frac{r-1}{r},-\frac{e x^r}{d}\right ) \left (a r+a+b (r+1) \log \left (c x^n\right )-b n\right )+d \left (a+b \log \left (c x^n\right )\right )}{d^2 r x \left (d+e x^r\right )} \]

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

[Out]

(-(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)*Hypergeometric2F1[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 [392] \[ \int \frac{(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx \]

[B]   time = 0.139764 (sec), size = 111 ,normalized size = 3.96 \[ \frac{x (f x)^m \left ((m+1) \left (a+b \log \left (c x^n\right )\right ) \text{Hypergeometric2F1}\left (1,\frac{m+1}{r},\frac{m+r+1}{r},-\frac{e x^r}{d}\right )-b n \text{HypergeometricPFQ}\left (\left \{1,\frac{m}{r}+\frac{1}{r},\frac{m}{r}+\frac{1}{r}\right \},\left \{\frac{m}{r}+\frac{1}{r}+1,\frac{m}{r}+\frac{1}{r}+1\right \},-\frac{e x^r}{d}\right )\right )}{d (m+1)^2} \]

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

[Out]

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

[B]   time = 0.523032 (sec), size = 177 ,normalized size = 6.32 \[ \frac{x (f x)^m \left (b n (m-r+1) \left (d+e x^r\right ) \text{HypergeometricPFQ}\left (\left \{1,\frac{m}{r}+\frac{1}{r},\frac{m}{r}+\frac{1}{r}\right \},\left \{\frac{m}{r}+\frac{1}{r}+1,\frac{m}{r}+\frac{1}{r}+1\right \},-\frac{e x^r}{d}\right )-(m+1) \left (\left (d+e x^r\right ) \text{Hypergeometric2F1}\left (1,\frac{m+1}{r},\frac{m+r+1}{r},-\frac{e x^r}{d}\right ) \left (a (m-r+1)+b (m-r+1) \log \left (c x^n\right )+b n\right )-d (m+1) \left (a+b \log \left (c x^n\right )\right )\right )\right )}{d^2 (m+1)^2 r \left (d+e x^r\right )} \]

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

[Out]

(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))

Integral number [731] \[ \int (g x)^q \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx \]

[B]   time = 0.400831 (sec), size = 304 ,normalized size = 9.81 \[ \frac{x (g x)^q \left (-b k m n \text{HypergeometricPFQ}\left (\left \{1,\frac{q}{m}+\frac{1}{m},\frac{q}{m}+\frac{1}{m}\right \},\left \{\frac{q}{m}+\frac{1}{m}+1,\frac{q}{m}+\frac{1}{m}+1\right \},-\frac{f x^m}{e}\right )+k m \text{Hypergeometric2F1}\left (1,\frac{q+1}{m},\frac{m+q+1}{m},-\frac{f x^m}{e}\right ) \left (a q+a+b (q+1) \log \left (c x^n\right )-b n\right )+a q^2 \log \left (d \left (e+f x^m\right )^k\right )+2 a q \log \left (d \left (e+f x^m\right )^k\right )+a \log \left (d \left (e+f x^m\right )^k\right )-a k m q-a k m+b q^2 \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )+2 b q \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )+b \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )-b k m q \log \left (c x^n\right )-b k m \log \left (c x^n\right )-b n q \log \left (d \left (e+f x^m\right )^k\right )-b n \log \left (d \left (e+f x^m\right )^k\right )+2 b k m n\right )}{(q+1)^3} \]

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

[Out]

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

[B]   time = 0.171137 (sec), size = 168 ,normalized size = 5.79 \[ \frac{1}{27} x^3 \left (-b k m n \text{HypergeometricPFQ}\left (\left \{1,\frac{3}{m},\frac{3}{m}\right \},\left \{\frac{3}{m}+1,\frac{3}{m}+1\right \},-\frac{f x^m}{e}\right )+k m \text{Hypergeometric2F1}\left (1,\frac{3}{m},\frac{m+3}{m},-\frac{f x^m}{e}\right ) \left (3 a+3 b \log \left (c x^n\right )-b n\right )+9 a \log \left (d \left (e+f x^m\right )^k\right )-3 a k m+9 b \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )-3 b k m \log \left (c x^n\right )-3 b n \log \left (d \left (e+f x^m\right )^k\right )+2 b k m n\right ) \]

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

[Out]

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

Integral number [738] \[ \int x \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx \]

[B]   time = 0.186015 (sec), size = 152 ,normalized size = 5.63 \[ \frac{1}{8} x^2 \left (-b k m n \text{HypergeometricPFQ}\left (\left \{1,\frac{2}{m},\frac{2}{m}\right \},\left \{\frac{2}{m}+1,\frac{2}{m}+1\right \},-\frac{f x^m}{e}\right )+k m \text{Hypergeometric2F1}\left (1,\frac{2}{m},\frac{m+2}{m},-\frac{f x^m}{e}\right ) \left (2 a+2 b \log \left (c x^n\right )-b n\right )-2 \left ((b n-2 a) \log \left (d \left (e+f x^m\right )^k\right )+k m (a-b n)+b \log \left (c x^n\right ) \left (k m-2 \log \left (d \left (e+f x^m\right )^k\right )\right )\right )\right ) \]

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

[Out]

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

Integral number [739] \[ \int \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx \]

[B]   time = 0.196002 (sec), size = 165 ,normalized size = 6.35 \[ x \left (-b k m n \text{HypergeometricPFQ}\left (\left \{1,\frac{1}{m},\frac{1}{m}\right \},\left \{\frac{1}{m}+1,\frac{1}{m}+1\right \},-\frac{f x^m}{e}\right )+k m \text{Hypergeometric2F1}\left (1,\frac{1}{m},\frac{1}{m}+1,-\frac{f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )-b n\right )+a \log \left (d \left (e+f x^m\right )^k\right )+b \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )-b n \log \left (d \left (e+f x^m\right )^k\right )-b k m n \log (x)+b k m n\right )-k m x \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )+b k m n x \]

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

[Out]

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*HypergeometricPFQ[{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 [741] \[ \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{x^2} \, dx \]

[B]   time = 0.159511 (sec), size = 158 ,normalized size = 5.45 \[ -\frac{-b k m n \text{HypergeometricPFQ}\left (\left \{1,-\frac{1}{m},-\frac{1}{m}\right \},\left \{1-\frac{1}{m},1-\frac{1}{m}\right \},-\frac{f x^m}{e}\right )-k m \text{Hypergeometric2F1}\left (1,-\frac{1}{m},\frac{m-1}{m},-\frac{f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )+b n\right )+a \log \left (d \left (e+f x^m\right )^k\right )+a k m+b \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )+b k m \log \left (c x^n\right )+b n \log \left (d \left (e+f x^m\right )^k\right )+2 b k m n}{x} \]

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

[Out]

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

Integral number [742] \[ \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{x^3} \, dx \]

[B]   time = 0.187937 (sec), size = 149 ,normalized size = 5.14 \[ \frac{b k m n \text{HypergeometricPFQ}\left (\left \{1,-\frac{2}{m},-\frac{2}{m}\right \},\left \{1-\frac{2}{m},1-\frac{2}{m}\right \},-\frac{f x^m}{e}\right )+k m \text{Hypergeometric2F1}\left (1,-\frac{2}{m},\frac{m-2}{m},-\frac{f x^m}{e}\right ) \left (2 a+2 b \log \left (c x^n\right )+b n\right )-2 \left ((2 a+b n) \log \left (d \left (e+f x^m\right )^k\right )+k m (a+b n)+b \log \left (c x^n\right ) \left (2 \log \left (d \left (e+f x^m\right )^k\right )+k m\right )\right )}{8 x^2} \]

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

[Out]

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

Integral number [813] \[ \int -(d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx \]

[B]   time = 0.381626 (sec), size = 266 ,normalized size = 8.87 \[ -\frac{x (d x)^m \left (-b n q \text{HypergeometricPFQ}\left (\left \{1,\frac{m}{q}+\frac{1}{q},\frac{m}{q}+\frac{1}{q}\right \},\left \{\frac{m}{q}+\frac{1}{q}+1,\frac{m}{q}+\frac{1}{q}+1\right \},e x^q\right )+q \text{Hypergeometric2F1}\left (1,\frac{m+1}{q},\frac{m+q+1}{q},e x^q\right ) \left (a m+a+b (m+1) \log \left (c x^n\right )-b n\right )+a m^2 \log \left (1-e x^q\right )+2 a m \log \left (1-e x^q\right )+a \log \left (1-e x^q\right )-a m q-a q+b m^2 \log \left (c x^n\right ) \log \left (1-e x^q\right )+2 b m \log \left (c x^n\right ) \log \left (1-e x^q\right )+b \log \left (c x^n\right ) \log \left (1-e x^q\right )-b m q \log \left (c x^n\right )-b q \log \left (c x^n\right )-b m n \log \left (1-e x^q\right )-b n \log \left (1-e x^q\right )+2 b n q\right )}{(m+1)^3} \]

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

[Out]

-((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*Log[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)

3.2.2 Maple

Integral number [813] \[ \int -(d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx \]

[B]   time = 0.719 (sec), size = 844 ,normalized size = 28.13 \[ -{\frac{ \left ( dx \right ) ^{m}{x}^{-m}a}{q} \left ( -e \right ) ^{-{\frac{m}{q}}-{q}^{-1}} \left ({\frac{q{x}^{1+m}\ln \left ( 1-e{x}^{q} \right ) }{1+m} \left ( -e \right ) ^{{\frac{m}{q}}+{q}^{-1}}}-{\frac{q{x}^{1+m+q}e \left ( -q-m-1 \right ) }{ \left ( 1+m \right ) \left ( 1+m+q \right ) } \left ( -e \right ) ^{{\frac{m}{q}}+{q}^{-1}}{\it LerchPhi} \left ( e{x}^{q},1,{\frac{1+m+q}{q}} \right ) } \right ) }-{\frac{ \left ( dx \right ) ^{m}{x}^{-m}b\ln \left ( c \right ) }{q} \left ( -e \right ) ^{-{\frac{m}{q}}-{q}^{-1}} \left ({\frac{q{x}^{1+m}\ln \left ( 1-e{x}^{q} \right ) }{1+m} \left ( -e \right ) ^{{\frac{m}{q}}+{q}^{-1}}}-{\frac{q{x}^{1+m+q}e \left ( -q-m-1 \right ) }{ \left ( 1+m \right ) \left ( 1+m+q \right ) } \left ( -e \right ) ^{{\frac{m}{q}}+{q}^{-1}}{\it LerchPhi} \left ( e{x}^{q},1,{\frac{1+m+q}{q}} \right ) } \right ) }+ \left ({\frac{\ln \left ( -e \right ) \left ( dx \right ) ^{m}{x}^{-m}bn}{{q}^{2}} \left ( -e \right ) ^{-{\frac{m}{q}}-{q}^{-1}} \left ({\frac{q{x}^{m}\ln \left ( 1-e{x}^{q} \right ) }{1+m} \left ( -e \right ) ^{{\frac{m}{q}}+{q}^{-1}}}-{\frac{q{x}^{q+m}e \left ( -q-m-1 \right ) }{ \left ( 1+m \right ) \left ( 1+m+q \right ) } \left ( -e \right ) ^{{\frac{m}{q}}+{q}^{-1}}{\it LerchPhi} \left ( e{x}^{q},1,{\frac{1+m+q}{q}} \right ) } \right ) }-{\frac{ \left ( dx \right ) ^{m}{x}^{-m}bn}{q} \left ( -e \right ) ^{-{\frac{m}{q}}-{q}^{-1}} \left ({\frac{q\ln \left ( x \right ){x}^{m}\ln \left ( 1-e{x}^{q} \right ) }{1+m} \left ( -e \right ) ^{{\frac{m}{q}}+{q}^{-1}}}+{\frac{\ln \left ( -e \right ){x}^{m}\ln \left ( 1-e{x}^{q} \right ) }{1+m} \left ( -e \right ) ^{{\frac{m}{q}}+{q}^{-1}}}-{\frac{q{x}^{m}\ln \left ( 1-e{x}^{q} \right ) }{ \left ( 1+m \right ) ^{2}} \left ( -e \right ) ^{{\frac{m}{q}}+{q}^{-1}}}+{\frac{q{x}^{q+m}e \left ( -q-m-1 \right ) }{ \left ( 1+m \right ) \left ( 1+m+q \right ) ^{2}} \left ( -e \right ) ^{{\frac{m}{q}}+{q}^{-1}}{\it LerchPhi} \left ( e{x}^{q},1,{\frac{1+m+q}{q}} \right ) }-{\frac{q{x}^{q+m}e\ln \left ( x \right ) \left ( -q-m-1 \right ) }{ \left ( 1+m \right ) \left ( 1+m+q \right ) } \left ( -e \right ) ^{{\frac{m}{q}}+{q}^{-1}}{\it LerchPhi} \left ( e{x}^{q},1,{\frac{1+m+q}{q}} \right ) }-{\frac{{x}^{q+m}e\ln \left ( -e \right ) \left ( -q-m-1 \right ) }{ \left ( 1+m \right ) \left ( 1+m+q \right ) } \left ( -e \right ) ^{{\frac{m}{q}}+{q}^{-1}}{\it LerchPhi} \left ( e{x}^{q},1,{\frac{1+m+q}{q}} \right ) }+{\frac{q{x}^{q+m}e}{ \left ( 1+m \right ) \left ( 1+m+q \right ) } \left ( -e \right ) ^{{\frac{m}{q}}+{q}^{-1}}{\it LerchPhi} \left ( e{x}^{q},1,{\frac{1+m+q}{q}} \right ) }+{\frac{q{x}^{q+m}e \left ( -q-m-1 \right ) }{ \left ( 1+m \right ) ^{2} \left ( 1+m+q \right ) } \left ( -e \right ) ^{{\frac{m}{q}}+{q}^{-1}}{\it LerchPhi} \left ( e{x}^{q},1,{\frac{1+m+q}{q}} \right ) }+{\frac{{x}^{q+m}e \left ( -q-m-1 \right ) }{ \left ( 1+m \right ) \left ( 1+m+q \right ) } \left ( -e \right ) ^{{\frac{m}{q}}+{q}^{-1}}{\it LerchPhi} \left ( e{x}^{q},2,{\frac{1+m+q}{q}} \right ) } \right ) } \right ) x \]

[In]  int(-(d*x)^m*(a+b*ln(c*x^n))*ln(1-e*x^q),x)

[Out]

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

Integral number [814] \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,e x^q\right ) \, dx \]

[B]   time = 0.402 (sec), size = 867 ,normalized size = 4.87 \[ \text{result too large to display} \]

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

[Out]

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

Integral number [815] \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (3,e x^q\right ) \, dx \]

[B]   time = 1.737 (sec), size = 1065 ,normalized size = 4.35 \[ \text{result too large to display} \]

[In]  int((d*x)^m*(a+b*ln(c*x^n))*polylog(3,e*x^q),x)

[Out]

-(d*x)^m*x^(-m)*(-e)^(-1/q*m-1/q)*a/q*(q^3*x^(1+m)*(-e)^(1/q*m+1/q)/(1+m)^3*ln(1-e*x^q)+q^2*x^(1+m)*(-e)^(1/q*
m+1/q)/(1+m)^2*polylog(2,e*x^q)-q*x^(1+m)*(-e)^(1/q*m+1/q)/(1+m)*polylog(3,e*x^q)+q^3*x^(1+m+q)*e*(-e)^(1/q*m+
1/q)/(1+m)^3*LerchPhi(e*x^q,1,(1+m+q)/q))-(d*x)^m*x^(-m)*(-e)^(-1/q*m-1/q)*b*ln(c)/q*(q^3*x^(1+m)*(-e)^(1/q*m+
1/q)/(1+m)^3*ln(1-e*x^q)+q^2*x^(1+m)*(-e)^(1/q*m+1/q)/(1+m)^2*polylog(2,e*x^q)-q*x^(1+m)*(-e)^(1/q*m+1/q)/(1+m
)*polylog(3,e*x^q)+q^3*x^(1+m+q)*e*(-e)^(1/q*m+1/q)/(1+m)^3*LerchPhi(e*x^q,1,(1+m+q)/q))+(1/q^2*ln(-e)*(-e)^(-
1/q*m-1/q)*(d*x)^m*x^(-m)*b*n*(q^3*x^m*(-e)^(1/q*m+1/q)/(1+m)^3*ln(1-e*x^q)+q^2*x^m*(-e)^(1/q*m+1/q)/(1+m)^2*p
olylog(2,e*x^q)-q*x^m*(-e)^(1/q*m+1/q)/(1+m)*polylog(3,e*x^q)+q^3*x^(q+m)*e*(-e)^(1/q*m+1/q)/(1+m)^3*LerchPhi(
e*x^q,1,(1+m+q)/q))-(-e)^(-1/q*m-1/q)*(d*x)^m*x^(-m)*b*n/q*(q^3*ln(x)*x^m*(-e)^(1/q*m+1/q)/(1+m)^3*ln(1-e*x^q)
+q^2*ln(-e)*x^m*(-e)^(1/q*m+1/q)/(1+m)^3*ln(1-e*x^q)-3*q^3*x^m*(-e)^(1/q*m+1/q)/(1+m)^4*ln(1-e*x^q)+q^2*ln(x)*
x^m*(-e)^(1/q*m+1/q)/(1+m)^2*polylog(2,e*x^q)+q*ln(-e)*x^m*(-e)^(1/q*m+1/q)/(1+m)^2*polylog(2,e*x^q)-2*q^2*x^m
*(-e)^(1/q*m+1/q)/(1+m)^3*polylog(2,e*x^q)-q*ln(x)*x^m*(-e)^(1/q*m+1/q)/(1+m)*polylog(3,e*x^q)-ln(-e)*x^m*(-e)
^(1/q*m+1/q)/(1+m)*polylog(3,e*x^q)+q*x^m*(-e)^(1/q*m+1/q)/(1+m)^2*polylog(3,e*x^q)+q^3*x^(q+m)*e*ln(x)*(-e)^(
1/q*m+1/q)/(1+m)^3*LerchPhi(e*x^q,1,(1+m+q)/q)+q^2*x^(q+m)*e*ln(-e)*(-e)^(1/q*m+1/q)/(1+m)^3*LerchPhi(e*x^q,1,
(1+m+q)/q)-3*q^3*x^(q+m)*e*(-e)^(1/q*m+1/q)/(1+m)^4*LerchPhi(e*x^q,1,(1+m+q)/q)-q^2*x^(q+m)*e*(-e)^(1/q*m+1/q)
/(1+m)^3*LerchPhi(e*x^q,2,(1+m+q)/q)))*x

3.3 Test file Number [55] 3_Logarithms/3.3_u-a+b_log-c-d+e_x^m-^n-^p

3.3.1 Mathematica

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

[B]   time = 8.96102 (sec), size = 1224 ,normalized size = 3.22 \[ \text{result too large to display} \]

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

[Out]

(2*a*p*x*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2)/b - (2*a^(3/2)*p*ArcTan[(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 - (96*a*Sqrt[b]*x - 8*b^(3/
2)*x^3 - 24*a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]] + (36*I)*a^(3/2)*Log[((-I)*Sqrt[a])/Sqrt[b] + x] - 36*a^(3/2)*
ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[((-I)*Sqrt[a])/Sqrt[b] + x] - (9*I)*a^(3/2)*Log[((-I)*Sqrt[a])/Sqrt[b] + x]^2
- (36*I)*a^(3/2)*Log[(I*Sqrt[a])/Sqrt[b] + x] - 36*a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[(I*Sqrt[a])/Sqrt[b]
 + x] + (9*I)*a^(3/2)*Log[(I*Sqrt[a])/Sqrt[b] + x]^2 + (18*I)*a^(3/2)*Log[((-I)*Sqrt[a])/Sqrt[b] + x]*Log[1/2
- ((I/2)*Sqrt[b]*x)/Sqrt[a]] - (18*I)*a^(3/2)*Log[(I*Sqrt[a])/Sqrt[b] + x]*Log[1/2 + ((I/2)*Sqrt[b]*x)/Sqrt[a]
] - 36*a*Sqrt[b]*x*Log[a + b*x^2] + 12*b^(3/2)*x^3*Log[a + b*x^2] + 36*a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log
[a + b*x^2] - (18*I)*a^(3/2)*PolyLog[2, 1/2 - ((I/2)*Sqrt[b]*x)/Sqrt[a]] + (18*I)*a^(3/2)*PolyLog[2, 1/2 + ((I
/2)*Sqrt[b]*x)/Sqrt[a]])/(27*b^(3/2))) + (p^3*(416*Sqrt[-a]*a^(3/2)*Sqrt[a + b*x^2]*Sqrt[1 - a/(a + b*x^2)]*Ar
cSin[Sqrt[a]/Sqrt[a + b*x^2]] + 36*Sqrt[-a]*a^(3/2)*Sqrt[1 - a/(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)] + 4*Sqrt[a]*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, a/
(a + b*x^2)]*Log[a + b*x^2] + Sqrt[a + b*x^2]*ArcSin[Sqrt[a]/Sqrt[a + b*x^2]]*Log[a + b*x^2]^2) - (2*Sqrt[-a]*
b*x^2*((a + b*x^2)*(16 - 24*Log[a + b*x^2] + 18*Log[a + b*x^2]^2 - 9*Log[a + b*x^2]^3) + a*(-640 + 312*Log[a +
 b*x^2] - 72*Log[a + b*x^2]^2 + 9*Log[a + b*x^2]^3)))/3 - 48*a^2*(4*Sqrt[b*x^2]*ArcTanh[Sqrt[b*x^2]/Sqrt[-a]]*
(Log[a + b*x^2] - Log[(a + b*x^2)/a]) - Sqrt[-a]*Sqrt[1 - (a + b*x^2)/a]*(Log[(a + b*x^2)/a]^2 - 4*Log[(a + b*
x^2)/a]*Log[(1 + Sqrt[1 - (a + b*x^2)/a])/2] + 2*Log[(1 + Sqrt[1 - (a + b*x^2)/a])/2]^2 - 4*PolyLog[2, 1/2 - S
qrt[1 - (a + b*x^2)/a]/2]))))/(18*Sqrt[-a]*b^2*x)

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

[B]   time = 9.28609 (sec), size = 1061 ,normalized size = 3.66 \[ \frac{\left (\sqrt{-a} b \left (\log ^3\left (b x^2+a\right )-6 \log ^2\left (b x^2+a\right )+24 \log \left (b x^2+a\right )-48\right ) x^2-48 \sqrt{-a^2} \sqrt{b x^2+a} \sqrt{1-\frac{a}{b x^2+a}} \sin ^{-1}\left (\frac{\sqrt{a}}{\sqrt{b x^2+a}}\right )-6 \sqrt{-a^2} \sqrt{1-\frac{a}{b x^2+a}} \left (\sqrt{b x^2+a} \sin ^{-1}\left (\frac{\sqrt{a}}{\sqrt{b x^2+a}}\right ) \log ^2\left (b x^2+a\right )+4 \sqrt{a} \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right \},\left \{\frac{3}{2},\frac{3}{2}\right \},\frac{a}{b x^2+a}\right ) \log \left (b x^2+a\right )+8 \sqrt{a} \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right \},\left \{\frac{3}{2},\frac{3}{2},\frac{3}{2}\right \},\frac{a}{b x^2+a}\right )\right )+24 a \sqrt{b x^2} \tanh ^{-1}\left (\frac{\sqrt{b x^2}}{\sqrt{-a}}\right ) \left (\log \left (b x^2+a\right )-\log \left (\frac{b x^2+a}{a}\right )\right )+6 (-a)^{3/2} \sqrt{1-\frac{b x^2+a}{a}} \left (\log ^2\left (\frac{b x^2+a}{a}\right )-4 \log \left (\frac{1}{2} \left (\sqrt{1-\frac{b x^2+a}{a}}+1\right )\right ) \log \left (\frac{b x^2+a}{a}\right )+2 \log ^2\left (\frac{1}{2} \left (\sqrt{1-\frac{b x^2+a}{a}}+1\right )\right )-4 \text{PolyLog}\left (2,\frac{1}{2}-\frac{1}{2} \sqrt{1-\frac{b x^2+a}{a}}\right )\right )\right ) p^3}{\sqrt{-a} b x}+3 \left (\log \left (c \left (b x^2+a\right )^p\right )-p \log \left (b x^2+a\right )\right ) \left (x \log ^2\left (b x^2+a\right )-\frac{i \sqrt{a} \log ^2\left (x-\frac{i \sqrt{a}}{\sqrt{b}}\right )+4 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (x-\frac{i \sqrt{a}}{\sqrt{b}}\right )-2 i \sqrt{a} \log \left (\frac{1}{2}-\frac{i \sqrt{b} x}{2 \sqrt{a}}\right ) \log \left (x-\frac{i \sqrt{a}}{\sqrt{b}}\right )-4 i \sqrt{a} \log \left (x-\frac{i \sqrt{a}}{\sqrt{b}}\right )-i \sqrt{a} \log ^2\left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )-8 \sqrt{b} x+4 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )+4 i \sqrt{a} \log \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )+2 i \sqrt{a} \log \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right ) \log \left (\frac{i \sqrt{b} x}{2 \sqrt{a}}+\frac{1}{2}\right )+4 \sqrt{b} x \log \left (b x^2+a\right )-4 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (b x^2+a\right )+2 i \sqrt{a} \text{PolyLog}\left (2,\frac{1}{2}-\frac{i \sqrt{b} x}{2 \sqrt{a}}\right )-2 i \sqrt{a} \text{PolyLog}\left (2,\frac{i \sqrt{b} x}{2 \sqrt{a}}+\frac{1}{2}\right )}{\sqrt{b}}\right ) p^2+\frac{6 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (\log \left (c \left (b x^2+a\right )^p\right )-p \log \left (b x^2+a\right )\right )^2 p}{\sqrt{b}}+3 x \log \left (b x^2+a\right ) \left (\log \left (c \left (b x^2+a\right )^p\right )-p \log \left (b x^2+a\right )\right )^2 p+x \left (\log \left (c \left (b x^2+a\right )^p\right )-p \log \left (b x^2+a\right )\right )^2 \left (-\log \left (b x^2+a\right ) p-6 p+\log \left (c \left (b x^2+a\right )^p\right )\right ) \]

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

[Out]

(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])*(x*Log[a +
 b*x^2]^2 - (-8*Sqrt[b]*x - (4*I)*Sqrt[a]*Log[((-I)*Sqrt[a])/Sqrt[b] + x] + 4*Sqrt[a]*ArcTan[(Sqrt[b]*x)/Sqrt[
a]]*Log[((-I)*Sqrt[a])/Sqrt[b] + x] + I*Sqrt[a]*Log[((-I)*Sqrt[a])/Sqrt[b] + x]^2 + (4*I)*Sqrt[a]*Log[(I*Sqrt[
a])/Sqrt[b] + x] + 4*Sqrt[a]*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[(I*Sqrt[a])/Sqrt[b] + x] - I*Sqrt[a]*Log[(I*Sqrt[
a])/Sqrt[b] + x]^2 - (2*I)*Sqrt[a]*Log[((-I)*Sqrt[a])/Sqrt[b] + x]*Log[1/2 - ((I/2)*Sqrt[b]*x)/Sqrt[a]] + (2*I
)*Sqrt[a]*Log[(I*Sqrt[a])/Sqrt[b] + x]*Log[1/2 + ((I/2)*Sqrt[b]*x)/Sqrt[a]] + 4*Sqrt[b]*x*Log[a + b*x^2] - 4*S
qrt[a]*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[a + b*x^2] + (2*I)*Sqrt[a]*PolyLog[2, 1/2 - ((I/2)*Sqrt[b]*x)/Sqrt[a]]
- (2*I)*Sqrt[a]*PolyLog[2, 1/2 + ((I/2)*Sqrt[b]*x)/Sqrt[a]])/Sqrt[b]) + (p^3*(-48*Sqrt[-a^2]*Sqrt[a + b*x^2]*S
qrt[1 - a/(a + b*x^2)]*ArcSin[Sqrt[a]/Sqrt[a + b*x^2]] - 6*Sqrt[-a^2]*Sqrt[1 - a/(a + b*x^2)]*(8*Sqrt[a]*Hyper
geometricPFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3/2, 3/2}, a/(a + b*x^2)] + 4*Sqrt[a]*HypergeometricPFQ[{1/2, 1/2, 1/
2}, {3/2, 3/2}, a/(a + b*x^2)]*Log[a + b*x^2] + Sqrt[a + b*x^2]*ArcSin[Sqrt[a]/Sqrt[a + b*x^2]]*Log[a + b*x^2]
^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) + 24*a*Sqrt[b*x^2]*ArcT
anh[Sqrt[b*x^2]/Sqrt[-a]]*(Log[a + b*x^2] - Log[(a + b*x^2)/a]) + 6*(-a)^(3/2)*Sqrt[1 - (a + b*x^2)/a]*(Log[(a
 + b*x^2)/a]^2 - 4*Log[(a + b*x^2)/a]*Log[(1 + Sqrt[1 - (a + b*x^2)/a])/2] + 2*Log[(1 + Sqrt[1 - (a + b*x^2)/a
])/2]^2 - 4*PolyLog[2, 1/2 - Sqrt[1 - (a + 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 = 1.91863 (sec), size = 669 ,normalized size = 13.12 \[ \frac{p^3 \left (-96 \sqrt{a} \sqrt{1-\frac{a}{a+b x^2}} \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right \},\left \{\frac{3}{2},\frac{3}{2},\frac{3}{2}\right \},\frac{a}{a+b x^2}\right )-48 \sqrt{a} \sqrt{1-\frac{a}{a+b x^2}} \log \left (a+b x^2\right ) \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right \},\left \{\frac{3}{2},\frac{3}{2}\right \},\frac{a}{a+b x^2}\right )-2 \log ^2\left (a+b x^2\right ) \left (\sqrt{a} \log \left (a+b x^2\right )+6 \sqrt{a+b x^2} \sqrt{1-\frac{a}{a+b x^2}} \sin ^{-1}\left (\frac{\sqrt{a}}{\sqrt{a+b x^2}}\right )\right )\right )}{2 \sqrt{a} x}+3 p^2 \left (\log \left (c \left (a+b x^2\right )^p\right )-p \log \left (a+b x^2\right )\right ) \left (-\frac{\log ^2\left (a+b x^2\right )}{x}-\frac{\sqrt{b} \left (2 i \text{PolyLog}\left (2,\frac{1}{2}-\frac{i \sqrt{b} x}{2 \sqrt{a}}\right )-2 i \text{PolyLog}\left (2,\frac{1}{2}+\frac{i \sqrt{b} x}{2 \sqrt{a}}\right )-4 \log \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )+i \log ^2\left (x-\frac{i \sqrt{a}}{\sqrt{b}}\right )-i \log ^2\left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )-2 i \log \left (\frac{1}{2}-\frac{i \sqrt{b} x}{2 \sqrt{a}}\right ) \log \left (x-\frac{i \sqrt{a}}{\sqrt{b}}\right )+2 i \log \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right ) \log \left (\frac{1}{2}+\frac{i \sqrt{b} x}{2 \sqrt{a}}\right )+4 \log \left (x-\frac{i \sqrt{a}}{\sqrt{b}}\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )+4 \log \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right )}{\sqrt{a}}\right )-\frac{3 p \log \left (a+b x^2\right ) \left (\log \left (c \left (a+b x^2\right )^p\right )-p \log \left (a+b x^2\right )\right )^2}{x}-\frac{\left (\log \left (c \left (a+b x^2\right )^p\right )-p \log \left (a+b x^2\right )\right )^3}{x}+\frac{6 \sqrt{b} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (\log \left (c \left (a+b x^2\right )^p\right )-p \log \left (a+b x^2\right )\right )^2}{\sqrt{a}} \]

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

[Out]

(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)]*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, a/(a + b*x^2)]*Log[a +
 b*x^2] - 2*Log[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[(Sqrt[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*Log[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) - (Sqrt[b]*(4*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[((-I)*Sqrt[a])/Sqrt[b] + x] + I*Log[((-I)*Sqrt[a])/Sqrt[b] +
 x]^2 + 4*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[(I*Sqrt[a])/Sqrt[b] + x] - I*Log[(I*Sqrt[a])/Sqrt[b] + x]^2 - (2*I)*
Log[((-I)*Sqrt[a])/Sqrt[b] + x]*Log[1/2 - ((I/2)*Sqrt[b]*x)/Sqrt[a]] + (2*I)*Log[(I*Sqrt[a])/Sqrt[b] + x]*Log[
1/2 + ((I/2)*Sqrt[b]*x)/Sqrt[a]] - 4*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[a + b*x^2] + (2*I)*PolyLog[2, 1/2 - ((I/2
)*Sqrt[b]*x)/Sqrt[a]] - (2*I)*PolyLog[2, 1/2 + ((I/2)*Sqrt[b]*x)/Sqrt[a]]))/Sqrt[a])

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

[B]   time = 6.69371 (sec), size = 1068 ,normalized size = 4.2 \[ \text{result too large to display} \]

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

[Out]

(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)/Sqrt[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 - b*x^2*(8*Sqrt[b]*x*ArcTan[(Sqrt[b]*x)/Sqrt[a]] + 4*Sqrt[b]*x*ArcTan[(Sq
rt[b]*x)/Sqrt[a]]*Log[((-I)*Sqrt[a])/Sqrt[b] + x] + I*Sqrt[b]*x*Log[((-I)*Sqrt[a])/Sqrt[b] + x]^2 + 4*Sqrt[b]*
x*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[(I*Sqrt[a])/Sqrt[b] + x] - I*Sqrt[b]*x*Log[(I*Sqrt[a])/Sqrt[b] + x]^2 - (2*I
)*Sqrt[b]*x*Log[((-I)*Sqrt[a])/Sqrt[b] + x]*Log[1/2 - ((I/2)*Sqrt[b]*x)/Sqrt[a]] + (2*I)*Sqrt[b]*x*Log[(I*Sqrt
[a])/Sqrt[b] + x]*Log[1/2 + ((I/2)*Sqrt[b]*x)/Sqrt[a]] - 4*Sqrt[a]*Log[a + b*x^2] - 4*Sqrt[b]*x*ArcTan[(Sqrt[b
]*x)/Sqrt[a]]*Log[a + b*x^2] + (2*I)*Sqrt[b]*x*PolyLog[2, 1/2 - ((I/2)*Sqrt[b]*x)/Sqrt[a]] - (2*I)*Sqrt[b]*x*P
olyLog[2, 1/2 + ((I/2)*Sqrt[b]*x)/Sqrt[a]])) + 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)*ArcTanh[Sqrt[b*x^2]/Sqrt[-a]]*L
og[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)*Arc
Sin[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 = 4.36339 (sec), size = 994 ,normalized size = 12.91 \[ \text{result too large to display} \]

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

[Out]

((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)*Hypergeo
metricPFQ[{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*HypergeometricPFQ[{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)*Hyper
geometricPFQ[{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))*Lo
g[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)*Hyperg
eometricPFQ[{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*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)) + x^2*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x
^2)^p])^3 + m*x^2*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^3))/((1 + m)^2*x)

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

[B]   time = 1.59117 (sec), size = 466 ,normalized size = 6.21 \[ \frac{(f x)^m \left (\frac{4 d (m+1) p^2 \left (\frac{e x^2}{d+e x^2}\right )^{\frac{1}{2}-\frac{m}{2}} \left ((m-1) \log \left (d+e x^2\right ) \text{Hypergeometric2F1}\left (\frac{1}{2}-\frac{m}{2},\frac{1}{2}-\frac{m}{2},\frac{3}{2}-\frac{m}{2},\frac{d}{d+e x^2}\right )-2 \text{HypergeometricPFQ}\left (\left \{\frac{1}{2}-\frac{m}{2},\frac{1}{2}-\frac{m}{2},\frac{1}{2}-\frac{m}{2}\right \},\left \{\frac{3}{2}-\frac{m}{2},\frac{3}{2}-\frac{m}{2}\right \},\frac{d}{d+e x^2}\right )\right )}{e (m-1)^2 x}+\frac{2 p \left (p \log \left (d+e x^2\right )-\log \left (c \left (d+e x^2\right )^p\right )\right ) \left (2 e x^3 \text{Hypergeometric2F1}\left (1,\frac{m+3}{2},\frac{m+5}{2},-\frac{e x^2}{d}\right )-d (m+3) x \log \left (d+e x^2\right )\right )}{d (m+3)}-\frac{2 m p \left (p \log \left (d+e x^2\right )-\log \left (c \left (d+e x^2\right )^p\right )\right ) \left (d (m+3) x \log \left (d+e x^2\right )-2 e x^3 \text{Hypergeometric2F1}\left (1,\frac{m+3}{2},\frac{m+5}{2},-\frac{e x^2}{d}\right )\right )}{d (m+3)}+4 p^2 x \left (\frac{2 e x^2 \text{Hypergeometric2F1}\left (1,\frac{m+3}{2},\frac{m+5}{2},-\frac{e x^2}{d}\right )}{d (m+3)}-\log \left (d+e x^2\right )\right )+m x \left (\log \left (c \left (d+e x^2\right )^p\right )-p \log \left (d+e x^2\right )\right )^2+x \left (\log \left (c \left (d+e x^2\right )^p\right )-p \log \left (d+e x^2\right )\right )^2+(m+1) p^2 x \log ^2\left (d+e x^2\right )\right )}{(m+1)^2} \]

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

[Out]

((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*Hypergeometric
PFQ[{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)^2*x) + (2*p*(2*e*x^3*Hypergeometric2
F1[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 = 13.0184 (sec), size = 2295 ,normalized size = 3.36 \[ \text{Result too large to show} \]

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

[Out]

(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)/Sqrt[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]) + Log[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 - (-8*Sqrt[e]*x - (4*I)*Sqrt[d]*Log[((-I)*Sqrt[d])/Sqrt[e] + x] + 4*Sqrt[d]*Ar
cTan[(Sqrt[e]*x)/Sqrt[d]]*Log[((-I)*Sqrt[d])/Sqrt[e] + x] + I*Sqrt[d]*Log[((-I)*Sqrt[d])/Sqrt[e] + x]^2 + (4*I
)*Sqrt[d]*Log[(I*Sqrt[d])/Sqrt[e] + x] + 4*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[(I*Sqrt[d])/Sqrt[e] + x] -
I*Sqrt[d]*Log[(I*Sqrt[d])/Sqrt[e] + x]^2 - (2*I)*Sqrt[d]*Log[((-I)*Sqrt[d])/Sqrt[e] + x]*Log[1/2 - ((I/2)*Sqrt
[e]*x)/Sqrt[d]] + (2*I)*Sqrt[d]*Log[(I*Sqrt[d])/Sqrt[e] + x]*Log[1/2 + ((I/2)*Sqrt[e]*x)/Sqrt[d]] + 4*Sqrt[e]*
x*Log[d + e*x^2] - 4*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[d + e*x^2] + (2*I)*Sqrt[d]*PolyLog[2, 1/2 - ((I/2
)*Sqrt[e]*x)/Sqrt[d]] - (2*I)*Sqrt[d]*PolyLog[2, 1/2 + ((I/2)*Sqrt[e]*x)/Sqrt[d]])/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 - (96*d*Sqrt[e]*x - 8*e^(3/2)*x^3 - 24*d^(3/2)*
ArcTan[(Sqrt[e]*x)/Sqrt[d]] + (36*I)*d^(3/2)*Log[((-I)*Sqrt[d])/Sqrt[e] + x] - 36*d^(3/2)*ArcTan[(Sqrt[e]*x)/S
qrt[d]]*Log[((-I)*Sqrt[d])/Sqrt[e] + x] - (9*I)*d^(3/2)*Log[((-I)*Sqrt[d])/Sqrt[e] + x]^2 - (36*I)*d^(3/2)*Log
[(I*Sqrt[d])/Sqrt[e] + x] - 36*d^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[(I*Sqrt[d])/Sqrt[e] + x] + (9*I)*d^(3/2
)*Log[(I*Sqrt[d])/Sqrt[e] + x]^2 + (18*I)*d^(3/2)*Log[((-I)*Sqrt[d])/Sqrt[e] + x]*Log[1/2 - ((I/2)*Sqrt[e]*x)/
Sqrt[d]] - (18*I)*d^(3/2)*Log[(I*Sqrt[d])/Sqrt[e] + x]*Log[1/2 + ((I/2)*Sqrt[e]*x)/Sqrt[d]] - 36*d*Sqrt[e]*x*L
og[d + e*x^2] + 12*e^(3/2)*x^3*Log[d + e*x^2] + 36*d^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[d + e*x^2] - (18*I)
*d^(3/2)*PolyLog[2, 1/2 - ((I/2)*Sqrt[e]*x)/Sqrt[d]] + (18*I)*d^(3/2)*PolyLog[2, 1/2 + ((I/2)*Sqrt[e]*x)/Sqrt[
d]])/(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, 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]*ArcSin[Sqrt[d]/Sqrt[d + e*x^2]]*Log[d + e*x^2]^2) - (2*Sqrt[-d]*e*x^2*((d + e*x^2)
*(16 - 24*Log[d + e*x^2] + 18*Log[d + e*x^2]^2 - 9*Log[d + e*x^2]^3) + d*(-640 + 312*Log[d + e*x^2] - 72*Log[d
 + e*x^2]^2 + 9*Log[d + e*x^2]^3)))/3 - 48*d^2*(4*Sqrt[e*x^2]*ArcTanh[Sqrt[e*x^2]/Sqrt[-d]]*(Log[d + e*x^2] -
Log[(d + e*x^2)/d]) - Sqrt[-d]*Sqrt[1 - (d + e*x^2)/d]*(Log[(d + e*x^2)/d]^2 - 4*Log[(d + e*x^2)/d]*Log[(1 + S
qrt[1 - (d + e*x^2)/d])/2] + 2*Log[(1 + Sqrt[1 - (d + e*x^2)/d])/2]^2 - 4*PolyLog[2, 1/2 - Sqrt[1 - (d + e*x^2
)/d]/2]))))/(18*Sqrt[-d]*e^2*x) + (f*p^3*(-48*Sqrt[-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]*ArcSin[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]*ArcTanh[Sqrt[e*x^2]/Sqrt[-d]]*(Log[d + e
*x^2] - Log[(d + e*x^2)/d]) + 6*(-d)^(3/2)*Sqrt[1 - (d + e*x^2)/d]*(Log[(d + e*x^2)/d]^2 - 4*Log[(d + e*x^2)/d
]*Log[(1 + Sqrt[1 - (d + e*x^2)/d])/2] + 2*Log[(1 + Sqrt[1 - (d + e*x^2)/d])/2]^2 - 4*PolyLog[2, 1/2 - Sqrt[1
- (d + e*x^2)/d]/2])))/(Sqrt[-d]*e*x)

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 = 14.0542 (sec), size = 2899 ,normalized size = 2.57 \[ \text{Result too large to show} \]

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

[Out]

(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*x^2*(6*d - e*x^2) + (-6*d^2 - 4*d*e*x^2 + 2*e^2*x^4)*Log[d + e*x^2] + 2*d^2*Log[d + e*x^2]^2)/(8*e^2))*(-(
p*Log[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]) + Log[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*Log[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 - (-8*Sqr
t[e]*x - (4*I)*Sqrt[d]*Log[((-I)*Sqrt[d])/Sqrt[e] + x] + 4*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[((-I)*Sqrt[
d])/Sqrt[e] + x] + I*Sqrt[d]*Log[((-I)*Sqrt[d])/Sqrt[e] + x]^2 + (4*I)*Sqrt[d]*Log[(I*Sqrt[d])/Sqrt[e] + x] +
4*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[(I*Sqrt[d])/Sqrt[e] + x] - I*Sqrt[d]*Log[(I*Sqrt[d])/Sqrt[e] + x]^2
- (2*I)*Sqrt[d]*Log[((-I)*Sqrt[d])/Sqrt[e] + x]*Log[1/2 - ((I/2)*Sqrt[e]*x)/Sqrt[d]] + (2*I)*Sqrt[d]*Log[(I*Sq
rt[d])/Sqrt[e] + x]*Log[1/2 + ((I/2)*Sqrt[e]*x)/Sqrt[d]] + 4*Sqrt[e]*x*Log[d + e*x^2] - 4*Sqrt[d]*ArcTan[(Sqrt
[e]*x)/Sqrt[d]]*Log[d + e*x^2] + (2*I)*Sqrt[d]*PolyLog[2, 1/2 - ((I/2)*Sqrt[e]*x)/Sqrt[d]] - (2*I)*Sqrt[d]*Pol
yLog[2, 1/2 + ((I/2)*Sqrt[e]*x)/Sqrt[d]])/Sqrt[e]) + 3*g^2*p^2*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])*((
x^7*Log[d + e*x^2]^2)/7 - (147840*d^3*Sqrt[e]*x - 19880*d^2*e^(3/2)*x^3 + 6048*d*e^(5/2)*x^5 - 1800*e^(7/2)*x^
7 - 59640*d^(7/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + (44100*I)*d^(7/2)*Log[((-I)*Sqrt[d])/Sqrt[e] + x] - 44100*d^(7
/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[((-I)*Sqrt[d])/Sqrt[e] + x] - (11025*I)*d^(7/2)*Log[((-I)*Sqrt[d])/Sqrt[e]
 + x]^2 - (44100*I)*d^(7/2)*Log[(I*Sqrt[d])/Sqrt[e] + x] - 44100*d^(7/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[(I*Sq
rt[d])/Sqrt[e] + x] + (11025*I)*d^(7/2)*Log[(I*Sqrt[d])/Sqrt[e] + x]^2 + (22050*I)*d^(7/2)*Log[((-I)*Sqrt[d])/
Sqrt[e] + x]*Log[1/2 - ((I/2)*Sqrt[e]*x)/Sqrt[d]] - (22050*I)*d^(7/2)*Log[(I*Sqrt[d])/Sqrt[e] + x]*Log[1/2 + (
(I/2)*Sqrt[e]*x)/Sqrt[d]] - 44100*d^3*Sqrt[e]*x*Log[d + e*x^2] + 14700*d^2*e^(3/2)*x^3*Log[d + e*x^2] - 8820*d
*e^(5/2)*x^5*Log[d + e*x^2] + 6300*e^(7/2)*x^7*Log[d + e*x^2] + 44100*d^(7/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[
d + e*x^2] - (22050*I)*d^(7/2)*PolyLog[2, 1/2 - ((I/2)*Sqrt[e]*x)/Sqrt[d]] + (22050*I)*d^(7/2)*PolyLog[2, 1/2
+ ((I/2)*Sqrt[e]*x)/Sqrt[d]])/(77175*e^(7/2))) + (g^2*p^3*(702272*Sqrt[-d]*d^(7/2)*Sqrt[d + e*x^2]*Sqrt[1 - d/
(d + e*x^2)]*ArcSin[Sqrt[d]/Sqrt[d + e*x^2]] + 44100*Sqrt[-d]*d^(7/2)*Sqrt[1 - d/(d + e*x^2)]*(8*Sqrt[d]*Hyper
geometricPFQ[{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]*ArcSin[Sqrt[d]/Sqrt[d + e*x^2]]*Log[d + e*x^2]
^2) - (2*Sqrt[-d]*e*x^2*(-1125*(d + e*x^2)^3*(-48 + 168*Log[d + e*x^2] - 294*Log[d + e*x^2]^2 + 343*Log[d + e*
x^2]^3) + 27*d*(d + e*x^2)^2*(-18208 + 44520*Log[d + e*x^2] - 53900*Log[d + e*x^2]^2 + 42875*Log[d + e*x^2]^3)
 + d^3*(-39193856 + 18434640*Log[d + e*x^2] - 3880800*Log[d + e*x^2]^2 + 385875*Log[d + e*x^2]^3) - d^2*(d + e
*x^2)*(-2762192 + 3924480*Log[d + e*x^2] - 2690100*Log[d + e*x^2]^2 + 1157625*Log[d + e*x^2]^3)))/105 - 73920*
d^4*(4*Sqrt[e*x^2]*ArcTanh[Sqrt[e*x^2]/Sqrt[-d]]*(Log[d + e*x^2] - Log[(d + e*x^2)/d]) - Sqrt[-d]*Sqrt[1 - (d
+ e*x^2)/d]*(Log[(d + e*x^2)/d]^2 - 4*Log[(d + e*x^2)/d]*Log[(1 + Sqrt[1 - (d + e*x^2)/d])/2] + 2*Log[(1 + Sqr
t[1 - (d + e*x^2)/d])/2]^2 - 4*PolyLog[2, 1/2 - Sqrt[1 - (d + e*x^2)/d]/2]))))/(51450*Sqrt[-d]*e^4*x) + (f^2*p
^3*(-48*Sqrt[-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*Sqr
t[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]*ArcSin[Sqr
t[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]*ArcTanh[Sqrt[e*x^2]/Sqrt[-d]]*(Log[d + e*x^2] - Log[(d + e*x^2)/d]) + 6*(-d)^(
3/2)*Sqrt[1 - (d + e*x^2)/d]*(Log[(d + e*x^2)/d]^2 - 4*Log[(d + e*x^2)/d]*Log[(1 + Sqrt[1 - (d + e*x^2)/d])/2]
 + 2*Log[(1 + Sqrt[1 - (d + e*x^2)/d])/2]^2 - 4*PolyLog[2, 1/2 - Sqrt[1 - (d + e*x^2)/d]/2])))/(Sqrt[-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 = 6.0612 (sec), size = 1590 ,normalized size = 3.07 \[ \text{result too large to display} \]

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

[Out]

(g*(-(d^2*p^3) + e^2*p^3*x^4)*Log[d + e*x^2]^3)/(4*e^2) + (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^2*Log[d + e*x^2]^2*(3*d^2*p + 2*d*e*p*x^2 - e^2*p*x^4 - 2*d^2*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^
2)^p]) + 2*e^2*x^4*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])))/(8*e^2) + (3*d*g*p*x^2*(7*p^2 - 6*p*(-(p*Log
[d + e*x^2]) + Log[c*(d + e*x^2)^p]) + 2*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2))/(8*e) - (3*d^2*g*p*L
og[d + e*x^2]*(7*p^2 - 6*p*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p]) + 2*(-(p*Log[d + e*x^2]) + Log[c*(d +
e*x^2)^p])^2))/(8*e^2) + (3*g*p*x^2*Log[d + e*x^2]*(-6*d*p^2 + e*p^2*x^2 + 4*d*p*(-(p*Log[d + e*x^2]) + Log[c*
(d + e*x^2)^p]) - 2*e*p*x^2*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p]) + 2*e*x^2*(-(p*Log[d + e*x^2]) + Log[
c*(d + e*x^2)^p])^2))/(8*e) + (g*x^4*(-3*p^3 + 6*p^2*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p]) - 6*p*(-(p*L
og[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2 + 4*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^3))/16 + 3*f*p^2*(-(
p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])*(x*Log[d + e*x^2]^2 - (-8*Sqrt[e]*x - (4*I)*Sqrt[d]*Log[((-I)*Sqrt[d
])/Sqrt[e] + x] + 4*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[((-I)*Sqrt[d])/Sqrt[e] + x] + I*Sqrt[d]*Log[((-I)*
Sqrt[d])/Sqrt[e] + x]^2 + (4*I)*Sqrt[d]*Log[(I*Sqrt[d])/Sqrt[e] + x] + 4*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*L
og[(I*Sqrt[d])/Sqrt[e] + x] - I*Sqrt[d]*Log[(I*Sqrt[d])/Sqrt[e] + x]^2 - (2*I)*Sqrt[d]*Log[((-I)*Sqrt[d])/Sqrt
[e] + x]*Log[1/2 - ((I/2)*Sqrt[e]*x)/Sqrt[d]] + (2*I)*Sqrt[d]*Log[(I*Sqrt[d])/Sqrt[e] + x]*Log[1/2 + ((I/2)*Sq
rt[e]*x)/Sqrt[d]] + 4*Sqrt[e]*x*Log[d + e*x^2] - 4*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[d + e*x^2] + (2*I)*
Sqrt[d]*PolyLog[2, 1/2 - ((I/2)*Sqrt[e]*x)/Sqrt[d]] - (2*I)*Sqrt[d]*PolyLog[2, 1/2 + ((I/2)*Sqrt[e]*x)/Sqrt[d]
])/Sqrt[e]) + (f*p^3*(-48*Sqrt[-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]*ArcSin[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]*ArcTanh[Sqrt[e*x^2]/Sqrt[-d]]*(Log[d + e*x^2] - Log[(d + e*x
^2)/d]) + 6*(-d)^(3/2)*Sqrt[1 - (d + e*x^2)/d]*(Log[(d + e*x^2)/d]^2 - 4*Log[(d + e*x^2)/d]*Log[(1 + Sqrt[1 -
(d + e*x^2)/d])/2] + 2*Log[(1 + Sqrt[1 - (d + e*x^2)/d])/2]^2 - 4*PolyLog[2, 1/2 - Sqrt[1 - (d + e*x^2)/d]/2])
))/(Sqrt[-d]*e*x)

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

[B]   time = 18.2793 (sec), size = 1744 ,normalized size = 2.2 \[ \text{result too large to display} \]

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

[Out]

(-2*b*d^4*n*x^(1/3)*(a + b*(-(n*Log[d + e*x^(2/3)]) + 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)]) + 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)]) + 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)]) + Log[c*(d
+ e*x^(2/3))^n]))^2)/(7*e) + (2*b*d^(9/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(a + b*(-(n*Log[d + e*x^(2/3)])
+ 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)]) + Log[c*(
d + e*x^(2/3))^n]))^2 + (x^3*(a + b*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n]))^2*(3*a - 2*b*n + 3*b
*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n])))/9 + (3*b^3*n^3*((-6951008*d^(9/2)*Sqrt[1 - d/(d + e*x^
(2/3))]*Sqrt[d + e*x^(2/3)]*ArcSin[Sqrt[d]/Sqrt[d + e*x^(2/3)]])/(297675*e*x^(1/3)) + (2*d^5*Sqrt[1 - d/(d + e
*x^(2/3))]*(-16*HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3/2, 3/2}, d/(d + e*x^(2/3))] - 8*Hypergeometric
PFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, d/(d + e*x^(2/3))]*Log[d + e*x^(2/3)] - (2*Sqrt[d + e*x^(2/3)]*ArcSin[Sqrt[d]
/Sqrt[d + e*x^(2/3)]]*Log[d + e*x^(2/3)]^2)/Sqrt[d]))/(3*e*x^(1/3)) + x^(1/3)*((2*(d + e*x^(2/3))^4*(-16 + 72*
Log[d + e*x^(2/3)] - 162*Log[d + e*x^(2/3)]^2 + 243*Log[d + e*x^(2/3)]^3))/2187 - (4*d*(d + e*x^(2/3))^3*(-248
72 + 85680*Log[d + e*x^(2/3)] - 146853*Log[d + e*x^(2/3)]^2 + 166698*Log[d + e*x^(2/3)]^3))/750141 + (2*d^4*(-
1179876752 + 547391880*Log[d + e*x^(2/3)] - 111727350*Log[d + e*x^(2/3)]^2 + 10418625*Log[d + e*x^(2/3)]^3))/9
3767625 - (8*d^3*(d + e*x^(2/3))*(-27010916 + 37647540*Log[d + e*x^(2/3)] - 25103925*Log[d + e*x^(2/3)]^2 + 10
418625*Log[d + e*x^(2/3)]^3))/93767625 + (4*d^2*(d + e*x^(2/3))^2*(-4747112 + 11406780*Log[d + e*x^(2/3)] - 13
494600*Log[d + e*x^(2/3)]^2 + 10418625*Log[d + e*x^(2/3)]^3))/31255875) - (2252*d^5*x^(1/3)*((-4*ArcTanh[Sqrt[
e*x^(2/3)]/Sqrt[-d]]*(Log[d + e*x^(2/3)] - Log[(d + e*x^(2/3))/d]))/Sqrt[-d] + (Sqrt[1 - (d + e*x^(2/3))/d]*(2
*Log[(1 + Sqrt[1 - (d + e*x^(2/3))/d])/2]^2 - 4*Log[(1 + Sqrt[1 - (d + e*x^(2/3))/d])/2]*Log[(d + e*x^(2/3))/d
] + Log[(d + e*x^(2/3))/d]^2 - 4*PolyLog[2, 1 + (-1 - Sqrt[1 - (d + e*x^(2/3))/d])/2]))/Sqrt[e*x^(2/3)]))/(945
*Sqrt[e*x^(2/3)])))/(2*e^4) + (9*b^2*n^2*(a + b*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n]))*((9008*d
^(9/2)*Sqrt[1 - d/(d + e*x^(2/3))]*Sqrt[d + e*x^(2/3)]*ArcSin[Sqrt[d]/Sqrt[d + e*x^(2/3)]])/(2835*e*x^(1/3)) +
 x^(1/3)*((2*(d + e*x^(2/3))^4*(8 - 36*Log[d + e*x^(2/3)] + 81*Log[d + e*x^(2/3)]^2))/729 - (8*d*(d + e*x^(2/3
))^3*(680 - 2331*Log[d + e*x^(2/3)] + 3969*Log[d + e*x^(2/3)]^2))/35721 + (2*d^4*(1737752 - 709380*Log[d + e*x
^(2/3)] + 99225*Log[d + e*x^(2/3)]^2))/893025 - (8*d^3*(d + e*x^(2/3))*(119516 - 159390*Log[d + e*x^(2/3)] + 9
9225*Log[d + e*x^(2/3)]^2))/893025 + (4*d^2*(d + e*x^(2/3))^2*(36212 - 85680*Log[d + e*x^(2/3)] + 99225*Log[d
+ e*x^(2/3)]^2))/297675) + (2*d^5*x^(1/3)*((-4*ArcTanh[Sqrt[e*x^(2/3)]/Sqrt[-d]]*(Log[d + e*x^(2/3)] - Log[(d
+ e*x^(2/3))/d]))/Sqrt[-d] + (Sqrt[1 - (d + e*x^(2/3))/d]*(2*Log[(1 + Sqrt[1 - (d + e*x^(2/3))/d])/2]^2 - 4*Lo
g[(1 + Sqrt[1 - (d + e*x^(2/3))/d])/2]*Log[(d + e*x^(2/3))/d] + Log[(d + e*x^(2/3))/d]^2 - 4*PolyLog[2, 1 + (-
1 - Sqrt[1 - (d + e*x^(2/3))/d])/2]))/Sqrt[e*x^(2/3)]))/(9*Sqrt[e*x^(2/3)])))/(2*e^4)

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

[B]   time = 16.0876 (sec), size = 1243 ,normalized size = 2.56 \[ \text{result too large to display} \]

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

[Out]

(6*b*d*n*x^(1/3)*(a + b*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n]))^2)/e - (6*b*d^(3/2)*n*ArcTan[(Sq
rt[e]*x^(1/3))/Sqrt[d]]*(a + b*(-(n*Log[d + e*x^(2/3)]) + 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)]) + Log[c*(d + e*x^(2/3))^n]))^2 + x*(a + b*(-(n*Log[d + e*x^(2/3
)]) + Log[c*(d + e*x^(2/3))^n]))^2*(a - 2*b*n + b*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n])) + (3*b
^3*n^3*((416*d^(3/2)*Sqrt[1 - d/(d + e*x^(2/3))]*Sqrt[d + e*x^(2/3)]*ArcSin[Sqrt[d]/Sqrt[d + e*x^(2/3)]])/(9*e
*x^(1/3)) - (2*d^2*Sqrt[1 - d/(d + e*x^(2/3))]*(-16*HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3/2, 3/2}, d
/(d + e*x^(2/3))] - 8*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, d/(d + e*x^(2/3))]*Log[d + e*x^(2/3)] - (
2*Sqrt[d + e*x^(2/3)]*ArcSin[Sqrt[d]/Sqrt[d + e*x^(2/3)]]*Log[d + e*x^(2/3)]^2)/Sqrt[d]))/(e*x^(1/3)) + x^(1/3
)*((-2*d*(-640 + 312*Log[d + e*x^(2/3)] - 72*Log[d + e*x^(2/3)]^2 + 9*Log[d + e*x^(2/3)]^3))/27 + (2*(d + e*x^
(2/3))*(-16 + 24*Log[d + e*x^(2/3)] - 18*Log[d + e*x^(2/3)]^2 + 9*Log[d + e*x^(2/3)]^3))/27) + (16*d^2*x^(1/3)
*((-4*ArcTanh[Sqrt[e*x^(2/3)]/Sqrt[-d]]*(Log[d + e*x^(2/3)] - Log[(d + e*x^(2/3))/d]))/Sqrt[-d] + (Sqrt[1 - (d
 + e*x^(2/3))/d]*(2*Log[(1 + Sqrt[1 - (d + e*x^(2/3))/d])/2]^2 - 4*Log[(1 + Sqrt[1 - (d + e*x^(2/3))/d])/2]*Lo
g[(d + e*x^(2/3))/d] + Log[(d + e*x^(2/3))/d]^2 - 4*PolyLog[2, 1 + (-1 - Sqrt[1 - (d + e*x^(2/3))/d])/2]))/Sqr
t[e*x^(2/3)]))/(3*Sqrt[e*x^(2/3)])))/(2*e) + (b^2*n^2*x^(1/3)*(a + b*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x
^(2/3))^n]))*((-96*d^(3/2)*Sqrt[1 - d/(d + e*x^(2/3))]*Sqrt[d + e*x^(2/3)]*ArcSin[Sqrt[d]/Sqrt[d + e*x^(2/3)]]
)/(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
[(d + e*x^(2/3))/d]))/Sqrt[e*x^(2/3)] - (9*d^2*Sqrt[1 - (d + e*x^(2/3))/d]*(2*Log[(1 + Sqrt[1 - (d + e*x^(2/3)
)/d])/2]^2 - 4*Log[(1 + Sqrt[1 - (d + e*x^(2/3))/d])/2]*Log[(d + e*x^(2/3))/d] + Log[(d + e*x^(2/3))/d]^2 - 4*
PolyLog[2, 1/2 - Sqrt[1 - (d + e*x^(2/3))/d]/2]))/(e*x^(2/3))))/(3*e)

Integral number [489] \[ \int \frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^2} \, dx \]

[B]   time = 14.7702 (sec), size = 1088 ,normalized size = 3.41 \[ \text{result too large to display} \]

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

[Out]

(-6*b*e^(3/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(a + b*(-(n*Log[d + e*x^(2/3)]) + 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)]) + Log[c*(d + e*x^(2/3))^n]))^2)/x - (a
 + b*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n]))^3/x - (6*(a^2*b*e*n + 2*a*b^2*e*n*(-(n*Log[d + e*x^
(2/3)]) + Log[c*(d + e*x^(2/3))^n]) + b^3*e*n*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n])^2))/(d*x^(1
/3)) + (3*b^3*e*n^3*((-2*Sqrt[1 - d/(d + e*x^(2/3))]*(-16*HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3/2, 3
/2}, d/(d + e*x^(2/3))] - 8*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, d/(d + e*x^(2/3))]*Log[d + e*x^(2/3
)] - (2*Sqrt[d + e*x^(2/3)]*ArcSin[Sqrt[d]/Sqrt[d + e*x^(2/3)]]*Log[d + e*x^(2/3)]^2)/Sqrt[d]))/(d*x^(1/3)) +
e*x^(1/3)*((-4*Log[d + e*x^(2/3)]^2)/(d*e*x^(2/3)) - (2*Log[d + e*x^(2/3)]^3)/(3*e^2*x^(4/3))) + (4*Sqrt[e*x^(
2/3)]*((-4*ArcTanh[Sqrt[e*x^(2/3)]/Sqrt[-d]]*(Log[d + e*x^(2/3)] - Log[(d + e*x^(2/3))/d]))/Sqrt[-d] + (Sqrt[1
 - (d + e*x^(2/3))/d]*(2*Log[(1 + Sqrt[1 - (d + e*x^(2/3))/d])/2]^2 - 4*Log[(1 + Sqrt[1 - (d + e*x^(2/3))/d])/
2]*Log[(d + e*x^(2/3))/d] + Log[(d + e*x^(2/3))/d]^2 - 4*PolyLog[2, 1 + (-1 - Sqrt[1 - (d + e*x^(2/3))/d])/2])
)/Sqrt[e*x^(2/3)]))/(d*x^(1/3))))/2 + (3*b^2*e*n^2*(a + b*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n])
)*((-16*Sqrt[1 - d/(d + e*x^(2/3))]*Sqrt[d + e*x^(2/3)]*ArcSin[Sqrt[d]/Sqrt[d + e*x^(2/3)]])/d^(3/2) - (2*(4*(
d + e*x^(2/3)) + d*(-4 + Log[d + e*x^(2/3)]))*Log[d + e*x^(2/3)])/(d*e*x^(2/3)) - (8*Sqrt[e*x^(2/3)]*ArcTanh[S
qrt[e*x^(2/3)]/Sqrt[-d]]*(Log[d + e*x^(2/3)] - Log[(d + e*x^(2/3))/d]))/(-d)^(3/2) - (2*Sqrt[1 - (d + e*x^(2/3
))/d]*(2*Log[(1 + Sqrt[1 - (d + e*x^(2/3))/d])/2]^2 - 4*Log[(1 + Sqrt[1 - (d + e*x^(2/3))/d])/2]*Log[(d + e*x^
(2/3))/d] + Log[(d + e*x^(2/3))/d]^2 - 4*PolyLog[2, 1/2 - Sqrt[1 - (d + e*x^(2/3))/d]/2]))/d))/(2*x^(1/3))

Integral number [490] \[ \int \frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^4} \, dx \]

[B]   time = 12.8392 (sec), size = 1633 ,normalized size = 2.58 \[ \text{result too large to display} \]

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

[Out]

(2*b*e^(9/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(a + b*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n]))^
2)/d^(9/2) - (b*n*Log[d + e*x^(2/3)]*(a + b*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n]))^2)/x^3 - (a
+ b*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n]))^3/(3*x^3) + (2*e^2*(a^2*b*n + 2*a*b^2*n*(-(n*Log[d +
 e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n]) + b^3*n*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n])^2))/(5*d
^2*x^(5/3)) - (2*e^3*(a^2*b*n + 2*a*b^2*n*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n]) + b^3*n*(-(n*Lo
g[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n])^2))/(3*d^3*x) + (2*e^4*(a^2*b*n + 2*a*b^2*n*(-(n*Log[d + e*x^(2/
3)]) + Log[c*(d + e*x^(2/3))^n]) + b^3*n*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n])^2))/(d^4*x^(1/3)
) - (2*(a^2*b*e*n + 2*a*b^2*e*n*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n]) + b^3*e*n*(-(n*Log[d + e*
x^(2/3)]) + Log[c*(d + e*x^(2/3))^n])^2))/(7*d*x^(7/3)) - (b^3*n^3*(1376*e^4*Sqrt[1 - d/(d + e*x^(2/3))]*Sqrt[
d + e*x^(2/3)]*x^(8/3)*ArcSin[Sqrt[d]/Sqrt[d + e*x^(2/3)]] + 210*e^4*Sqrt[1 - d/(d + e*x^(2/3))]*x^(8/3)*(8*Sq
rt[d]*HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3/2, 3/2}, d/(d + e*x^(2/3))] + 4*Sqrt[d]*HypergeometricPF
Q[{1/2, 1/2, 1/2}, {3/2, 3/2}, d/(d + e*x^(2/3))]*Log[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)]^2) + Sqrt[d]*(6*d^2*e^2*x^(4/3)*(4 - 7*Log[d + e*x^(2/3)])*Log[d + e*x^(2/
3)] + 30*d^3*e*x^(2/3)*Log[d + e*x^(2/3)]^2 + 35*d^4*Log[d + e*x^(2/3)]^3 + 2*d*e^3*x^2*(8 - 48*Log[d + e*x^(2
/3)] + 35*Log[d + e*x^(2/3)]^2) - 2*e^4*x^(8/3)*(120 - 284*Log[d + e*x^(2/3)] + 105*Log[d + e*x^(2/3)]^2)) + (
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[(d + e*
x^(2/3))/d]) - Sqrt[-d]*Sqrt[1 - (d + e*x^(2/3))/d]*(2*Log[(1 + Sqrt[1 - (d + e*x^(2/3))/d])/2]^2 - 4*Log[(1 +
 Sqrt[1 - (d + e*x^(2/3))/d])/2]*Log[(d + e*x^(2/3))/d] + Log[(d + e*x^(2/3))/d]^2 - 4*PolyLog[2, 1/2 - Sqrt[1
 - (d + e*x^(2/3))/d]/2])))/(-d)^(3/2)))/(105*d^(9/2)*x^3) + (b^2*e^5*n^2*x^(1/3)*(a + b*(-(n*Log[d + e*x^(2/3
)]) + Log[c*(d + e*x^(2/3))^n]))*((2816*Sqrt[1 - d/(d + e*x^(2/3))]*Sqrt[d + e*x^(2/3)]*ArcSin[Sqrt[d]/Sqrt[d
+ e*x^(2/3)]])/(d^(9/2)*e*x^(2/3)) - (120*Log[d + e*x^(2/3)])/(d*e^4*x^(8/3)) - (210*Log[d + e*x^(2/3)]^2)/(e^
5*x^(10/3)) + (24*(-2 + 7*Log[d + e*x^(2/3)]))/(d^2*e^3*x^2) - (8*(-24 + 35*Log[d + e*x^(2/3)]))/(d^3*e^2*x^(4
/3)) + (8*(-142 + 105*Log[d + e*x^(2/3)]))/(d^4*e*x^(2/3)) - (840*ArcTanh[Sqrt[e*x^(2/3)]/Sqrt[-d]]*(Log[d + e
*x^(2/3)] - Log[(d + e*x^(2/3))/d]))/((-d)^(9/2)*Sqrt[e*x^(2/3)]) + (210*Sqrt[1 - (d + e*x^(2/3))/d]*(2*Log[(1
 + Sqrt[1 - (d + e*x^(2/3))/d])/2]^2 - 4*Log[(1 + Sqrt[1 - (d + e*x^(2/3))/d])/2]*Log[(d + e*x^(2/3))/d] + Log
[(d + e*x^(2/3))/d]^2 - 4*PolyLog[2, 1/2 - Sqrt[1 - (d + e*x^(2/3))/d]/2]))/(d^4*e*x^(2/3))))/210

Integral number [491] \[ \int \frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^6} \, dx \]

[B]   time = 17.8633 (sec), size = 2147 ,normalized size = 2.34 \[ \text{Result too large to show} \]

[In]  Integrate[(a + b*Log[c*(d + e*x^(2/3))^n])^3/x^6,x]

[Out]

(-6*b*e^(15/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(a + b*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n])
)^2)/(5*d^(15/2)) - (3*b*n*Log[d + e*x^(2/3)]*(a + b*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n]))^2)/
(5*x^5) - (a + b*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n]))^3/(5*x^5) + (6*e^2*(a^2*b*n + 2*a*b^2*n
*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n]) + b^3*n*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))
^n])^2))/(55*d^2*x^(11/3)) - (2*e^3*(a^2*b*n + 2*a*b^2*n*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n])
+ b^3*n*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n])^2))/(15*d^3*x^3) + (6*e^4*(a^2*b*n + 2*a*b^2*n*(-
(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n]) + b^3*n*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n]
)^2))/(35*d^4*x^(7/3)) - (6*e^5*(a^2*b*n + 2*a*b^2*n*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n]) + b^
3*n*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n])^2))/(25*d^5*x^(5/3)) + (2*e^6*(a^2*b*n + 2*a*b^2*n*(-
(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n]) + b^3*n*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n]
)^2))/(5*d^6*x) - (6*e^7*(a^2*b*n + 2*a*b^2*n*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n]) + b^3*n*(-(
n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n])^2))/(5*d^7*x^(1/3)) - (6*(a^2*b*e*n + 2*a*b^2*e*n*(-(n*Log[d
 + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n]) + b^3*e*n*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n])^2))/
(65*d*x^(13/3)) + (9*b^2*e^8*n^2*(a + b*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n]))*((-1409104*Sqrt[
1 - d/(d + e*x^(2/3))]*Sqrt[d + e*x^(2/3)]*ArcSin[Sqrt[d]/Sqrt[d + e*x^(2/3)]])/(675675*d^(15/2)*e*x^(1/3)) +
x^(1/3)*((-8*Log[d + e*x^(2/3)])/(195*d*e^7*x^(14/3)) - (2*Log[d + e*x^(2/3)]^2)/(15*e^8*x^(16/3)) + (8*(-2 +
13*Log[d + e*x^(2/3)]))/(2145*d^2*e^6*x^4) - (8*(-48 + 143*Log[d + e*x^(2/3)]))/(19305*d^3*e^5*x^(10/3)) + (8*
(-718 + 1287*Log[d + e*x^(2/3)]))/(135135*d^4*e^4*x^(8/3)) - (8*(-7600 + 9009*Log[d + e*x^(2/3)]))/(675675*d^5
*e^3*x^2) - (8*(-86048 + 45045*Log[d + e*x^(2/3)]))/(675675*d^7*e*x^(2/3)) + (8*(-56018 + 45045*Log[d + e*x^(2
/3)]))/(2027025*d^6*e^2*x^(4/3))) - (2*x^(1/3)*((-4*ArcTanh[Sqrt[e*x^(2/3)]/Sqrt[-d]]*(Log[d + e*x^(2/3)] - Lo
g[(d + e*x^(2/3))/d]))/Sqrt[-d] + (Sqrt[1 - (d + e*x^(2/3))/d]*(2*Log[(1 + Sqrt[1 - (d + e*x^(2/3))/d])/2]^2 -
 4*Log[(1 + Sqrt[1 - (d + e*x^(2/3))/d])/2]*Log[(d + e*x^(2/3))/d] + Log[(d + e*x^(2/3))/d]^2 - 4*PolyLog[2, 1
 + (-1 - Sqrt[1 - (d + e*x^(2/3))/d])/2]))/Sqrt[e*x^(2/3)]))/(15*d^7*Sqrt[e*x^(2/3)])))/2 + (b^3*n^3*(2837296*
e^7*Sqrt[1 - d/(d + e*x^(2/3))]*Sqrt[d + e*x^(2/3)]*x^(14/3)*ArcSin[Sqrt[d]/Sqrt[d + e*x^(2/3)]] + 270270*e^7*
Sqrt[1 - d/(d + e*x^(2/3))]*x^(14/3)*(8*Sqrt[d]*HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3/2, 3/2}, d/(d
+ e*x^(2/3))] + 4*Sqrt[d]*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, d/(d + e*x^(2/3))]*Log[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)]^2) + Sqrt[d]*(-20790*d^6*e*x^(2/
3)*Log[d + e*x^(2/3)]^2 - 45045*d^7*Log[d + e*x^(2/3)]^3 + 1890*d^5*e^2*x^(4/3)*Log[d + e*x^(2/3)]*(-4 + 13*Lo
g[d + e*x^(2/3)]) - 210*d^4*e^3*x^2*(8 - 96*Log[d + e*x^(2/3)] + 143*Log[d + e*x^(2/3)]^2) + 30*d^3*e^4*x^(8/3
)*(264 - 1436*Log[d + e*x^(2/3)] + 1287*Log[d + e*x^(2/3)]^2) - 6*d^2*e^5*x^(10/3)*(4720 - 15200*Log[d + e*x^(
2/3)] + 9009*Log[d + e*x^(2/3)]^2) + 2*d*e^6*x^4*(54000 - 112036*Log[d + e*x^(2/3)] + 45045*Log[d + e*x^(2/3)]
^2) - 2*e^7*x^(14/3)*(386072 - 516288*Log[d + e*x^(2/3)] + 135135*Log[d + e*x^(2/3)]^2)) - (528414*Sqrt[d]*e^7
*x^(14/3)*(4*Sqrt[e*x^(2/3)]*ArcTanh[Sqrt[e*x^(2/3)]/Sqrt[-d]]*(Log[d + e*x^(2/3)] - Log[(d + e*x^(2/3))/d]) -
 Sqrt[-d]*Sqrt[1 - (d + e*x^(2/3))/d]*(2*Log[(1 + Sqrt[1 - (d + e*x^(2/3))/d])/2]^2 - 4*Log[(1 + Sqrt[1 - (d +
 e*x^(2/3))/d])/2]*Log[(d + e*x^(2/3))/d] + Log[(d + e*x^(2/3))/d]^2 - 4*PolyLog[2, 1/2 - Sqrt[1 - (d + e*x^(2
/3))/d]/2])))/Sqrt[-d]))/(225225*d^(15/2)*x^5)

Integral number [532] \[ \int x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3 \, dx \]

[A]   time = 12.7899 (sec), size = 1464 ,normalized size = 1.15 \[ \text{result too large to display} \]

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

[Out]

(-2*b*e^4*n*x^(1/3)*(a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^(2/3))^n]))^2)/d^4 + (2*b*e^3*n*x*(a + b*
(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^(2/3))^n]))^2)/(3*d^3) - (2*b*e^2*n*x^(5/3)*(a + b*(-(n*Log[d + e/x^
(2/3)]) + Log[c*(d + e/x^(2/3))^n]))^2)/(5*d^2) + (2*b*e*n*x^(7/3)*(a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d
+ e/x^(2/3))^n]))^2)/(7*d) + (2*b*e^(9/2)*n*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*(a + b*(-(n*Log[d + e/x^(2/3)])
+ Log[c*(d + e/x^(2/3))^n]))^2)/d^(9/2) + b*n*x^3*Log[d + e/x^(2/3)]*(a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(
d + e/x^(2/3))^n]))^2 + (x^3*(a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^(2/3))^n]))^3)/3 + (b^3*n^3*x^3*
((1376*e^4*Sqrt[1 - d/(d + e/x^(2/3))]*Sqrt[d + e/x^(2/3)]*ArcSin[Sqrt[d]/Sqrt[d + e/x^(2/3)]])/x^(8/3) + (210
*e^4*Sqrt[1 - d/(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))] + 4*Sqrt[d]*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, d/(d + e/x^(2/3))]*Log[d + e/x^(2/3)] + S
qrt[d + e/x^(2/3)]*ArcSin[Sqrt[d]/Sqrt[d + e/x^(2/3)]]*Log[d + e/x^(2/3)]^2))/x^(8/3) + Sqrt[d]*((6*d^2*e^2*(4
 - 7*Log[d + e/x^(2/3)])*Log[d + e/x^(2/3)])/x^(4/3) + (30*d^3*e*Log[d + e/x^(2/3)]^2)/x^(2/3) + 35*d^4*Log[d
+ e/x^(2/3)]^3 + (2*d*e^3*(8 - 48*Log[d + e/x^(2/3)] + 35*Log[d + e/x^(2/3)]^2))/x^2 - (2*e^4*(120 - 284*Log[d
 + e/x^(2/3)] + 105*Log[d + e/x^(2/3)]^2))/x^(8/3)) + (352*d^(3/2)*e^4*(4*Sqrt[e/x^(2/3)]*ArcTanh[Sqrt[e/x^(2/
3)]/Sqrt[-d]]*(Log[d + e/x^(2/3)] - Log[(d + e/x^(2/3))/d]) - Sqrt[-d]*Sqrt[1 - (d + e/x^(2/3))/d]*(2*Log[(1 +
 Sqrt[1 - (d + e/x^(2/3))/d])/2]^2 - 4*Log[(1 + Sqrt[1 - (d + e/x^(2/3))/d])/2]*Log[(d + e/x^(2/3))/d] + Log[(
d + e/x^(2/3))/d]^2 - 4*PolyLog[2, 1/2 - Sqrt[1 - (d + e/x^(2/3))/d]/2])))/((-d)^(3/2)*x^(8/3))))/(105*d^(9/2)
) - (b^2*e^5*n^2*(a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^(2/3))^n]))*((2816*Sqrt[1 - d/(d + e/x^(2/3)
)]*Sqrt[d + e/x^(2/3)]*x^(2/3)*ArcSin[Sqrt[d]/Sqrt[d + e/x^(2/3)]])/(d^(9/2)*e) - (120*x^(8/3)*Log[d + e/x^(2/
3)])/(d*e^4) - (210*x^(10/3)*Log[d + e/x^(2/3)]^2)/e^5 + (24*x^2*(-2 + 7*Log[d + e/x^(2/3)]))/(d^2*e^3) - (8*x
^(4/3)*(-24 + 35*Log[d + e/x^(2/3)]))/(d^3*e^2) + (8*x^(2/3)*(-142 + 105*Log[d + e/x^(2/3)]))/(d^4*e) - (840*A
rcTanh[Sqrt[e/x^(2/3)]/Sqrt[-d]]*(Log[d + e/x^(2/3)] - Log[(d + e/x^(2/3))/d]))/((-d)^(9/2)*Sqrt[e/x^(2/3)]) +
 (210*Sqrt[1 - (d + e/x^(2/3))/d]*x^(2/3)*(2*Log[(1 + Sqrt[1 - (d + e/x^(2/3))/d])/2]^2 - 4*Log[(1 + Sqrt[1 -
(d + e/x^(2/3))/d])/2]*Log[(d + e/x^(2/3))/d] + Log[(d + e/x^(2/3))/d]^2 - 4*PolyLog[2, 1/2 - Sqrt[1 - (d + e/
x^(2/3))/d]/2]))/(d^4*e)))/(210*x^(1/3))

Integral number [533] \[ \int \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3 \, dx \]

[A]   time = 14.7652 (sec), size = 1040 ,normalized size = 1.41 \[ \text{result too large to display} \]

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

[Out]

(6*b*e*n*x^(1/3)*(a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^(2/3))^n]))^2)/d - (6*b*e^(3/2)*n*ArcTan[(Sq
rt[d]*x^(1/3))/Sqrt[e]]*(a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^(2/3))^n]))^2)/d^(3/2) + 3*b*n*x*Log[
d + e/x^(2/3)]*(a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^(2/3))^n]))^2 + x*(a + b*(-(n*Log[d + e/x^(2/3
)]) + Log[c*(d + e/x^(2/3))^n]))^3 - (3*b^3*e*n^3*((-2*Sqrt[1 - d/(d + e/x^(2/3))]*x^(1/3)*(-16*Hypergeometric
PFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3/2, 3/2}, d/(d + e/x^(2/3))] - 8*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2
}, d/(d + e/x^(2/3))]*Log[d + e/x^(2/3)] - (2*Sqrt[d + e/x^(2/3)]*ArcSin[Sqrt[d]/Sqrt[d + e/x^(2/3)]]*Log[d +
e/x^(2/3)]^2)/Sqrt[d]))/d + (e*((-4*x^(2/3)*Log[d + e/x^(2/3)]^2)/(d*e) - (2*x^(4/3)*Log[d + e/x^(2/3)]^3)/(3*
e^2)))/x^(1/3) + (4*Sqrt[e/x^(2/3)]*x^(1/3)*((-4*ArcTanh[Sqrt[e/x^(2/3)]/Sqrt[-d]]*(Log[d + e/x^(2/3)] - Log[(
d + e/x^(2/3))/d]))/Sqrt[-d] + (Sqrt[1 - (d + e/x^(2/3))/d]*(2*Log[(1 + Sqrt[1 - (d + e/x^(2/3))/d])/2]^2 - 4*
Log[(1 + Sqrt[1 - (d + e/x^(2/3))/d])/2]*Log[(d + e/x^(2/3))/d] + Log[(d + e/x^(2/3))/d]^2 - 4*PolyLog[2, 1 +
(-1 - Sqrt[1 - (d + e/x^(2/3))/d])/2]))/Sqrt[e/x^(2/3)]))/d))/2 - (3*b^2*e*n^2*x^(1/3)*(a + b*(-(n*Log[d + e/x
^(2/3)]) + Log[c*(d + e/x^(2/3))^n]))*((-16*Sqrt[1 - d/(d + e/x^(2/3))]*Sqrt[d + e/x^(2/3)]*ArcSin[Sqrt[d]/Sqr
t[d + e/x^(2/3)]])/d^(3/2) - (2*x^(2/3)*(4*(d + e/x^(2/3)) + d*(-4 + Log[d + e/x^(2/3)]))*Log[d + e/x^(2/3)])/
(d*e) - (8*Sqrt[e/x^(2/3)]*ArcTanh[Sqrt[e/x^(2/3)]/Sqrt[-d]]*(Log[d + e/x^(2/3)] - Log[(d + e/x^(2/3))/d]))/(-
d)^(3/2) - (2*Sqrt[1 - (d + e/x^(2/3))/d]*(2*Log[(1 + Sqrt[1 - (d + e/x^(2/3))/d])/2]^2 - 4*Log[(1 + Sqrt[1 -
(d + e/x^(2/3))/d])/2]*Log[(d + e/x^(2/3))/d] + Log[(d + e/x^(2/3))/d]^2 - 4*PolyLog[2, 1/2 - Sqrt[1 - (d + e/
x^(2/3))/d]/2]))/d))/2

Integral number [534] \[ \int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{x^2} \, dx \]

[B]   time = 12.6948 (sec), size = 1551 ,normalized size = 3.21 \[ \text{result too large to display} \]

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

[Out]

(-6*b*d*n*(a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^(2/3))^n]))^2)/(e*x^(1/3)) - (6*b*d^(3/2)*n*ArcTan[
(Sqrt[d]*x^(1/3))/Sqrt[e]]*(a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^(2/3))^n]))^2)/e^(3/2) - (3*b*n*Lo
g[d + e/x^(2/3)]*(a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^(2/3))^n]))^2)/x - ((a + b*(-(n*Log[d + e/x^
(2/3)]) + Log[c*(d + e/x^(2/3))^n]))^2*(a - 2*b*n + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^(2/3))^n])))/x
 - (3*b^3*n^3*((416*d^(3/2)*Sqrt[1 - d/(d + e/x^(2/3))]*Sqrt[d + e/x^(2/3)]*x^(1/3)*ArcSin[Sqrt[d]/Sqrt[d + e/
x^(2/3)]])/(9*e) - (2*d^2*Sqrt[1 - d/(d + e/x^(2/3))]*x^(1/3)*(-16*HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {3/
2, 3/2, 3/2}, d/(d + e/x^(2/3))] - 8*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, d/(d + e/x^(2/3))]*Log[d +
 e/x^(2/3)] - (2*Sqrt[d + e/x^(2/3)]*ArcSin[Sqrt[d]/Sqrt[d + e/x^(2/3)]]*Log[d + e/x^(2/3)]^2)/Sqrt[d]))/e + (
(-2*d*(-640 + 312*Log[d + e/x^(2/3)] - 72*Log[d + e/x^(2/3)]^2 + 9*Log[d + e/x^(2/3)]^3))/27 + (2*(d + e/x^(2/
3))*(-16 + 24*Log[d + e/x^(2/3)] - 18*Log[d + e/x^(2/3)]^2 + 9*Log[d + e/x^(2/3)]^3))/27)/x^(1/3) + (16*d^2*((
-4*ArcTanh[Sqrt[e/x^(2/3)]/Sqrt[-d]]*(Log[d + e/x^(2/3)] - Log[(d + e/x^(2/3))/d]))/Sqrt[-d] + (Sqrt[1 - (d +
e/x^(2/3))/d]*(2*Log[(1 + Sqrt[1 - (d + e/x^(2/3))/d])/2]^2 - 4*Log[(1 + Sqrt[1 - (d + e/x^(2/3))/d])/2]*Log[(
d + e/x^(2/3))/d] + Log[(d + e/x^(2/3))/d]^2 - 4*PolyLog[2, 1 + (-1 - Sqrt[1 - (d + e/x^(2/3))/d])/2]))/Sqrt[e
/x^(2/3)]))/(3*Sqrt[e/x^(2/3)]*x^(1/3))))/(2*e) + 9*b^2*n^2*(a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^(
2/3))^n]))*(-Log[(e + d*x^(2/3))/x^(2/3)]^2/(3*x) - (8*e^(3/2) - 96*d*Sqrt[e]*x^(2/3) - 96*d^(3/2)*x*ArcTan[(S
qrt[d]*x^(1/3))/Sqrt[e]] - 12*e^(3/2)*Log[d + e/x^(2/3)] + 36*d*Sqrt[e]*x^(2/3)*Log[d + e/x^(2/3)] + 36*d^(3/2
)*x*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*Log[d + e/x^(2/3)] - 36*d^(3/2)*x*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*Log[
((-I)*Sqrt[e])/Sqrt[d] + x^(1/3)] - (9*I)*d^(3/2)*x*Log[((-I)*Sqrt[e])/Sqrt[d] + x^(1/3)]^2 - 36*d^(3/2)*x*Arc
Tan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*Log[(I*Sqrt[e])/Sqrt[d] + x^(1/3)] + (9*I)*d^(3/2)*x*Log[(I*Sqrt[e])/Sqrt[d] +
x^(1/3)]^2 + (18*I)*d^(3/2)*x*Log[((-I)*Sqrt[e])/Sqrt[d] + x^(1/3)]*Log[1/2 - ((I/2)*Sqrt[d]*x^(1/3))/Sqrt[e]]
 - (18*I)*d^(3/2)*x*Log[(I*Sqrt[e])/Sqrt[d] + x^(1/3)]*Log[1/2 + ((I/2)*Sqrt[d]*x^(1/3))/Sqrt[e]] + 72*d^(3/2)
*x*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*Log[x^(1/3)] - (36*I)*d^(3/2)*x*Log[1 - (I*Sqrt[d]*x^(1/3))/Sqrt[e]]*Log[
x^(1/3)] + (36*I)*d^(3/2)*x*Log[1 + (I*Sqrt[d]*x^(1/3))/Sqrt[e]]*Log[x^(1/3)] - (18*I)*d^(3/2)*x*PolyLog[2, 1/
2 - ((I/2)*Sqrt[d]*x^(1/3))/Sqrt[e]] + (18*I)*d^(3/2)*x*PolyLog[2, 1/2 + ((I/2)*Sqrt[d]*x^(1/3))/Sqrt[e]] + (3
6*I)*d^(3/2)*x*PolyLog[2, ((-I)*Sqrt[d]*x^(1/3))/Sqrt[e]] - (36*I)*d^(3/2)*x*PolyLog[2, (I*Sqrt[d]*x^(1/3))/Sq
rt[e]])/(27*e^(3/2)*x))

Integral number [535] \[ \int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{x^4} \, dx \]

[B]   time = 13.4033 (sec), size = 1993 ,normalized size = 2.54 \[ \text{result too large to display} \]

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

[Out]

(-2*b*d*n*(a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^(2/3))^n]))^2)/(7*e*x^(7/3)) + (2*b*d^2*n*(a + b*(-
(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^(2/3))^n]))^2)/(5*e^2*x^(5/3)) - (2*b*d^3*n*(a + b*(-(n*Log[d + e/x^(2
/3)]) + Log[c*(d + e/x^(2/3))^n]))^2)/(3*e^3*x) + (2*b*d^4*n*(a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^
(2/3))^n]))^2)/(e^4*x^(1/3)) + (2*b*d^(9/2)*n*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*(a + b*(-(n*Log[d + e/x^(2/3)]
) + Log[c*(d + e/x^(2/3))^n]))^2)/e^(9/2) - (b*n*Log[d + e/x^(2/3)]*(a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d
 + e/x^(2/3))^n]))^2)/x^3 - ((a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^(2/3))^n]))^2*(3*a - 2*b*n + 3*b
*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^(2/3))^n])))/(9*x^3) - (3*b^3*n^3*((-6951008*d^(9/2)*Sqrt[1 - d/(d
+ e/x^(2/3))]*Sqrt[d + e/x^(2/3)]*x^(1/3)*ArcSin[Sqrt[d]/Sqrt[d + e/x^(2/3)]])/(297675*e) + (2*d^5*Sqrt[1 - d/
(d + e/x^(2/3))]*x^(1/3)*(-16*HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3/2, 3/2}, d/(d + e/x^(2/3))] - 8*
HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, d/(d + e/x^(2/3))]*Log[d + e/x^(2/3)] - (2*Sqrt[d + e/x^(2/3)]*
ArcSin[Sqrt[d]/Sqrt[d + e/x^(2/3)]]*Log[d + e/x^(2/3)]^2)/Sqrt[d]))/(3*e) + ((2*(d + e/x^(2/3))^4*(-16 + 72*Lo
g[d + e/x^(2/3)] - 162*Log[d + e/x^(2/3)]^2 + 243*Log[d + e/x^(2/3)]^3))/2187 - (4*d*(d + e/x^(2/3))^3*(-24872
 + 85680*Log[d + e/x^(2/3)] - 146853*Log[d + e/x^(2/3)]^2 + 166698*Log[d + e/x^(2/3)]^3))/750141 + (2*d^4*(-11
79876752 + 547391880*Log[d + e/x^(2/3)] - 111727350*Log[d + e/x^(2/3)]^2 + 10418625*Log[d + e/x^(2/3)]^3))/937
67625 - (8*d^3*(d + e/x^(2/3))*(-27010916 + 37647540*Log[d + e/x^(2/3)] - 25103925*Log[d + e/x^(2/3)]^2 + 1041
8625*Log[d + e/x^(2/3)]^3))/93767625 + (4*d^2*(d + e/x^(2/3))^2*(-4747112 + 11406780*Log[d + e/x^(2/3)] - 1349
4600*Log[d + e/x^(2/3)]^2 + 10418625*Log[d + e/x^(2/3)]^3))/31255875)/x^(1/3) - (2252*d^5*((-4*ArcTanh[Sqrt[e/
x^(2/3)]/Sqrt[-d]]*(Log[d + e/x^(2/3)] - Log[(d + e/x^(2/3))/d]))/Sqrt[-d] + (Sqrt[1 - (d + e/x^(2/3))/d]*(2*L
og[(1 + Sqrt[1 - (d + e/x^(2/3))/d])/2]^2 - 4*Log[(1 + Sqrt[1 - (d + e/x^(2/3))/d])/2]*Log[(d + e/x^(2/3))/d]
+ Log[(d + e/x^(2/3))/d]^2 - 4*PolyLog[2, 1 + (-1 - Sqrt[1 - (d + e/x^(2/3))/d])/2]))/Sqrt[e/x^(2/3)]))/(945*S
qrt[e/x^(2/3)]*x^(1/3))))/(2*e^4) + 9*b^2*n^2*(a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^(2/3))^n]))*(-L
og[(e + d*x^(2/3))/x^(2/3)]^2/(9*x^3) - ((9800*e^(9/2))/x^3 - (28800*d*e^(7/2))/x^(7/3) + (72072*d^2*e^(5/2))/
x^(5/3) - (208320*d^3*e^(3/2))/x + (1418760*d^4*Sqrt[e])/x^(1/3) + 1418760*d^(9/2)*ArcTan[(Sqrt[d]*x^(1/3))/Sq
rt[e]] - (44100*e^(9/2)*Log[d + e/x^(2/3)])/x^3 + (56700*d*e^(7/2)*Log[d + e/x^(2/3)])/x^(7/3) - (79380*d^2*e^
(5/2)*Log[d + e/x^(2/3)])/x^(5/3) + (132300*d^3*e^(3/2)*Log[d + e/x^(2/3)])/x - (396900*d^4*Sqrt[e]*Log[d + e/
x^(2/3)])/x^(1/3) - 396900*d^(9/2)*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*Log[d + e/x^(2/3)] + 396900*d^(9/2)*ArcTa
n[(Sqrt[d]*x^(1/3))/Sqrt[e]]*Log[((-I)*Sqrt[e])/Sqrt[d] + x^(1/3)] + (99225*I)*d^(9/2)*Log[((-I)*Sqrt[e])/Sqrt
[d] + x^(1/3)]^2 + 396900*d^(9/2)*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*Log[(I*Sqrt[e])/Sqrt[d] + x^(1/3)] - (9922
5*I)*d^(9/2)*Log[(I*Sqrt[e])/Sqrt[d] + x^(1/3)]^2 - (198450*I)*d^(9/2)*Log[((-I)*Sqrt[e])/Sqrt[d] + x^(1/3)]*L
og[1/2 - ((I/2)*Sqrt[d]*x^(1/3))/Sqrt[e]] + (198450*I)*d^(9/2)*Log[(I*Sqrt[e])/Sqrt[d] + x^(1/3)]*Log[1/2 + ((
I/2)*Sqrt[d]*x^(1/3))/Sqrt[e]] - 793800*d^(9/2)*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*Log[x^(1/3)] + (396900*I)*d^
(9/2)*Log[1 - (I*Sqrt[d]*x^(1/3))/Sqrt[e]]*Log[x^(1/3)] - (396900*I)*d^(9/2)*Log[1 + (I*Sqrt[d]*x^(1/3))/Sqrt[
e]]*Log[x^(1/3)] + (198450*I)*d^(9/2)*PolyLog[2, 1/2 - ((I/2)*Sqrt[d]*x^(1/3))/Sqrt[e]] - (198450*I)*d^(9/2)*P
olyLog[2, 1/2 + ((I/2)*Sqrt[d]*x^(1/3))/Sqrt[e]] - (396900*I)*d^(9/2)*PolyLog[2, ((-I)*Sqrt[d]*x^(1/3))/Sqrt[e
]] + (396900*I)*d^(9/2)*PolyLog[2, (I*Sqrt[d]*x^(1/3))/Sqrt[e]])/(893025*e^(9/2)))

Integral number [536] \[ \int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{x^6} \, dx \]

[B]   time = 14.4156 (sec), size = 2527 ,normalized size = 2.36 \[ \text{Result too large to show} \]

[In]  Integrate[(a + b*Log[c*(d + e/x^(2/3))^n])^3/x^6,x]

[Out]

(-6*b*d*n*(a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^(2/3))^n]))^2)/(65*e*x^(13/3)) + (6*b*d^2*n*(a + b*
(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^(2/3))^n]))^2)/(55*e^2*x^(11/3)) - (2*b*d^3*n*(a + b*(-(n*Log[d + e/
x^(2/3)]) + Log[c*(d + e/x^(2/3))^n]))^2)/(15*e^3*x^3) + (6*b*d^4*n*(a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d
 + e/x^(2/3))^n]))^2)/(35*e^4*x^(7/3)) - (6*b*d^5*n*(a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^(2/3))^n]
))^2)/(25*e^5*x^(5/3)) + (2*b*d^6*n*(a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^(2/3))^n]))^2)/(5*e^6*x)
- (6*b*d^7*n*(a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^(2/3))^n]))^2)/(5*e^7*x^(1/3)) - (6*b*d^(15/2)*n
*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*(a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^(2/3))^n]))^2)/(5*e^(15/2)
) - (3*b*n*Log[d + e/x^(2/3)]*(a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^(2/3))^n]))^2)/(5*x^5) - ((a +
b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^(2/3))^n]))^2*(5*a - 2*b*n + 5*b*(-(n*Log[d + e/x^(2/3)]) + Log[c*
(d + e/x^(2/3))^n])))/(25*x^5) - (3*b^3*n^3*((171744416384*d^(15/2)*Sqrt[1 - d/(d + e/x^(2/3))]*Sqrt[d + e/x^(
2/3)]*x^(1/3)*ArcSin[Sqrt[d]/Sqrt[d + e/x^(2/3)]])/(10145260125*e) - (2*d^8*Sqrt[1 - d/(d + e/x^(2/3))]*x^(1/3
)*(-16*HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3/2, 3/2}, d/(d + e/x^(2/3))] - 8*HypergeometricPFQ[{1/2,
 1/2, 1/2}, {3/2, 3/2}, d/(d + e/x^(2/3))]*Log[d + e/x^(2/3)] - (2*Sqrt[d + e/x^(2/3)]*ArcSin[Sqrt[d]/Sqrt[d +
 e/x^(2/3)]]*Log[d + e/x^(2/3)]^2)/Sqrt[d]))/(5*e) + ((2*(d + e/x^(2/3))^7*(-16 + 120*Log[d + e/x^(2/3)] - 450
*Log[d + e/x^(2/3)]^2 + 1125*Log[d + e/x^(2/3)]^3))/16875 - (2*d*(d + e/x^(2/3))^6*(-387424 + 2500680*Log[d +
e/x^(2/3)] - 8061300*Log[d + e/x^(2/3)]^2 + 17301375*Log[d + e/x^(2/3)]^3))/37074375 + (2*d^2*(d + e/x^(2/3))^
5*(-879939664 + 4761522480*Log[d + e/x^(2/3)] - 12846061800*Log[d + e/x^(2/3)]^2 + 23028130125*Log[d + e/x^(2/
3)]^3))/16448664375 - (2*d^3*(d + e/x^(2/3))^4*(-45474969536 + 198775811520*Log[d + e/x^(2/3)] - 431981035200*
Log[d + e/x^(2/3)]^2 + 621759513375*Log[d + e/x^(2/3)]^3))/266468362875 - (2*d^7*(-860839342730240 + 386811361
800864*Log[d + e/x^(2/3)] - 73915930408320*Log[d + e/x^(2/3)]^2 + 6093243231075*Log[d + e/x^(2/3)]^3))/9139864
8466125 + (2*d^6*(d + e/x^(2/3))*(-129377615035232 + 172985924599488*Log[d + e/x^(2/3)] - 109279872341640*Log[
d + e/x^(2/3)]^2 + 42652702617525*Log[d + e/x^(2/3)]^3))/91398648466125 - (2*d^5*(d + e/x^(2/3))^2*(-220693737
31264 + 51165580113648*Log[d + e/x^(2/3)] - 57985437149640*Log[d + e/x^(2/3)]^2 + 42652702617525*Log[d + e/x^(
2/3)]^3))/30466216155375 + (2*d^4*(d + e/x^(2/3))^3*(-35624015583008 + 118895609392560*Log[d + e/x^(2/3)] - 19
6258028066100*Log[d + e/x^(2/3)]^2 + 213263513087625*Log[d + e/x^(2/3)]^3))/91398648466125)/x^(1/3) + (364288*
d^8*((-4*ArcTanh[Sqrt[e/x^(2/3)]/Sqrt[-d]]*(Log[d + e/x^(2/3)] - Log[(d + e/x^(2/3))/d]))/Sqrt[-d] + (Sqrt[1 -
 (d + e/x^(2/3))/d]*(2*Log[(1 + Sqrt[1 - (d + e/x^(2/3))/d])/2]^2 - 4*Log[(1 + Sqrt[1 - (d + e/x^(2/3))/d])/2]
*Log[(d + e/x^(2/3))/d] + Log[(d + e/x^(2/3))/d]^2 - 4*PolyLog[2, 1 + (-1 - Sqrt[1 - (d + e/x^(2/3))/d])/2]))/
Sqrt[e/x^(2/3)]))/(225225*Sqrt[e/x^(2/3)]*x^(1/3))))/(2*e^7) + 9*b^2*n^2*(a + b*(-(n*Log[d + e/x^(2/3)]) + Log
[c*(d + e/x^(2/3))^n]))*(-Log[(e + d*x^(2/3))/x^(2/3)]^2/(15*x^5) - (4*e*(2/(225*e*x^5) - (56*d)/(2535*e^2*x^(
13/3)) + (1006*d^2)/(23595*e^3*x^(11/3)) - (4448*d^3)/(57915*e^4*x^3) + (44006*d^4)/(315315*e^5*x^(7/3)) - (62
024*d^5)/(225225*e^6*x^(5/3)) + (92054*d^6)/(135135*e^7*x) - (182144*d^7)/(45045*e^8*x^(1/3)) - (182144*d^(15/
2)*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]])/(45045*e^(17/2)) - Log[d + e/x^(2/3)]/(15*e*x^5) + (d*Log[d + e/x^(2/3)]
)/(13*e^2*x^(13/3)) - (d^2*Log[d + e/x^(2/3)])/(11*e^3*x^(11/3)) + (d^3*Log[d + e/x^(2/3)])/(9*e^4*x^3) - (d^4
*Log[d + e/x^(2/3)])/(7*e^5*x^(7/3)) + (d^5*Log[d + e/x^(2/3)])/(5*e^6*x^(5/3)) - (d^6*Log[d + e/x^(2/3)])/(3*
e^7*x) + (d^7*Log[d + e/x^(2/3)])/(e^8*x^(1/3)) + (d^(15/2)*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*Log[d + e/x^(2/3
)])/e^(17/2) - (d^(15/2)*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*Log[((-I)*Sqrt[e])/Sqrt[d] + x^(1/3)])/e^(17/2) - (
(I/4)*d^(15/2)*Log[((-I)*Sqrt[e])/Sqrt[d] + x^(1/3)]^2)/e^(17/2) - (d^(15/2)*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]
*Log[(I*Sqrt[e])/Sqrt[d] + x^(1/3)])/e^(17/2) + ((I/4)*d^(15/2)*Log[(I*Sqrt[e])/Sqrt[d] + x^(1/3)]^2)/e^(17/2)
 + ((I/2)*d^(15/2)*Log[((-I)*Sqrt[e])/Sqrt[d] + x^(1/3)]*Log[1/2 - ((I/2)*Sqrt[d]*x^(1/3))/Sqrt[e]])/e^(17/2)
- ((I/2)*d^(15/2)*Log[(I*Sqrt[e])/Sqrt[d] + x^(1/3)]*Log[1/2 + ((I/2)*Sqrt[d]*x^(1/3))/Sqrt[e]])/e^(17/2) + (2
*d^(15/2)*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*Log[x^(1/3)])/e^(17/2) - (I*d^(15/2)*Log[1 - (I*Sqrt[d]*x^(1/3))/S
qrt[e]]*Log[x^(1/3)])/e^(17/2) + (I*d^(15/2)*Log[1 + (I*Sqrt[d]*x^(1/3))/Sqrt[e]]*Log[x^(1/3)])/e^(17/2) - ((I
/2)*d^(15/2)*PolyLog[2, 1/2 - ((I/2)*Sqrt[d]*x^(1/3))/Sqrt[e]])/e^(17/2) + ((I/2)*d^(15/2)*PolyLog[2, 1/2 + ((
I/2)*Sqrt[d]*x^(1/3))/Sqrt[e]])/e^(17/2) + (I*d^(15/2)*PolyLog[2, ((-I)*Sqrt[d]*x^(1/3))/Sqrt[e]])/e^(17/2) -
(I*d^(15/2)*PolyLog[2, (I*Sqrt[d]*x^(1/3))/Sqrt[e]])/e^(17/2)))/15)

3.4 Test file Number [72] 4_Trig_functions/4.1a_Sine/4.1.7(dtrig)^m(a+b(csin)^n)^p

3.4.1 Mathematica

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

[C]   time = 0.509955 (sec), size = 394 ,normalized size = 15.15 \[ \frac{\frac{24 \cos (c+d x) (a+b \sin (c+d x))}{4 a+3 b \sin (c+d x)-b \sin (3 (c+d x))}-i \text{RootSum}\left [i \text{$\#$1}^6 b-3 i \text{$\#$1}^4 b+8 \text{$\#$1}^3 a+3 i \text{$\#$1}^2 b-i b\&,\frac{2 \text{$\#$1}^4 b \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )-4 i \text{$\#$1}^3 a \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )+2 \text{$\#$1} a \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )-6 i \text{$\#$1}^2 b \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )-i b \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )+12 \text{$\#$1}^2 b \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )-i \text{$\#$1}^4 b \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )-2 \text{$\#$1}^3 a \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )+4 i \text{$\#$1} a \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )+2 b \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )}{\text{$\#$1}^5 b-2 \text{$\#$1}^3 b-4 i \text{$\#$1}^2 a+\text{$\#$1} b}\&\right ]}{18 a b d} \]

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

[Out]

((-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*Cos[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*S
in[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 = 0.367314 (sec), size = 273 ,normalized size = 10.5 \[ \frac{\frac{12 \sin (2 (c+d x))}{4 a+3 b \sin (c+d x)-b \sin (3 (c+d x))}-i \text{RootSum}\left [i \text{$\#$1}^6 b-3 i \text{$\#$1}^4 b+8 \text{$\#$1}^3 a+3 i \text{$\#$1}^2 b-i b\&,\frac{2 \text{$\#$1}^4 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )-6 i \text{$\#$1}^2 \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )-i \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )+12 \text{$\#$1}^2 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )-i \text{$\#$1}^4 \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )+2 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )}{\text{$\#$1}^5 b-2 \text{$\#$1}^3 b-4 i \text{$\#$1}^2 a+\text{$\#$1} b}\&\right ]}{18 a d} \]

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

[Out]

((-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)*Lo
g[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*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) & ] + (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 = 0.685194 (sec), size = 502 ,normalized size = 29.53 \[ \frac{-\frac{12 b \cos (c+d x) (a \cos (2 (c+d x))-3 a+2 b \sin (c+d x))}{(a-b) (a+b) (4 a+3 b \sin (c+d x)-b \sin (3 (c+d x)))}+\frac{i \text{RootSum}\left [i \text{$\#$1}^6 b-3 i \text{$\#$1}^4 b+8 \text{$\#$1}^3 a+3 i \text{$\#$1}^2 b-i b\&,\frac{2 \text{$\#$1}^4 b^2 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )-4 i \text{$\#$1}^3 a b \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )+12 i \text{$\#$1}^2 a^2 \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )-24 \text{$\#$1}^2 a^2 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )+2 \text{$\#$1} a b \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )-6 i \text{$\#$1}^2 b^2 \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )-i b^2 \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )+12 \text{$\#$1}^2 b^2 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )-i \text{$\#$1}^4 b^2 \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )-2 \text{$\#$1}^3 a b \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )+4 i \text{$\#$1} a b \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )+2 b^2 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )}{\text{$\#$1}^5 b-2 \text{$\#$1}^3 b-4 i \text{$\#$1}^2 a+\text{$\#$1} b}\&\right ]}{a^2-b^2}}{18 a d} \]

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

[Out]

((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*Log[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*Log[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 = 4.24594 (sec), size = 845 ,normalized size = 32.5 \[ \frac{-\frac{i b \text{RootSum}\left [i b \text{$\#$1}^6-3 i b \text{$\#$1}^4+8 a \text{$\#$1}^3+3 i b \text{$\#$1}^2-i b\&,\frac{2 b^3 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right ) \text{$\#$1}^4+16 a^2 b \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right ) \text{$\#$1}^4-i b^3 \log \left (\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right ) \text{$\#$1}^4-8 i a^2 b \log \left (\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right ) \text{$\#$1}^4-20 i a^3 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right ) \text{$\#$1}^3-16 i a b^2 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right ) \text{$\#$1}^3-10 a^3 \log \left (\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right ) \text{$\#$1}^3-8 a b^2 \log \left (\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right ) \text{$\#$1}^3+12 b^3 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right ) \text{$\#$1}^2-120 a^2 b \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right ) \text{$\#$1}^2-6 i b^3 \log \left (\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right ) \text{$\#$1}^2+60 i a^2 b \log \left (\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right ) \text{$\#$1}^2+20 i a^3 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right ) \text{$\#$1}+16 i a b^2 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right ) \text{$\#$1}+10 a^3 \log \left (\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right ) \text{$\#$1}+8 a b^2 \log \left (\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right ) \text{$\#$1}+2 b^3 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )+16 a^2 b \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )-i b^3 \log \left (\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right )-8 i a^2 b \log \left (\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right )}{b \text{$\#$1}^5-2 b \text{$\#$1}^3-4 i a \text{$\#$1}^2+b \text{$\#$1}}\&\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 b^2 a+3 b^2 \cos (2 (c+d x)) a+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]

[Out]

(((-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*ArcT
an[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*Lo
g[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[Sin[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] - Sin[(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 = 3.84205 (sec), size = 1158 ,normalized size = 44.54 \[ \text{result too large to display} \]

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

[Out]

((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*ArcTan[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*Ar
cTan[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*ArcTan[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 + 14*a*b^5 + (78*a^5*b + 606*a^3*b^3 + 81*a*b^5)*Cos[2*(c + d*x)] + 18*a*b^3*(4*a^2 + b^2)*Cos[
4*(c + d*x)] + 2*a^5*b*Cos[6*(c + d*x)] - 30*a^3*b^3*Cos[6*(c + d*x)] - 17*a*b^5*Cos[6*(c + d*x)] + 48*a^6*Sin
[c + d*x] - 244*a^4*b^2*Sin[c + d*x] + 20*a^2*b^4*Sin[c + d*x] - 4*b^6*Sin[c + d*x] + 16*a^6*Sin[3*(c + d*x)]
- 194*a^4*b^2*Sin[3*(c + d*x)] - 86*a^2*b^4*Sin[3*(c + d*x)] - 6*b^6*Sin[3*(c + d*x)] - 14*a^4*b^2*Sin[5*(c +
d*x)] - 74*a^2*b^4*Sin[5*(c + d*x)] - 2*b^6*Sin[5*(c + d*x)]))/(4*a + 3*b*Sin[c + d*x] - b*Sin[3*(c + d*x)]))/
(72*a*(a^2 - b^2)^3*d)

3.4.2 Maple

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

[B]   time = 0.191 (sec), size = 550 ,normalized size = 21.15 \[ -{\frac{2}{3\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5} \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}a+3\,a \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+8\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}b+3\,a \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+a \right ) ^{-1}}+{\frac{2}{3\,bd} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4} \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}a+3\,a \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+8\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}b+3\,a \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+a \right ) ^{-1}}+{\frac{8}{3\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}a+3\,a \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+8\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}b+3\,a \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+a \right ) ^{-1}}+{\frac{4}{3\,bd} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}a+3\,a \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+8\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}b+3\,a \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+a \right ) ^{-1}}+{\frac{2}{3\,da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}a+3\,a \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+8\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}b+3\,a \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+a \right ) ^{-1}}+{\frac{2}{3\,bd} \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}a+3\,a \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+8\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}b+3\,a \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+a \right ) ^{-1}}+{\frac{2}{9\,bda}\sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{6}+3\,a{{\it \_Z}}^{4}+8\,b{{\it \_Z}}^{3}+3\,a{{\it \_Z}}^{2}+a \right ) }{\frac{{{\it \_R}}^{4}b+{{\it \_R}}^{3}a+{\it \_R}\,a+b}{{{\it \_R}}^{5}a+2\,{{\it \_R}}^{3}a+4\,{{\it \_R}}^{2}b+{\it \_R}\,a}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -{\it \_R} \right ) }} \]

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

[Out]

-2/3/d/(tan(1/2*d*x+1/2*c)^6*a+3*a*tan(1/2*d*x+1/2*c)^4+8*tan(1/2*d*x+1/2*c)^3*b+3*a*tan(1/2*d*x+1/2*c)^2+a)/a
*tan(1/2*d*x+1/2*c)^5+2/3/d/(tan(1/2*d*x+1/2*c)^6*a+3*a*tan(1/2*d*x+1/2*c)^4+8*tan(1/2*d*x+1/2*c)^3*b+3*a*tan(
1/2*d*x+1/2*c)^2+a)*tan(1/2*d*x+1/2*c)^4/b+8/3/d/(tan(1/2*d*x+1/2*c)^6*a+3*a*tan(1/2*d*x+1/2*c)^4+8*tan(1/2*d*
x+1/2*c)^3*b+3*a*tan(1/2*d*x+1/2*c)^2+a)/a*tan(1/2*d*x+1/2*c)^3+4/3/d/(tan(1/2*d*x+1/2*c)^6*a+3*a*tan(1/2*d*x+
1/2*c)^4+8*tan(1/2*d*x+1/2*c)^3*b+3*a*tan(1/2*d*x+1/2*c)^2+a)*tan(1/2*d*x+1/2*c)^2/b+2/3/d/(tan(1/2*d*x+1/2*c)
^6*a+3*a*tan(1/2*d*x+1/2*c)^4+8*tan(1/2*d*x+1/2*c)^3*b+3*a*tan(1/2*d*x+1/2*c)^2+a)/a*tan(1/2*d*x+1/2*c)+2/3/d/
(tan(1/2*d*x+1/2*c)^6*a+3*a*tan(1/2*d*x+1/2*c)^4+8*tan(1/2*d*x+1/2*c)^3*b+3*a*tan(1/2*d*x+1/2*c)^2+a)/b+2/9/d/
a/b*sum((_R^4*b+_R^3*a+_R*a+b)/(_R^5*a+2*_R^3*a+4*_R^2*b+_R*a)*ln(tan(1/2*d*x+1/2*c)-_R),_R=RootOf(_Z^6*a+3*_Z
^4*a+8*_Z^3*b+3*_Z^2*a+a))

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

[B]   time = 0.169 (sec), size = 236 ,normalized size = 9.08 \[ -{\frac{2}{3\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5} \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}a+3\,a \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+8\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}b+3\,a \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+a \right ) ^{-1}}+{\frac{2}{3\,da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}a+3\,a \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+8\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}b+3\,a \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+a \right ) ^{-1}}+{\frac{2}{9\,da}\sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{6}+3\,a{{\it \_Z}}^{4}+8\,b{{\it \_Z}}^{3}+3\,a{{\it \_Z}}^{2}+a \right ) }{\frac{{{\it \_R}}^{4}+1}{{{\it \_R}}^{5}a+2\,{{\it \_R}}^{3}a+4\,{{\it \_R}}^{2}b+{\it \_R}\,a}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -{\it \_R} \right ) }} \]

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

[Out]

-2/3/d/(tan(1/2*d*x+1/2*c)^6*a+3*a*tan(1/2*d*x+1/2*c)^4+8*tan(1/2*d*x+1/2*c)^3*b+3*a*tan(1/2*d*x+1/2*c)^2+a)/a
*tan(1/2*d*x+1/2*c)^5+2/3/d/(tan(1/2*d*x+1/2*c)^6*a+3*a*tan(1/2*d*x+1/2*c)^4+8*tan(1/2*d*x+1/2*c)^3*b+3*a*tan(
1/2*d*x+1/2*c)^2+a)/a*tan(1/2*d*x+1/2*c)+2/9/d/a*sum((_R^4+1)/(_R^5*a+2*_R^3*a+4*_R^2*b+_R*a)*ln(tan(1/2*d*x+1
/2*c)-_R),_R=RootOf(_Z^6*a+3*_Z^4*a+8*_Z^3*b+3*_Z^2*a+a))

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

[B]   time = 0.136 (sec), size = 658 ,normalized size = 38.71 \[{\frac{2\,{b}^{2}}{3\,da \left ({a}^{2}-{b}^{2} \right ) } \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5} \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}a+3\,a \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+8\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}b+3\,a \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+a \right ) ^{-1}}-{\frac{2\,b}{3\,d \left ({a}^{2}-{b}^{2} \right ) } \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4} \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}a+3\,a \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+8\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}b+3\,a \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+a \right ) ^{-1}}+{\frac{8\,{b}^{2}}{3\,da \left ({a}^{2}-{b}^{2} \right ) } \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}a+3\,a \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+8\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}b+3\,a \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+a \right ) ^{-1}}+{\frac{8\,b}{3\,d \left ({a}^{2}-{b}^{2} \right ) } \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}a+3\,a \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+8\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}b+3\,a \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+a \right ) ^{-1}}-{\frac{2\,{b}^{2}}{3\,da \left ({a}^{2}-{b}^{2} \right ) }\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}a+3\,a \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+8\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}b+3\,a \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+a \right ) ^{-1}}+{\frac{2\,b}{3\,d \left ({a}^{2}-{b}^{2} \right ) } \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}a+3\,a \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+8\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}b+3\,a \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+a \right ) ^{-1}}+{\frac{1}{9\,da}\sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{6}+3\,a{{\it \_Z}}^{4}+8\,b{{\it \_Z}}^{3}+3\,a{{\it \_Z}}^{2}+a \right ) }{\frac{ \left ( 3\,{a}^{2}-2\,{b}^{2} \right ){{\it \_R}}^{4}-2\,ab{{\it \_R}}^{3}+6\,{a}^{2}{{\it \_R}}^{2}-2\,{\it \_R}\,ab+3\,{a}^{2}-2\,{b}^{2}}{ \left ({a}^{2}-{b}^{2} \right ) \left ({{\it \_R}}^{5}a+2\,{{\it \_R}}^{3}a+4\,{{\it \_R}}^{2}b+{\it \_R}\,a \right ) }\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -{\it \_R} \right ) }} \]

[In]  int(1/(a+b*sin(d*x+c)^3)^2,x)

[Out]

2/3/d/(tan(1/2*d*x+1/2*c)^6*a+3*a*tan(1/2*d*x+1/2*c)^4+8*tan(1/2*d*x+1/2*c)^3*b+3*a*tan(1/2*d*x+1/2*c)^2+a)*b^
2/a/(a^2-b^2)*tan(1/2*d*x+1/2*c)^5-2/3/d/(tan(1/2*d*x+1/2*c)^6*a+3*a*tan(1/2*d*x+1/2*c)^4+8*tan(1/2*d*x+1/2*c)
^3*b+3*a*tan(1/2*d*x+1/2*c)^2+a)*b/(a^2-b^2)*tan(1/2*d*x+1/2*c)^4+8/3/d/(tan(1/2*d*x+1/2*c)^6*a+3*a*tan(1/2*d*
x+1/2*c)^4+8*tan(1/2*d*x+1/2*c)^3*b+3*a*tan(1/2*d*x+1/2*c)^2+a)*b^2/a/(a^2-b^2)*tan(1/2*d*x+1/2*c)^3+8/3/d/(ta
n(1/2*d*x+1/2*c)^6*a+3*a*tan(1/2*d*x+1/2*c)^4+8*tan(1/2*d*x+1/2*c)^3*b+3*a*tan(1/2*d*x+1/2*c)^2+a)*b/(a^2-b^2)
*tan(1/2*d*x+1/2*c)^2-2/3/d/(tan(1/2*d*x+1/2*c)^6*a+3*a*tan(1/2*d*x+1/2*c)^4+8*tan(1/2*d*x+1/2*c)^3*b+3*a*tan(
1/2*d*x+1/2*c)^2+a)*b^2/a/(a^2-b^2)*tan(1/2*d*x+1/2*c)+2/3/d/(tan(1/2*d*x+1/2*c)^6*a+3*a*tan(1/2*d*x+1/2*c)^4+
8*tan(1/2*d*x+1/2*c)^3*b+3*a*tan(1/2*d*x+1/2*c)^2+a)*b/(a^2-b^2)+1/9/d/a*sum(((3*a^2-2*b^2)*_R^4-2*a*b*_R^3+6*
a^2*_R^2-2*_R*a*b+3*a^2-2*b^2)/(a^2-b^2)/(_R^5*a+2*_R^3*a+4*_R^2*b+_R*a)*ln(tan(1/2*d*x+1/2*c)-_R),_R=RootOf(_
Z^6*a+3*_Z^4*a+8*_Z^3*b+3*_Z^2*a+a))

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

[B]   time = 0.203 (sec), size = 1276 ,normalized size = 49.08 \[ \text{result too large to display} \]

[In]  int(sec(d*x+c)^2/(a+b*sin(d*x+c)^3)^2,x)

[Out]

-4/3/d*b^2/(a-b)^2/(a+b)^2/(tan(1/2*d*x+1/2*c)^6*a+3*a*tan(1/2*d*x+1/2*c)^4+8*tan(1/2*d*x+1/2*c)^3*b+3*a*tan(1
/2*d*x+1/2*c)^2+a)*a*tan(1/2*d*x+1/2*c)^5-2/3/d*b^4/(a-b)^2/(a+b)^2/(tan(1/2*d*x+1/2*c)^6*a+3*a*tan(1/2*d*x+1/
2*c)^4+8*tan(1/2*d*x+1/2*c)^3*b+3*a*tan(1/2*d*x+1/2*c)^2+a)/a*tan(1/2*d*x+1/2*c)^5-2/3/d*b/(a-b)^2/(a+b)^2/(ta
n(1/2*d*x+1/2*c)^6*a+3*a*tan(1/2*d*x+1/2*c)^4+8*tan(1/2*d*x+1/2*c)^3*b+3*a*tan(1/2*d*x+1/2*c)^2+a)*tan(1/2*d*x
+1/2*c)^4*a^2+8/3/d*b^3/(a-b)^2/(a+b)^2/(tan(1/2*d*x+1/2*c)^6*a+3*a*tan(1/2*d*x+1/2*c)^4+8*tan(1/2*d*x+1/2*c)^
3*b+3*a*tan(1/2*d*x+1/2*c)^2+a)*tan(1/2*d*x+1/2*c)^4-8/3/d*b^2/(a-b)^2/(a+b)^2/(tan(1/2*d*x+1/2*c)^6*a+3*a*tan
(1/2*d*x+1/2*c)^4+8*tan(1/2*d*x+1/2*c)^3*b+3*a*tan(1/2*d*x+1/2*c)^2+a)*a*tan(1/2*d*x+1/2*c)^3-16/3/d*b^4/(a-b)
^2/(a+b)^2/(tan(1/2*d*x+1/2*c)^6*a+3*a*tan(1/2*d*x+1/2*c)^4+8*tan(1/2*d*x+1/2*c)^3*b+3*a*tan(1/2*d*x+1/2*c)^2+
a)/a*tan(1/2*d*x+1/2*c)^3-4/3/d*b/(a-b)^2/(a+b)^2/(tan(1/2*d*x+1/2*c)^6*a+3*a*tan(1/2*d*x+1/2*c)^4+8*tan(1/2*d
*x+1/2*c)^3*b+3*a*tan(1/2*d*x+1/2*c)^2+a)*tan(1/2*d*x+1/2*c)^2*a^2-20/3/d*b^3/(a-b)^2/(a+b)^2/(tan(1/2*d*x+1/2
*c)^6*a+3*a*tan(1/2*d*x+1/2*c)^4+8*tan(1/2*d*x+1/2*c)^3*b+3*a*tan(1/2*d*x+1/2*c)^2+a)*tan(1/2*d*x+1/2*c)^2+4/3
/d*b^2/(a-b)^2/(a+b)^2/(tan(1/2*d*x+1/2*c)^6*a+3*a*tan(1/2*d*x+1/2*c)^4+8*tan(1/2*d*x+1/2*c)^3*b+3*a*tan(1/2*d
*x+1/2*c)^2+a)*a*tan(1/2*d*x+1/2*c)+2/3/d*b^4/(a-b)^2/(a+b)^2/(tan(1/2*d*x+1/2*c)^6*a+3*a*tan(1/2*d*x+1/2*c)^4
+8*tan(1/2*d*x+1/2*c)^3*b+3*a*tan(1/2*d*x+1/2*c)^2+a)/a*tan(1/2*d*x+1/2*c)-2/3/d*b/(a-b)^2/(a+b)^2/(tan(1/2*d*
x+1/2*c)^6*a+3*a*tan(1/2*d*x+1/2*c)^4+8*tan(1/2*d*x+1/2*c)^3*b+3*a*tan(1/2*d*x+1/2*c)^2+a)*a^2-4/3/d*b^3/(a-b)
^2/(a+b)^2/(tan(1/2*d*x+1/2*c)^6*a+3*a*tan(1/2*d*x+1/2*c)^4+8*tan(1/2*d*x+1/2*c)^3*b+3*a*tan(1/2*d*x+1/2*c)^2+
a)-1/9/d*b/(a-b)^2/(a+b)^2/a*sum((b*(11*a^2-2*b^2)*_R^4+2*a*(-5*a^2-4*b^2)*_R^3+54*_R^2*a^2*b+2*a*(-5*a^2-4*b^
2)*_R+11*a^2*b-2*b^3)/(_R^5*a+2*_R^3*a+4*_R^2*b+_R*a)*ln(tan(1/2*d*x+1/2*c)-_R),_R=RootOf(_Z^6*a+3*_Z^4*a+8*_Z
^3*b+3*_Z^2*a+a))-1/d/(a-b)^2/(tan(1/2*d*x+1/2*c)+1)-1/d/(a+b)^2/(tan(1/2*d*x+1/2*c)-1)

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

[B]   time = 0.279 (sec), size = 1549 ,normalized size = 59.58 \[ \text{result too large to display} \]

[In]  int(sec(d*x+c)^4/(a+b*sin(d*x+c)^3)^2,x)

[Out]

2/3/d*b^2/(a-b)^3/(a+b)^3/(tan(1/2*d*x+1/2*c)^6*a+3*a*tan(1/2*d*x+1/2*c)^4+8*tan(1/2*d*x+1/2*c)^3*b+3*a*tan(1/
2*d*x+1/2*c)^2+a)*a^3*tan(1/2*d*x+1/2*c)^5+14/3/d*b^4/(a-b)^3/(a+b)^3/(tan(1/2*d*x+1/2*c)^6*a+3*a*tan(1/2*d*x+
1/2*c)^4+8*tan(1/2*d*x+1/2*c)^3*b+3*a*tan(1/2*d*x+1/2*c)^2+a)*a*tan(1/2*d*x+1/2*c)^5+2/3/d*b^6/(a-b)^3/(a+b)^3
/(tan(1/2*d*x+1/2*c)^6*a+3*a*tan(1/2*d*x+1/2*c)^4+8*tan(1/2*d*x+1/2*c)^3*b+3*a*tan(1/2*d*x+1/2*c)^2+a)/a*tan(1
/2*d*x+1/2*c)^5-6/d*b^5/(a-b)^3/(a+b)^3/(tan(1/2*d*x+1/2*c)^6*a+3*a*tan(1/2*d*x+1/2*c)^4+8*tan(1/2*d*x+1/2*c)^
3*b+3*a*tan(1/2*d*x+1/2*c)^2+a)*tan(1/2*d*x+1/2*c)^4+16/d*b^4/(a-b)^3/(a+b)^3/(tan(1/2*d*x+1/2*c)^6*a+3*a*tan(
1/2*d*x+1/2*c)^4+8*tan(1/2*d*x+1/2*c)^3*b+3*a*tan(1/2*d*x+1/2*c)^2+a)*a*tan(1/2*d*x+1/2*c)^3+8/d*b^6/(a-b)^3/(
a+b)^3/(tan(1/2*d*x+1/2*c)^6*a+3*a*tan(1/2*d*x+1/2*c)^4+8*tan(1/2*d*x+1/2*c)^3*b+3*a*tan(1/2*d*x+1/2*c)^2+a)/a
*tan(1/2*d*x+1/2*c)^3+12/d*b^3/(a-b)^3/(a+b)^3/(tan(1/2*d*x+1/2*c)^6*a+3*a*tan(1/2*d*x+1/2*c)^4+8*tan(1/2*d*x+
1/2*c)^3*b+3*a*tan(1/2*d*x+1/2*c)^2+a)*tan(1/2*d*x+1/2*c)^2*a^2+12/d*b^5/(a-b)^3/(a+b)^3/(tan(1/2*d*x+1/2*c)^6
*a+3*a*tan(1/2*d*x+1/2*c)^4+8*tan(1/2*d*x+1/2*c)^3*b+3*a*tan(1/2*d*x+1/2*c)^2+a)*tan(1/2*d*x+1/2*c)^2-2/3/d*b^
2/(a-b)^3/(a+b)^3/(tan(1/2*d*x+1/2*c)^6*a+3*a*tan(1/2*d*x+1/2*c)^4+8*tan(1/2*d*x+1/2*c)^3*b+3*a*tan(1/2*d*x+1/
2*c)^2+a)*a^3*tan(1/2*d*x+1/2*c)-14/3/d*b^4/(a-b)^3/(a+b)^3/(tan(1/2*d*x+1/2*c)^6*a+3*a*tan(1/2*d*x+1/2*c)^4+8
*tan(1/2*d*x+1/2*c)^3*b+3*a*tan(1/2*d*x+1/2*c)^2+a)*a*tan(1/2*d*x+1/2*c)-2/3/d*b^6/(a-b)^3/(a+b)^3/(tan(1/2*d*
x+1/2*c)^6*a+3*a*tan(1/2*d*x+1/2*c)^4+8*tan(1/2*d*x+1/2*c)^3*b+3*a*tan(1/2*d*x+1/2*c)^2+a)/a*tan(1/2*d*x+1/2*c
)+4/d*b^3/(a-b)^3/(a+b)^3/(tan(1/2*d*x+1/2*c)^6*a+3*a*tan(1/2*d*x+1/2*c)^4+8*tan(1/2*d*x+1/2*c)^3*b+3*a*tan(1/
2*d*x+1/2*c)^2+a)*a^2+2/d*b^5/(a-b)^3/(a+b)^3/(tan(1/2*d*x+1/2*c)^6*a+3*a*tan(1/2*d*x+1/2*c)^4+8*tan(1/2*d*x+1
/2*c)^3*b+3*a*tan(1/2*d*x+1/2*c)^2+a)+1/9/d*b^2/(a-b)^3/(a+b)^3/a*sum(((19*a^4+28*a^2*b^2-2*b^4)*_R^4+18*a*b*(
-4*a^2-b^2)*_R^3+6*a^2*(11*a^2+34*b^2)*_R^2+18*a*b*(-4*a^2-b^2)*_R+19*a^4+28*a^2*b^2-2*b^4)/(_R^5*a+2*_R^3*a+4
*_R^2*b+_R*a)*ln(tan(1/2*d*x+1/2*c)-_R),_R=RootOf(_Z^6*a+3*_Z^4*a+8*_Z^3*b+3*_Z^2*a+a))-1/3/d/(a-b)^2/(tan(1/2
*d*x+1/2*c)+1)^3+1/2/d/(a-b)^2/(tan(1/2*d*x+1/2*c)+1)^2-1/d/(a-b)^3/(tan(1/2*d*x+1/2*c)+1)*a+4/d/(a-b)^3/(tan(
1/2*d*x+1/2*c)+1)*b-1/3/d/(a+b)^2/(tan(1/2*d*x+1/2*c)-1)^3-1/2/d/(a+b)^2/(tan(1/2*d*x+1/2*c)-1)^2-1/d/(a+b)^3/
(tan(1/2*d*x+1/2*c)-1)*a-4/d/(a+b)^3/(tan(1/2*d*x+1/2*c)-1)*b

3.5 Test file Number [112] 4_Trig_functions/4.3a_Secant/4.3.1.3(dsin)^n(a+bsec)^m

3.5.1 Mathematica

Integral number [276] \[ \int \csc ^4(c+d x) (a+b \sec (c+d x))^n \, dx \]

[B]   time = 29.36 (sec), size = 8963 ,normalized size = 373.46 \[ \text{Result too large to show} \]

[In]  Integrate[Csc[c + d*x]^4*(a + b*Sec[c + d*x])^n,x]

[Out]

Result too large to show

3.6 Test file Number [142] 5_Inverse_trig_functions/5.2a_Inverse_tangent/5.2.2u(a+barctan(c+dx))^p

3.6.1 Mathematica

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

[B]   time = 0.626653 (sec), size = 163 ,normalized size = 7.09 \[ \frac{\frac{5 \sqrt [3]{2} \sqrt{\pi } \text{Gamma}\left (\frac{5}{3}\right ) \, _3F_2\left (1,\frac{4}{3},\frac{4}{3};\frac{11}{6},\frac{7}{3};\frac{1}{(a+b x)^2+1}\right )}{(a+b x)^2+1}+6 \text{Gamma}\left (\frac{11}{6}\right ) \text{Gamma}\left (\frac{7}{3}\right ) \left (\frac{4 (a+b x) \, _2F_1\left (1,\frac{4}{3};\frac{11}{6};\frac{1}{(a+b x)^2+1}\right ) \tan ^{-1}(a+b x)}{(a+b x)^2+1}+10 (a+b x) \tan ^{-1}(a+b x)+15\right )}{20 b \text{Gamma}\left (\frac{11}{6}\right ) \text{Gamma}\left (\frac{7}{3}\right ) \sqrt [3]{a^2+2 a b x+b^2 x^2+1}} \]

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

[Out]

(6*Gamma[11/6]*Gamma[7/3]*(15 + 10*(a + b*x)*ArcTan[a + b*x] + (4*(a + b*x)*ArcTan[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] \[ \int \frac{\tan ^{-1}(a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx \]

[B]   time = 0.246852 (sec), size = 165 ,normalized size = 6.6 \[ \frac{\frac{5 \sqrt [3]{2} \sqrt{\pi } \text{Gamma}\left (\frac{5}{3}\right ) \, _3F_2\left (1,\frac{4}{3},\frac{4}{3};\frac{11}{6},\frac{7}{3};\frac{1}{(a+b x)^2+1}\right )}{(a+b x)^2+1}+6 \text{Gamma}\left (\frac{11}{6}\right ) \text{Gamma}\left (\frac{7}{3}\right ) \left (\frac{4 (a+b x) \, _2F_1\left (1,\frac{4}{3};\frac{11}{6};\frac{1}{(a+b x)^2+1}\right ) \tan ^{-1}(a+b x)}{(a+b x)^2+1}+10 (a+b x) \tan ^{-1}(a+b x)+15\right )}{20 b \text{Gamma}\left (\frac{11}{6}\right ) \text{Gamma}\left (\frac{7}{3}\right ) \sqrt [3]{c \left (a^2+2 a b x+b^2 x^2+1\right )}} \]

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

[Out]

(6*Gamma[11/6]*Gamma[7/3]*(15 + 10*(a + b*x)*ArcTan[a + b*x] + (4*(a + b*x)*ArcTan[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] \[ \int \frac{(a+b x)^2 \tan ^{-1}(a+b x)}{\sqrt [3]{1+a^2+2 a b x+b^2 x^2}} \, dx \]

[B]   time = 2.75314 (sec), size = 181 ,normalized size = 6.03 \[ -\frac{3 \left ((a+b x)^2+1\right )^{2/3} \left (\frac{5 \sqrt [3]{2} \sqrt{\pi } \text{Gamma}\left (\frac{5}{3}\right ) \, _3F_2\left (1,\frac{4}{3},\frac{4}{3};\frac{11}{6},\frac{7}{3};\frac{1}{(a+b x)^2+1}\right )}{\left ((a+b x)^2+1\right )^2}+\text{Gamma}\left (\frac{11}{6}\right ) \text{Gamma}\left (\frac{7}{3}\right ) \left (\frac{24 (a+b x) \, _2F_1\left (1,\frac{4}{3};\frac{11}{6};\frac{1}{(a+b x)^2+1}\right ) \tan ^{-1}(a+b x)}{\left ((a+b x)^2+1\right )^2}+\frac{90}{(a+b x)^2+1}+5 \tan ^{-1}(a+b x) \left (6 \sin \left (2 \tan ^{-1}(a+b x)\right )-4 (a+b x)\right )+15\right )\right )}{140 b \text{Gamma}\left (\frac{11}{6}\right ) \text{Gamma}\left (\frac{7}{3}\right )} \]

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

[Out]

(-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 + Gamma[11/6]*Gamma[7/3]*(15 + 90/(1 + (a + b*x)^2) + (24*(a + b*x)*A
rcTan[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] \[ \int \frac{(a+b x)^2 \tan ^{-1}(a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx \]

[B]   time = 1.69328 (sec), size = 225 ,normalized size = 7.03 \[ -\frac{3 \sqrt [3]{a^2+2 a b x+b^2 x^2+1} \left ((a+b x)^2+1\right )^{2/3} \left (\frac{5 \sqrt [3]{2} \sqrt{\pi } \text{Gamma}\left (\frac{5}{3}\right ) \, _3F_2\left (1,\frac{4}{3},\frac{4}{3};\frac{11}{6},\frac{7}{3};\frac{1}{(a+b x)^2+1}\right )}{\left ((a+b x)^2+1\right )^2}+\text{Gamma}\left (\frac{11}{6}\right ) \text{Gamma}\left (\frac{7}{3}\right ) \left (\frac{24 (a+b x) \, _2F_1\left (1,\frac{4}{3};\frac{11}{6};\frac{1}{(a+b x)^2+1}\right ) \tan ^{-1}(a+b x)}{\left ((a+b x)^2+1\right )^2}+\frac{90}{(a+b x)^2+1}+5 \tan ^{-1}(a+b x) \left (6 \sin \left (2 \tan ^{-1}(a+b x)\right )-4 (a+b x)\right )+15\right )\right )}{140 b \text{Gamma}\left (\frac{11}{6}\right ) \text{Gamma}\left (\frac{7}{3}\right ) \sqrt [3]{c \left (a^2+2 a b x+b^2 x^2+1\right )}} \]

[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]

[Out]

(-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)*Sqrt[Pi]*Gamma[5/3]*Hypergeometric
PFQ[{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])

3.7 Test file Number [146] 5_Inverse_trig_functions/5.2b_Inverse_cotangent/5.2.1Inversecotangentfunctions

3.7.1 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 = 0.530724 (sec), size = 177 ,normalized size = 7.7 \[ \frac{6 \text{Gamma}\left (\frac{11}{6}\right ) \text{Gamma}\left (\frac{7}{3}\right ) \left (4 (a+b x) \, _2F_1\left (1,\frac{4}{3};\frac{11}{6};\frac{1}{a^2+2 b x a+b^2 x^2+1}\right ) \cot ^{-1}(a+b x)+5 \left (a^2+2 a b x+b^2 x^2+1\right ) \left (2 (a+b x) \cot ^{-1}(a+b x)-3\right )\right )-5 \sqrt [3]{2} \sqrt{\pi } \text{Gamma}\left (\frac{5}{3}\right ) \, _3F_2\left (1,\frac{4}{3},\frac{4}{3};\frac{11}{6},\frac{7}{3};\frac{1}{a^2+2 b x a+b^2 x^2+1}\right )}{20 b \text{Gamma}\left (\frac{11}{6}\right ) \text{Gamma}\left (\frac{7}{3}\right ) \left (a^2+2 a b x+b^2 x^2+1\right )^{4/3}} \]

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

[Out]

(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)*Ar
cCot[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]*HypergeometricPFQ[{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 = 0.218273 (sec), size = 180 ,normalized size = 7.2 \[ \frac{c \left (6 \text{Gamma}\left (\frac{11}{6}\right ) \text{Gamma}\left (\frac{7}{3}\right ) \left (4 (a+b x) \, _2F_1\left (1,\frac{4}{3};\frac{11}{6};\frac{1}{a^2+2 b x a+b^2 x^2+1}\right ) \cot ^{-1}(a+b x)+5 \left (a^2+2 a b x+b^2 x^2+1\right ) \left (2 (a+b x) \cot ^{-1}(a+b x)-3\right )\right )-5 \sqrt [3]{2} \sqrt{\pi } \text{Gamma}\left (\frac{5}{3}\right ) \, _3F_2\left (1,\frac{4}{3},\frac{4}{3};\frac{11}{6},\frac{7}{3};\frac{1}{a^2+2 b x a+b^2 x^2+1}\right )\right )}{20 b \text{Gamma}\left (\frac{11}{6}\right ) \text{Gamma}\left (\frac{7}{3}\right ) \left (c \left (a^2+2 a b x+b^2 x^2+1\right )\right )^{4/3}} \]

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

[Out]

(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]*Gam
ma[5/3]*HypergeometricPFQ[{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 = 1.34172 (sec), size = 198 ,normalized size = 6.6 \[ \frac{3 \left (5 \sqrt [3]{2} \sqrt{\pi } \text{Gamma}\left (\frac{5}{3}\right ) \, _3F_2\left (1,\frac{4}{3},\frac{4}{3};\frac{11}{6},\frac{7}{3};\frac{1}{a^2+2 b x a+b^2 x^2+1}\right )+\text{Gamma}\left (\frac{11}{6}\right ) \text{Gamma}\left (\frac{7}{3}\right ) \left (5 \left ((a+b x)^2+1\right ) \left (3 \left ((a+b x)^2+7\right )+4 (a+b x) \left ((a+b x)^2-2\right ) \cot ^{-1}(a+b x)\right )-24 (a+b x) \, _2F_1\left (1,\frac{4}{3};\frac{11}{6};\frac{1}{a^2+2 b x a+b^2 x^2+1}\right ) \cot ^{-1}(a+b x)\right )\right )}{140 b \text{Gamma}\left (\frac{11}{6}\right ) \text{Gamma}\left (\frac{7}{3}\right ) \sqrt [3]{a^2+2 a b x+b^2 x^2+1} \left ((a+b x)^2+1\right )} \]

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

[Out]

(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]*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]*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 = 0.451458 (sec), size = 200 ,normalized size = 6.25 \[ \frac{3 \left (5 \sqrt [3]{2} \sqrt{\pi } \text{Gamma}\left (\frac{5}{3}\right ) \, _3F_2\left (1,\frac{4}{3},\frac{4}{3};\frac{11}{6},\frac{7}{3};\frac{1}{a^2+2 b x a+b^2 x^2+1}\right )+\text{Gamma}\left (\frac{11}{6}\right ) \text{Gamma}\left (\frac{7}{3}\right ) \left (5 \left ((a+b x)^2+1\right ) \left (3 \left ((a+b x)^2+7\right )+4 (a+b x) \left ((a+b x)^2-2\right ) \cot ^{-1}(a+b x)\right )-24 (a+b x) \, _2F_1\left (1,\frac{4}{3};\frac{11}{6};\frac{1}{a^2+2 b x a+b^2 x^2+1}\right ) \cot ^{-1}(a+b x)\right )\right )}{140 b \text{Gamma}\left (\frac{11}{6}\right ) \text{Gamma}\left (\frac{7}{3}\right ) \left ((a+b x)^2+1\right ) \sqrt [3]{c \left (a^2+2 a b x+b^2 x^2+1\right )}} \]

[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]

[Out]

(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]*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]*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])

3.8 Test file Number [165] 6_Hyperbolic_functions/6.2a_Hyperbolic_tangent/6.2.7(dhyper)^m(a+b(ctanh)^n)^p

3.8.1 Mathematica

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

[B]   time = 0.673355 (sec), size = 826 ,normalized size = 25.03 \[ \frac{\cosh (3 (c+d x)) a^3+27 b \sinh (c+d x) a^2-b \sinh (3 (c+d x)) a^2-9 \left (a^2+3 b^2\right ) \cosh (c+d x) a-b^2 \cosh (3 (c+d x)) a-2 b \text{RootSum}\left [a \text{$\#$1}^6+b \text{$\#$1}^6+3 a \text{$\#$1}^4-3 b \text{$\#$1}^4+3 a \text{$\#$1}^2+3 b \text{$\#$1}^2+a-b\&,\frac{3 a^2 c \text{$\#$1}^4+3 b^2 c \text{$\#$1}^4-3 a b c \text{$\#$1}^4+3 a^2 d x \text{$\#$1}^4+3 b^2 d x \text{$\#$1}^4-3 a b d x \text{$\#$1}^4+6 a^2 \log \left (\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right ) \text{$\#$1}\right ) \text{$\#$1}^4+6 b^2 \log \left (\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right ) \text{$\#$1}\right ) \text{$\#$1}^4-6 a b \log \left (\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right ) \text{$\#$1}\right ) \text{$\#$1}^4+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 (\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right ) \text{$\#$1}\right ) \text{$\#$1}^2-4 b^2 \log \left (\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right ) \text{$\#$1}\right ) \text{$\#$1}^2+3 a^2 c+3 b^2 c+3 a b c+3 a^2 d x+3 b^2 d x+3 a b d x+6 a^2 \log \left (\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right ) \text{$\#$1}\right )+6 b^2 \log \left (\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right ) \text{$\#$1}\right )+6 a b \log \left (\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right ) \text{$\#$1}\right )}{a \text{$\#$1}^5+b \text{$\#$1}^5+2 a \text{$\#$1}^3-2 b \text{$\#$1}^3+a \text{$\#$1}+b \text{$\#$1}}\&\right ] a+9 b^3 \sinh (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]

[Out]

(-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] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 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] + 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[-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*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 = 0.294725 (sec), size = 409 ,normalized size = 13.19 \[ \frac{b \text{RootSum}\left [\text{$\#$1}^6 a+\text{$\#$1}^6 b+3 \text{$\#$1}^4 a-3 \text{$\#$1}^4 b+3 \text{$\#$1}^2 a+3 \text{$\#$1}^2 b+a-b\&,\frac{4 \text{$\#$1}^4 a \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )+2 \text{$\#$1}^4 a c+2 \text{$\#$1}^4 a d x-2 \text{$\#$1}^4 b \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )-\text{$\#$1}^4 b c-\text{$\#$1}^4 b d x+4 a \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )+2 b \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )+2 a c+2 a d x+b c+b d x}{\text{$\#$1}^5 a+\text{$\#$1}^5 b+2 \text{$\#$1}^3 a-2 \text{$\#$1}^3 b+\text{$\#$1} a+\text{$\#$1} b}\&\right ]+6 a \cosh (c+d x)-6 b \sinh (c+d x)}{6 d (a-b) (a+b)} \]

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

[Out]

(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[-Cosh[(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 = 0.228452 (sec), size = 331 ,normalized size = 10.68 \[ -\frac{b \text{RootSum}\left [\text{$\#$1}^6 a+\text{$\#$1}^6 b+3 \text{$\#$1}^4 a-3 \text{$\#$1}^4 b+3 \text{$\#$1}^2 a+3 \text{$\#$1}^2 b+a-b\&,\frac{2 \text{$\#$1}^4 \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )+\text{$\#$1}^4 c+\text{$\#$1}^4 d x-4 \text{$\#$1}^2 \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )-2 \text{$\#$1}^2 c-2 \text{$\#$1}^2 d x+2 \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )+c+d x}{\text{$\#$1}^5 a+\text{$\#$1}^5 b+2 \text{$\#$1}^3 a-2 \text{$\#$1}^3 b+\text{$\#$1} a+\text{$\#$1} b}\&\right ]-6 \log \left (\sinh \left (\frac{1}{2} (c+d x)\right )\right )+6 \log \left (\cosh \left (\frac{1}{2} (c+d x)\right )\right )}{6 a d} \]

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

[Out]

-(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*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - S
inh[(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) & ])/(6*a*d)

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

[B]   time = 0.607256 (sec), size = 214 ,normalized size = 6.48 \[ -\frac{16 b \text{RootSum}\left [\text{$\#$1}^6 a+\text{$\#$1}^6 b+3 \text{$\#$1}^4 a-3 \text{$\#$1}^4 b+3 \text{$\#$1}^2 a+3 \text{$\#$1}^2 b+a-b\&,\frac{2 \text{$\#$1} \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )+\text{$\#$1} c+\text{$\#$1} d x}{\text{$\#$1}^4 a+\text{$\#$1}^4 b+2 \text{$\#$1}^2 a-2 \text{$\#$1}^2 b+a+b}\&\right ]+3 \left (\text{csch}^2\left (\frac{1}{2} (c+d x)\right )+\text{sech}^2\left (\frac{1}{2} (c+d x)\right )+4 \log \left (\sinh \left (\frac{1}{2} (c+d x)\right )\right )-4 \log \left (\cosh \left (\frac{1}{2} (c+d x)\right )\right )\right )}{24 a d} \]

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

[Out]

-(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))/(24*a*d)

3.8.2 Maple

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

[B]   time = 0.136 (sec), size = 346 ,normalized size = 10.48 \[ -8\,{\frac{1}{d \left ( -16\,b+16\,a \right ) \left ( \tanh \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{2}}}+{\frac{16}{3\,d \left ( -16\,b+16\,a \right ) } \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-3}}-{\frac{b}{d \left ( -b+a \right ) ^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-{\frac{a}{2\,d \left ( -b+a \right ) ^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-{\frac{ab}{3\,d \left ( -b+a \right ) ^{2} \left ( a+b \right ) ^{2}}\sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{6}+3\,a{{\it \_Z}}^{4}+8\,b{{\it \_Z}}^{3}+3\,a{{\it \_Z}}^{2}+a \right ) }{\frac{ \left ( 2\,{a}^{2}+{b}^{2} \right ){{\it \_R}}^{4}-6\,b{{\it \_R}}^{3}a+2\, \left ( 4\,{a}^{2}+5\,{b}^{2} \right ){{\it \_R}}^{2}-6\,ab{\it \_R}+2\,{a}^{2}+{b}^{2}}{{{\it \_R}}^{5}a+2\,{{\it \_R}}^{3}a+4\,{{\it \_R}}^{2}b+{\it \_R}\,a}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -{\it \_R} \right ) }}-{\frac{16}{3\,d \left ( 16\,a+16\,b \right ) } \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-3}}-8\,{\frac{1}{d \left ( 16\,a+16\,b \right ) \left ( \tanh \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{2}}}-{\frac{b}{d \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}+{\frac{a}{2\,d \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}} \]

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

[Out]

-8/d/(-16*b+16*a)/(tanh(1/2*d*x+1/2*c)+1)^2+16/3/d/(tanh(1/2*d*x+1/2*c)+1)^3/(-16*b+16*a)-1/d/(-b+a)^2/(tanh(1
/2*d*x+1/2*c)+1)*b-1/2/d/(-b+a)^2/(tanh(1/2*d*x+1/2*c)+1)*a-1/3/d*a*b/(-b+a)^2/(a+b)^2*sum(((2*a^2+b^2)*_R^4-6
*b*_R^3*a+2*(4*a^2+5*b^2)*_R^2-6*a*b*_R+2*a^2+b^2)/(_R^5*a+2*_R^3*a+4*_R^2*b+_R*a)*ln(tanh(1/2*d*x+1/2*c)-_R),
_R=RootOf(_Z^6*a+3*_Z^4*a+8*_Z^3*b+3*_Z^2*a+a))-16/3/d/(tanh(1/2*d*x+1/2*c)-1)^3/(16*a+16*b)-8/d/(16*a+16*b)/(
tanh(1/2*d*x+1/2*c)-1)^2-1/d/(a+b)^2/(tanh(1/2*d*x+1/2*c)-1)*b+1/2/d/(a+b)^2/(tanh(1/2*d*x+1/2*c)-1)*a

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

[B]   time = 0.128 (sec), size = 164 ,normalized size = 5.29 \[ -4\,{\frac{1}{d \left ( 4\,a+4\,b \right ) \left ( \tanh \left ( 1/2\,dx+c/2 \right ) -1 \right ) }}+4\,{\frac{1}{d \left ( 4\,a-4\,b \right ) \left ( \tanh \left ( 1/2\,dx+c/2 \right ) +1 \right ) }}+{\frac{b}{3\,d \left ( -b+a \right ) \left ( a+b \right ) }\sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{6}+3\,a{{\it \_Z}}^{4}+8\,b{{\it \_Z}}^{3}+3\,a{{\it \_Z}}^{2}+a \right ) }{\frac{{{\it \_R}}^{4}a-2\,{{\it \_R}}^{3}b+6\,{{\it \_R}}^{2}a-2\,{\it \_R}\,b+a}{{{\it \_R}}^{5}a+2\,{{\it \_R}}^{3}a+4\,{{\it \_R}}^{2}b+{\it \_R}\,a}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -{\it \_R} \right ) }} \]

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

[Out]

-4/d/(4*a+4*b)/(tanh(1/2*d*x+1/2*c)-1)+4/d/(4*a-4*b)/(tanh(1/2*d*x+1/2*c)+1)+1/3/d*b/(-b+a)/(a+b)*sum((_R^4*a-
2*_R^3*b+6*_R^2*a-2*_R*b+a)/(_R^5*a+2*_R^3*a+4*_R^2*b+_R*a)*ln(tanh(1/2*d*x+1/2*c)-_R),_R=RootOf(_Z^6*a+3*_Z^4
*a+8*_Z^3*b+3*_Z^2*a+a))

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

[B]   time = 0.122 (sec), size = 98 ,normalized size = 3.16 \[{\frac{1}{da}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{4\,b}{3\,da}\sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{6}+3\,a{{\it \_Z}}^{4}+8\,b{{\it \_Z}}^{3}+3\,a{{\it \_Z}}^{2}+a \right ) }{\frac{{{\it \_R}}^{2}}{{{\it \_R}}^{5}a+2\,{{\it \_R}}^{3}a+4\,{{\it \_R}}^{2}b+{\it \_R}\,a}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -{\it \_R} \right ) }} \]

[In]  int(csch(d*x+c)/(a+b*tanh(d*x+c)^3),x)

[Out]

1/d/a*ln(tanh(1/2*d*x+1/2*c))-4/3/d/a*b*sum(_R^2/(_R^5*a+2*_R^3*a+4*_R^2*b+_R*a)*ln(tanh(1/2*d*x+1/2*c)-_R),_R
=RootOf(_Z^6*a+3*_Z^4*a+8*_Z^3*b+3*_Z^2*a+a))

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

[B]   time = 0.149 (sec), size = 144 ,normalized size = 4.36 \[{\frac{1}{8\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}-{\frac{1}{8\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}}-{\frac{1}{2\,da}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{b}{3\,da}\sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{6}+3\,a{{\it \_Z}}^{4}+8\,b{{\it \_Z}}^{3}+3\,a{{\it \_Z}}^{2}+a \right ) }{\frac{{{\it \_R}}^{4}-2\,{{\it \_R}}^{2}+1}{{{\it \_R}}^{5}a+2\,{{\it \_R}}^{3}a+4\,{{\it \_R}}^{2}b+{\it \_R}\,a}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -{\it \_R} \right ) }} \]

[In]  int(csch(d*x+c)^3/(a+b*tanh(d*x+c)^3),x)

[Out]

1/8/d/a*tanh(1/2*d*x+1/2*c)^2-1/8/d/tanh(1/2*d*x+1/2*c)^2/a-1/2/d/a*ln(tanh(1/2*d*x+1/2*c))-1/3/d/a*b*sum((_R^
4-2*_R^2+1)/(_R^5*a+2*_R^3*a+4*_R^2*b+_R*a)*ln(tanh(1/2*d*x+1/2*c)-_R),_R=RootOf(_Z^6*a+3*_Z^4*a+8*_Z^3*b+3*_Z
^2*a+a))