\(\int \frac {(c+d x)^3}{(a+b \tanh (e+f x))^2} \, dx\) [73]

Optimal result

Integrand size = 20, antiderivative size = 642 \[ \int \frac {(c+d x)^3}{(a+b \tanh (e+f x))^2} \, dx=-\frac {2 b^2 (c+d x)^3}{\left (a^2-b^2\right )^2 f}+\frac {2 b^2 (c+d x)^3}{(a-b) (a+b)^2 \left (a-b+(a+b) e^{2 e+2 f x}\right ) f}+\frac {(c+d x)^4}{4 (a-b)^2 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^2}-\frac {2 b (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f}+\frac {2 b^2 (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f}+\frac {3 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^3}-\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^2}+\frac {3 b^2 d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^2}-\frac {3 b^2 d^3 \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 \left (a^2-b^2\right )^2 f^4}+\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^3}-\frac {3 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^3}-\frac {3 b d^3 \operatorname {PolyLog}\left (4,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 (a-b)^2 (a+b) f^4}+\frac {3 b^2 d^3 \operatorname {PolyLog}\left (4,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 \left (a^2-b^2\right )^2 f^4} \] Output:

-2*b^2*(d*x+c)^3/(a^2-b^2)^2/f+2*b^2*(d*x+c)^3/(a-b)/(a+b)^2/(a-b+(a+b)*ex 
p(2*f*x+2*e))/f+1/4*(d*x+c)^4/(a-b)^2/d+3*b^2*d*(d*x+c)^2*ln(1+(a+b)*exp(2 
*f*x+2*e)/(a-b))/(a^2-b^2)^2/f^2-2*b*(d*x+c)^3*ln(1+(a+b)*exp(2*f*x+2*e)/( 
a-b))/(a-b)^2/(a+b)/f+2*b^2*(d*x+c)^3*ln(1+(a+b)*exp(2*f*x+2*e)/(a-b))/(a^ 
2-b^2)^2/f+3*b^2*d^2*(d*x+c)*polylog(2,-(a+b)*exp(2*f*x+2*e)/(a-b))/(a^2-b 
^2)^2/f^3-3*b*d*(d*x+c)^2*polylog(2,-(a+b)*exp(2*f*x+2*e)/(a-b))/(a-b)^2/( 
a+b)/f^2+3*b^2*d*(d*x+c)^2*polylog(2,-(a+b)*exp(2*f*x+2*e)/(a-b))/(a^2-b^2 
)^2/f^2-3/2*b^2*d^3*polylog(3,-(a+b)*exp(2*f*x+2*e)/(a-b))/(a^2-b^2)^2/f^4 
+3*b*d^2*(d*x+c)*polylog(3,-(a+b)*exp(2*f*x+2*e)/(a-b))/(a-b)^2/(a+b)/f^3- 
3*b^2*d^2*(d*x+c)*polylog(3,-(a+b)*exp(2*f*x+2*e)/(a-b))/(a^2-b^2)^2/f^3-3 
/2*b*d^3*polylog(4,-(a+b)*exp(2*f*x+2*e)/(a-b))/(a-b)^2/(a+b)/f^4+3/2*b^2* 
d^3*polylog(4,-(a+b)*exp(2*f*x+2*e)/(a-b))/(a^2-b^2)^2/f^4
 

Mathematica [A] (warning: unable to verify)

Time = 5.10 (sec) , antiderivative size = 656, normalized size of antiderivative = 1.02 \[ \int \frac {(c+d x)^3}{(a+b \tanh (e+f x))^2} \, dx=\frac {16 b c^2 f^3 (-3 b d+2 a c f) x+\frac {16 (a-b) b^2 f^3 (c+d x)^3}{b \left (-1+e^{2 e}\right )+a \left (1+e^{2 e}\right )}-\frac {8 a (a-b) b f^4 (c+d x)^4}{d \left (b \left (-1+e^{2 e}\right )+a \left (1+e^{2 e}\right )\right )}+48 b c d f^2 (b d-a c f) x \log \left (1+\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )+24 b d^2 f^2 (b d-2 a c f) x^2 \log \left (1+\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )-16 a b d^3 f^3 x^3 \log \left (1+\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )+8 b c^2 f^2 (3 b d-2 a c f) \log \left (a-b+(a+b) e^{2 (e+f x)}\right )+24 b c d f (-b d+a c f) \operatorname {PolyLog}\left (2,\frac {(-a+b) e^{-2 (e+f x)}}{a+b}\right )-12 b d^2 (b d-2 a c f) \left (2 f x \operatorname {PolyLog}\left (2,\frac {(-a+b) e^{-2 (e+f x)}}{a+b}\right )+\operatorname {PolyLog}\left (3,\frac {(-a+b) e^{-2 (e+f x)}}{a+b}\right )\right )+12 a b d^3 \left (2 f^2 x^2 \operatorname {PolyLog}\left (2,\frac {(-a+b) e^{-2 (e+f x)}}{a+b}\right )+2 f x \operatorname {PolyLog}\left (3,\frac {(-a+b) e^{-2 (e+f x)}}{a+b}\right )+\operatorname {PolyLog}\left (4,\frac {(-a+b) e^{-2 (e+f x)}}{a+b}\right )\right )+\frac {(a-b) (a+b) f^3 \left (\left (a^2+b^2\right ) f x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) \cosh (f x)+\left (a^2-b^2\right ) f x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) \cosh (2 e+f x)+2 b \left (-4 b (c+d x)^3+a f x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )\right ) \sinh (f x)\right )}{(a \cosh (e)+b \sinh (e)) (a \cosh (e+f x)+b \sinh (e+f x))}}{8 (a-b)^2 (a+b)^2 f^4} \] Input:

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

Output:

(16*b*c^2*f^3*(-3*b*d + 2*a*c*f)*x + (16*(a - b)*b^2*f^3*(c + d*x)^3)/(b*( 
-1 + E^(2*e)) + a*(1 + E^(2*e))) - (8*a*(a - b)*b*f^4*(c + d*x)^4)/(d*(b*( 
-1 + E^(2*e)) + a*(1 + E^(2*e)))) + 48*b*c*d*f^2*(b*d - a*c*f)*x*Log[1 + ( 
a - b)/((a + b)*E^(2*(e + f*x)))] + 24*b*d^2*f^2*(b*d - 2*a*c*f)*x^2*Log[1 
 + (a - b)/((a + b)*E^(2*(e + f*x)))] - 16*a*b*d^3*f^3*x^3*Log[1 + (a - b) 
/((a + b)*E^(2*(e + f*x)))] + 8*b*c^2*f^2*(3*b*d - 2*a*c*f)*Log[a - b + (a 
 + b)*E^(2*(e + f*x))] + 24*b*c*d*f*(-(b*d) + a*c*f)*PolyLog[2, (-a + b)/( 
(a + b)*E^(2*(e + f*x)))] - 12*b*d^2*(b*d - 2*a*c*f)*(2*f*x*PolyLog[2, (-a 
 + b)/((a + b)*E^(2*(e + f*x)))] + PolyLog[3, (-a + b)/((a + b)*E^(2*(e + 
f*x)))]) + 12*a*b*d^3*(2*f^2*x^2*PolyLog[2, (-a + b)/((a + b)*E^(2*(e + f* 
x)))] + 2*f*x*PolyLog[3, (-a + b)/((a + b)*E^(2*(e + f*x)))] + PolyLog[4, 
(-a + b)/((a + b)*E^(2*(e + f*x)))]) + ((a - b)*(a + b)*f^3*((a^2 + b^2)*f 
*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)*Cosh[f*x] + (a^2 - b^2)*f*x 
*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)*Cosh[2*e + f*x] + 2*b*(-4*b*( 
c + d*x)^3 + a*f*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3))*Sinh[f*x]) 
)/((a*Cosh[e] + b*Sinh[e])*(a*Cosh[e + f*x] + b*Sinh[e + f*x])))/(8*(a - b 
)^2*(a + b)^2*f^4)
 

Rubi [A] (verified)

Time = 2.46 (sec) , antiderivative size = 642, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3042, 4217, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(c+d x)^3}{(a+b \tanh (e+f x))^2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(c+d x)^3}{(a-i b \tan (i e+i f x))^2}dx\)

\(\Big \downarrow \) 4217

\(\displaystyle \int \left (\frac {4 b^2 (c+d x)^3 e^{4 e+4 f x}}{(a-b)^2 \left (a \left (\frac {b}{a}+1\right ) e^{2 e+2 f x}+a \left (1-\frac {b}{a}\right )\right )^2}+\frac {4 b (c+d x)^3 e^{2 e+2 f x}}{(a-b)^2 \left (-a \left (\frac {b}{a}+1\right ) e^{2 e+2 f x}-a \left (1-\frac {b}{a}\right )\right )}+\frac {(c+d x)^3}{(a-b)^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^3 \left (a^2-b^2\right )^2}-\frac {3 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^3 \left (a^2-b^2\right )^2}+\frac {3 b^2 d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^2 \left (a^2-b^2\right )^2}+\frac {3 b^2 d (c+d x)^2 \log \left (\frac {(a+b) e^{2 e+2 f x}}{a-b}+1\right )}{f^2 \left (a^2-b^2\right )^2}+\frac {2 b^2 (c+d x)^3 \log \left (\frac {(a+b) e^{2 e+2 f x}}{a-b}+1\right )}{f \left (a^2-b^2\right )^2}-\frac {2 b^2 (c+d x)^3}{f \left (a^2-b^2\right )^2}-\frac {3 b^2 d^3 \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 f^4 \left (a^2-b^2\right )^2}+\frac {3 b^2 d^3 \operatorname {PolyLog}\left (4,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 f^4 \left (a^2-b^2\right )^2}+\frac {2 b^2 (c+d x)^3}{f (a-b) (a+b)^2 \left ((a+b) e^{2 e+2 f x}+a-b\right )}+\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^3 (a-b)^2 (a+b)}-\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^2 (a-b)^2 (a+b)}-\frac {2 b (c+d x)^3 \log \left (\frac {(a+b) e^{2 e+2 f x}}{a-b}+1\right )}{f (a-b)^2 (a+b)}+\frac {(c+d x)^4}{4 d (a-b)^2}-\frac {3 b d^3 \operatorname {PolyLog}\left (4,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 f^4 (a-b)^2 (a+b)}\)

Input:

Int[(c + d*x)^3/(a + b*Tanh[e + f*x])^2,x]
 

Output:

(-2*b^2*(c + d*x)^3)/((a^2 - b^2)^2*f) + (2*b^2*(c + d*x)^3)/((a - b)*(a + 
 b)^2*(a - b + (a + b)*E^(2*e + 2*f*x))*f) + (c + d*x)^4/(4*(a - b)^2*d) + 
 (3*b^2*d*(c + d*x)^2*Log[1 + ((a + b)*E^(2*e + 2*f*x))/(a - b)])/((a^2 - 
b^2)^2*f^2) - (2*b*(c + d*x)^3*Log[1 + ((a + b)*E^(2*e + 2*f*x))/(a - b)]) 
/((a - b)^2*(a + b)*f) + (2*b^2*(c + d*x)^3*Log[1 + ((a + b)*E^(2*e + 2*f* 
x))/(a - b)])/((a^2 - b^2)^2*f) + (3*b^2*d^2*(c + d*x)*PolyLog[2, -(((a + 
b)*E^(2*e + 2*f*x))/(a - b))])/((a^2 - b^2)^2*f^3) - (3*b*d*(c + d*x)^2*Po 
lyLog[2, -(((a + b)*E^(2*e + 2*f*x))/(a - b))])/((a - b)^2*(a + b)*f^2) + 
(3*b^2*d*(c + d*x)^2*PolyLog[2, -(((a + b)*E^(2*e + 2*f*x))/(a - b))])/((a 
^2 - b^2)^2*f^2) - (3*b^2*d^3*PolyLog[3, -(((a + b)*E^(2*e + 2*f*x))/(a - 
b))])/(2*(a^2 - b^2)^2*f^4) + (3*b*d^2*(c + d*x)*PolyLog[3, -(((a + b)*E^( 
2*e + 2*f*x))/(a - b))])/((a - b)^2*(a + b)*f^3) - (3*b^2*d^2*(c + d*x)*Po 
lyLog[3, -(((a + b)*E^(2*e + 2*f*x))/(a - b))])/((a^2 - b^2)^2*f^3) - (3*b 
*d^3*PolyLog[4, -(((a + b)*E^(2*e + 2*f*x))/(a - b))])/(2*(a - b)^2*(a + b 
)*f^4) + (3*b^2*d^3*PolyLog[4, -(((a + b)*E^(2*e + 2*f*x))/(a - b))])/(2*( 
a^2 - b^2)^2*f^4)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4217

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), 
 x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (1/(a - I*b) - 2*I*(b/(a^2 + 
 b^2 + (a - I*b)^2*E^(2*I*(e + f*x)))))^(-n), x], x] /; FreeQ[{a, b, c, d, 
e, f}, x] && NeQ[a^2 + b^2, 0] && ILtQ[n, 0] && IGtQ[m, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2682\) vs. \(2(622)=1244\).

Time = 1.02 (sec) , antiderivative size = 2683, normalized size of antiderivative = 4.18

method	result	size
risch	\(\text {Expression too large to display}\)	\(2683\)

Input:

int((d*x+c)^3/(a+b*tanh(f*x+e))^2,x,method=_RETURNVERBOSE)
 

Output:

12/(a^2+2*a*b+b^2)/f^3*b/(a-b)^2*e^2*d^2*c*a*ln(exp(f*x+e))+3/(a^2+2*a*b+b 
^2)/f^2*b/(a-b)*a*c^2*d/(-a+b)*polylog(2,(a+b)*exp(2*f*x+2*e)/(-a+b))-3/(a 
^2+2*a*b+b^2)/f^3*b/(a-b)/(-a+b)*d^2*c*a*polylog(3,(a+b)*exp(2*f*x+2*e)/(- 
a+b))+2/(a^2+2*a*b+b^2)/f*b/(a-b)/(-a+b)*d^3*a*ln(1-(a+b)*exp(2*f*x+2*e)/( 
-a+b))*x^3+3/(a^2+2*a*b+b^2)/f^2*b/(a-b)/(-a+b)*d^3*a*polylog(2,(a+b)*exp( 
2*f*x+2*e)/(-a+b))*x^2+2/(a^2+2*a*b+b^2)/f^4*b/(a-b)/(-a+b)*d^3*a*ln(1-(a+ 
b)*exp(2*f*x+2*e)/(-a+b))*e^3-3/(a^2+2*a*b+b^2)/f^3*b/(a-b)/(-a+b)*d^3*a*p 
olylog(3,(a+b)*exp(2*f*x+2*e)/(-a+b))*x+6/(a^2+2*a*b+b^2)/f^2*b/(a-b)^2*e* 
a*c^2*d*ln(a*exp(2*f*x+2*e)+exp(2*f*x+2*e)*b+a-b)-12/(a^2+2*a*b+b^2)/f^2*b 
/(a-b)^2*e*a*c^2*d*ln(exp(f*x+e))-6/(a^2+2*a*b+b^2)/f^2*b^2/(a-b)/(-a+b)*c 
*d^2*ln(1-(a+b)*exp(2*f*x+2*e)/(-a+b))*x-6/(a^2+2*a*b+b^2)/f^3*b^2/(a-b)/( 
-a+b)*c*d^2*ln(1-(a+b)*exp(2*f*x+2*e)/(-a+b))*e-6/(a^2+2*a*b+b^2)/f^2*b/(a 
-b)*a*c^2*d/(-a+b)*e^2-4/(a^2+2*a*b+b^2)*b/(a-b)/(-a+b)*d^2*c*a*x^3+8/(a^2 
+2*a*b+b^2)/f^3*b/(a-b)/(-a+b)*d^2*c*a*e^3-6/(a^2+2*a*b+b^2)*b/(a-b)*a*c^2 
*d/(-a+b)*x^2-4/(a^2+2*a*b+b^2)/f^3*b/(a-b)/(-a+b)*d^3*a*e^3*x-6/(a^2+2*a* 
b+b^2)/f^3*b/(a-b)^2*e^2*d^2*c*a*ln(a*exp(2*f*x+2*e)+exp(2*f*x+2*e)*b+a-b) 
-6/(a^2+2*a*b+b^2)/f^3*b^2/(a-b)/(-a+b)*d^3*e^2*x-3/(a^2+2*a*b+b^2)/f^3*b^ 
2/(a-b)/(-a+b)*c*d^2*polylog(2,(a+b)*exp(2*f*x+2*e)/(-a+b))-3/(a^2+2*a*b+b 
^2)/f^2*b^2/(a-b)/(-a+b)*d^3*ln(1-(a+b)*exp(2*f*x+2*e)/(-a+b))*x^2+3/(a^2+ 
2*a*b+b^2)/f^4*b^2/(a-b)/(-a+b)*d^3*ln(1-(a+b)*exp(2*f*x+2*e)/(-a+b))*e...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6160 vs. \(2 (619) = 1238\).

Time = 0.22 (sec) , antiderivative size = 6160, normalized size of antiderivative = 9.60 \[ \int \frac {(c+d x)^3}{(a+b \tanh (e+f x))^2} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^3/(a+b*tanh(f*x+e))^2,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {(c+d x)^3}{(a+b \tanh (e+f x))^2} \, dx=\int \frac {\left (c + d x\right )^{3}}{\left (a + b \tanh {\left (e + f x \right )}\right )^{2}}\, dx \] Input:

integrate((d*x+c)**3/(a+b*tanh(f*x+e))**2,x)
 

Output:

Integral((c + d*x)**3/(a + b*tanh(e + f*x))**2, x)
 

Maxima [A] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 1060, normalized size of antiderivative = 1.65 \[ \int \frac {(c+d x)^3}{(a+b \tanh (e+f x))^2} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^3/(a+b*tanh(f*x+e))^2,x, algorithm="maxima")
 

Output:

-6*b^2*c^2*d*f*x/(a^4*f^2 - 2*a^2*b^2*f^2 + b^4*f^2) - 2/3*(4*f^3*x^3*log( 
(a*e^(2*e) + b*e^(2*e))*e^(2*f*x)/(a - b) + 1) + 6*f^2*x^2*dilog(-(a*e^(2* 
e) + b*e^(2*e))*e^(2*f*x)/(a - b)) - 6*f*x*polylog(3, -(a*e^(2*e) + b*e^(2 
*e))*e^(2*f*x)/(a - b)) + 3*polylog(4, -(a*e^(2*e) + b*e^(2*e))*e^(2*f*x)/ 
(a - b)))*a*b*d^3/(a^4*f^4 - 2*a^2*b^2*f^4 + b^4*f^4) + 3*b^2*c^2*d*log((a 
*e^(2*e) + b*e^(2*e))*e^(2*f*x) + a - b)/(a^4*f^2 - 2*a^2*b^2*f^2 + b^4*f^ 
2) - c^3*(2*a*b*log(-(a - b)*e^(-2*f*x - 2*e) - a - b)/((a^4 - 2*a^2*b^2 + 
 b^4)*f) + 2*b^2/((a^4 - 2*a^2*b^2 + b^4 + (a^4 - 2*a^3*b + 2*a*b^3 - b^4) 
*e^(-2*f*x - 2*e))*f) - (f*x + e)/((a^2 + 2*a*b + b^2)*f)) - 3/2*(2*a*b*c* 
d^2*f - b^2*d^3)*(2*f^2*x^2*log((a*e^(2*e) + b*e^(2*e))*e^(2*f*x)/(a - b) 
+ 1) + 2*f*x*dilog(-(a*e^(2*e) + b*e^(2*e))*e^(2*f*x)/(a - b)) - polylog(3 
, -(a*e^(2*e) + b*e^(2*e))*e^(2*f*x)/(a - b)))/(a^4*f^4 - 2*a^2*b^2*f^4 + 
b^4*f^4) - 3*(a*b*c^2*d*f - b^2*c*d^2)*(2*f*x*log((a*e^(2*e) + b*e^(2*e))* 
e^(2*f*x)/(a - b) + 1) + dilog(-(a*e^(2*e) + b*e^(2*e))*e^(2*f*x)/(a - b)) 
)/(a^4*f^3 - 2*a^2*b^2*f^3 + b^4*f^3) + (a*b*d^3*f^4*x^4 + 2*(2*a*b*c*d^2* 
f - b^2*d^3)*f^3*x^3 + 6*(a*b*c^2*d*f^2 - b^2*c*d^2*f)*f^2*x^2)/(a^4*f^4 - 
 2*a^2*b^2*f^4 + b^4*f^4) + 1/4*(24*b^2*c^2*d*x + (a^2*d^3*f - 2*a*b*d^3*f 
 + b^2*d^3*f)*x^4 + 4*(a^2*c*d^2*f - 2*a*b*c*d^2*f + (c*d^2*f + 2*d^3)*b^2 
)*x^3 + 6*(a^2*c^2*d*f - 2*a*b*c^2*d*f + (c^2*d*f + 4*c*d^2)*b^2)*x^2 + (( 
a^2*d^3*f*e^(2*e) - b^2*d^3*f*e^(2*e))*x^4 + 4*(a^2*c*d^2*f*e^(2*e) - b...
 

Giac [F]

\[ \int \frac {(c+d x)^3}{(a+b \tanh (e+f x))^2} \, dx=\int { \frac {{\left (d x + c\right )}^{3}}{{\left (b \tanh \left (f x + e\right ) + a\right )}^{2}} \,d x } \] Input:

integrate((d*x+c)^3/(a+b*tanh(f*x+e))^2,x, algorithm="giac")
 

Output:

integrate((d*x + c)^3/(b*tanh(f*x + e) + a)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^3}{(a+b \tanh (e+f x))^2} \, dx=\int \frac {{\left (c+d\,x\right )}^3}{{\left (a+b\,\mathrm {tanh}\left (e+f\,x\right )\right )}^2} \,d x \] Input:

int((c + d*x)^3/(a + b*tanh(e + f*x))^2,x)
 

Output:

int((c + d*x)^3/(a + b*tanh(e + f*x))^2, x)
 

Reduce [F]

\[ \int \frac {(c+d x)^3}{(a+b \tanh (e+f x))^2} \, dx=\text {too large to display} \] Input:

int((d*x+c)^3/(a+b*tanh(f*x+e))^2,x)
                                                                                    
                                                                                    
 

Output:

(16*e**(2*e + 2*f*x)*int(x**3/(e**(4*e + 4*f*x)*a**4 + 4*e**(4*e + 4*f*x)* 
a**3*b + 6*e**(4*e + 4*f*x)*a**2*b**2 + 4*e**(4*e + 4*f*x)*a*b**3 + e**(4* 
e + 4*f*x)*b**4 + 2*e**(2*e + 2*f*x)*a**4 + 4*e**(2*e + 2*f*x)*a**3*b - 4* 
e**(2*e + 2*f*x)*a*b**3 - 2*e**(2*e + 2*f*x)*b**4 + a**4 - 2*a**2*b**2 + b 
**4),x)*a**6*b*d**3*f**4 + 16*e**(2*e + 2*f*x)*int(x**3/(e**(4*e + 4*f*x)* 
a**4 + 4*e**(4*e + 4*f*x)*a**3*b + 6*e**(4*e + 4*f*x)*a**2*b**2 + 4*e**(4* 
e + 4*f*x)*a*b**3 + e**(4*e + 4*f*x)*b**4 + 2*e**(2*e + 2*f*x)*a**4 + 4*e* 
*(2*e + 2*f*x)*a**3*b - 4*e**(2*e + 2*f*x)*a*b**3 - 2*e**(2*e + 2*f*x)*b** 
4 + a**4 - 2*a**2*b**2 + b**4),x)*a**5*b**2*d**3*f**4 - 32*e**(2*e + 2*f*x 
)*int(x**3/(e**(4*e + 4*f*x)*a**4 + 4*e**(4*e + 4*f*x)*a**3*b + 6*e**(4*e 
+ 4*f*x)*a**2*b**2 + 4*e**(4*e + 4*f*x)*a*b**3 + e**(4*e + 4*f*x)*b**4 + 2 
*e**(2*e + 2*f*x)*a**4 + 4*e**(2*e + 2*f*x)*a**3*b - 4*e**(2*e + 2*f*x)*a* 
b**3 - 2*e**(2*e + 2*f*x)*b**4 + a**4 - 2*a**2*b**2 + b**4),x)*a**4*b**3*d 
**3*f**4 - 32*e**(2*e + 2*f*x)*int(x**3/(e**(4*e + 4*f*x)*a**4 + 4*e**(4*e 
 + 4*f*x)*a**3*b + 6*e**(4*e + 4*f*x)*a**2*b**2 + 4*e**(4*e + 4*f*x)*a*b** 
3 + e**(4*e + 4*f*x)*b**4 + 2*e**(2*e + 2*f*x)*a**4 + 4*e**(2*e + 2*f*x)*a 
**3*b - 4*e**(2*e + 2*f*x)*a*b**3 - 2*e**(2*e + 2*f*x)*b**4 + a**4 - 2*a** 
2*b**2 + b**4),x)*a**3*b**4*d**3*f**4 + 16*e**(2*e + 2*f*x)*int(x**3/(e**( 
4*e + 4*f*x)*a**4 + 4*e**(4*e + 4*f*x)*a**3*b + 6*e**(4*e + 4*f*x)*a**2*b* 
*2 + 4*e**(4*e + 4*f*x)*a*b**3 + e**(4*e + 4*f*x)*b**4 + 2*e**(2*e + 2*...