3.1.57 \(\int \frac {(c+d x)^3}{(a+b \coth (e+f x))^2} \, dx\) [57]

3.1.57.1 Optimal result

Integrand size = 20, antiderivative size = 638 \[ \int \frac {(c+d x)^3}{(a+b \coth (e+f x))^2} \, dx=-\frac {2 b^2 (c+d x)^3}{\left (a^2-b^2\right )^2 f}+\frac {2 b^2 (c+d x)^3}{(a-b) (a+b)^2 \left (a-b-(a+b) e^{2 e+2 f x}\right ) f}+\frac {(c+d x)^4}{4 (a-b)^2 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^2}-\frac {2 b (c+d x)^3 \log \left (1-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f}+\frac {2 b^2 (c+d x)^3 \log \left (1-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f}+\frac {3 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^3}-\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^2}+\frac {3 b^2 d (c+d x)^2 \operatorname {PolyLog}\left (2,\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^2}-\frac {3 b^2 d^3 \operatorname {PolyLog}\left (3,\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 \left (a^2-b^2\right )^2 f^4}+\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^3}-\frac {3 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (3,\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^3}-\frac {3 b d^3 \operatorname {PolyLog}\left (4,\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 (a-b)^2 (a+b) f^4}+\frac {3 b^2 d^3 \operatorname {PolyLog}\left (4,\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 \left (a^2-b^2\right )^2 f^4} \]

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*pol 
ylog(4,(a+b)*exp(2*f*x+2*e)/(a-b))/(a^2-b^2)^2/f^4
 

3.1.57.2 Mathematica [A] (verified)

Time = 7.97 (sec) , antiderivative size = 659, normalized size of antiderivative = 1.03 \[ \int \frac {(c+d x)^3}{(a+b \coth (e+f x))^2} \, dx=\frac {16 b c^2 f^3 (-3 b d+2 a c f) x-\frac {16 (a-b) b^2 f^3 (c+d x)^3}{a \left (-1+e^{2 e}\right )+b \left (1+e^{2 e}\right )}+\frac {8 a (a-b) b f^4 (c+d x)^4}{d \left (a \left (-1+e^{2 e}\right )+b \left (1+e^{2 e}\right )\right )}+48 b c d f^2 (b d-a c f) x \log \left (1+\frac {(-a+b) e^{-2 (e+f x)}}{a+b}\right )+24 b d^2 f^2 (b d-2 a c f) x^2 \log \left (1+\frac {(-a+b) e^{-2 (e+f x)}}{a+b}\right )-16 a b d^3 f^3 x^3 \log \left (1+\frac {(-a+b) e^{-2 (e+f x)}}{a+b}\right )+8 b c^2 f^2 (3 b d-2 a c f) \log \left (a-b-(a+b) e^{2 (e+f x)}\right )+24 b c d f (-b d+a c f) \operatorname {PolyLog}\left (2,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )-12 b d^2 (b d-2 a c f) \left (2 f x \operatorname {PolyLog}\left (2,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )+\operatorname {PolyLog}\left (3,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )\right )+12 a b d^3 \left (2 f^2 x^2 \operatorname {PolyLog}\left (2,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )+2 f x \operatorname {PolyLog}\left (3,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )+\operatorname {PolyLog}\left (4,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )\right )-\frac {(a-b) (a+b) f^3 \left (\left (a^2+b^2\right ) f x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) \cosh (f x)-\left (a^2-b^2\right ) f x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) \cosh (2 e+f x)+2 b \left (-4 b (c+d x)^3+a f x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )\right ) \sinh (f x)\right )}{(b \cosh (e)+a \sinh (e)) (b \cosh (e+f x)+a \sinh (e+f x))}}{8 (a-b)^2 (a+b)^2 f^4} \]

input

Integrate[(c + d*x)^3/(a + b*Coth[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)/(a*( 
-1 + E^(2*e)) + b*(1 + E^(2*e))) + (8*a*(a - b)*b*f^4*(c + d*x)^4)/(d*(a*( 
-1 + E^(2*e)) + b*(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]))/( 
(b*Cosh[e] + a*Sinh[e])*(b*Cosh[e + f*x] + a*Sinh[e + f*x])))/(8*(a - b)^2 
*(a + b)^2*f^4)
 

3.1.57.3 Rubi [A] (verified)

Time = 2.44 (sec) , antiderivative size = 638, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4217

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

\(\Big \downarrow \) 2009

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

input

Int[(c + d*x)^3/(a + b*Coth[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*PolyL 
og[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)*PolyLog[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*P 
olyLog[4, ((a + b)*E^(2*e + 2*f*x))/(a - b)])/(2*(a^2 - b^2)^2*f^4)
 

3.1.57.3.1 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]
 

3.1.57.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2443\) vs. \(2(618)=1236\).

Time = 0.58 (sec) , antiderivative size = 2444, normalized size of antiderivative = 3.83

method	result	size
risch	\(\text {Expression too large to display}\)	\(2444\)

input

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

output

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

3.1.57.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6171 vs. \(2 (614) = 1228\).

Time = 0.36 (sec) , antiderivative size = 6171, normalized size of antiderivative = 9.67 \[ \int \frac {(c+d x)^3}{(a+b \coth (e+f x))^2} \, dx=\text {Too large to display} \]

input

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

output

Too large to include
 

3.1.57.6 Sympy [F(-2)]

Exception generated. \[ \int \frac {(c+d x)^3}{(a+b \coth (e+f x))^2} \, dx=\text {Exception raised: TypeError} \]

input

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

output

Exception raised: TypeError >> Invalid NaN comparison
 

3.1.57.7 Maxima [A] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 1056, normalized size of antiderivative = 1.66 \[ \int \frac {(c+d x)^3}{(a+b \coth (e+f x))^2} \, dx=\text {Too large to display} \]

input

integrate((d*x+c)^3/(a+b*coth(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^2*...
 

3.1.57.8 Giac [F]

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

input

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

output

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

3.1.57.9 Mupad [F(-1)]

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

input

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

output

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