\(\int (c+d x) (a+b \coth (e+f x))^3 \, dx\) [49]

Optimal result

Integrand size = 18, antiderivative size = 259 \[ \int (c+d x) (a+b \coth (e+f x))^3 \, dx=\frac {b^3 d x}{2 f}+\frac {a^3 (c+d x)^2}{2 d}-\frac {3 a^2 b (c+d x)^2}{2 d}+\frac {3 a b^2 (c+d x)^2}{2 d}-\frac {b^3 (c+d x)^2}{2 d}-\frac {b^3 d \coth (e+f x)}{2 f^2}-\frac {3 a b^2 (c+d x) \coth (e+f x)}{f}-\frac {b^3 (c+d x) \coth ^2(e+f x)}{2 f}+\frac {3 a^2 b (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {3 a b^2 d \log (\sinh (e+f x))}{f^2}+\frac {3 a^2 b d \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{2 f^2}+\frac {b^3 d \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{2 f^2} \] Output:

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

Mathematica [A] (verified)

Time = 7.47 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.23 \[ \int (c+d x) (a+b \coth (e+f x))^3 \, dx=\frac {(a+b \coth (e+f x))^3 \sinh (e+f x) \left (-b^3 f (c+d x)-a \left (a^2+3 b^2\right ) (e+f x) (-2 c f+d (e-f x)) \sinh ^2(e+f x)+2 b \left (\frac {3 a^2 f^2 (c+d x)^2}{2 d}+\frac {b^2 f^2 (c+d x)^2}{2 d}+3 a b d (e+f x)+\left (3 a b d+3 a^2 f (c+d x)+b^2 f (c+d x)\right ) \log \left (1-e^{-e-f x}\right )+\left (3 a b d+3 a^2 f (c+d x)+b^2 f (c+d x)\right ) \log \left (1+e^{-e-f x}\right )-\left (3 a^2+b^2\right ) d \operatorname {PolyLog}\left (2,-e^{-e-f x}\right )-\left (3 a^2+b^2\right ) d \operatorname {PolyLog}\left (2,e^{-e-f x}\right )\right ) \sinh ^2(e+f x)-\frac {1}{2} b^2 (b d+6 a f (c+d x)) \sinh (2 (e+f x))\right )}{2 f^2 (b \cosh (e+f x)+a \sinh (e+f x))^3} \] Input:

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

Output:

((a + b*Coth[e + f*x])^3*Sinh[e + f*x]*(-(b^3*f*(c + d*x)) - a*(a^2 + 3*b^ 
2)*(e + f*x)*(-2*c*f + d*(e - f*x))*Sinh[e + f*x]^2 + 2*b*((3*a^2*f^2*(c + 
 d*x)^2)/(2*d) + (b^2*f^2*(c + d*x)^2)/(2*d) + 3*a*b*d*(e + f*x) + (3*a*b* 
d + 3*a^2*f*(c + d*x) + b^2*f*(c + d*x))*Log[1 - E^(-e - f*x)] + (3*a*b*d 
+ 3*a^2*f*(c + d*x) + b^2*f*(c + d*x))*Log[1 + E^(-e - f*x)] - (3*a^2 + b^ 
2)*d*PolyLog[2, -E^(-e - f*x)] - (3*a^2 + b^2)*d*PolyLog[2, E^(-e - f*x)]) 
*Sinh[e + f*x]^2 - (b^2*(b*d + 6*a*f*(c + d*x))*Sinh[2*(e + f*x)])/2))/(2* 
f^2*(b*Cosh[e + f*x] + a*Sinh[e + f*x])^3)
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3042, 4205, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4205

\(\displaystyle \int \left (a^3 (c+d x)+3 a^2 b (c+d x) \coth (e+f x)+3 a b^2 (c+d x) \coth ^2(e+f x)+b^3 (c+d x) \coth ^3(e+f x)\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 4205

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(650\) vs. \(2(241)=482\).

Time = 0.30 (sec) , antiderivative size = 651, normalized size of antiderivative = 2.51

method	result	size
risch	\(-\frac {b^{3} d \,e^{2}}{f^{2}}+\frac {b^{3} c \ln \left ({\mathrm e}^{f x +e}-1\right )}{f}-\frac {2 b^{3} c \ln \left ({\mathrm e}^{f x +e}\right )}{f}+\frac {b^{3} c \ln \left ({\mathrm e}^{f x +e}+1\right )}{f}+\frac {b^{3} d \operatorname {polylog}\left (2, {\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {b^{3} d \operatorname {polylog}\left (2, -{\mathrm e}^{f x +e}\right )}{f^{2}}-\frac {6 b \,a^{2} d e x}{f}-\frac {3 b e \,a^{2} d \ln \left ({\mathrm e}^{f x +e}-1\right )}{f^{2}}+\frac {6 b e \,a^{2} d \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {3 b \,a^{2} d \ln \left (1-{\mathrm e}^{f x +e}\right ) e}{f^{2}}+\frac {3 b \,a^{2} d \ln \left (1-{\mathrm e}^{f x +e}\right ) x}{f}+\frac {3 b \,a^{2} d \ln \left ({\mathrm e}^{f x +e}+1\right ) x}{f}-\frac {b^{2} \left (6 a d f x \,{\mathrm e}^{2 f x +2 e}+2 b d f x \,{\mathrm e}^{2 f x +2 e}+6 a c f \,{\mathrm e}^{2 f x +2 e}+2 b c f \,{\mathrm e}^{2 f x +2 e}-6 a d f x +b d \,{\mathrm e}^{2 f x +2 e}-6 a c f -b d \right )}{f^{2} \left ({\mathrm e}^{2 f x +2 e}-1\right )^{2}}+\frac {a^{3} d \,x^{2}}{2}-\frac {b^{3} d \,x^{2}}{2}+a^{3} c x +b^{3} c x -\frac {3 a^{2} b d \,x^{2}}{2}+\frac {3 a \,b^{2} d \,x^{2}}{2}+3 a^{2} b c x +3 a \,b^{2} c x -\frac {3 b \,a^{2} d \,e^{2}}{f^{2}}-\frac {2 b^{3} d e x}{f}+\frac {b^{3} d \ln \left (1-{\mathrm e}^{f x +e}\right ) e}{f^{2}}+\frac {3 b^{2} a d \ln \left ({\mathrm e}^{f x +e}-1\right )}{f^{2}}-\frac {6 b^{2} a d \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {3 b^{2} a d \ln \left ({\mathrm e}^{f x +e}+1\right )}{f^{2}}+\frac {3 b \,a^{2} c \ln \left ({\mathrm e}^{f x +e}-1\right )}{f}-\frac {6 b \,a^{2} c \ln \left ({\mathrm e}^{f x +e}\right )}{f}+\frac {3 b \,a^{2} c \ln \left ({\mathrm e}^{f x +e}+1\right )}{f}-\frac {b^{3} e d \ln \left ({\mathrm e}^{f x +e}-1\right )}{f^{2}}+\frac {2 b^{3} e d \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {3 b \,a^{2} d \operatorname {polylog}\left (2, {\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {3 b \,a^{2} d \operatorname {polylog}\left (2, -{\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {b^{3} d \ln \left (1-{\mathrm e}^{f x +e}\right ) x}{f}+\frac {b^{3} d \ln \left ({\mathrm e}^{f x +e}+1\right ) x}{f}\)	\(651\)

Input:

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

Output:

-1/f^2*b^3*d*e^2+1/f*b^3*c*ln(exp(f*x+e)-1)-2/f*b^3*c*ln(exp(f*x+e))+1/f*b 
^3*c*ln(exp(f*x+e)+1)+1/f^2*b^3*d*polylog(2,exp(f*x+e))+1/f^2*b^3*d*polylo 
g(2,-exp(f*x+e))-6/f*b*a^2*d*e*x-3/f^2*b*e*a^2*d*ln(exp(f*x+e)-1)+6/f^2*b* 
e*a^2*d*ln(exp(f*x+e))+3/f^2*b*a^2*d*ln(1-exp(f*x+e))*e+3/f*b*a^2*d*ln(1-e 
xp(f*x+e))*x+3/f*b*a^2*d*ln(exp(f*x+e)+1)*x-b^2*(6*a*d*f*x*exp(2*f*x+2*e)+ 
2*b*d*f*x*exp(2*f*x+2*e)+6*a*c*f*exp(2*f*x+2*e)+2*b*c*f*exp(2*f*x+2*e)-6*a 
*d*f*x+b*d*exp(2*f*x+2*e)-6*a*c*f-b*d)/f^2/(exp(2*f*x+2*e)-1)^2+1/2*a^3*d* 
x^2-1/2*b^3*d*x^2+a^3*c*x+b^3*c*x-3/2*a^2*b*d*x^2+3/2*a*b^2*d*x^2+3*a^2*b* 
c*x+3*a*b^2*c*x-3/f^2*b*a^2*d*e^2-2/f*b^3*d*e*x+1/f^2*b^3*d*ln(1-exp(f*x+e 
))*e+3/f^2*b^2*a*d*ln(exp(f*x+e)-1)-6/f^2*b^2*a*d*ln(exp(f*x+e))+3/f^2*b^2 
*a*d*ln(exp(f*x+e)+1)+3/f*b*a^2*c*ln(exp(f*x+e)-1)-6/f*b*a^2*c*ln(exp(f*x+ 
e))+3/f*b*a^2*c*ln(exp(f*x+e)+1)-1/f^2*b^3*e*d*ln(exp(f*x+e)-1)+2/f^2*b^3* 
e*d*ln(exp(f*x+e))+3/f^2*b*a^2*d*polylog(2,exp(f*x+e))+3/f^2*b*a^2*d*polyl 
og(2,-exp(f*x+e))+1/f*b^3*d*ln(1-exp(f*x+e))*x+1/f*b^3*d*ln(exp(f*x+e)+1)* 
x
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2907 vs. \(2 (239) = 478\).

Time = 0.15 (sec) , antiderivative size = 2907, normalized size of antiderivative = 11.22 \[ \int (c+d x) (a+b \coth (e+f x))^3 \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*f^2*x^2 - 12*a*b^2*d*e + 2*(a^3 - 3 
*a^2*b + 3*a*b^2 - b^3)*c*f^2*x + ((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*f^2*x 
^2 - 12*a*b^2*d*e + 2*(3*a^2*b + b^3)*d*e^2 - 4*(3*a^2*b + b^3)*c*e*f - 2* 
(6*a*b^2*d*f - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c*f^2)*x)*cosh(f*x + e)^4 + 
 4*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*f^2*x^2 - 12*a*b^2*d*e + 2*(3*a^2*b 
+ b^3)*d*e^2 - 4*(3*a^2*b + b^3)*c*e*f - 2*(6*a*b^2*d*f - (a^3 - 3*a^2*b + 
 3*a*b^2 - b^3)*c*f^2)*x)*cosh(f*x + e)*sinh(f*x + e)^3 + ((a^3 - 3*a^2*b 
+ 3*a*b^2 - b^3)*d*f^2*x^2 - 12*a*b^2*d*e + 2*(3*a^2*b + b^3)*d*e^2 - 4*(3 
*a^2*b + b^3)*c*e*f - 2*(6*a*b^2*d*f - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c*f 
^2)*x)*sinh(f*x + e)^4 + 2*b^3*d + 2*(3*a^2*b + b^3)*d*e^2 - 2*((a^3 - 3*a 
^2*b + 3*a*b^2 - b^3)*d*f^2*x^2 - 12*a*b^2*d*e + b^3*d + 2*(3*a^2*b + b^3) 
*d*e^2 - 2*(2*(3*a^2*b + b^3)*c*e - (3*a*b^2 + b^3)*c)*f + 2*((a^3 - 3*a^2 
*b + 3*a*b^2 - b^3)*c*f^2 - (3*a*b^2 - b^3)*d*f)*x)*cosh(f*x + e)^2 - 2*(( 
a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*f^2*x^2 - 12*a*b^2*d*e + b^3*d + 2*(3*a^2 
*b + b^3)*d*e^2 - 3*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*f^2*x^2 - 12*a*b^2* 
d*e + 2*(3*a^2*b + b^3)*d*e^2 - 4*(3*a^2*b + b^3)*c*e*f - 2*(6*a*b^2*d*f - 
 (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c*f^2)*x)*cosh(f*x + e)^2 - 2*(2*(3*a^2*b 
 + b^3)*c*e - (3*a*b^2 + b^3)*c)*f + 2*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c* 
f^2 - (3*a*b^2 - b^3)*d*f)*x)*sinh(f*x + e)^2 + 4*(3*a*b^2*c - (3*a^2*b + 
b^3)*c*e)*f + 2*((3*a^2*b + b^3)*d*cosh(f*x + e)^4 + 4*(3*a^2*b + b^3)*...
 

Sympy [F]

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

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

Output:

Integral((a + b*coth(e + f*x))**3*(c + d*x), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 528 vs. \(2 (239) = 478\).

Time = 0.18 (sec) , antiderivative size = 528, normalized size of antiderivative = 2.04 \[ \int (c+d x) (a+b \coth (e+f x))^3 \, dx=\frac {1}{2} \, a^{3} d x^{2} + a^{3} c x + \frac {3 \, a^{2} b c \log \left (\sinh \left (f x + e\right )\right )}{f} - {\left (3 \, a^{2} b d + b^{3} d\right )} x^{2} - \frac {2 \, {\left (b^{3} c f + 3 \, a b^{2} d\right )} x}{f} + \frac {12 \, a b^{2} c f + 2 \, b^{3} d + {\left (3 \, a^{2} b d f^{2} + 3 \, a b^{2} d f^{2} + b^{3} d f^{2}\right )} x^{2} + 2 \, {\left (b^{3} c f^{2} + 3 \, {\left (c f^{2} + 2 \, d f\right )} a b^{2}\right )} x + {\left ({\left (3 \, a^{2} b d f^{2} e^{\left (4 \, e\right )} + 3 \, a b^{2} d f^{2} e^{\left (4 \, e\right )} + b^{3} d f^{2} e^{\left (4 \, e\right )}\right )} x^{2} + 2 \, {\left (3 \, a b^{2} c f^{2} e^{\left (4 \, e\right )} + b^{3} c f^{2} e^{\left (4 \, e\right )}\right )} x\right )} e^{\left (4 \, f x\right )} - 2 \, {\left (6 \, a b^{2} c f e^{\left (2 \, e\right )} + {\left (2 \, c f e^{\left (2 \, e\right )} + d e^{\left (2 \, e\right )}\right )} b^{3} + {\left (3 \, a^{2} b d f^{2} e^{\left (2 \, e\right )} + 3 \, a b^{2} d f^{2} e^{\left (2 \, e\right )} + b^{3} d f^{2} e^{\left (2 \, e\right )}\right )} x^{2} + 2 \, {\left (3 \, {\left (c f^{2} e^{\left (2 \, e\right )} + d f e^{\left (2 \, e\right )}\right )} a b^{2} + {\left (c f^{2} e^{\left (2 \, e\right )} + d f e^{\left (2 \, e\right )}\right )} b^{3}\right )} x\right )} e^{\left (2 \, f x\right )}}{2 \, {\left (f^{2} e^{\left (4 \, f x + 4 \, e\right )} - 2 \, f^{2} e^{\left (2 \, f x + 2 \, e\right )} + f^{2}\right )}} + \frac {{\left (3 \, a^{2} b d + b^{3} d\right )} {\left (f x \log \left (e^{\left (f x + e\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (f x + e\right )}\right )\right )}}{f^{2}} + \frac {{\left (3 \, a^{2} b d + b^{3} d\right )} {\left (f x \log \left (-e^{\left (f x + e\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (f x + e\right )}\right )\right )}}{f^{2}} + \frac {{\left (b^{3} c f + 3 \, a b^{2} d\right )} \log \left (e^{\left (f x + e\right )} + 1\right )}{f^{2}} + \frac {{\left (b^{3} c f + 3 \, a b^{2} d\right )} \log \left (e^{\left (f x + e\right )} - 1\right )}{f^{2}} \] Input:

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

Output:

1/2*a^3*d*x^2 + a^3*c*x + 3*a^2*b*c*log(sinh(f*x + e))/f - (3*a^2*b*d + b^ 
3*d)*x^2 - 2*(b^3*c*f + 3*a*b^2*d)*x/f + 1/2*(12*a*b^2*c*f + 2*b^3*d + (3* 
a^2*b*d*f^2 + 3*a*b^2*d*f^2 + b^3*d*f^2)*x^2 + 2*(b^3*c*f^2 + 3*(c*f^2 + 2 
*d*f)*a*b^2)*x + ((3*a^2*b*d*f^2*e^(4*e) + 3*a*b^2*d*f^2*e^(4*e) + b^3*d*f 
^2*e^(4*e))*x^2 + 2*(3*a*b^2*c*f^2*e^(4*e) + b^3*c*f^2*e^(4*e))*x)*e^(4*f* 
x) - 2*(6*a*b^2*c*f*e^(2*e) + (2*c*f*e^(2*e) + d*e^(2*e))*b^3 + (3*a^2*b*d 
*f^2*e^(2*e) + 3*a*b^2*d*f^2*e^(2*e) + b^3*d*f^2*e^(2*e))*x^2 + 2*(3*(c*f^ 
2*e^(2*e) + d*f*e^(2*e))*a*b^2 + (c*f^2*e^(2*e) + d*f*e^(2*e))*b^3)*x)*e^( 
2*f*x))/(f^2*e^(4*f*x + 4*e) - 2*f^2*e^(2*f*x + 2*e) + f^2) + (3*a^2*b*d + 
 b^3*d)*(f*x*log(e^(f*x + e) + 1) + dilog(-e^(f*x + e)))/f^2 + (3*a^2*b*d 
+ b^3*d)*(f*x*log(-e^(f*x + e) + 1) + dilog(e^(f*x + e)))/f^2 + (b^3*c*f + 
 3*a*b^2*d)*log(e^(f*x + e) + 1)/f^2 + (b^3*c*f + 3*a*b^2*d)*log(e^(f*x + 
e) - 1)/f^2
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(48*e**(4*e + 4*f*x)*int(x/(e**(6*e + 6*f*x) - 3*e**(4*e + 4*f*x) + 3*e**( 
2*e + 2*f*x) - 1),x)*a**2*b*d*f**2 + 16*e**(4*e + 4*f*x)*int(x/(e**(6*e + 
6*f*x) - 3*e**(4*e + 4*f*x) + 3*e**(2*e + 2*f*x) - 1),x)*b**3*d*f**2 + 24* 
e**(4*e + 4*f*x)*log(e**(e + f*x) - 1)*a**2*b*c*f + 18*e**(4*e + 4*f*x)*lo 
g(e**(e + f*x) - 1)*a**2*b*d + 24*e**(4*e + 4*f*x)*log(e**(e + f*x) - 1)*a 
*b**2*d + 8*e**(4*e + 4*f*x)*log(e**(e + f*x) - 1)*b**3*c*f + 6*e**(4*e + 
4*f*x)*log(e**(e + f*x) - 1)*b**3*d + 24*e**(4*e + 4*f*x)*log(e**(e + f*x) 
 + 1)*a**2*b*c*f + 18*e**(4*e + 4*f*x)*log(e**(e + f*x) + 1)*a**2*b*d + 24 
*e**(4*e + 4*f*x)*log(e**(e + f*x) + 1)*a*b**2*d + 8*e**(4*e + 4*f*x)*log( 
e**(e + f*x) + 1)*b**3*c*f + 6*e**(4*e + 4*f*x)*log(e**(e + f*x) + 1)*b**3 
*d + 8*e**(4*e + 4*f*x)*a**3*c*f**2*x + 4*e**(4*e + 4*f*x)*a**3*d*f**2*x** 
2 - 24*e**(4*e + 4*f*x)*a**2*b*c*f**2*x + 12*e**(4*e + 4*f*x)*a**2*b*d*f** 
2*x**2 - 36*e**(4*e + 4*f*x)*a**2*b*d*f*x + 3*e**(4*e + 4*f*x)*a**2*b*d + 
24*e**(4*e + 4*f*x)*a*b**2*c*f**2*x - 24*e**(4*e + 4*f*x)*a*b**2*c*f + 12* 
e**(4*e + 4*f*x)*a*b**2*d*f**2*x**2 - 48*e**(4*e + 4*f*x)*a*b**2*d*f*x - 8 
*e**(4*e + 4*f*x)*b**3*c*f**2*x - 8*e**(4*e + 4*f*x)*b**3*c*f + 4*e**(4*e 
+ 4*f*x)*b**3*d*f**2*x**2 - 12*e**(4*e + 4*f*x)*b**3*d*f*x - 3*e**(4*e + 4 
*f*x)*b**3*d - 96*e**(2*e + 2*f*x)*int(x/(e**(6*e + 6*f*x) - 3*e**(4*e + 4 
*f*x) + 3*e**(2*e + 2*f*x) - 1),x)*a**2*b*d*f**2 - 32*e**(2*e + 2*f*x)*int 
(x/(e**(6*e + 6*f*x) - 3*e**(4*e + 4*f*x) + 3*e**(2*e + 2*f*x) - 1),x)*...