\(\int (c+d x)^2 (a+b \coth (e+f x))^2 \, dx\) [43]

Optimal result

Integrand size = 20, antiderivative size = 209 \[ \int (c+d x)^2 (a+b \coth (e+f x))^2 \, dx=-\frac {b^2 (c+d x)^2}{f}+\frac {a^2 (c+d x)^3}{3 d}-\frac {2 a b (c+d x)^3}{3 d}+\frac {b^2 (c+d x)^3}{3 d}-\frac {b^2 (c+d x)^2 \coth (e+f x)}{f}+\frac {2 b^2 d (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^2}+\frac {2 a b (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^2 d^2 \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^3}+\frac {2 a b d (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}-\frac {a b d^2 \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{f^3} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(473\) vs. \(2(209)=418\).

Time = 4.59 (sec) , antiderivative size = 473, normalized size of antiderivative = 2.26 \[ \int (c+d x)^2 (a+b \coth (e+f x))^2 \, dx=-\frac {4 b c (b d+a c f) x}{f}-\frac {4 b c (b d+a c f) x}{\left (-1+e^{2 e}\right ) f}-\frac {2 b d (b d+2 a c f) x^2}{\left (-1+e^{2 e}\right ) f}+\frac {4 a b d^2 x^3}{3-3 e^{2 e}}+\frac {1}{3} x \left (3 c^2+3 c d x+d^2 x^2\right ) \left (a^2+b^2+2 a b \coth (e)\right )+\frac {2 b d (b d+2 a c f) x \log \left (1-e^{-e-f x}\right )}{f^2}+\frac {2 a b d^2 x^2 \log \left (1-e^{-e-f x}\right )}{f}+\frac {2 b d (b d+2 a c f) x \log \left (1+e^{-e-f x}\right )}{f^2}+\frac {2 a b d^2 x^2 \log \left (1+e^{-e-f x}\right )}{f}+\frac {2 b c (b d+a c f) \log \left (1-e^{e+f x}\right )}{f^2}+\frac {2 b c (b d+a c f) \log \left (1+e^{e+f x}\right )}{f^2}-\frac {2 b d (b d+2 a c f) \operatorname {PolyLog}\left (2,-e^{-e-f x}\right )}{f^3}-\frac {4 a b d^2 x \operatorname {PolyLog}\left (2,-e^{-e-f x}\right )}{f^2}-\frac {2 b d (b d+2 a c f) \operatorname {PolyLog}\left (2,e^{-e-f x}\right )}{f^3}-\frac {4 a b d^2 x \operatorname {PolyLog}\left (2,e^{-e-f x}\right )}{f^2}-\frac {4 a b d^2 \operatorname {PolyLog}\left (3,-e^{-e-f x}\right )}{f^3}-\frac {4 a b d^2 \operatorname {PolyLog}\left (3,e^{-e-f x}\right )}{f^3}+\frac {b^2 (c+d x)^2 \text {csch}(e) \text {csch}(e+f x) \sinh (f x)}{f} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 209, 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, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4205

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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. \(824\) vs. \(2(203)=406\).

Time = 0.26 (sec) , antiderivative size = 825, normalized size of antiderivative = 3.95

method	result	size
risch	\(\frac {d^{2} a^{2} x^{3}}{3}+\frac {d^{2} x^{3} b^{2}}{3}+\frac {c^{3} a^{2}}{3 d}+\frac {c^{3} b^{2}}{3 d}+\frac {8 b a \,d^{2} e^{3}}{3 f^{3}}-\frac {4 b^{2} d^{2} e x}{f^{2}}+\frac {2 b a \,c^{2} \ln \left ({\mathrm e}^{f x +e}-1\right )}{f}-\frac {4 b a \,c^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f}+\frac {2 b a \,c^{2} \ln \left ({\mathrm e}^{f x +e}+1\right )}{f}+\frac {2 b^{2} c d \ln \left ({\mathrm e}^{f x +e}-1\right )}{f^{2}}-\frac {4 b^{2} c d \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {2 b^{2} c d \ln \left ({\mathrm e}^{f x +e}+1\right )}{f^{2}}-\frac {4 b a \,d^{2} \operatorname {polylog}\left (3, {\mathrm e}^{f x +e}\right )}{f^{3}}-\frac {4 b a \,d^{2} \operatorname {polylog}\left (3, -{\mathrm e}^{f x +e}\right )}{f^{3}}-\frac {2 b^{2} e \,d^{2} \ln \left ({\mathrm e}^{f x +e}-1\right )}{f^{3}}+\frac {4 b^{2} e \,d^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{3}}+\frac {2 b^{2} d^{2} \ln \left (1-{\mathrm e}^{f x +e}\right ) x}{f^{2}}+\frac {2 b^{2} d^{2} \ln \left (1-{\mathrm e}^{f x +e}\right ) e}{f^{3}}+\frac {2 b^{2} d^{2} \ln \left ({\mathrm e}^{f x +e}+1\right ) x}{f^{2}}+\frac {4 b a \,d^{2} e^{2} x}{f^{2}}-\frac {4 b \,e^{2} a \,d^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{3}}+\frac {2 b a \,d^{2} \ln \left (1-{\mathrm e}^{f x +e}\right ) x^{2}}{f}-\frac {2 b a \,d^{2} \ln \left (1-{\mathrm e}^{f x +e}\right ) e^{2}}{f^{3}}+\frac {4 b a \,d^{2} \operatorname {polylog}\left (2, {\mathrm e}^{f x +e}\right ) x}{f^{2}}+\frac {2 b a \,d^{2} \ln \left ({\mathrm e}^{f x +e}+1\right ) x^{2}}{f}+\frac {4 b a \,d^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{f x +e}\right ) x}{f^{2}}+\frac {4 b a c d \operatorname {polylog}\left (2, {\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {4 b a c d \operatorname {polylog}\left (2, -{\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {2 b \,e^{2} a \,d^{2} \ln \left ({\mathrm e}^{f x +e}-1\right )}{f^{3}}-\frac {2 b^{2} d^{2} x^{2}}{f}-\frac {2 b^{2} d^{2} e^{2}}{f^{3}}-\frac {4 b a c d \,e^{2}}{f^{2}}+\frac {2 b^{2} d^{2} \operatorname {polylog}\left (2, {\mathrm e}^{f x +e}\right )}{f^{3}}+\frac {2 b^{2} d^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{f x +e}\right )}{f^{3}}-\frac {2 d^{2} a b \,x^{3}}{3}+d \,a^{2} c \,x^{2}+d \,x^{2} c \,b^{2}+a^{2} c^{2} x +x \,c^{2} b^{2}+\frac {2 c^{3} a b}{3 d}-\frac {2 b^{2} \left (d^{2} x^{2}+2 c d x +c^{2}\right )}{f \left ({\mathrm e}^{2 f x +2 e}-1\right )}-\frac {8 b a c d e x}{f}+\frac {4 b a c d \ln \left (1-{\mathrm e}^{f x +e}\right ) x}{f}+\frac {4 b a c d \ln \left ({\mathrm e}^{f x +e}+1\right ) x}{f}-\frac {4 b e a c d \ln \left ({\mathrm e}^{f x +e}-1\right )}{f^{2}}+\frac {8 b e a c d \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {4 b a c d \ln \left (1-{\mathrm e}^{f x +e}\right ) e}{f^{2}}-2 d a b c \,x^{2}+2 a b \,c^{2} x\)	\(825\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1854 vs. \(2 (201) = 402\).

Time = 0.12 (sec) , antiderivative size = 1854, normalized size of antiderivative = 8.87 \[ \int (c+d x)^2 (a+b \coth (e+f x))^2 \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

Integral((a + b*coth(e + f*x))**2*(c + d*x)**2, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 494 vs. \(2 (201) = 402\).

Time = 0.18 (sec) , antiderivative size = 494, normalized size of antiderivative = 2.36 \[ \int (c+d x)^2 (a+b \coth (e+f x))^2 \, dx=\frac {1}{3} \, a^{2} d^{2} x^{3} + a^{2} c d x^{2} + a^{2} c^{2} x - \frac {4 \, b^{2} c d x}{f} + \frac {2 \, a b c^{2} \log \left (\sinh \left (f x + e\right )\right )}{f} + \frac {2 \, b^{2} c d \log \left (e^{\left (f x + e\right )} + 1\right )}{f^{2}} + \frac {2 \, b^{2} c d \log \left (e^{\left (f x + e\right )} - 1\right )}{f^{2}} + \frac {2 \, {\left (f^{2} x^{2} \log \left (e^{\left (f x + e\right )} + 1\right ) + 2 \, f x {\rm Li}_2\left (-e^{\left (f x + e\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (f x + e\right )})\right )} a b d^{2}}{f^{3}} + \frac {2 \, {\left (f^{2} x^{2} \log \left (-e^{\left (f x + e\right )} + 1\right ) + 2 \, f x {\rm Li}_2\left (e^{\left (f x + e\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (f x + e\right )})\right )} a b d^{2}}{f^{3}} - \frac {6 \, b^{2} c^{2} + 3 \, {\left (c^{2} f + 4 \, c d\right )} b^{2} x + {\left (2 \, a b d^{2} f + b^{2} d^{2} f\right )} x^{3} + 3 \, {\left (2 \, a b c d f + {\left (c d f + 2 \, d^{2}\right )} b^{2}\right )} x^{2} - {\left (3 \, b^{2} c^{2} f x e^{\left (2 \, e\right )} + {\left (2 \, a b d^{2} f e^{\left (2 \, e\right )} + b^{2} d^{2} f e^{\left (2 \, e\right )}\right )} x^{3} + 3 \, {\left (2 \, a b c d f e^{\left (2 \, e\right )} + b^{2} c d f e^{\left (2 \, e\right )}\right )} x^{2}\right )} e^{\left (2 \, f x\right )}}{3 \, {\left (f e^{\left (2 \, f x + 2 \, e\right )} - f\right )}} + \frac {2 \, {\left (2 \, a b c d f + b^{2} d^{2}\right )} {\left (f x \log \left (e^{\left (f x + e\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (f x + e\right )}\right )\right )}}{f^{3}} + \frac {2 \, {\left (2 \, a b c d f + b^{2} d^{2}\right )} {\left (f x \log \left (-e^{\left (f x + e\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (f x + e\right )}\right )\right )}}{f^{3}} - \frac {2 \, {\left (2 \, a b d^{2} f^{3} x^{3} + 3 \, {\left (2 \, a b c d f + b^{2} d^{2}\right )} f^{2} x^{2}\right )}}{3 \, f^{3}} \] Input:

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 12*e**(2*e + 2*f*x)*int(x**2/(e**(4*e + 4*f*x) - 2*e**(2*e + 2*f*x) + 
1),x)*a*b*d**2*f**3 - 24*e**(2*e + 2*f*x)*int(x/(e**(4*e + 4*f*x) - 2*e**( 
2*e + 2*f*x) + 1),x)*a*b*c*d*f**3 - 12*e**(2*e + 2*f*x)*int(x/(e**(4*e + 4 
*f*x) - 2*e**(2*e + 2*f*x) + 1),x)*a*b*d**2*f**2 - 12*e**(2*e + 2*f*x)*int 
(x/(e**(4*e + 4*f*x) - 2*e**(2*e + 2*f*x) + 1),x)*b**2*d**2*f**2 + 6*e**(2 
*e + 2*f*x)*log(e**(e + f*x) - 1)*a*b*c**2*f**2 + 6*e**(2*e + 2*f*x)*log(e 
**(e + f*x) - 1)*a*b*c*d*f + 3*e**(2*e + 2*f*x)*log(e**(e + f*x) - 1)*a*b* 
d**2 + 6*e**(2*e + 2*f*x)*log(e**(e + f*x) - 1)*b**2*c*d*f + 3*e**(2*e + 2 
*f*x)*log(e**(e + f*x) - 1)*b**2*d**2 + 6*e**(2*e + 2*f*x)*log(e**(e + f*x 
) + 1)*a*b*c**2*f**2 + 6*e**(2*e + 2*f*x)*log(e**(e + f*x) + 1)*a*b*c*d*f 
+ 3*e**(2*e + 2*f*x)*log(e**(e + f*x) + 1)*a*b*d**2 + 6*e**(2*e + 2*f*x)*l 
og(e**(e + f*x) + 1)*b**2*c*d*f + 3*e**(2*e + 2*f*x)*log(e**(e + f*x) + 1) 
*b**2*d**2 + 3*e**(2*e + 2*f*x)*a**2*c**2*f**3*x + 3*e**(2*e + 2*f*x)*a**2 
*c*d*f**3*x**2 + e**(2*e + 2*f*x)*a**2*d**2*f**3*x**3 - 6*e**(2*e + 2*f*x) 
*a*b*c**2*f**3*x + 6*e**(2*e + 2*f*x)*a*b*c*d*f**3*x**2 - 12*e**(2*e + 2*f 
*x)*a*b*c*d*f**2*x + 2*e**(2*e + 2*f*x)*a*b*d**2*f**3*x**3 - 6*e**(2*e + 2 
*f*x)*a*b*d**2*f*x + 3*e**(2*e + 2*f*x)*b**2*c**2*f**3*x - 6*e**(2*e + 2*f 
*x)*b**2*c**2*f**2 + 3*e**(2*e + 2*f*x)*b**2*c*d*f**3*x**2 - 12*e**(2*e + 
2*f*x)*b**2*c*d*f**2*x + e**(2*e + 2*f*x)*b**2*d**2*f**3*x**3 - 6*e**(2*e 
+ 2*f*x)*b**2*d**2*f*x + 12*int(x**2/(e**(4*e + 4*f*x) - 2*e**(2*e + 2*...