\(\int (c+d x)^3 \text {csch}^2(a+b x) \, dx\) [28]

Optimal result

Integrand size = 16, antiderivative size = 103 \[ \int (c+d x)^3 \text {csch}^2(a+b x) \, dx=-\frac {(c+d x)^3}{b}-\frac {(c+d x)^3 \coth (a+b x)}{b}+\frac {3 d (c+d x)^2 \log \left (1-e^{2 (a+b x)}\right )}{b^2}+\frac {3 d^2 (c+d x) \operatorname {PolyLog}\left (2,e^{2 (a+b x)}\right )}{b^3}-\frac {3 d^3 \operatorname {PolyLog}\left (3,e^{2 (a+b x)}\right )}{2 b^4} \] Output:

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

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.80 \[ \int (c+d x)^3 \text {csch}^2(a+b x) \, dx=\frac {-\frac {2 (c+d x)^3}{-1+e^{2 a}}+\frac {3 d (c+d x)^2 \log \left (1-e^{-a-b x}\right )}{b}+\frac {3 d (c+d x)^2 \log \left (1+e^{-a-b x}\right )}{b}-\frac {6 d^2 \left (b (c+d x) \operatorname {PolyLog}\left (2,-e^{-a-b x}\right )+d \operatorname {PolyLog}\left (3,-e^{-a-b x}\right )\right )}{b^3}-\frac {6 d^2 \left (b (c+d x) \operatorname {PolyLog}\left (2,e^{-a-b x}\right )+d \operatorname {PolyLog}\left (3,e^{-a-b x}\right )\right )}{b^3}+(c+d x)^3 \text {csch}(a) \text {csch}(a+b x) \sinh (b x)}{b} \] Input:

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

Output:

((-2*(c + d*x)^3)/(-1 + E^(2*a)) + (3*d*(c + d*x)^2*Log[1 - E^(-a - b*x)]) 
/b + (3*d*(c + d*x)^2*Log[1 + E^(-a - b*x)])/b - (6*d^2*(b*(c + d*x)*PolyL 
og[2, -E^(-a - b*x)] + d*PolyLog[3, -E^(-a - b*x)]))/b^3 - (6*d^2*(b*(c + 
d*x)*PolyLog[2, E^(-a - b*x)] + d*PolyLog[3, E^(-a - b*x)]))/b^3 + (c + d* 
x)^3*Csch[a]*Csch[a + b*x]*Sinh[b*x])/b
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.72 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.42, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {3042, 25, 4672, 26, 3042, 26, 4201, 2620, 3011, 2720, 7143}

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.

\(\displaystyle \int (c+d x)^3 \text {csch}^2(a+b x) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -(c+d x)^3 \csc (i a+i b x)^2dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\int (c+d x)^3 \csc (i a+i b x)^2dx\)

\(\Big \downarrow \) 4672

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

\(\displaystyle -\frac {(c+d x)^3 \coth (a+b x)}{b}-\frac {3 i d \int (c+d x)^2 \tan \left (\frac {1}{2} (2 i a+\pi )+i b x\right )dx}{b}\)

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

\(\displaystyle -\frac {(c+d x)^3 \coth (a+b x)}{b}-\frac {3 i d \left (2 i \left (\frac {(c+d x)^2 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {d \int (c+d x) \log \left (1+e^{2 a+2 b x-i \pi }\right )dx}{b}\right )-\frac {i (c+d x)^3}{3 d}\right )}{b}\)

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

Int[(c + d*x)^3*Csch[a + b*x]^2,x]
 

Output:

-(((c + d*x)^3*Coth[a + b*x])/b) - ((3*I)*d*(((-1/3*I)*(c + d*x)^3)/d + (2 
*I)*(((c + d*x)^2*Log[1 + E^(2*a - I*Pi + 2*b*x)])/(2*b) - (d*(-1/2*((c + 
d*x)*PolyLog[2, -E^(2*a - I*Pi + 2*b*x)])/b + (d*PolyLog[3, -E^(2*a - I*Pi 
 + 2*b*x)])/(4*b^2)))/b)))/b
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

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

rule 4201

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x 
_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I   Int[ 
(c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] 
 /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
 

rule 4672

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp 
[(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1) 
*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(472\) vs. \(2(101)=202\).

Time = 0.13 (sec) , antiderivative size = 473, normalized size of antiderivative = 4.59

method	result	size
risch	\(-\frac {12 d^{2} c a x}{b^{2}}-\frac {6 d^{2} c a \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{3}}+\frac {12 d^{2} c a \ln \left ({\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {6 d^{2} c \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b^{2}}+\frac {6 d^{2} c \ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{3}}+\frac {6 d^{2} c \ln \left ({\mathrm e}^{b x +a}+1\right ) x}{b^{2}}-\frac {2 \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}\right )}{\left ({\mathrm e}^{2 b x +2 a}-1\right ) b}-\frac {6 d^{3} \operatorname {polylog}\left (3, {\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {6 d^{3} \operatorname {polylog}\left (3, -{\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {2 d^{3} x^{3}}{b}+\frac {4 d^{3} a^{3}}{b^{4}}-\frac {6 d^{2} c \,x^{2}}{b}-\frac {6 d^{2} c \,a^{2}}{b^{3}}+\frac {6 d^{3} a^{2} x}{b^{3}}+\frac {3 d \,c^{2} \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{2}}-\frac {6 d \,c^{2} \ln \left ({\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {3 d \,c^{2} \ln \left ({\mathrm e}^{b x +a}+1\right )}{b^{2}}+\frac {6 d^{2} c \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {6 d^{2} c \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {3 d^{3} a^{2} \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{4}}-\frac {6 d^{3} a^{2} \ln \left ({\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {3 d^{3} \ln \left (1-{\mathrm e}^{b x +a}\right ) x^{2}}{b^{2}}-\frac {3 d^{3} \ln \left (1-{\mathrm e}^{b x +a}\right ) a^{2}}{b^{4}}+\frac {6 d^{3} \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right ) x}{b^{3}}+\frac {3 d^{3} \ln \left ({\mathrm e}^{b x +a}+1\right ) x^{2}}{b^{2}}+\frac {6 d^{3} \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right ) x}{b^{3}}\)	\(473\)

Input:

int((d*x+c)^3*csch(b*x+a)^2,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1159 vs. \(2 (100) = 200\).

Time = 0.09 (sec) , antiderivative size = 1159, normalized size of antiderivative = 11.25 \[ \int (c+d x)^3 \text {csch}^2(a+b x) \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^3*csch(b*x+a)^2,x, algorithm="fricas")
 

Output:

-(2*b^3*c^3 - 6*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 2*a^3*d^3 + 2*(b^3*d^3*x^3 + 
 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3 
)*cosh(b*x + a)^2 + 4*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + 3*a 
*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*cosh(b*x + a)*sinh(b*x + a) + 2*(b^3 
*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 
 + a^3*d^3)*sinh(b*x + a)^2 + 6*(b*d^3*x + b*c*d^2 - (b*d^3*x + b*c*d^2)*c 
osh(b*x + a)^2 - 2*(b*d^3*x + b*c*d^2)*cosh(b*x + a)*sinh(b*x + a) - (b*d^ 
3*x + b*c*d^2)*sinh(b*x + a)^2)*dilog(cosh(b*x + a) + sinh(b*x + a)) + 6*( 
b*d^3*x + b*c*d^2 - (b*d^3*x + b*c*d^2)*cosh(b*x + a)^2 - 2*(b*d^3*x + b*c 
*d^2)*cosh(b*x + a)*sinh(b*x + a) - (b*d^3*x + b*c*d^2)*sinh(b*x + a)^2)*d 
ilog(-cosh(b*x + a) - sinh(b*x + a)) + 3*(b^2*d^3*x^2 + 2*b^2*c*d^2*x + b^ 
2*c^2*d - (b^2*d^3*x^2 + 2*b^2*c*d^2*x + b^2*c^2*d)*cosh(b*x + a)^2 - 2*(b 
^2*d^3*x^2 + 2*b^2*c*d^2*x + b^2*c^2*d)*cosh(b*x + a)*sinh(b*x + a) - (b^2 
*d^3*x^2 + 2*b^2*c*d^2*x + b^2*c^2*d)*sinh(b*x + a)^2)*log(cosh(b*x + a) + 
 sinh(b*x + a) + 1) + 3*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3 - (b^2*c^2*d - 
2*a*b*c*d^2 + a^2*d^3)*cosh(b*x + a)^2 - 2*(b^2*c^2*d - 2*a*b*c*d^2 + a^2* 
d^3)*cosh(b*x + a)*sinh(b*x + a) - (b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*sin 
h(b*x + a)^2)*log(cosh(b*x + a) + sinh(b*x + a) - 1) + 3*(b^2*d^3*x^2 + 2* 
b^2*c*d^2*x + 2*a*b*c*d^2 - a^2*d^3 - (b^2*d^3*x^2 + 2*b^2*c*d^2*x + 2*a*b 
*c*d^2 - a^2*d^3)*cosh(b*x + a)^2 - 2*(b^2*d^3*x^2 + 2*b^2*c*d^2*x + 2*...
 

Sympy [F]

\[ \int (c+d x)^3 \text {csch}^2(a+b x) \, dx=\int \left (c + d x\right )^{3} \operatorname {csch}^{2}{\left (a + b x \right )}\, dx \] Input:

integrate((d*x+c)**3*csch(b*x+a)**2,x)
 

Output:

Integral((c + d*x)**3*csch(a + b*x)**2, x)
                                                                                    
                                                                                    
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 320 vs. \(2 (100) = 200\).

Time = 0.22 (sec) , antiderivative size = 320, normalized size of antiderivative = 3.11 \[ \int (c+d x)^3 \text {csch}^2(a+b x) \, dx=-3 \, c^{2} d {\left (\frac {2 \, x e^{\left (2 \, b x + 2 \, a\right )}}{b e^{\left (2 \, b x + 2 \, a\right )} - b} - \frac {\log \left ({\left (e^{\left (b x + a\right )} + 1\right )} e^{\left (-a\right )}\right )}{b^{2}} - \frac {\log \left ({\left (e^{\left (b x + a\right )} - 1\right )} e^{\left (-a\right )}\right )}{b^{2}}\right )} + \frac {6 \, {\left (b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )\right )} c d^{2}}{b^{3}} + \frac {6 \, {\left (b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )\right )} c d^{2}}{b^{3}} + \frac {2 \, c^{3}}{b {\left (e^{\left (-2 \, b x - 2 \, a\right )} - 1\right )}} - \frac {2 \, {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2}\right )}}{b e^{\left (2 \, b x + 2 \, a\right )} - b} + \frac {3 \, {\left (b^{2} x^{2} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (b x + a\right )})\right )} d^{3}}{b^{4}} + \frac {3 \, {\left (b^{2} x^{2} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (b x + a\right )})\right )} d^{3}}{b^{4}} - \frac {2 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2}\right )}}{b^{4}} \] Input:

integrate((d*x+c)^3*csch(b*x+a)^2,x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int (c+d x)^3 \text {csch}^2(a+b x) \, dx=\int { {\left (d x + c\right )}^{3} \operatorname {csch}\left (b x + a\right )^{2} \,d x } \] Input:

integrate((d*x+c)^3*csch(b*x+a)^2,x, algorithm="giac")
 

Output:

integrate((d*x + c)^3*csch(b*x + a)^2, x)
 

Mupad [F(-1)]

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

int((c + d*x)^3/sinh(a + b*x)^2,x)
 

Output:

int((c + d*x)^3/sinh(a + b*x)^2, x)
 

Reduce [F]

\[ \int (c+d x)^3 \text {csch}^2(a+b x) \, dx=\frac {-4 b^{3} d^{3} x^{3}-6 b^{2} d^{3} x^{2}+3 e^{2 b x +2 a} \mathrm {log}\left (e^{b x +a}-1\right ) d^{3}+3 e^{2 b x +2 a} \mathrm {log}\left (e^{b x +a}+1\right ) d^{3}-12 e^{2 b x +2 a} \left (\int \frac {x^{2}}{e^{4 b x +4 a}-2 e^{2 b x +2 a}+1}d x \right ) b^{3} d^{3}-12 e^{2 b x +2 a} \left (\int \frac {x}{e^{4 b x +4 a}-2 e^{2 b x +2 a}+1}d x \right ) b^{2} d^{3}+24 \left (\int \frac {x}{e^{4 b x +4 a}-2 e^{2 b x +2 a}+1}d x \right ) b^{3} c \,d^{2}-6 \,\mathrm {log}\left (e^{b x +a}-1\right ) b^{2} c^{2} d -6 \,\mathrm {log}\left (e^{b x +a}-1\right ) b c \,d^{2}-6 \,\mathrm {log}\left (e^{b x +a}+1\right ) b^{2} c^{2} d -6 \,\mathrm {log}\left (e^{b x +a}+1\right ) b c \,d^{2}-6 e^{2 b x +2 a} b \,d^{3} x -12 b^{3} c \,d^{2} x^{2}-4 e^{2 b x +2 a} b^{3} c^{3}-12 e^{2 b x +2 a} b^{2} c \,d^{2} x -3 \,\mathrm {log}\left (e^{b x +a}-1\right ) d^{3}-3 \,\mathrm {log}\left (e^{b x +a}+1\right ) d^{3}-24 e^{2 b x +2 a} \left (\int \frac {x}{e^{4 b x +4 a}-2 e^{2 b x +2 a}+1}d x \right ) b^{3} c \,d^{2}-12 e^{2 b x +2 a} b^{3} c^{2} d x +12 \left (\int \frac {x}{e^{4 b x +4 a}-2 e^{2 b x +2 a}+1}d x \right ) b^{2} d^{3}+6 e^{2 b x +2 a} \mathrm {log}\left (e^{b x +a}-1\right ) b^{2} c^{2} d +6 e^{2 b x +2 a} \mathrm {log}\left (e^{b x +a}-1\right ) b c \,d^{2}+6 e^{2 b x +2 a} \mathrm {log}\left (e^{b x +a}+1\right ) b^{2} c^{2} d +6 e^{2 b x +2 a} \mathrm {log}\left (e^{b x +a}+1\right ) b c \,d^{2}+12 \left (\int \frac {x^{2}}{e^{4 b x +4 a}-2 e^{2 b x +2 a}+1}d x \right ) b^{3} d^{3}}{2 b^{4} \left (e^{2 b x +2 a}-1\right )} \] Input:

int((d*x+c)^3*csch(b*x+a)^2,x)
 

Output:

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