3.217 \(\int \frac{(e+f x)^3 \text{csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx\)

Optimal. Leaf size=546 \[ -\frac{12 i f^2 (e+f x) \text{PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac{3 i f^2 (e+f x) \text{PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac{9 f^2 (e+f x) \text{PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac{9 f^2 (e+f x) \text{PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac{9 f (e+f x)^2 \text{PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}-\frac{9 f (e+f x)^2 \text{PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}-\frac{3 f^3 \text{PolyLog}\left (2,-e^{c+d x}\right )}{a d^4}+\frac{3 f^3 \text{PolyLog}\left (2,e^{c+d x}\right )}{a d^4}+\frac{12 i f^3 \text{PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}+\frac{3 i f^3 \text{PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^4}+\frac{9 f^3 \text{PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}-\frac{9 f^3 \text{PolyLog}\left (4,e^{c+d x}\right )}{a d^4}-\frac{6 f^2 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^3}-\frac{6 i f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac{3 i f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}-\frac{3 f (e+f x)^2 \text{csch}(c+d x)}{2 a d^2}+\frac{3 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac{i (e+f x)^3 \tanh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right )}{a d}+\frac{i (e+f x)^3 \coth (c+d x)}{a d}-\frac{(e+f x)^3 \coth (c+d x) \text{csch}(c+d x)}{2 a d}+\frac{2 i (e+f x)^3}{a d} \]

[Out]

((2*I)*(e + f*x)^3)/(a*d) - (6*f^2*(e + f*x)*ArcTanh[E^(c + d*x)])/(a*d^3) + (3*(e + f*x)^3*ArcTanh[E^(c + d*x
)])/(a*d) + (I*(e + f*x)^3*Coth[c + d*x])/(a*d) - (3*f*(e + f*x)^2*Csch[c + d*x])/(2*a*d^2) - ((e + f*x)^3*Cot
h[c + d*x]*Csch[c + d*x])/(2*a*d) - ((6*I)*f*(e + f*x)^2*Log[1 + I*E^(c + d*x)])/(a*d^2) - ((3*I)*f*(e + f*x)^
2*Log[1 - E^(2*(c + d*x))])/(a*d^2) - (3*f^3*PolyLog[2, -E^(c + d*x)])/(a*d^4) + (9*f*(e + f*x)^2*PolyLog[2, -
E^(c + d*x)])/(2*a*d^2) - ((12*I)*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^3) + (3*f^3*PolyLog[2, E^(c
+ d*x)])/(a*d^4) - (9*f*(e + f*x)^2*PolyLog[2, E^(c + d*x)])/(2*a*d^2) - ((3*I)*f^2*(e + f*x)*PolyLog[2, E^(2
*(c + d*x))])/(a*d^3) - (9*f^2*(e + f*x)*PolyLog[3, -E^(c + d*x)])/(a*d^3) + ((12*I)*f^3*PolyLog[3, (-I)*E^(c
+ d*x)])/(a*d^4) + (9*f^2*(e + f*x)*PolyLog[3, E^(c + d*x)])/(a*d^3) + (((3*I)/2)*f^3*PolyLog[3, E^(2*(c + d*x
))])/(a*d^4) + (9*f^3*PolyLog[4, -E^(c + d*x)])/(a*d^4) - (9*f^3*PolyLog[4, E^(c + d*x)])/(a*d^4) + (I*(e + f*
x)^3*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(a*d)

________________________________________________________________________________________

Rubi [A]  time = 1.21183, antiderivative size = 546, normalized size of antiderivative = 1., number of steps used = 40, number of rules used = 13, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.419, Rules used = {5575, 4186, 4182, 2279, 2391, 2531, 6609, 2282, 6589, 4184, 3716, 2190, 3318} \[ -\frac{12 i f^2 (e+f x) \text{PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac{3 i f^2 (e+f x) \text{PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac{9 f^2 (e+f x) \text{PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac{9 f^2 (e+f x) \text{PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac{9 f (e+f x)^2 \text{PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}-\frac{9 f (e+f x)^2 \text{PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}-\frac{3 f^3 \text{PolyLog}\left (2,-e^{c+d x}\right )}{a d^4}+\frac{3 f^3 \text{PolyLog}\left (2,e^{c+d x}\right )}{a d^4}+\frac{12 i f^3 \text{PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}+\frac{3 i f^3 \text{PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^4}+\frac{9 f^3 \text{PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}-\frac{9 f^3 \text{PolyLog}\left (4,e^{c+d x}\right )}{a d^4}-\frac{6 f^2 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^3}-\frac{6 i f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac{3 i f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}-\frac{3 f (e+f x)^2 \text{csch}(c+d x)}{2 a d^2}+\frac{3 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac{i (e+f x)^3 \tanh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right )}{a d}+\frac{i (e+f x)^3 \coth (c+d x)}{a d}-\frac{(e+f x)^3 \coth (c+d x) \text{csch}(c+d x)}{2 a d}+\frac{2 i (e+f x)^3}{a d} \]

Antiderivative was successfully verified.

[In]

Int[((e + f*x)^3*Csch[c + d*x]^3)/(a + I*a*Sinh[c + d*x]),x]

[Out]

((2*I)*(e + f*x)^3)/(a*d) - (6*f^2*(e + f*x)*ArcTanh[E^(c + d*x)])/(a*d^3) + (3*(e + f*x)^3*ArcTanh[E^(c + d*x
)])/(a*d) + (I*(e + f*x)^3*Coth[c + d*x])/(a*d) - (3*f*(e + f*x)^2*Csch[c + d*x])/(2*a*d^2) - ((e + f*x)^3*Cot
h[c + d*x]*Csch[c + d*x])/(2*a*d) - ((6*I)*f*(e + f*x)^2*Log[1 + I*E^(c + d*x)])/(a*d^2) - ((3*I)*f*(e + f*x)^
2*Log[1 - E^(2*(c + d*x))])/(a*d^2) - (3*f^3*PolyLog[2, -E^(c + d*x)])/(a*d^4) + (9*f*(e + f*x)^2*PolyLog[2, -
E^(c + d*x)])/(2*a*d^2) - ((12*I)*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^3) + (3*f^3*PolyLog[2, E^(c
+ d*x)])/(a*d^4) - (9*f*(e + f*x)^2*PolyLog[2, E^(c + d*x)])/(2*a*d^2) - ((3*I)*f^2*(e + f*x)*PolyLog[2, E^(2
*(c + d*x))])/(a*d^3) - (9*f^2*(e + f*x)*PolyLog[3, -E^(c + d*x)])/(a*d^3) + ((12*I)*f^3*PolyLog[3, (-I)*E^(c
+ d*x)])/(a*d^4) + (9*f^2*(e + f*x)*PolyLog[3, E^(c + d*x)])/(a*d^3) + (((3*I)/2)*f^3*PolyLog[3, E^(2*(c + d*x
))])/(a*d^4) + (9*f^3*PolyLog[4, -E^(c + d*x)])/(a*d^4) - (9*f^3*PolyLog[4, E^(c + d*x)])/(a*d^4) + (I*(e + f*
x)^3*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(a*d)

Rule 5575

Int[(Csch[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[1/a, Int[(e + f*x)^m*Csch[c + d*x]^n, x], x] - Dist[b/a, Int[((e + f*x)^m*Csch[c + d*x]^(n - 1))/
(a + b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]

Rule 4186

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

Rule 4182

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

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2531

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] + Dist[(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 6609

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

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{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 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> 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]

Rule 4184

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp[((c + d*x)^m*Cot[e + f*x])/f, x]
+ Dist[(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 3716

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c
+ d*x)^(m + 1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(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 3318

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(2*a)^n, Int[(c
+ d*x)^m*Sin[(1*(e + (Pi*a)/(2*b)))/2 + (f*x)/2]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2
- b^2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[m, 0])

Rubi steps

\begin{align*} \int \frac{(e+f x)^3 \text{csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\left (i \int \frac{(e+f x)^3 \text{csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx\right )+\frac{\int (e+f x)^3 \text{csch}^3(c+d x) \, dx}{a}\\ &=-\frac{3 f (e+f x)^2 \text{csch}(c+d x)}{2 a d^2}-\frac{(e+f x)^3 \coth (c+d x) \text{csch}(c+d x)}{2 a d}-\frac{i \int (e+f x)^3 \text{csch}^2(c+d x) \, dx}{a}-\frac{\int (e+f x)^3 \text{csch}(c+d x) \, dx}{2 a}+\frac{\left (3 f^2\right ) \int (e+f x) \text{csch}(c+d x) \, dx}{a d^2}-\int \frac{(e+f x)^3 \text{csch}(c+d x)}{a+i a \sinh (c+d x)} \, dx\\ &=-\frac{6 f^2 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^3}+\frac{(e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac{i (e+f x)^3 \coth (c+d x)}{a d}-\frac{3 f (e+f x)^2 \text{csch}(c+d x)}{2 a d^2}-\frac{(e+f x)^3 \coth (c+d x) \text{csch}(c+d x)}{2 a d}+i \int \frac{(e+f x)^3}{a+i a \sinh (c+d x)} \, dx-\frac{\int (e+f x)^3 \text{csch}(c+d x) \, dx}{a}-\frac{(3 i f) \int (e+f x)^2 \coth (c+d x) \, dx}{a d}+\frac{(3 f) \int (e+f x)^2 \log \left (1-e^{c+d x}\right ) \, dx}{2 a d}-\frac{(3 f) \int (e+f x)^2 \log \left (1+e^{c+d x}\right ) \, dx}{2 a d}-\frac{\left (3 f^3\right ) \int \log \left (1-e^{c+d x}\right ) \, dx}{a d^3}+\frac{\left (3 f^3\right ) \int \log \left (1+e^{c+d x}\right ) \, dx}{a d^3}\\ &=\frac{i (e+f x)^3}{a d}-\frac{6 f^2 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^3}+\frac{3 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac{i (e+f x)^3 \coth (c+d x)}{a d}-\frac{3 f (e+f x)^2 \text{csch}(c+d x)}{2 a d^2}-\frac{(e+f x)^3 \coth (c+d x) \text{csch}(c+d x)}{2 a d}+\frac{3 f (e+f x)^2 \text{Li}_2\left (-e^{c+d x}\right )}{2 a d^2}-\frac{3 f (e+f x)^2 \text{Li}_2\left (e^{c+d x}\right )}{2 a d^2}+\frac{i \int (e+f x)^3 \csc ^2\left (\frac{1}{2} \left (i c+\frac{\pi }{2}\right )+\frac{i d x}{2}\right ) \, dx}{2 a}+\frac{(6 i f) \int \frac{e^{2 (c+d x)} (e+f x)^2}{1-e^{2 (c+d x)}} \, dx}{a d}+\frac{(3 f) \int (e+f x)^2 \log \left (1-e^{c+d x}\right ) \, dx}{a d}-\frac{(3 f) \int (e+f x)^2 \log \left (1+e^{c+d x}\right ) \, dx}{a d}-\frac{\left (3 f^2\right ) \int (e+f x) \text{Li}_2\left (-e^{c+d x}\right ) \, dx}{a d^2}+\frac{\left (3 f^2\right ) \int (e+f x) \text{Li}_2\left (e^{c+d x}\right ) \, dx}{a d^2}-\frac{\left (3 f^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}+\frac{\left (3 f^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}\\ &=\frac{i (e+f x)^3}{a d}-\frac{6 f^2 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^3}+\frac{3 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac{i (e+f x)^3 \coth (c+d x)}{a d}-\frac{3 f (e+f x)^2 \text{csch}(c+d x)}{2 a d^2}-\frac{(e+f x)^3 \coth (c+d x) \text{csch}(c+d x)}{2 a d}-\frac{3 i f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}-\frac{3 f^3 \text{Li}_2\left (-e^{c+d x}\right )}{a d^4}+\frac{9 f (e+f x)^2 \text{Li}_2\left (-e^{c+d x}\right )}{2 a d^2}+\frac{3 f^3 \text{Li}_2\left (e^{c+d x}\right )}{a d^4}-\frac{9 f (e+f x)^2 \text{Li}_2\left (e^{c+d x}\right )}{2 a d^2}-\frac{3 f^2 (e+f x) \text{Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac{3 f^2 (e+f x) \text{Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac{i (e+f x)^3 \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{a d}-\frac{(3 i f) \int (e+f x)^2 \coth \left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \, dx}{a d}+\frac{\left (6 i f^2\right ) \int (e+f x) \log \left (1-e^{2 (c+d x)}\right ) \, dx}{a d^2}-\frac{\left (6 f^2\right ) \int (e+f x) \text{Li}_2\left (-e^{c+d x}\right ) \, dx}{a d^2}+\frac{\left (6 f^2\right ) \int (e+f x) \text{Li}_2\left (e^{c+d x}\right ) \, dx}{a d^2}+\frac{\left (3 f^3\right ) \int \text{Li}_3\left (-e^{c+d x}\right ) \, dx}{a d^3}-\frac{\left (3 f^3\right ) \int \text{Li}_3\left (e^{c+d x}\right ) \, dx}{a d^3}\\ &=\frac{2 i (e+f x)^3}{a d}-\frac{6 f^2 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^3}+\frac{3 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac{i (e+f x)^3 \coth (c+d x)}{a d}-\frac{3 f (e+f x)^2 \text{csch}(c+d x)}{2 a d^2}-\frac{(e+f x)^3 \coth (c+d x) \text{csch}(c+d x)}{2 a d}-\frac{3 i f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}-\frac{3 f^3 \text{Li}_2\left (-e^{c+d x}\right )}{a d^4}+\frac{9 f (e+f x)^2 \text{Li}_2\left (-e^{c+d x}\right )}{2 a d^2}+\frac{3 f^3 \text{Li}_2\left (e^{c+d x}\right )}{a d^4}-\frac{9 f (e+f x)^2 \text{Li}_2\left (e^{c+d x}\right )}{2 a d^2}-\frac{3 i f^2 (e+f x) \text{Li}_2\left (e^{2 (c+d x)}\right )}{a d^3}-\frac{9 f^2 (e+f x) \text{Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac{9 f^2 (e+f x) \text{Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac{i (e+f x)^3 \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{a d}+\frac{(6 f) \int \frac{e^{2 \left (\frac{c}{2}+\frac{d x}{2}\right )} (e+f x)^2}{1+i e^{2 \left (\frac{c}{2}+\frac{d x}{2}\right )}} \, dx}{a d}+\frac{\left (3 f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}-\frac{\left (3 f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}+\frac{\left (3 i f^3\right ) \int \text{Li}_2\left (e^{2 (c+d x)}\right ) \, dx}{a d^3}+\frac{\left (6 f^3\right ) \int \text{Li}_3\left (-e^{c+d x}\right ) \, dx}{a d^3}-\frac{\left (6 f^3\right ) \int \text{Li}_3\left (e^{c+d x}\right ) \, dx}{a d^3}\\ &=\frac{2 i (e+f x)^3}{a d}-\frac{6 f^2 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^3}+\frac{3 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac{i (e+f x)^3 \coth (c+d x)}{a d}-\frac{3 f (e+f x)^2 \text{csch}(c+d x)}{2 a d^2}-\frac{(e+f x)^3 \coth (c+d x) \text{csch}(c+d x)}{2 a d}-\frac{6 i f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac{3 i f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}-\frac{3 f^3 \text{Li}_2\left (-e^{c+d x}\right )}{a d^4}+\frac{9 f (e+f x)^2 \text{Li}_2\left (-e^{c+d x}\right )}{2 a d^2}+\frac{3 f^3 \text{Li}_2\left (e^{c+d x}\right )}{a d^4}-\frac{9 f (e+f x)^2 \text{Li}_2\left (e^{c+d x}\right )}{2 a d^2}-\frac{3 i f^2 (e+f x) \text{Li}_2\left (e^{2 (c+d x)}\right )}{a d^3}-\frac{9 f^2 (e+f x) \text{Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac{9 f^2 (e+f x) \text{Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac{3 f^3 \text{Li}_4\left (-e^{c+d x}\right )}{a d^4}-\frac{3 f^3 \text{Li}_4\left (e^{c+d x}\right )}{a d^4}+\frac{i (e+f x)^3 \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{a d}+\frac{\left (12 i f^2\right ) \int (e+f x) \log \left (1+i e^{2 \left (\frac{c}{2}+\frac{d x}{2}\right )}\right ) \, dx}{a d^2}+\frac{\left (3 i f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a d^4}+\frac{\left (6 f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}-\frac{\left (6 f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}\\ &=\frac{2 i (e+f x)^3}{a d}-\frac{6 f^2 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^3}+\frac{3 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac{i (e+f x)^3 \coth (c+d x)}{a d}-\frac{3 f (e+f x)^2 \text{csch}(c+d x)}{2 a d^2}-\frac{(e+f x)^3 \coth (c+d x) \text{csch}(c+d x)}{2 a d}-\frac{6 i f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac{3 i f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}-\frac{3 f^3 \text{Li}_2\left (-e^{c+d x}\right )}{a d^4}+\frac{9 f (e+f x)^2 \text{Li}_2\left (-e^{c+d x}\right )}{2 a d^2}-\frac{12 i f^2 (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{a d^3}+\frac{3 f^3 \text{Li}_2\left (e^{c+d x}\right )}{a d^4}-\frac{9 f (e+f x)^2 \text{Li}_2\left (e^{c+d x}\right )}{2 a d^2}-\frac{3 i f^2 (e+f x) \text{Li}_2\left (e^{2 (c+d x)}\right )}{a d^3}-\frac{9 f^2 (e+f x) \text{Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac{9 f^2 (e+f x) \text{Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac{3 i f^3 \text{Li}_3\left (e^{2 (c+d x)}\right )}{2 a d^4}+\frac{9 f^3 \text{Li}_4\left (-e^{c+d x}\right )}{a d^4}-\frac{9 f^3 \text{Li}_4\left (e^{c+d x}\right )}{a d^4}+\frac{i (e+f x)^3 \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{a d}+\frac{\left (12 i f^3\right ) \int \text{Li}_2\left (-i e^{2 \left (\frac{c}{2}+\frac{d x}{2}\right )}\right ) \, dx}{a d^3}\\ &=\frac{2 i (e+f x)^3}{a d}-\frac{6 f^2 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^3}+\frac{3 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac{i (e+f x)^3 \coth (c+d x)}{a d}-\frac{3 f (e+f x)^2 \text{csch}(c+d x)}{2 a d^2}-\frac{(e+f x)^3 \coth (c+d x) \text{csch}(c+d x)}{2 a d}-\frac{6 i f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac{3 i f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}-\frac{3 f^3 \text{Li}_2\left (-e^{c+d x}\right )}{a d^4}+\frac{9 f (e+f x)^2 \text{Li}_2\left (-e^{c+d x}\right )}{2 a d^2}-\frac{12 i f^2 (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{a d^3}+\frac{3 f^3 \text{Li}_2\left (e^{c+d x}\right )}{a d^4}-\frac{9 f (e+f x)^2 \text{Li}_2\left (e^{c+d x}\right )}{2 a d^2}-\frac{3 i f^2 (e+f x) \text{Li}_2\left (e^{2 (c+d x)}\right )}{a d^3}-\frac{9 f^2 (e+f x) \text{Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac{9 f^2 (e+f x) \text{Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac{3 i f^3 \text{Li}_3\left (e^{2 (c+d x)}\right )}{2 a d^4}+\frac{9 f^3 \text{Li}_4\left (-e^{c+d x}\right )}{a d^4}-\frac{9 f^3 \text{Li}_4\left (e^{c+d x}\right )}{a d^4}+\frac{i (e+f x)^3 \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{a d}+\frac{\left (12 i f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{2 \left (\frac{c}{2}+\frac{d x}{2}\right )}\right )}{a d^4}\\ &=\frac{2 i (e+f x)^3}{a d}-\frac{6 f^2 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^3}+\frac{3 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac{i (e+f x)^3 \coth (c+d x)}{a d}-\frac{3 f (e+f x)^2 \text{csch}(c+d x)}{2 a d^2}-\frac{(e+f x)^3 \coth (c+d x) \text{csch}(c+d x)}{2 a d}-\frac{6 i f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac{3 i f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}-\frac{3 f^3 \text{Li}_2\left (-e^{c+d x}\right )}{a d^4}+\frac{9 f (e+f x)^2 \text{Li}_2\left (-e^{c+d x}\right )}{2 a d^2}-\frac{12 i f^2 (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{a d^3}+\frac{3 f^3 \text{Li}_2\left (e^{c+d x}\right )}{a d^4}-\frac{9 f (e+f x)^2 \text{Li}_2\left (e^{c+d x}\right )}{2 a d^2}-\frac{3 i f^2 (e+f x) \text{Li}_2\left (e^{2 (c+d x)}\right )}{a d^3}-\frac{9 f^2 (e+f x) \text{Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac{12 i f^3 \text{Li}_3\left (-i e^{c+d x}\right )}{a d^4}+\frac{9 f^2 (e+f x) \text{Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac{3 i f^3 \text{Li}_3\left (e^{2 (c+d x)}\right )}{2 a d^4}+\frac{9 f^3 \text{Li}_4\left (-e^{c+d x}\right )}{a d^4}-\frac{9 f^3 \text{Li}_4\left (e^{c+d x}\right )}{a d^4}+\frac{i (e+f x)^3 \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{a d}\\ \end{align*}

Mathematica [B]  time = 68.6696, size = 2479, normalized size = 4.54 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((e + f*x)^3*Csch[c + d*x]^3)/(a + I*a*Sinh[c + d*x]),x]

[Out]

(-3*e^3*Log[Tanh[(c + d*x)/2]])/(2*a*d) + (3*e*f^2*Log[Tanh[(c + d*x)/2]])/(a*d^3) - (9*e^2*f*(-(c*Log[Tanh[(c
+ d*x)/2]]) - I*((I*c + I*d*x)*(Log[1 - E^(I*(I*c + I*d*x))] - Log[1 + E^(I*(I*c + I*d*x))]) + I*(PolyLog[2,
-E^(I*(I*c + I*d*x))] - PolyLog[2, E^(I*(I*c + I*d*x))]))))/(2*a*d^2) + (3*f^3*(-(c*Log[Tanh[(c + d*x)/2]]) -
I*((I*c + I*d*x)*(Log[1 - E^(I*(I*c + I*d*x))] - Log[1 + E^(I*(I*c + I*d*x))]) + I*(PolyLog[2, -E^(I*(I*c + I*
d*x))] - PolyLog[2, E^(I*(I*c + I*d*x))]))))/(a*d^4) - (2*(d^3*(e + f*x)^3 + 3*d^2*(1 + I*E^c)*f*(e + f*x)^2*L
og[1 - I*E^(-c - d*x)] + (6*I)*(I - E^c)*f^2*(d*(e + f*x)*PolyLog[2, I*E^(-c - d*x)] + f*PolyLog[3, I*E^(-c -
d*x)])))/(a*d^4*(-I + E^c)) + ((I/2)*E^c*f^3*Csch[c]*((2*d^3*x^3)/E^(2*c) - 3*d^2*(1 - E^(-2*c))*x^2*Log[1 - E
^(-c - d*x)] - 3*d^2*(1 - E^(-2*c))*x^2*Log[1 + E^(-c - d*x)] + 6*(1 - E^(-2*c))*(d*x*PolyLog[2, -E^(-c - d*x)
] + PolyLog[3, -E^(-c - d*x)]) + 6*(1 - E^(-2*c))*(d*x*PolyLog[2, E^(-c - d*x)] + PolyLog[3, E^(-c - d*x)])))/
(a*d^4) + (9*e*f^2*(d^2*x^2*ArcTanh[Cosh[c + d*x] + Sinh[c + d*x]] + d*x*PolyLog[2, -Cosh[c + d*x] - Sinh[c +
d*x]] - d*x*PolyLog[2, Cosh[c + d*x] + Sinh[c + d*x]] - PolyLog[3, -Cosh[c + d*x] - Sinh[c + d*x]] + PolyLog[3
, Cosh[c + d*x] + Sinh[c + d*x]]))/(a*d^3) - (3*f^3*(-2*d^3*x^3*ArcTanh[Cosh[c + d*x] + Sinh[c + d*x]] - 3*d^2
*x^2*PolyLog[2, -Cosh[c + d*x] - Sinh[c + d*x]] + 3*d^2*x^2*PolyLog[2, Cosh[c + d*x] + Sinh[c + d*x]] + 6*d*x*
PolyLog[3, -Cosh[c + d*x] - Sinh[c + d*x]] - 6*d*x*PolyLog[3, Cosh[c + d*x] + Sinh[c + d*x]] - 6*PolyLog[4, -C
osh[c + d*x] - Sinh[c + d*x]] + 6*PolyLog[4, Cosh[c + d*x] + Sinh[c + d*x]]))/(2*a*d^4) + ((3*I)*e^2*f*Csch[c]
*(-(d*x*Cosh[c]) + Log[Cosh[d*x]*Sinh[c] + Cosh[c]*Sinh[d*x]]*Sinh[c]))/(a*d^2*(-Cosh[c]^2 + Sinh[c]^2)) + (Cs
ch[c]*Csch[c + d*x]^2*(3*e^2*f*Cosh[(d*x)/2] + 6*e*f^2*x*Cosh[(d*x)/2] + 3*f^3*x^2*Cosh[(d*x)/2] + 3*e^2*f*Cos
h[(3*d*x)/2] + 6*e*f^2*x*Cosh[(3*d*x)/2] + 3*f^3*x^2*Cosh[(3*d*x)/2] + (5*I)*d*e^3*Cosh[c - (d*x)/2] + (15*I)*
d*e^2*f*x*Cosh[c - (d*x)/2] + (15*I)*d*e*f^2*x^2*Cosh[c - (d*x)/2] + (5*I)*d*f^3*x^3*Cosh[c - (d*x)/2] - I*d*e
^3*Cosh[c + (d*x)/2] - (3*I)*d*e^2*f*x*Cosh[c + (d*x)/2] - (3*I)*d*e*f^2*x^2*Cosh[c + (d*x)/2] - I*d*f^3*x^3*C
osh[c + (d*x)/2] - 3*e^2*f*Cosh[2*c + (d*x)/2] - 6*e*f^2*x*Cosh[2*c + (d*x)/2] - 3*f^3*x^2*Cosh[2*c + (d*x)/2]
+ I*d*e^3*Cosh[c + (3*d*x)/2] + (3*I)*d*e^2*f*x*Cosh[c + (3*d*x)/2] + (3*I)*d*e*f^2*x^2*Cosh[c + (3*d*x)/2] +
I*d*f^3*x^3*Cosh[c + (3*d*x)/2] - 3*e^2*f*Cosh[2*c + (3*d*x)/2] - 6*e*f^2*x*Cosh[2*c + (3*d*x)/2] - 3*f^3*x^2
*Cosh[2*c + (3*d*x)/2] - (3*I)*d*e^3*Cosh[3*c + (3*d*x)/2] - (9*I)*d*e^2*f*x*Cosh[3*c + (3*d*x)/2] - (9*I)*d*e
*f^2*x^2*Cosh[3*c + (3*d*x)/2] - (3*I)*d*f^3*x^3*Cosh[3*c + (3*d*x)/2] - (4*I)*d*e^3*Cosh[c + (5*d*x)/2] - (12
*I)*d*e^2*f*x*Cosh[c + (5*d*x)/2] - (12*I)*d*e*f^2*x^2*Cosh[c + (5*d*x)/2] - (4*I)*d*f^3*x^3*Cosh[c + (5*d*x)/
2] + (2*I)*d*e^3*Cosh[3*c + (5*d*x)/2] + (6*I)*d*e^2*f*x*Cosh[3*c + (5*d*x)/2] + (6*I)*d*e*f^2*x^2*Cosh[3*c +
(5*d*x)/2] + (2*I)*d*f^3*x^3*Cosh[3*c + (5*d*x)/2] - d*e^3*Sinh[(d*x)/2] - 3*d*e^2*f*x*Sinh[(d*x)/2] - 3*d*e*f
^2*x^2*Sinh[(d*x)/2] - d*f^3*x^3*Sinh[(d*x)/2] - d*e^3*Sinh[(3*d*x)/2] - 3*d*e^2*f*x*Sinh[(3*d*x)/2] - 3*d*e*f
^2*x^2*Sinh[(3*d*x)/2] - d*f^3*x^3*Sinh[(3*d*x)/2] + (3*I)*e^2*f*Sinh[c - (d*x)/2] + (6*I)*e*f^2*x*Sinh[c - (d
*x)/2] + (3*I)*f^3*x^2*Sinh[c - (d*x)/2] + (3*I)*e^2*f*Sinh[c + (d*x)/2] + (6*I)*e*f^2*x*Sinh[c + (d*x)/2] + (
3*I)*f^3*x^2*Sinh[c + (d*x)/2] - 3*d*e^3*Sinh[2*c + (d*x)/2] - 9*d*e^2*f*x*Sinh[2*c + (d*x)/2] - 9*d*e*f^2*x^2
*Sinh[2*c + (d*x)/2] - 3*d*f^3*x^3*Sinh[2*c + (d*x)/2] + (3*I)*e^2*f*Sinh[c + (3*d*x)/2] + (6*I)*e*f^2*x*Sinh[
c + (3*d*x)/2] + (3*I)*f^3*x^2*Sinh[c + (3*d*x)/2] - d*e^3*Sinh[2*c + (3*d*x)/2] - 3*d*e^2*f*x*Sinh[2*c + (3*d
*x)/2] - 3*d*e*f^2*x^2*Sinh[2*c + (3*d*x)/2] - d*f^3*x^3*Sinh[2*c + (3*d*x)/2] - (3*I)*e^2*f*Sinh[3*c + (3*d*x
)/2] - (6*I)*e*f^2*x*Sinh[3*c + (3*d*x)/2] - (3*I)*f^3*x^2*Sinh[3*c + (3*d*x)/2] + 2*d*e^3*Sinh[2*c + (5*d*x)/
2] + 6*d*e^2*f*x*Sinh[2*c + (5*d*x)/2] + 6*d*e*f^2*x^2*Sinh[2*c + (5*d*x)/2] + 2*d*f^3*x^3*Sinh[2*c + (5*d*x)/
2]))/(8*a*d^2*(Cosh[c/2] + I*Sinh[c/2])*(Cosh[c/2 + (d*x)/2] + I*Sinh[c/2 + (d*x)/2])) - ((3*I)*e*f^2*Csch[c]*
Sech[c]*(-((d^2*x^2)/E^ArcTanh[Tanh[c]]) + (I*(-(d*x*(-Pi + (2*I)*ArcTanh[Tanh[c]])) - Pi*Log[1 + E^(2*d*x)] -
2*(I*d*x + I*ArcTanh[Tanh[c]])*Log[1 - E^((2*I)*(I*d*x + I*ArcTanh[Tanh[c]]))] + Pi*Log[Cosh[d*x]] + (2*I)*Ar
cTanh[Tanh[c]]*Log[I*Sinh[d*x + ArcTanh[Tanh[c]]]] + I*PolyLog[2, E^((2*I)*(I*d*x + I*ArcTanh[Tanh[c]]))])*Tan
h[c])/Sqrt[1 - Tanh[c]^2]))/(a*d^3*Sqrt[Sech[c]^2*(Cosh[c]^2 - Sinh[c]^2)])

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Maple [B]  time = 0.256, size = 2058, normalized size = 3.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)^3*csch(d*x+c)^3/(a+I*a*sinh(d*x+c)),x)

[Out]

-6*I/a/d^2*ln(1-exp(d*x+c))*e*f^2*x-6*I/a/d^2*ln(exp(d*x+c)+1)*e*f^2*x+24*I/a/d^2*c*e*f^2*x-6*I/a/d^3*ln(1-exp
(d*x+c))*c*e*f^2+6*I/a/d^3*e*f^2*c*ln(exp(d*x+c)-1)-24*I/a/d^3*e*f^2*c*ln(exp(d*x+c))+6*I/d^4/a*f^3*c^2*ln(1+I
*exp(d*x+c))-6*I/d^2/a*ln(exp(d*x+c)-I)*e^2*f-12*I/d^3/a*f^3*polylog(2,-I*exp(d*x+c))*x-6*I/d^2/a*f^3*ln(1+I*e
xp(d*x+c))*x^2-12*I/d^3/a*e*f^2*polylog(2,-I*exp(d*x+c))-6*I/d^4/a*f^3*c^2*ln(exp(d*x+c)-I)-12*I/d^2/a*e*f^2*l
n(1+I*exp(d*x+c))*x-12*I/d^3/a*e*f^2*ln(1+I*exp(d*x+c))*c+12*I/d^3/a*e*f^2*c*ln(exp(d*x+c)-I)-3*f^3*polylog(2,
-exp(d*x+c))/a/d^4+3*f^3*polylog(2,exp(d*x+c))/a/d^4+9*f^3*polylog(4,-exp(d*x+c))/a/d^4-9*f^3*polylog(4,exp(d*
x+c))/a/d^4+12*I*f^3*polylog(3,-I*exp(d*x+c))/a/d^4-(-3*I*f^3*x^2*exp(3*d*x+3*c)-3*I*d*e^3*exp(3*d*x+3*c)-3*I*
e^2*f*exp(3*d*x+3*c)+4*d*e^3+3*I*d*e*f^2*x^2*exp(d*x+c)-5*d*f^3*x^3*exp(2*d*x+2*c)-6*e*f^2*x*exp(2*d*x+2*c)+3*
d*f^3*x^3*exp(4*d*x+4*c)+6*e*f^2*x*exp(4*d*x+4*c)+I*d*f^3*x^3*exp(d*x+c)+9*d*e*f^2*x^2*exp(4*d*x+4*c)+9*d*e^2*
f*x*exp(4*d*x+4*c)-6*I*e*f^2*x*exp(3*d*x+3*c)-3*I*d*f^3*x^3*exp(3*d*x+3*c)+4*d*f^3*x^3+3*I*d*e^2*f*x*exp(d*x+c
)-15*d*e*f^2*x^2*exp(2*d*x+2*c)-15*d*e^2*f*x*exp(2*d*x+2*c)+12*d*e*f^2*x^2+12*d*e^2*f*x+3*I*exp(d*x+c)*e^2*f+3
*I*f^3*x^2*exp(d*x+c)+6*I*e*f^2*x*exp(d*x+c)+I*d*e^3*exp(d*x+c)-9*I*d*e*f^2*x^2*exp(3*d*x+3*c)-9*I*d*e^2*f*x*e
xp(3*d*x+3*c)-3*f^3*x^2*exp(2*d*x+2*c)-3*e^2*f*exp(2*d*x+2*c)+3*f^3*x^2*exp(4*d*x+4*c)+3*d*e^3*exp(4*d*x+4*c)+
3*e^2*f*exp(4*d*x+4*c)-5*d*e^3*exp(2*d*x+2*c))/(exp(2*d*x+2*c)-1)^2/d^2/(exp(d*x+c)-I)/a-9/2/d^3/a*e*f^2*c^2*l
n(exp(d*x+c)-1)+9/2/d^2/a*e^2*f*c*ln(exp(d*x+c)-1)-9/2/d^2/a*ln(1-exp(d*x+c))*c*e^2*f-9/2/d/a*ln(1-exp(d*x+c))
*e^2*f*x+9/2/d/a*ln(exp(d*x+c)+1)*e^2*f*x+9/2/d^3/a*e*f^2*c^2*ln(1-exp(d*x+c))-9/2/d/a*e*f^2*ln(1-exp(d*x+c))*
x^2-9/d^2/a*e*f^2*polylog(2,exp(d*x+c))*x+9/2/d/a*e*f^2*ln(exp(d*x+c)+1)*x^2+9/d^2/a*e*f^2*polylog(2,-exp(d*x+
c))*x-3*I/a/d^4*f^3*c^2*ln(exp(d*x+c)-1)-12*I/a/d^3*f^3*c^2*x+12*I/a/d*e*f^2*x^2+3*I/a/d^4*f^3*c^2*ln(1-exp(d*
x+c))-3*I/a/d^2*f^3*ln(1-exp(d*x+c))*x^2-3*I/a/d^2*f^3*ln(exp(d*x+c)+1)*x^2-6*I/a/d^3*f^3*polylog(2,-exp(d*x+c
))*x-6*I/a/d^3*f^3*polylog(2,exp(d*x+c))*x+12*I/a/d^4*f^3*c^2*ln(exp(d*x+c))-3*I/a/d^2*e^2*f*ln(exp(d*x+c)-1)-
3*I/a/d^2*e^2*f*ln(exp(d*x+c)+1)+12*I/a/d^2*e^2*f*ln(exp(d*x+c))+12*I/a/d^3*c^2*e*f^2-6*I/a/d^3*e*f^2*polylog(
2,exp(d*x+c))-6*I/a/d^3*e*f^2*polylog(2,-exp(d*x+c))+3/a/d^3*e*f^2*ln(exp(d*x+c)-1)-3/a/d^3*e*f^2*ln(exp(d*x+c
)+1)+3/a/d^3*f^3*ln(1-exp(d*x+c))*x-3/2/d/a*e^3*ln(exp(d*x+c)-1)+3/2/d/a*e^3*ln(exp(d*x+c)+1)-3/a/d^3*f^3*ln(e
xp(d*x+c)+1)*x+3/a/d^4*f^3*c*ln(1-exp(d*x+c))-3/a/d^4*f^3*c*ln(exp(d*x+c)-1)+4*I/a/d*f^3*x^3-8*I/a/d^4*f^3*c^3
+6*I/a/d^4*f^3*polylog(3,exp(d*x+c))+6*I/a/d^4*f^3*polylog(3,-exp(d*x+c))-9/2/d^2/a*f^3*polylog(2,exp(d*x+c))*
x^2+9/d^3/a*f^3*polylog(3,exp(d*x+c))*x-9/2/d^2/a*e^2*f*polylog(2,exp(d*x+c))+9/2/d^2/a*e^2*f*polylog(2,-exp(d
*x+c))+9/d^3/a*e*f^2*polylog(3,exp(d*x+c))-9/d^3/a*e*f^2*polylog(3,-exp(d*x+c))+3/2/d^4/a*f^3*c^3*ln(exp(d*x+c
)-1)+3/2/d/a*f^3*ln(exp(d*x+c)+1)*x^3+9/2/d^2/a*f^3*polylog(2,-exp(d*x+c))*x^2-9/d^3/a*f^3*polylog(3,-exp(d*x+
c))*x-3/2/d/a*f^3*ln(1-exp(d*x+c))*x^3-3/2/d^4/a*f^3*ln(1-exp(d*x+c))*c^3

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Maxima [B]  time = 2.74902, size = 1777, normalized size = 3.25 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*csch(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="maxima")

[Out]

-1/2*e^3*(16*(-I*e^(-d*x - c) - 5*e^(-2*d*x - 2*c) + 3*I*e^(-3*d*x - 3*c) + 3*e^(-4*d*x - 4*c) + 4)/((8*a*e^(-
d*x - c) - 16*I*a*e^(-2*d*x - 2*c) - 16*a*e^(-3*d*x - 3*c) + 8*I*a*e^(-4*d*x - 4*c) + 8*a*e^(-5*d*x - 5*c) + 8
*I*a)*d) - 3*log(e^(-d*x - c) + 1)/(a*d) + 3*log(e^(-d*x - c) - 1)/(a*d)) + 6*I*e^2*f*x/(a*d) - 6*I*e^2*f*log(
I*e^(d*x + c) + 1)/(a*d^2) - (4*d*f^3*x^3 + 12*d*e*f^2*x^2 + 12*d*e^2*f*x + 3*(d*f^3*x^3*e^(4*c) + e^2*f*e^(4*
c) + (3*d*e*f^2 + f^3)*x^2*e^(4*c) + (3*d*e^2*f + 2*e*f^2)*x*e^(4*c))*e^(4*d*x) + (-3*I*d*f^3*x^3*e^(3*c) - 3*
I*e^2*f*e^(3*c) + (-9*I*d*e*f^2 - 3*I*f^3)*x^2*e^(3*c) + (-9*I*d*e^2*f - 6*I*e*f^2)*x*e^(3*c))*e^(3*d*x) - (5*
d*f^3*x^3*e^(2*c) + 3*e^2*f*e^(2*c) + 3*(5*d*e*f^2 + f^3)*x^2*e^(2*c) + 3*(5*d*e^2*f + 2*e*f^2)*x*e^(2*c))*e^(
2*d*x) + (I*d*f^3*x^3*e^c + 3*I*e^2*f*e^c + (3*I*d*e*f^2 + 3*I*f^3)*x^2*e^c + (3*I*d*e^2*f + 6*I*e*f^2)*x*e^c)
*e^(d*x))/(a*d^2*e^(5*d*x + 5*c) - I*a*d^2*e^(4*d*x + 4*c) - 2*a*d^2*e^(3*d*x + 3*c) + 2*I*a*d^2*e^(2*d*x + 2*
c) + a*d^2*e^(d*x + c) - I*a*d^2) - 12*I*(d*x*log(I*e^(d*x + c) + 1) + dilog(-I*e^(d*x + c)))*e*f^2/(a*d^3) +
3/2*(d^3*x^3*log(e^(d*x + c) + 1) + 3*d^2*x^2*dilog(-e^(d*x + c)) - 6*d*x*polylog(3, -e^(d*x + c)) + 6*polylog
(4, -e^(d*x + c)))*f^3/(a*d^4) - 3/2*(d^3*x^3*log(-e^(d*x + c) + 1) + 3*d^2*x^2*dilog(e^(d*x + c)) - 6*d*x*pol
ylog(3, e^(d*x + c)) + 6*polylog(4, e^(d*x + c)))*f^3/(a*d^4) - 6*I*(d^2*x^2*log(I*e^(d*x + c) + 1) + 2*d*x*di
log(-I*e^(d*x + c)) - 2*polylog(3, -I*e^(d*x + c)))*f^3/(a*d^4) - 3*(-I*d*e^2*f + e*f^2)*x/(a*d^2) - 3*(-I*d*e
^2*f - e*f^2)*x/(a*d^2) + 3*(-I*d*e^2*f - e*f^2)*log(e^(d*x + c) + 1)/(a*d^3) + 3*(-I*d*e^2*f + e*f^2)*log(e^(
d*x + c) - 1)/(a*d^3) - 3/2*(d^2*x^2*log(-e^(d*x + c) + 1) + 2*d*x*dilog(e^(d*x + c)) - 2*polylog(3, e^(d*x +
c)))*(3*d*e*f^2 + 2*I*f^3)/(a*d^4) + 3/2*(d^2*x^2*log(e^(d*x + c) + 1) + 2*d*x*dilog(-e^(d*x + c)) - 2*polylog
(3, -e^(d*x + c)))*(3*d*e*f^2 - 2*I*f^3)/(a*d^4) + 1/2*(9*d^2*e^2*f - 12*I*d*e*f^2 - 6*f^3)*(d*x*log(e^(d*x +
c) + 1) + dilog(-e^(d*x + c)))/(a*d^4) - 1/2*(9*d^2*e^2*f + 12*I*d*e*f^2 - 6*f^3)*(d*x*log(-e^(d*x + c) + 1) +
dilog(e^(d*x + c)))/(a*d^4) + 1/8*(3*d^4*f^3*x^4 + 4*(3*d*e*f^2 + 2*I*f^3)*d^3*x^3 + (18*d^2*e^2*f + 24*I*d*e
*f^2 - 12*f^3)*d^2*x^2)/(a*d^4) - 1/8*(3*d^4*f^3*x^4 + 4*(3*d*e*f^2 - 2*I*f^3)*d^3*x^3 + (18*d^2*e^2*f - 24*I*
d*e*f^2 - 12*f^3)*d^2*x^2)/(a*d^4) + (2*I*d^3*f^3*x^3 + 6*I*d^3*e*f^2*x^2)/(a*d^4)

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Fricas [C]  time = 3.72538, size = 10116, normalized size = 18.53 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*csch(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-(8*d^3*e^3 - 24*c*d^2*e^2*f + 24*c^2*d*e*f^2 - 8*c^3*f^3 + (24*d*f^3*x + 24*d*e*f^2 - (-24*I*d*f^3*x - 24*I*d
*e*f^2)*e^(5*d*x + 5*c) + 24*(d*f^3*x + d*e*f^2)*e^(4*d*x + 4*c) - (48*I*d*f^3*x + 48*I*d*e*f^2)*e^(3*d*x + 3*
c) - 48*(d*f^3*x + d*e*f^2)*e^(2*d*x + 2*c) - (-24*I*d*f^3*x - 24*I*d*e*f^2)*e^(d*x + c))*dilog(-I*e^(d*x + c)
) - (-9*I*d^2*f^3*x^2 - 9*I*d^2*e^2*f - 12*d*e*f^2 + 6*I*f^3 - 6*(3*I*d^2*e*f^2 + 2*d*f^3)*x + (9*d^2*f^3*x^2
+ 9*d^2*e^2*f - 12*I*d*e*f^2 - 6*f^3 + (18*d^2*e*f^2 - 12*I*d*f^3)*x)*e^(5*d*x + 5*c) + (-9*I*d^2*f^3*x^2 - 9*
I*d^2*e^2*f - 12*d*e*f^2 + 6*I*f^3 - 6*(3*I*d^2*e*f^2 + 2*d*f^3)*x)*e^(4*d*x + 4*c) - (18*d^2*f^3*x^2 + 18*d^2
*e^2*f - 24*I*d*e*f^2 - 12*f^3 + (36*d^2*e*f^2 - 24*I*d*f^3)*x)*e^(3*d*x + 3*c) + (18*I*d^2*f^3*x^2 + 18*I*d^2
*e^2*f + 24*d*e*f^2 - 12*I*f^3 - 12*(-3*I*d^2*e*f^2 - 2*d*f^3)*x)*e^(2*d*x + 2*c) + (9*d^2*f^3*x^2 + 9*d^2*e^2
*f - 12*I*d*e*f^2 - 6*f^3 + (18*d^2*e*f^2 - 12*I*d*f^3)*x)*e^(d*x + c))*dilog(-e^(d*x + c)) - (9*I*d^2*f^3*x^2
+ 9*I*d^2*e^2*f - 12*d*e*f^2 - 6*I*f^3 - 6*(-3*I*d^2*e*f^2 + 2*d*f^3)*x - (9*d^2*f^3*x^2 + 9*d^2*e^2*f + 12*I
*d*e*f^2 - 6*f^3 + (18*d^2*e*f^2 + 12*I*d*f^3)*x)*e^(5*d*x + 5*c) + (9*I*d^2*f^3*x^2 + 9*I*d^2*e^2*f - 12*d*e*
f^2 - 6*I*f^3 - 6*(-3*I*d^2*e*f^2 + 2*d*f^3)*x)*e^(4*d*x + 4*c) + (18*d^2*f^3*x^2 + 18*d^2*e^2*f + 24*I*d*e*f^
2 - 12*f^3 + (36*d^2*e*f^2 + 24*I*d*f^3)*x)*e^(3*d*x + 3*c) + (-18*I*d^2*f^3*x^2 - 18*I*d^2*e^2*f + 24*d*e*f^2
+ 12*I*f^3 - 12*(3*I*d^2*e*f^2 - 2*d*f^3)*x)*e^(2*d*x + 2*c) - (9*d^2*f^3*x^2 + 9*d^2*e^2*f + 12*I*d*e*f^2 -
6*f^3 + (18*d^2*e*f^2 + 12*I*d*f^3)*x)*e^(d*x + c))*dilog(e^(d*x + c)) - (8*I*d^3*f^3*x^3 + 24*I*d^3*e*f^2*x^2
+ 24*I*d^3*e^2*f*x + 24*I*c*d^2*e^2*f - 24*I*c^2*d*e*f^2 + 8*I*c^3*f^3)*e^(5*d*x + 5*c) - 2*(d^3*f^3*x^3 - 3*
d^3*e^3 + 3*(4*c - 1)*d^2*e^2*f - 12*c^2*d*e*f^2 + 4*c^3*f^3 + 3*(d^3*e*f^2 - d^2*f^3)*x^2 + 3*(d^3*e^2*f - 2*
d^2*e*f^2)*x)*e^(4*d*x + 4*c) - (-10*I*d^3*f^3*x^3 + 6*I*d^3*e^3 + (-48*I*c + 6*I)*d^2*e^2*f + 48*I*c^2*d*e*f^
2 - 16*I*c^3*f^3 + (-30*I*d^3*e*f^2 + 6*I*d^2*f^3)*x^2 + (-30*I*d^3*e^2*f + 12*I*d^2*e*f^2)*x)*e^(3*d*x + 3*c)
+ 2*(3*d^3*f^3*x^3 - 5*d^3*e^3 + 3*(8*c - 1)*d^2*e^2*f - 24*c^2*d*e*f^2 + 8*c^3*f^3 + 3*(3*d^3*e*f^2 - d^2*f^
3)*x^2 + 3*(3*d^3*e^2*f - 2*d^2*e*f^2)*x)*e^(2*d*x + 2*c) - (6*I*d^3*f^3*x^3 - 2*I*d^3*e^3 + (24*I*c - 6*I)*d^
2*e^2*f - 24*I*c^2*d*e*f^2 + 8*I*c^3*f^3 + (18*I*d^3*e*f^2 - 6*I*d^2*f^3)*x^2 + (18*I*d^3*e^2*f - 12*I*d^2*e*f
^2)*x)*e^(d*x + c) - (-3*I*d^3*f^3*x^3 - 3*I*d^3*e^3 - 6*d^2*e^2*f + 6*I*d*e*f^2 - 3*(3*I*d^3*e*f^2 + 2*d^2*f^
3)*x^2 + (-9*I*d^3*e^2*f - 12*d^2*e*f^2 + 6*I*d*f^3)*x + (3*d^3*f^3*x^3 + 3*d^3*e^3 - 6*I*d^2*e^2*f - 6*d*e*f^
2 + (9*d^3*e*f^2 - 6*I*d^2*f^3)*x^2 + 3*(3*d^3*e^2*f - 4*I*d^2*e*f^2 - 2*d*f^3)*x)*e^(5*d*x + 5*c) + (-3*I*d^3
*f^3*x^3 - 3*I*d^3*e^3 - 6*d^2*e^2*f + 6*I*d*e*f^2 - 3*(3*I*d^3*e*f^2 + 2*d^2*f^3)*x^2 + (-9*I*d^3*e^2*f - 12*
d^2*e*f^2 + 6*I*d*f^3)*x)*e^(4*d*x + 4*c) - (6*d^3*f^3*x^3 + 6*d^3*e^3 - 12*I*d^2*e^2*f - 12*d*e*f^2 + (18*d^3
*e*f^2 - 12*I*d^2*f^3)*x^2 + 6*(3*d^3*e^2*f - 4*I*d^2*e*f^2 - 2*d*f^3)*x)*e^(3*d*x + 3*c) + (6*I*d^3*f^3*x^3 +
6*I*d^3*e^3 + 12*d^2*e^2*f - 12*I*d*e*f^2 - 6*(-3*I*d^3*e*f^2 - 2*d^2*f^3)*x^2 + (18*I*d^3*e^2*f + 24*d^2*e*f
^2 - 12*I*d*f^3)*x)*e^(2*d*x + 2*c) + (3*d^3*f^3*x^3 + 3*d^3*e^3 - 6*I*d^2*e^2*f - 6*d*e*f^2 + (9*d^3*e*f^2 -
6*I*d^2*f^3)*x^2 + 3*(3*d^3*e^2*f - 4*I*d^2*e*f^2 - 2*d*f^3)*x)*e^(d*x + c))*log(e^(d*x + c) + 1) + (12*d^2*e^
2*f - 24*c*d*e*f^2 + 12*c^2*f^3 - (-12*I*d^2*e^2*f + 24*I*c*d*e*f^2 - 12*I*c^2*f^3)*e^(5*d*x + 5*c) + 12*(d^2*
e^2*f - 2*c*d*e*f^2 + c^2*f^3)*e^(4*d*x + 4*c) - (24*I*d^2*e^2*f - 48*I*c*d*e*f^2 + 24*I*c^2*f^3)*e^(3*d*x + 3
*c) - 24*(d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*e^(2*d*x + 2*c) - (-12*I*d^2*e^2*f + 24*I*c*d*e*f^2 - 12*I*c^2*f^
3)*e^(d*x + c))*log(e^(d*x + c) - I) - (3*I*d^3*e^3 - 3*(3*I*c + 2)*d^2*e^2*f + (9*I*c^2 + 12*c - 6*I)*d*e*f^2
+ (-3*I*c^3 - 6*c^2 + 6*I*c)*f^3 - (3*d^3*e^3 - (9*c - 6*I)*d^2*e^2*f + 3*(3*c^2 - 4*I*c - 2)*d*e*f^2 - (3*c^
3 - 6*I*c^2 - 6*c)*f^3)*e^(5*d*x + 5*c) + (3*I*d^3*e^3 - 3*(3*I*c + 2)*d^2*e^2*f + (9*I*c^2 + 12*c - 6*I)*d*e*
f^2 + (-3*I*c^3 - 6*c^2 + 6*I*c)*f^3)*e^(4*d*x + 4*c) + (6*d^3*e^3 - (18*c - 12*I)*d^2*e^2*f + 6*(3*c^2 - 4*I*
c - 2)*d*e*f^2 - (6*c^3 - 12*I*c^2 - 12*c)*f^3)*e^(3*d*x + 3*c) + (-6*I*d^3*e^3 - 6*(-3*I*c - 2)*d^2*e^2*f + (
-18*I*c^2 - 24*c + 12*I)*d*e*f^2 + (6*I*c^3 + 12*c^2 - 12*I*c)*f^3)*e^(2*d*x + 2*c) - (3*d^3*e^3 - (9*c - 6*I)
*d^2*e^2*f + 3*(3*c^2 - 4*I*c - 2)*d*e*f^2 - (3*c^3 - 6*I*c^2 - 6*c)*f^3)*e^(d*x + c))*log(e^(d*x + c) - 1) +
(12*d^2*f^3*x^2 + 24*d^2*e*f^2*x + 24*c*d*e*f^2 - 12*c^2*f^3 - (-12*I*d^2*f^3*x^2 - 24*I*d^2*e*f^2*x - 24*I*c*
d*e*f^2 + 12*I*c^2*f^3)*e^(5*d*x + 5*c) + 12*(d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3)*e^(4*d*x +
4*c) - (24*I*d^2*f^3*x^2 + 48*I*d^2*e*f^2*x + 48*I*c*d*e*f^2 - 24*I*c^2*f^3)*e^(3*d*x + 3*c) - 24*(d^2*f^3*x^2
+ 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3)*e^(2*d*x + 2*c) - (-12*I*d^2*f^3*x^2 - 24*I*d^2*e*f^2*x - 24*I*c*d*e
*f^2 + 12*I*c^2*f^3)*e^(d*x + c))*log(I*e^(d*x + c) + 1) - (3*I*d^3*f^3*x^3 + 9*I*c*d^2*e^2*f + (-9*I*c^2 - 12
*c)*d*e*f^2 + (3*I*c^3 + 6*c^2 - 6*I*c)*f^3 - 3*(-3*I*d^3*e*f^2 + 2*d^2*f^3)*x^2 + (9*I*d^3*e^2*f - 12*d^2*e*f
^2 - 6*I*d*f^3)*x - (3*d^3*f^3*x^3 + 9*c*d^2*e^2*f - 3*(3*c^2 - 4*I*c)*d*e*f^2 + (3*c^3 - 6*I*c^2 - 6*c)*f^3 +
(9*d^3*e*f^2 + 6*I*d^2*f^3)*x^2 + 3*(3*d^3*e^2*f + 4*I*d^2*e*f^2 - 2*d*f^3)*x)*e^(5*d*x + 5*c) + (3*I*d^3*f^3
*x^3 + 9*I*c*d^2*e^2*f + (-9*I*c^2 - 12*c)*d*e*f^2 + (3*I*c^3 + 6*c^2 - 6*I*c)*f^3 - 3*(-3*I*d^3*e*f^2 + 2*d^2
*f^3)*x^2 + (9*I*d^3*e^2*f - 12*d^2*e*f^2 - 6*I*d*f^3)*x)*e^(4*d*x + 4*c) + (6*d^3*f^3*x^3 + 18*c*d^2*e^2*f -
6*(3*c^2 - 4*I*c)*d*e*f^2 + (6*c^3 - 12*I*c^2 - 12*c)*f^3 + (18*d^3*e*f^2 + 12*I*d^2*f^3)*x^2 + 6*(3*d^3*e^2*f
+ 4*I*d^2*e*f^2 - 2*d*f^3)*x)*e^(3*d*x + 3*c) + (-6*I*d^3*f^3*x^3 - 18*I*c*d^2*e^2*f + (18*I*c^2 + 24*c)*d*e*
f^2 + (-6*I*c^3 - 12*c^2 + 12*I*c)*f^3 - 6*(3*I*d^3*e*f^2 - 2*d^2*f^3)*x^2 + (-18*I*d^3*e^2*f + 24*d^2*e*f^2 +
12*I*d*f^3)*x)*e^(2*d*x + 2*c) - (3*d^3*f^3*x^3 + 9*c*d^2*e^2*f - 3*(3*c^2 - 4*I*c)*d*e*f^2 + (3*c^3 - 6*I*c^
2 - 6*c)*f^3 + (9*d^3*e*f^2 + 6*I*d^2*f^3)*x^2 + 3*(3*d^3*e^2*f + 4*I*d^2*e*f^2 - 2*d*f^3)*x)*e^(d*x + c))*log
(-e^(d*x + c) + 1) - (18*f^3*e^(5*d*x + 5*c) - 18*I*f^3*e^(4*d*x + 4*c) - 36*f^3*e^(3*d*x + 3*c) + 36*I*f^3*e^
(2*d*x + 2*c) + 18*f^3*e^(d*x + c) - 18*I*f^3)*polylog(4, -e^(d*x + c)) + (18*f^3*e^(5*d*x + 5*c) - 18*I*f^3*e
^(4*d*x + 4*c) - 36*f^3*e^(3*d*x + 3*c) + 36*I*f^3*e^(2*d*x + 2*c) + 18*f^3*e^(d*x + c) - 18*I*f^3)*polylog(4,
e^(d*x + c)) - (24*I*f^3*e^(5*d*x + 5*c) + 24*f^3*e^(4*d*x + 4*c) - 48*I*f^3*e^(3*d*x + 3*c) - 48*f^3*e^(2*d*
x + 2*c) + 24*I*f^3*e^(d*x + c) + 24*f^3)*polylog(3, -I*e^(d*x + c)) - (18*I*d*f^3*x + 18*I*d*e*f^2 + 12*f^3 -
6*(3*d*f^3*x + 3*d*e*f^2 - 2*I*f^3)*e^(5*d*x + 5*c) + (18*I*d*f^3*x + 18*I*d*e*f^2 + 12*f^3)*e^(4*d*x + 4*c)
+ 12*(3*d*f^3*x + 3*d*e*f^2 - 2*I*f^3)*e^(3*d*x + 3*c) + (-36*I*d*f^3*x - 36*I*d*e*f^2 - 24*f^3)*e^(2*d*x + 2*
c) - 6*(3*d*f^3*x + 3*d*e*f^2 - 2*I*f^3)*e^(d*x + c))*polylog(3, -e^(d*x + c)) - (-18*I*d*f^3*x - 18*I*d*e*f^2
+ 12*f^3 + 6*(3*d*f^3*x + 3*d*e*f^2 + 2*I*f^3)*e^(5*d*x + 5*c) + (-18*I*d*f^3*x - 18*I*d*e*f^2 + 12*f^3)*e^(4
*d*x + 4*c) - 12*(3*d*f^3*x + 3*d*e*f^2 + 2*I*f^3)*e^(3*d*x + 3*c) + (36*I*d*f^3*x + 36*I*d*e*f^2 - 24*f^3)*e^
(2*d*x + 2*c) + 6*(3*d*f^3*x + 3*d*e*f^2 + 2*I*f^3)*e^(d*x + c))*polylog(3, e^(d*x + c)))/(2*a*d^4*e^(5*d*x +
5*c) - 2*I*a*d^4*e^(4*d*x + 4*c) - 4*a*d^4*e^(3*d*x + 3*c) + 4*I*a*d^4*e^(2*d*x + 2*c) + 2*a*d^4*e^(d*x + c) -
2*I*a*d^4)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)**3*csch(d*x+c)**3/(a+I*a*sinh(d*x+c)),x)

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*csch(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out