\(\int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx\) [217]

Optimal result

Integrand size = 31, antiderivative size = 546 \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {2 i (e+f x)^3}{a d}-\frac {6 f^2 (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d^3}+\frac {3 (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}+\frac {i (e+f x)^3 \coth (c+d x)}{a d}-\frac {3 f (e+f x)^2 \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {6 i f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}-\frac {3 f^3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^4}+\frac {9 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}-\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^4}-\frac {9 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}-\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {12 i f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}+\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac {3 i f^3 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {9 f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}-\frac {9 f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a d^4}+\frac {i (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \]

[Out]

12*I*f^3*polylog(3,-I*exp(d*x+c))/a/d^4-6*f^2*(f*x+e)*arctanh(exp(d*x+c))/a/d^3+3*(f*x+e)^3*arctanh(exp(d*x+c)
)/a/d-3*I*f*(f*x+e)^2*ln(1-exp(2*d*x+2*c))/a/d^2-3/2*f*(f*x+e)^2*csch(d*x+c)/a/d^2-1/2*(f*x+e)^3*coth(d*x+c)*c
sch(d*x+c)/a/d+I*(f*x+e)^3*coth(d*x+c)/a/d-6*I*f*(f*x+e)^2*ln(1+I*exp(d*x+c))/a/d^2-3*f^3*polylog(2,-exp(d*x+c
))/a/d^4+9/2*f*(f*x+e)^2*polylog(2,-exp(d*x+c))/a/d^2-3*I*f^2*(f*x+e)*polylog(2,exp(2*d*x+2*c))/a/d^3+3*f^3*po
lylog(2,exp(d*x+c))/a/d^4-9/2*f*(f*x+e)^2*polylog(2,exp(d*x+c))/a/d^2+I*(f*x+e)^3*tanh(1/2*c+1/4*I*Pi+1/2*d*x)
/a/d-9*f^2*(f*x+e)*polylog(3,-exp(d*x+c))/a/d^3+2*I*(f*x+e)^3/a/d+9*f^2*(f*x+e)*polylog(3,exp(d*x+c))/a/d^3-12
*I*f^2*(f*x+e)*polylog(2,-I*exp(d*x+c))/a/d^3+9*f^3*polylog(4,-exp(d*x+c))/a/d^4-9*f^3*polylog(4,exp(d*x+c))/a
/d^4+3/2*I*f^3*polylog(3,exp(2*d*x+2*c))/a/d^4

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 546, normalized size of antiderivative = 1.00, number of steps used = 40, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {5694, 4271, 4267, 2317, 2438, 2611, 6744, 2320, 6724, 4269, 3797, 2221, 3399} \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {6 f^2 (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d^3}+\frac {3 (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {3 f^3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^4}+\frac {3 f^3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^4}+\frac {12 i f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}+\frac {3 i f^3 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {9 f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}-\frac {9 f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a d^4}-\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac {9 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}-\frac {9 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}-\frac {6 i f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}-\frac {3 f (e+f x)^2 \text {csch}(c+d x)}{2 a d^2}+\frac {i (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{a d}+\frac {i (e+f x)^3 \coth (c+d x)}{a d}-\frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 a d}+\frac {2 i (e+f x)^3}{a d} \]

[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 2221

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]
- 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 2317

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 2320

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 2438

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 2611

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 3399

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(2*a)^n, Int[(c
+ d*x)^m*Sin[(1/2)*(e + Pi*(a/(2*b))) + 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])

Rule 3797

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))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 4267

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 4269

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 4271

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]) /; Free
Q[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rule 5694

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 6724

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 6744

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]

Rubi steps \begin{align*} \text {integral}& = -\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) \text {arctanh}\left (e^{c+d x}\right )}{a d^3}+\frac {(e+f x)^3 \text {arctanh}\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) \text {arctanh}\left (e^{c+d x}\right )}{a d^3}+\frac {3 (e+f x)^3 \text {arctanh}\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 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}-\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,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) \operatorname {PolyLog}\left (2,-e^{c+d x}\right ) \, dx}{a d^2}+\frac {\left (3 f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right ) \, dx}{a d^2}-\frac {\left (3 f^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}+\frac {\left (3 f^3\right ) \text {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) \text {arctanh}\left (e^{c+d x}\right )}{a d^3}+\frac {3 (e+f x)^3 \text {arctanh}\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 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^4}+\frac {9 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}+\frac {3 f^3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^4}-\frac {9 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}-\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,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) \operatorname {PolyLog}\left (2,-e^{c+d x}\right ) \, dx}{a d^2}+\frac {\left (6 f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right ) \, dx}{a d^2}+\frac {\left (3 f^3\right ) \int \operatorname {PolyLog}\left (3,-e^{c+d x}\right ) \, dx}{a d^3}-\frac {\left (3 f^3\right ) \int \operatorname {PolyLog}\left (3,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) \text {arctanh}\left (e^{c+d x}\right )}{a d^3}+\frac {3 (e+f x)^3 \text {arctanh}\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 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^4}+\frac {9 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}+\frac {3 f^3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^4}-\frac {9 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}-\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,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 ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}-\frac {\left (3 f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}+\frac {\left (3 i f^3\right ) \int \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right ) \, dx}{a d^3}+\frac {\left (6 f^3\right ) \int \operatorname {PolyLog}\left (3,-e^{c+d x}\right ) \, dx}{a d^3}-\frac {\left (6 f^3\right ) \int \operatorname {PolyLog}\left (3,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) \text {arctanh}\left (e^{c+d x}\right )}{a d^3}+\frac {3 (e+f x)^3 \text {arctanh}\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 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^4}+\frac {9 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}+\frac {3 f^3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^4}-\frac {9 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}-\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}-\frac {3 f^3 \operatorname {PolyLog}\left (4,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 ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {\left (6 f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}-\frac {\left (6 f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(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) \text {arctanh}\left (e^{c+d x}\right )}{a d^3}+\frac {3 (e+f x)^3 \text {arctanh}\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 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^4}+\frac {9 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}-\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^4}-\frac {9 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}-\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac {3 i f^3 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {9 f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}-\frac {9 f^3 \operatorname {PolyLog}\left (4,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 \operatorname {PolyLog}\left (2,-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) \text {arctanh}\left (e^{c+d x}\right )}{a d^3}+\frac {3 (e+f x)^3 \text {arctanh}\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 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^4}+\frac {9 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}-\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^4}-\frac {9 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}-\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac {3 i f^3 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {9 f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}-\frac {9 f^3 \operatorname {PolyLog}\left (4,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 ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(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) \text {arctanh}\left (e^{c+d x}\right )}{a d^3}+\frac {3 (e+f x)^3 \text {arctanh}\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 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^4}+\frac {9 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}-\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^4}-\frac {9 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}-\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {12 i f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}+\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac {3 i f^3 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {9 f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}-\frac {9 f^3 \operatorname {PolyLog}\left (4,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] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2585\) vs. \(2(546)=1092\).

Time = 90.16 (sec) , antiderivative size = 2585, normalized size of antiderivative = 4.73 \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

(-6*E^c*f*((e + f*x)^3/(3*E^c*f) + ((I + E^(-c))*(e + f*x)^2*Log[1 - I*E^(-c - d*x)])/d - ((2*I)*(-I + E^c)*f*
(d*(e + f*x)*PolyLog[2, I*E^(-c - d*x)] + f*PolyLog[3, I*E^(-c - d*x)]))/(d^3*E^c)))/(a*d*(-I + E^c)) + ((12*I
)*d^3*e^2*E^(2*c)*f*x + (12*I)*d^3*e*E^(2*c)*f^2*x^2 + (4*I)*d^3*E^(2*c)*f^3*x^3 - 6*d^3*e^3*ArcTanh[E^(c + d*
x)] + 6*d^3*e^3*E^(2*c)*ArcTanh[E^(c + d*x)] + 12*d*e*f^2*ArcTanh[E^(c + d*x)] - 12*d*e*E^(2*c)*f^2*ArcTanh[E^
(c + d*x)] + 9*d^3*e^2*f*x*Log[1 - E^(c + d*x)] - 9*d^3*e^2*E^(2*c)*f*x*Log[1 - E^(c + d*x)] - 6*d*f^3*x*Log[1
- E^(c + d*x)] + 6*d*E^(2*c)*f^3*x*Log[1 - E^(c + d*x)] + 9*d^3*e*f^2*x^2*Log[1 - E^(c + d*x)] - 9*d^3*e*E^(2
*c)*f^2*x^2*Log[1 - E^(c + d*x)] + 3*d^3*f^3*x^3*Log[1 - E^(c + d*x)] - 3*d^3*E^(2*c)*f^3*x^3*Log[1 - E^(c + d
*x)] - 9*d^3*e^2*f*x*Log[1 + E^(c + d*x)] + 9*d^3*e^2*E^(2*c)*f*x*Log[1 + E^(c + d*x)] + 6*d*f^3*x*Log[1 + E^(
c + d*x)] - 6*d*E^(2*c)*f^3*x*Log[1 + E^(c + d*x)] - 9*d^3*e*f^2*x^2*Log[1 + E^(c + d*x)] + 9*d^3*e*E^(2*c)*f^
2*x^2*Log[1 + E^(c + d*x)] - 3*d^3*f^3*x^3*Log[1 + E^(c + d*x)] + 3*d^3*E^(2*c)*f^3*x^3*Log[1 + E^(c + d*x)] +
(6*I)*d^2*e^2*f*Log[1 - E^(2*(c + d*x))] - (6*I)*d^2*e^2*E^(2*c)*f*Log[1 - E^(2*(c + d*x))] + (12*I)*d^2*e*f^
2*x*Log[1 - E^(2*(c + d*x))] - (12*I)*d^2*e*E^(2*c)*f^2*x*Log[1 - E^(2*(c + d*x))] + (6*I)*d^2*f^3*x^2*Log[1 -
E^(2*(c + d*x))] - (6*I)*d^2*E^(2*c)*f^3*x^2*Log[1 - E^(2*(c + d*x))] + 3*(-1 + E^(2*c))*f*(-2*f^2 + 3*d^2*(e
+ f*x)^2)*PolyLog[2, -E^(c + d*x)] - 3*(-1 + E^(2*c))*f*(-2*f^2 + 3*d^2*(e + f*x)^2)*PolyLog[2, E^(c + d*x)]
+ (6*I)*d*e*f^2*PolyLog[2, E^(2*(c + d*x))] - (6*I)*d*e*E^(2*c)*f^2*PolyLog[2, E^(2*(c + d*x))] + (6*I)*d*f^3*
x*PolyLog[2, E^(2*(c + d*x))] - (6*I)*d*E^(2*c)*f^3*x*PolyLog[2, E^(2*(c + d*x))] + 18*d*e*f^2*PolyLog[3, -E^(
c + d*x)] - 18*d*e*E^(2*c)*f^2*PolyLog[3, -E^(c + d*x)] + 18*d*f^3*x*PolyLog[3, -E^(c + d*x)] - 18*d*E^(2*c)*f
^3*x*PolyLog[3, -E^(c + d*x)] - 18*d*e*f^2*PolyLog[3, E^(c + d*x)] + 18*d*e*E^(2*c)*f^2*PolyLog[3, E^(c + d*x)
] - 18*d*f^3*x*PolyLog[3, E^(c + d*x)] + 18*d*E^(2*c)*f^3*x*PolyLog[3, E^(c + d*x)] - (3*I)*f^3*PolyLog[3, E^(
2*(c + d*x))] + (3*I)*E^(2*c)*f^3*PolyLog[3, E^(2*(c + d*x))] - 18*f^3*PolyLog[4, -E^(c + d*x)] + 18*E^(2*c)*f
^3*PolyLog[4, -E^(c + d*x)] + 18*f^3*PolyLog[4, E^(c + d*x)] - 18*E^(2*c)*f^3*PolyLog[4, E^(c + d*x)])/(2*a*d^
4*(-1 + E^(2*c))) + (Csch[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*Cosh[(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*Co
sh[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*Cosh[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*Co
sh[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*Si
nh[(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]))

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2123 vs. \(2 (504 ) = 1008\).

Time = 3.28 (sec) , antiderivative size = 2124, normalized size of antiderivative = 3.89

method	result	size
risch	\(\text {Expression too large to display}\)	\(2124\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 4252 vs. \(2 (485) = 970\).

Time = 0.32 (sec) , antiderivative size = 4252, normalized size of antiderivative = 7.79 \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1320 vs. \(2 (485) = 970\).

Time = 0.52 (sec) , antiderivative size = 1320, normalized size of antiderivative = 2.42 \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Too large to display} \]

[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*(2*(-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)/((a*e^(-d*x
- c) - 2*I*a*e^(-2*d*x - 2*c) - 2*a*e^(-3*d*x - 3*c) + I*a*e^(-4*d*x - 4*c) + a*e^(-5*d*x - 5*c) + 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) + I*e^2*f*e^(3*c
) + (3*I*d*e*f^2 + I*f^3)*x^2*e^(3*c) + (3*I*d*e^2*f + 2*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 - I*f^3)*x^2*e^c - 3*(-I*d*e^2*f - 2*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*polylog(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*dilog(-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) + 3/2*(3*d^2*e^2*f - 4*I*d*e*f^2 - 2*f^3)*(d*x*log(e^(d*x + c) + 1) + dilog(-e^
(d*x + c)))/(a*d^4) - 3/2*(3*d^2*e^2*f + 4*I*d*e*f^2 - 2*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 + 6*(3*d^2*e^2*f + 4*I*d*e*f^2 - 2*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 + 6*(3*d^2*e^2*f - 4*I*d*e*f^2 - 2*f^3)*d^2*
x^2)/(a*d^4) - 2*(-I*d^3*f^3*x^3 - 3*I*d^3*e*f^2*x^2)/(a*d^4)

Giac [F]

\[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \operatorname {csch}\left (d x + c\right )^{3}}{i \, a \sinh \left (d x + c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^3}{{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]

[In]

int((e + f*x)^3/(sinh(c + d*x)^3*(a + a*sinh(c + d*x)*1i)),x)

[Out]

int((e + f*x)^3/(sinh(c + d*x)^3*(a + a*sinh(c + d*x)*1i)), x)