\(\int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx\) [211]

Optimal result

Integrand size = 31, antiderivative size = 419 \[ \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {2 (e+f x)^3}{a d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {6 f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {12 f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac {12 f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}+\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {6 i f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}-\frac {6 i f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a d^4}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {5694, 4269, 3797, 2221, 2611, 2320, 6724, 4267, 6744, 3399} \[ \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {12 f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}-\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {6 i f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}-\frac {6 i f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a d^4}+\frac {12 f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac {6 f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{a d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}-\frac {2 (e+f x)^3}{a d} \]

[In]

Int[((e + f*x)^3*Csch[c + d*x]^2)/(a + I*a*Sinh[c + d*x]),x]

[Out]

(-2*(e + f*x)^3)/(a*d) + ((2*I)*(e + f*x)^3*ArcTanh[E^(c + d*x)])/(a*d) - ((e + f*x)^3*Coth[c + d*x])/(a*d) +
(6*f*(e + f*x)^2*Log[1 + I*E^(c + d*x)])/(a*d^2) + (3*f*(e + f*x)^2*Log[1 - E^(2*(c + d*x))])/(a*d^2) + ((3*I)
*f*(e + f*x)^2*PolyLog[2, -E^(c + d*x)])/(a*d^2) + (12*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^3) - (
(3*I)*f*(e + f*x)^2*PolyLog[2, E^(c + d*x)])/(a*d^2) + (3*f^2*(e + f*x)*PolyLog[2, E^(2*(c + d*x))])/(a*d^3) -
((6*I)*f^2*(e + f*x)*PolyLog[3, -E^(c + d*x)])/(a*d^3) - (12*f^3*PolyLog[3, (-I)*E^(c + d*x)])/(a*d^4) + ((6*
I)*f^2*(e + f*x)*PolyLog[3, E^(c + d*x)])/(a*d^3) - (3*f^3*PolyLog[3, E^(2*(c + d*x))])/(2*a*d^4) + ((6*I)*f^3
*PolyLog[4, -E^(c + d*x)])/(a*d^4) - ((6*I)*f^3*PolyLog[4, E^(c + d*x)])/(a*d^4) - ((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 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 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 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}(c+d x)}{a+i a \sinh (c+d x)} \, dx\right )+\frac {\int (e+f x)^3 \text {csch}^2(c+d x) \, dx}{a} \\ & = -\frac {(e+f x)^3 \coth (c+d x)}{a d}-\frac {i \int (e+f x)^3 \text {csch}(c+d x) \, dx}{a}+\frac {(3 f) \int (e+f x)^2 \coth (c+d x) \, dx}{a d}-\int \frac {(e+f x)^3}{a+i a \sinh (c+d x)} \, dx \\ & = -\frac {(e+f x)^3}{a d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}-\frac {\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 {(3 i f) \int (e+f x)^2 \log \left (1-e^{c+d x}\right ) \, dx}{a d}-\frac {(3 i f) \int (e+f x)^2 \log \left (1+e^{c+d x}\right ) \, dx}{a d}-\frac {(6 f) \int \frac {e^{2 (c+d x)} (e+f x)^2}{1-e^{2 (c+d x)}} \, dx}{a d} \\ & = -\frac {(e+f x)^3}{a d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(3 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) \operatorname {PolyLog}\left (2,-e^{c+d x}\right ) \, dx}{a d^2}+\frac {\left (6 i 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) \log \left (1-e^{2 (c+d x)}\right ) \, dx}{a d^2} \\ & = -\frac {2 (e+f x)^3}{a d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(6 i 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 (6 i f^3\right ) \int \operatorname {PolyLog}\left (3,-e^{c+d x}\right ) \, dx}{a d^3}-\frac {\left (6 i 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 (2,e^{2 (c+d x)}\right ) \, dx}{a d^3} \\ & = -\frac {2 (e+f x)^3}{a d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {6 f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {\left (12 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 (6 i 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 i 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}(2,x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a d^4} \\ & = -\frac {2 (e+f x)^3}{a d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {6 f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {12 f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {6 i f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}-\frac {6 i f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a d^4}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {\left (12 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 (e+f x)^3}{a d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {6 f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {12 f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {6 i f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}-\frac {6 i f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a d^4}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {\left (12 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 (e+f x)^3}{a d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {6 f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {12 f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac {12 f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}+\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {6 i f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}-\frac {6 i f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a d^4}-\frac {(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. \(1205\) vs. \(2(419)=838\).

Time = 8.70 (sec) , antiderivative size = 1205, normalized size of antiderivative = 2.88 \[ \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {6 i e^c f \left (\frac {e^{-c} (e+f x)^3}{3 f}+\frac {\left (i+e^{-c}\right ) (e+f x)^2 \log \left (1-i e^{-c-d x}\right )}{d}-\frac {2 i e^{-c} \left (-i+e^c\right ) f \left (d (e+f x) \operatorname {PolyLog}\left (2,i e^{-c-d x}\right )+f \operatorname {PolyLog}\left (3,i e^{-c-d x}\right )\right )}{d^3}\right )}{a d \left (-i+e^c\right )}+\frac {i d^3 e^2 \left (-1+e^{2 c}\right ) (d e+3 i f) x+d^3 e^2 \left (1-e^{2 c}\right ) (i d e+3 f) x-2 d^3 (e+f x)^3+3 d^2 e \left (-1+e^{2 c}\right ) f (-i d e+2 f) x \log \left (1-e^{-c-d x}\right )+3 d^2 \left (-1+e^{2 c}\right ) f^2 (-i d e+f) x^2 \log \left (1-e^{-c-d x}\right )-i d^3 \left (-1+e^{2 c}\right ) f^3 x^3 \log \left (1-e^{-c-d x}\right )+3 d^2 e \left (-1+e^{2 c}\right ) f (i d e+2 f) x \log \left (1+e^{-c-d x}\right )+3 d^2 \left (-1+e^{2 c}\right ) f^2 (i d e+f) x^2 \log \left (1+e^{-c-d x}\right )+i d^3 \left (-1+e^{2 c}\right ) f^3 x^3 \log \left (1+e^{-c-d x}\right )+d^2 e^2 \left (-1+e^{2 c}\right ) (-i d e+3 f) \log \left (1-e^{c+d x}\right )+d^2 e^2 \left (-1+e^{2 c}\right ) (i d e+3 f) \log \left (1+e^{c+d x}\right )+3 d e \left (1-e^{2 c}\right ) f (i d e+2 f) \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )+6 d \left (1-e^{2 c}\right ) f^2 (i d e+f) x \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )-3 i d^2 \left (-1+e^{2 c}\right ) f^3 x^2 \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )+3 i d e \left (-1+e^{2 c}\right ) (d e+2 i f) f \operatorname {PolyLog}\left (2,e^{-c-d x}\right )+6 i d \left (-1+e^{2 c}\right ) (d e+i f) f^2 x \operatorname {PolyLog}\left (2,e^{-c-d x}\right )+3 i d^2 \left (-1+e^{2 c}\right ) f^3 x^2 \operatorname {PolyLog}\left (2,e^{-c-d x}\right )-6 \left (-1+e^{2 c}\right ) f^2 (i d e+f) \operatorname {PolyLog}\left (3,-e^{-c-d x}\right )-6 i d \left (-1+e^{2 c}\right ) f^3 x \operatorname {PolyLog}\left (3,-e^{-c-d x}\right )+6 i \left (-1+e^{2 c}\right ) (d e+i f) f^2 \operatorname {PolyLog}\left (3,e^{-c-d x}\right )+6 i d \left (-1+e^{2 c}\right ) f^3 x \operatorname {PolyLog}\left (3,e^{-c-d x}\right )-6 i \left (-1+e^{2 c}\right ) f^3 \operatorname {PolyLog}\left (4,-e^{-c-d x}\right )+6 i \left (-1+e^{2 c}\right ) f^3 \operatorname {PolyLog}\left (4,e^{-c-d x}\right )}{a d^4 \left (-1+e^{2 c}\right )}+\frac {\text {sech}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (-e^3 \sinh \left (\frac {d x}{2}\right )-3 e^2 f x \sinh \left (\frac {d x}{2}\right )-3 e f^2 x^2 \sinh \left (\frac {d x}{2}\right )-f^3 x^3 \sinh \left (\frac {d x}{2}\right )\right )}{2 a d}+\frac {\text {csch}\left (\frac {c}{2}\right ) \text {csch}\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (e^3 \sinh \left (\frac {d x}{2}\right )+3 e^2 f x \sinh \left (\frac {d x}{2}\right )+3 e f^2 x^2 \sinh \left (\frac {d x}{2}\right )+f^3 x^3 \sinh \left (\frac {d x}{2}\right )\right )}{2 a d}-\frac {2 \left (e^3 \sinh \left (\frac {d x}{2}\right )+3 e^2 f x \sinh \left (\frac {d x}{2}\right )+3 e f^2 x^2 \sinh \left (\frac {d x}{2}\right )+f^3 x^3 \sinh \left (\frac {d x}{2}\right )\right )}{a d \left (\cosh \left (\frac {c}{2}\right )+i \sinh \left (\frac {c}{2}\right )\right ) \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )+i \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )} \]

[In]

Integrate[((e + f*x)^3*Csch[c + d*x]^2)/(a + I*a*Sinh[c + d*x]),x]

[Out]

((-6*I)*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)) + (I
*d^3*e^2*(-1 + E^(2*c))*(d*e + (3*I)*f)*x + d^3*e^2*(1 - E^(2*c))*(I*d*e + 3*f)*x - 2*d^3*(e + f*x)^3 + 3*d^2*
e*(-1 + E^(2*c))*f*((-I)*d*e + 2*f)*x*Log[1 - E^(-c - d*x)] + 3*d^2*(-1 + E^(2*c))*f^2*((-I)*d*e + f)*x^2*Log[
1 - E^(-c - d*x)] - I*d^3*(-1 + E^(2*c))*f^3*x^3*Log[1 - E^(-c - d*x)] + 3*d^2*e*(-1 + E^(2*c))*f*(I*d*e + 2*f
)*x*Log[1 + E^(-c - d*x)] + 3*d^2*(-1 + E^(2*c))*f^2*(I*d*e + f)*x^2*Log[1 + E^(-c - d*x)] + I*d^3*(-1 + E^(2*
c))*f^3*x^3*Log[1 + E^(-c - d*x)] + d^2*e^2*(-1 + E^(2*c))*((-I)*d*e + 3*f)*Log[1 - E^(c + d*x)] + d^2*e^2*(-1
+ E^(2*c))*(I*d*e + 3*f)*Log[1 + E^(c + d*x)] + 3*d*e*(1 - E^(2*c))*f*(I*d*e + 2*f)*PolyLog[2, -E^(-c - d*x)]
+ 6*d*(1 - E^(2*c))*f^2*(I*d*e + f)*x*PolyLog[2, -E^(-c - d*x)] - (3*I)*d^2*(-1 + E^(2*c))*f^3*x^2*PolyLog[2,
-E^(-c - d*x)] + (3*I)*d*e*(-1 + E^(2*c))*(d*e + (2*I)*f)*f*PolyLog[2, E^(-c - d*x)] + (6*I)*d*(-1 + E^(2*c))
*(d*e + I*f)*f^2*x*PolyLog[2, E^(-c - d*x)] + (3*I)*d^2*(-1 + E^(2*c))*f^3*x^2*PolyLog[2, E^(-c - d*x)] - 6*(-
1 + E^(2*c))*f^2*(I*d*e + f)*PolyLog[3, -E^(-c - d*x)] - (6*I)*d*(-1 + E^(2*c))*f^3*x*PolyLog[3, -E^(-c - d*x)
] + (6*I)*(-1 + E^(2*c))*(d*e + I*f)*f^2*PolyLog[3, E^(-c - d*x)] + (6*I)*d*(-1 + E^(2*c))*f^3*x*PolyLog[3, E^
(-c - d*x)] - (6*I)*(-1 + E^(2*c))*f^3*PolyLog[4, -E^(-c - d*x)] + (6*I)*(-1 + E^(2*c))*f^3*PolyLog[4, E^(-c -
d*x)])/(a*d^4*(-1 + E^(2*c))) + (Sech[c/2]*Sech[c/2 + (d*x)/2]*(-(e^3*Sinh[(d*x)/2]) - 3*e^2*f*x*Sinh[(d*x)/2
] - 3*e*f^2*x^2*Sinh[(d*x)/2] - f^3*x^3*Sinh[(d*x)/2]))/(2*a*d) + (Csch[c/2]*Csch[c/2 + (d*x)/2]*(e^3*Sinh[(d*
x)/2] + 3*e^2*f*x*Sinh[(d*x)/2] + 3*e*f^2*x^2*Sinh[(d*x)/2] + f^3*x^3*Sinh[(d*x)/2]))/(2*a*d) - (2*(e^3*Sinh[(
d*x)/2] + 3*e^2*f*x*Sinh[(d*x)/2] + 3*e*f^2*x^2*Sinh[(d*x)/2] + f^3*x^3*Sinh[(d*x)/2]))/(a*d*(Cosh[c/2] + I*Si
nh[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. 1603 vs. \(2 (390 ) = 780\).

Time = 3.17 (sec) , antiderivative size = 1604, normalized size of antiderivative = 3.83

method	result	size
risch	\(\text {Expression too large to display}\)	\(1604\)

[In]

int((f*x+e)^3*csch(d*x+c)^2/(a+I*a*sinh(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-6*I*f^3*polylog(4,exp(d*x+c))/a/d^4-6*f^3*polylog(3,-exp(d*x+c))/a/d^4-6*f^3*polylog(3,exp(d*x+c))/a/d^4-3*I/
d/a*e*f^2*ln(1-exp(d*x+c))*x^2+6*I*f^3*polylog(4,-exp(d*x+c))/a/d^4-12*f^3*polylog(3,-I*exp(d*x+c))/a/d^4-2*I*
(f^3*x^3*exp(2*d*x+2*c)+3*e*f^2*x^2*exp(2*d*x+2*c)+3*e^2*f*x*exp(2*d*x+2*c)-2*f^3*x^3-I*exp(d*x+c)*f^3*x^3+e^3
*exp(2*d*x+2*c)-6*e*f^2*x^2-3*I*exp(d*x+c)*e*f^2*x^2-6*e^2*f*x-3*I*exp(d*x+c)*e^2*f*x-2*e^3-I*exp(d*x+c)*e^3)/
(exp(2*d*x+2*c)-1)/(exp(d*x+c)-I)/a/d-12/a/d^3*f^2*e*c^2-12/a/d*f^2*e*x^2-6*I/d^2/a*e*f^2*polylog(2,exp(d*x+c)
)*x+3*I/d/a*e*f^2*ln(exp(d*x+c)+1)*x^2+6*I/d^2/a*e*f^2*polylog(2,-exp(d*x+c))*x+3*I/d^3/a*c^2*e*f^2*ln(1-exp(d
*x+c))-3*I/d^3/a*c^2*e*f^2*ln(exp(d*x+c)-1)-12*I/d^3/a*c*e*f^2*arctan(exp(d*x+c))+I/d/a*f^3*ln(exp(d*x+c)+1)*x
^3+3*I/d^2/a*f^3*polylog(2,-exp(d*x+c))*x^2-6*I/d^3/a*f^3*polylog(3,-exp(d*x+c))*x-I/d/a*f^3*ln(1-exp(d*x+c))*
x^3-3*I/d^2/a*e^2*f*polylog(2,exp(d*x+c))+3*I/d^2/a*e^2*f*polylog(2,-exp(d*x+c))+6*I/d^3/a*e*f^2*polylog(3,exp
(d*x+c))-6*I/d^3/a*e*f^2*polylog(3,-exp(d*x+c))+6*I/d^2/a*e^2*f*arctan(exp(d*x+c))+6*I/d^4/a*c^2*f^3*arctan(ex
p(d*x+c))-3*I/d^2/a*f^3*polylog(2,exp(d*x+c))*x^2+6*I/d^3/a*f^3*polylog(3,exp(d*x+c))*x-I/d^4/a*f^3*ln(1-exp(d
*x+c))*c^3-4/a/d*f^3*x^3-24/a/d^2*f^2*e*c*x-3*I/d/a*e^2*f*ln(1-exp(d*x+c))*x+3*I/d/a*e^2*f*ln(exp(d*x+c)+1)*x-
3*I/d^2/a*e^2*f*ln(1-exp(d*x+c))*c+3*I/d^2/a*e^2*c*f*ln(exp(d*x+c)-1)+24/a/d^3*f^2*e*c*ln(exp(d*x+c))+12/d^3/a
*f^2*e*polylog(2,-I*exp(d*x+c))+6/d^3/a*f^2*e*polylog(2,exp(d*x+c))+6/d^3/a*f^2*e*polylog(2,-exp(d*x+c))+3/d^4
/a*f^3*c^2*ln(exp(d*x+c)-1)+3/d^4/a*f^3*c^2*ln(1+exp(2*d*x+2*c))-12/d^4/a*c^2*f^3*ln(exp(d*x+c))+3/d^2/a*e^2*f
*ln(exp(d*x+c)-1)+3/d^2/a*e^2*f*ln(exp(d*x+c)+1)+3/d^2/a*e^2*f*ln(1+exp(2*d*x+2*c))-12/d^2/a*e^2*f*ln(exp(d*x+
c))-6/d^4/a*f^3*ln(1+I*exp(d*x+c))*c^2-3/d^4/a*f^3*ln(1-exp(d*x+c))*c^2+12/d^3/a*f^3*x*c^2-I/d/a*e^3*ln(exp(d*
x+c)-1)+6/d^2/a*f^3*ln(1+I*exp(d*x+c))*x^2+12/d^3/a*f^3*polylog(2,-I*exp(d*x+c))*x+3/d^2/a*f^3*ln(1-exp(d*x+c)
)*x^2+6/d^3/a*f^3*polylog(2,exp(d*x+c))*x+3/d^2/a*f^3*ln(exp(d*x+c)+1)*x^2+6/d^3/a*f^3*polylog(2,-exp(d*x+c))*
x+I/d/a*e^3*ln(exp(d*x+c)+1)+8/d^4/a*f^3*c^3-6/d^3/a*e*f^2*c*ln(exp(d*x+c)-1)+12/d^3/a*f^2*e*ln(1+I*exp(d*x+c)
)*c+6/d^3/a*f^2*e*ln(1-exp(d*x+c))*c+12/d^2/a*f^2*e*ln(1+I*exp(d*x+c))*x+6/d^2/a*f^2*e*ln(1-exp(d*x+c))*x+6/d^
2/a*f^2*e*ln(exp(d*x+c)+1)*x-6/d^3/a*e*f^2*c*ln(1+exp(2*d*x+2*c))+I/d^4/a*c^3*f^3*ln(exp(d*x+c)-1)

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. 2562 vs. \(2 (375) = 750\).

Time = 0.30 (sec) , antiderivative size = 2562, normalized size of antiderivative = 6.11 \[ \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate((f*x+e)^3*csch(d*x+c)^2/(a+I*a*sinh(d*x+c)),x, algorithm="fricas")

[Out]

(4*I*d^3*e^3 - 12*I*c*d^2*e^2*f + 12*I*c^2*d*e*f^2 - 4*I*c^3*f^3 - 12*(-I*d*f^3*x - I*d*e*f^2 - (d*f^3*x + d*e
*f^2)*e^(3*d*x + 3*c) + (I*d*f^3*x + I*d*e*f^2)*e^(2*d*x + 2*c) + (d*f^3*x + d*e*f^2)*e^(d*x + c))*dilog(-I*e^
(d*x + c)) - 3*(d^2*f^3*x^2 + d^2*e^2*f - 2*I*d*e*f^2 + 2*(d^2*e*f^2 - I*d*f^3)*x + (-I*d^2*f^3*x^2 - I*d^2*e^
2*f - 2*d*e*f^2 + 2*(-I*d^2*e*f^2 - d*f^3)*x)*e^(3*d*x + 3*c) - (d^2*f^3*x^2 + d^2*e^2*f - 2*I*d*e*f^2 + 2*(d^
2*e*f^2 - I*d*f^3)*x)*e^(2*d*x + 2*c) + (I*d^2*f^3*x^2 + I*d^2*e^2*f + 2*d*e*f^2 + 2*(I*d^2*e*f^2 + d*f^3)*x)*
e^(d*x + c))*dilog(-e^(d*x + c)) + 3*(d^2*f^3*x^2 + d^2*e^2*f + 2*I*d*e*f^2 + 2*(d^2*e*f^2 + I*d*f^3)*x - (I*d
^2*f^3*x^2 + I*d^2*e^2*f - 2*d*e*f^2 + 2*(I*d^2*e*f^2 - d*f^3)*x)*e^(3*d*x + 3*c) - (d^2*f^3*x^2 + d^2*e^2*f +
2*I*d*e*f^2 + 2*(d^2*e*f^2 + I*d*f^3)*x)*e^(2*d*x + 2*c) - (-I*d^2*f^3*x^2 - I*d^2*e^2*f + 2*d*e*f^2 + 2*(-I*
d^2*e*f^2 + d*f^3)*x)*e^(d*x + c))*dilog(e^(d*x + c)) - 4*(d^3*f^3*x^3 + 3*d^3*e*f^2*x^2 + 3*d^3*e^2*f*x + 3*c
*d^2*e^2*f - 3*c^2*d*e*f^2 + c^3*f^3)*e^(3*d*x + 3*c) - 2*(-I*d^3*f^3*x^3 - 3*I*d^3*e*f^2*x^2 - 3*I*d^3*e^2*f*
x + I*d^3*e^3 - 6*I*c*d^2*e^2*f + 6*I*c^2*d*e*f^2 - 2*I*c^3*f^3)*e^(2*d*x + 2*c) + 2*(d^3*f^3*x^3 + 3*d^3*e*f^
2*x^2 + 3*d^3*e^2*f*x - d^3*e^3 + 6*c*d^2*e^2*f - 6*c^2*d*e*f^2 + 2*c^3*f^3)*e^(d*x + c) - (d^3*f^3*x^3 + d^3*
e^3 - 3*I*d^2*e^2*f + 3*(d^3*e*f^2 - I*d^2*f^3)*x^2 + 3*(d^3*e^2*f - 2*I*d^2*e*f^2)*x - (I*d^3*f^3*x^3 + I*d^3
*e^3 + 3*d^2*e^2*f - 3*(-I*d^3*e*f^2 - d^2*f^3)*x^2 - 3*(-I*d^3*e^2*f - 2*d^2*e*f^2)*x)*e^(3*d*x + 3*c) - (d^3
*f^3*x^3 + d^3*e^3 - 3*I*d^2*e^2*f + 3*(d^3*e*f^2 - I*d^2*f^3)*x^2 + 3*(d^3*e^2*f - 2*I*d^2*e*f^2)*x)*e^(2*d*x
+ 2*c) - (-I*d^3*f^3*x^3 - I*d^3*e^3 - 3*d^2*e^2*f - 3*(I*d^3*e*f^2 + d^2*f^3)*x^2 - 3*(I*d^3*e^2*f + 2*d^2*e
*f^2)*x)*e^(d*x + c))*log(e^(d*x + c) + 1) - 6*(-I*d^2*e^2*f + 2*I*c*d*e*f^2 - I*c^2*f^3 - (d^2*e^2*f - 2*c*d*
e*f^2 + c^2*f^3)*e^(3*d*x + 3*c) + (I*d^2*e^2*f - 2*I*c*d*e*f^2 + I*c^2*f^3)*e^(2*d*x + 2*c) + (d^2*e^2*f - 2*
c*d*e*f^2 + c^2*f^3)*e^(d*x + c))*log(e^(d*x + c) - I) + (d^3*e^3 - 3*(c - I)*d^2*e^2*f + 3*(c^2 - 2*I*c)*d*e*
f^2 - (c^3 - 3*I*c^2)*f^3 + (-I*d^3*e^3 - 3*(-I*c - 1)*d^2*e^2*f - 3*(I*c^2 + 2*c)*d*e*f^2 + (I*c^3 + 3*c^2)*f
^3)*e^(3*d*x + 3*c) - (d^3*e^3 - 3*(c - I)*d^2*e^2*f + 3*(c^2 - 2*I*c)*d*e*f^2 - (c^3 - 3*I*c^2)*f^3)*e^(2*d*x
+ 2*c) + (I*d^3*e^3 - 3*(I*c + 1)*d^2*e^2*f - 3*(-I*c^2 - 2*c)*d*e*f^2 + (-I*c^3 - 3*c^2)*f^3)*e^(d*x + c))*l
og(e^(d*x + c) - 1) - 6*(-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 - (d^2*f^3*x^2 + 2*d^2*e
*f^2*x + 2*c*d*e*f^2 - c^2*f^3)*e^(3*d*x + 3*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^(2*d*x + 2*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^(d*x + c))*log(I*e^(d*x + c) + 1)
+ (d^3*f^3*x^3 + 3*c*d^2*e^2*f - 3*(c^2 - 2*I*c)*d*e*f^2 + (c^3 - 3*I*c^2)*f^3 + 3*(d^3*e*f^2 + I*d^2*f^3)*x^2
+ 3*(d^3*e^2*f + 2*I*d^2*e*f^2)*x + (-I*d^3*f^3*x^3 - 3*I*c*d^2*e^2*f - 3*(-I*c^2 - 2*c)*d*e*f^2 + (-I*c^3 -
3*c^2)*f^3 - 3*(I*d^3*e*f^2 - d^2*f^3)*x^2 - 3*(I*d^3*e^2*f - 2*d^2*e*f^2)*x)*e^(3*d*x + 3*c) - (d^3*f^3*x^3 +
3*c*d^2*e^2*f - 3*(c^2 - 2*I*c)*d*e*f^2 + (c^3 - 3*I*c^2)*f^3 + 3*(d^3*e*f^2 + I*d^2*f^3)*x^2 + 3*(d^3*e^2*f
+ 2*I*d^2*e*f^2)*x)*e^(2*d*x + 2*c) + (I*d^3*f^3*x^3 + 3*I*c*d^2*e^2*f - 3*(I*c^2 + 2*c)*d*e*f^2 + (I*c^3 + 3*
c^2)*f^3 - 3*(-I*d^3*e*f^2 + d^2*f^3)*x^2 - 3*(-I*d^3*e^2*f + 2*d^2*e*f^2)*x)*e^(d*x + c))*log(-e^(d*x + c) +
1) - 6*(-I*f^3*e^(3*d*x + 3*c) - f^3*e^(2*d*x + 2*c) + I*f^3*e^(d*x + c) + f^3)*polylog(4, -e^(d*x + c)) - 6*(
I*f^3*e^(3*d*x + 3*c) + f^3*e^(2*d*x + 2*c) - I*f^3*e^(d*x + c) - f^3)*polylog(4, e^(d*x + c)) - 12*(f^3*e^(3*
d*x + 3*c) - I*f^3*e^(2*d*x + 2*c) - f^3*e^(d*x + c) + I*f^3)*polylog(3, -I*e^(d*x + c)) + 6*(d*f^3*x + d*e*f^
2 - I*f^3 - (I*d*f^3*x + I*d*e*f^2 + f^3)*e^(3*d*x + 3*c) - (d*f^3*x + d*e*f^2 - I*f^3)*e^(2*d*x + 2*c) - (-I*
d*f^3*x - I*d*e*f^2 - f^3)*e^(d*x + c))*polylog(3, -e^(d*x + c)) - 6*(d*f^3*x + d*e*f^2 + I*f^3 + (-I*d*f^3*x
- I*d*e*f^2 + f^3)*e^(3*d*x + 3*c) - (d*f^3*x + d*e*f^2 + I*f^3)*e^(2*d*x + 2*c) + (I*d*f^3*x + I*d*e*f^2 - f^
3)*e^(d*x + c))*polylog(3, e^(d*x + c)))/(a*d^4*e^(3*d*x + 3*c) - I*a*d^4*e^(2*d*x + 2*c) - a*d^4*e^(d*x + c)
+ I*a*d^4)

Sympy [F]

\[ \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=- \frac {i \left (\int \frac {e^{3} \operatorname {csch}^{2}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {f^{3} x^{3} \operatorname {csch}^{2}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {3 e f^{2} x^{2} \operatorname {csch}^{2}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {3 e^{2} f x \operatorname {csch}^{2}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx\right )}{a} \]

[In]

integrate((f*x+e)**3*csch(d*x+c)**2/(a+I*a*sinh(d*x+c)),x)

[Out]

-I*(Integral(e**3*csch(c + d*x)**2/(sinh(c + d*x) - I), x) + Integral(f**3*x**3*csch(c + d*x)**2/(sinh(c + d*x
) - I), x) + Integral(3*e*f**2*x**2*csch(c + d*x)**2/(sinh(c + d*x) - I), x) + Integral(3*e**2*f*x*csch(c + d*
x)**2/(sinh(c + d*x) - I), x))/a

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. 939 vs. \(2 (375) = 750\).

Time = 0.42 (sec) , antiderivative size = 939, normalized size of antiderivative = 2.24 \[ \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate((f*x+e)^3*csch(d*x+c)^2/(a+I*a*sinh(d*x+c)),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \operatorname {csch}\left (d x + c\right )^{2}}{i \, a \sinh \left (d x + c\right ) + a} \,d x } \]

[In]

integrate((f*x+e)^3*csch(d*x+c)^2/(a+I*a*sinh(d*x+c)),x, algorithm="giac")

[Out]

integrate((f*x + e)^3*csch(d*x + c)^2/(I*a*sinh(d*x + c) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^3}{{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]

[In]

int((e + f*x)^3/(sinh(c + d*x)^2*(a + a*sinh(c + d*x)*1i)),x)

[Out]

int((e + f*x)^3/(sinh(c + d*x)^2*(a + a*sinh(c + d*x)*1i)), x)