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

Optimal. Leaf size=419 \[ -\frac {2 (e+f x)^3}{a d}+\frac {2 i (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {6 f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 i f (e+f x)^2 \text {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {12 f^2 (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {3 i f (e+f x)^2 \text {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac {3 f^2 (e+f x) \text {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \text {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac {12 f^3 \text {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}+\frac {6 i f^2 (e+f x) \text {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac {3 f^3 \text {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {6 i f^3 \text {PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}-\frac {6 i f^3 \text {PolyLog}\left (4,e^{c+d x}\right )}{a d^4}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.60, antiderivative size = 419, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 10, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {5694, 4269, 3797, 2221, 2611, 2320, 6724, 4267, 6744, 3399} \begin {gather*} -\frac {12 f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{a d^4}-\frac {3 f^3 \text {Li}_3\left (e^{2 (c+d x)}\right )}{2 a d^4}+\frac {6 i f^3 \text {Li}_4\left (-e^{c+d x}\right )}{a d^4}-\frac {6 i f^3 \text {Li}_4\left (e^{c+d x}\right )}{a d^4}+\frac {12 f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}+\frac {3 f^2 (e+f x) \text {Li}_2\left (e^{2 (c+d x)}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac {6 i f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (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 {2 i (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\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} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\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 \tanh ^{-1}\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 \tanh ^{-1}\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 \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (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) \text {Li}_2\left (-e^{c+d x}\right ) \, dx}{a d^2}+\frac {\left (6 i 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) \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 \tanh ^{-1}\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 \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {3 f^2 (e+f x) \text {Li}_2\left (e^{2 (c+d x)}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac {6 i f^2 (e+f x) \text {Li}_3\left (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 \text {Li}_3\left (-e^{c+d x}\right ) \, dx}{a d^3}-\frac {\left (6 i 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}_2\left (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 \tanh ^{-1}\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 \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {3 f^2 (e+f x) \text {Li}_2\left (e^{2 (c+d x)}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac {6 i f^2 (e+f x) \text {Li}_3\left (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 {\text {Li}_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 {\text {Li}_3(x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}-\frac {\left (3 f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_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 \tanh ^{-1}\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 \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac {12 f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {3 f^2 (e+f x) \text {Li}_2\left (e^{2 (c+d x)}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac {6 i f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac {3 f^3 \text {Li}_3\left (e^{2 (c+d x)}\right )}{2 a d^4}+\frac {6 i f^3 \text {Li}_4\left (-e^{c+d x}\right )}{a d^4}-\frac {6 i f^3 \text {Li}_4\left (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 \text {Li}_2\left (-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 \tanh ^{-1}\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 \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac {12 f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {3 f^2 (e+f x) \text {Li}_2\left (e^{2 (c+d x)}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac {6 i f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac {3 f^3 \text {Li}_3\left (e^{2 (c+d x)}\right )}{2 a d^4}+\frac {6 i f^3 \text {Li}_4\left (-e^{c+d x}\right )}{a d^4}-\frac {6 i f^3 \text {Li}_4\left (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 {\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 (e+f x)^3}{a d}+\frac {2 i (e+f x)^3 \tanh ^{-1}\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 \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac {12 f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {3 f^2 (e+f x) \text {Li}_2\left (e^{2 (c+d x)}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}-\frac {12 f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{a d^4}+\frac {6 i f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac {3 f^3 \text {Li}_3\left (e^{2 (c+d x)}\right )}{2 a d^4}+\frac {6 i f^3 \text {Li}_4\left (-e^{c+d x}\right )}{a d^4}-\frac {6 i f^3 \text {Li}_4\left (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*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1005\) vs. \(2(419)=838\).
time = 13.54, size = 1005, normalized size = 2.40 \begin {gather*} -\frac {2 i f \left (d^2 \left (-i d e^c x \left (3 e^2+3 e f x+f^2 x^2\right )+3 \left (1+i e^c\right ) (e+f x)^2 \log \left (1+i e^{c+d x}\right )\right )+6 d \left (1+i e^c\right ) f (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right )-6 i \left (-i+e^c\right ) f^2 \text {PolyLog}\left (3,-i e^{c+d x}\right )\right )}{a d^4 \left (-i+e^c\right )}-\frac {12 d^3 e^2 e^{2 c} f x-12 d^3 e^2 \left (-1+e^{2 c}\right ) f x+12 d^3 e f^2 x^2+4 d^3 f^3 x^3-4 i d^3 e^3 \left (-1+e^{2 c}\right ) \tanh ^{-1}\left (e^{c+d x}\right )+6 d^2 e^2 \left (-1+e^{2 c}\right ) f \left (2 d x-\log \left (1-e^{2 (c+d x)}\right )\right )+6 i d^2 e^2 \left (-1+e^{2 c}\right ) f \left (d x \left (\log \left (1-e^{c+d x}\right )-\log \left (1+e^{c+d x}\right )\right )-\text {PolyLog}\left (2,-e^{c+d x}\right )+\text {PolyLog}\left (2,e^{c+d x}\right )\right )+6 d e \left (-1+e^{2 c}\right ) f^2 \left (2 d x \left (d x-\log \left (1-e^{2 (c+d x)}\right )\right )-\text {PolyLog}\left (2,e^{2 (c+d x)}\right )\right )+6 i d e \left (-1+e^{2 c}\right ) f^2 \left (d^2 x^2 \log \left (1-e^{c+d x}\right )-d^2 x^2 \log \left (1+e^{c+d x}\right )-2 d x \text {PolyLog}\left (2,-e^{c+d x}\right )+2 d x \text {PolyLog}\left (2,e^{c+d x}\right )+2 \text {PolyLog}\left (3,-e^{c+d x}\right )-2 \text {PolyLog}\left (3,e^{c+d x}\right )\right )+\left (-1+e^{2 c}\right ) f^3 \left (2 d^2 x^2 \left (2 d x-3 \log \left (1-e^{2 (c+d x)}\right )\right )-6 d x \text {PolyLog}\left (2,e^{2 (c+d x)}\right )+3 \text {PolyLog}\left (3,e^{2 (c+d x)}\right )\right )+2 i \left (-1+e^{2 c}\right ) f^3 \left (d^3 x^3 \log \left (1-e^{c+d x}\right )-d^3 x^3 \log \left (1+e^{c+d x}\right )-3 d^2 x^2 \text {PolyLog}\left (2,-e^{c+d x}\right )+3 d^2 x^2 \text {PolyLog}\left (2,e^{c+d x}\right )+6 d x \text {PolyLog}\left (3,-e^{c+d x}\right )-6 d x \text {PolyLog}\left (3,e^{c+d x}\right )-6 \text {PolyLog}\left (4,-e^{c+d x}\right )+6 \text {PolyLog}\left (4,e^{c+d x}\right )\right )}{2 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 )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-2*I)*f*(d^2*((-I)*d*E^c*x*(3*e^2 + 3*e*f*x + f^2*x^2) + 3*(1 + I*E^c)*(e + f*x)^2*Log[1 + I*E^(c + d*x)]) +
6*d*(1 + I*E^c)*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)] - (6*I)*(-I + E^c)*f^2*PolyLog[3, (-I)*E^(c + d*x)])
)/(a*d^4*(-I + E^c)) - (12*d^3*e^2*E^(2*c)*f*x - 12*d^3*e^2*(-1 + E^(2*c))*f*x + 12*d^3*e*f^2*x^2 + 4*d^3*f^3*
x^3 - (4*I)*d^3*e^3*(-1 + E^(2*c))*ArcTanh[E^(c + d*x)] + 6*d^2*e^2*(-1 + E^(2*c))*f*(2*d*x - Log[1 - E^(2*(c
+ d*x))]) + (6*I)*d^2*e^2*(-1 + E^(2*c))*f*(d*x*(Log[1 - E^(c + d*x)] - Log[1 + E^(c + d*x)]) - PolyLog[2, -E^
(c + d*x)] + PolyLog[2, E^(c + d*x)]) + 6*d*e*(-1 + E^(2*c))*f^2*(2*d*x*(d*x - Log[1 - E^(2*(c + d*x))]) - Pol
yLog[2, E^(2*(c + d*x))]) + (6*I)*d*e*(-1 + E^(2*c))*f^2*(d^2*x^2*Log[1 - E^(c + d*x)] - d^2*x^2*Log[1 + E^(c
+ d*x)] - 2*d*x*PolyLog[2, -E^(c + d*x)] + 2*d*x*PolyLog[2, E^(c + d*x)] + 2*PolyLog[3, -E^(c + d*x)] - 2*Poly
Log[3, E^(c + d*x)]) + (-1 + E^(2*c))*f^3*(2*d^2*x^2*(2*d*x - 3*Log[1 - E^(2*(c + d*x))]) - 6*d*x*PolyLog[2, E
^(2*(c + d*x))] + 3*PolyLog[3, E^(2*(c + d*x))]) + (2*I)*(-1 + E^(2*c))*f^3*(d^3*x^3*Log[1 - E^(c + d*x)] - d^
3*x^3*Log[1 + E^(c + d*x)] - 3*d^2*x^2*PolyLog[2, -E^(c + d*x)] + 3*d^2*x^2*PolyLog[2, E^(c + d*x)] + 6*d*x*Po
lyLog[3, -E^(c + d*x)] - 6*d*x*PolyLog[3, E^(c + d*x)] - 6*PolyLog[4, -E^(c + d*x)] + 6*PolyLog[4, E^(c + d*x)
]))/(2*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*Sinh
[c/2])*(Cosh[c/2 + (d*x)/2] + I*Sinh[c/2 + (d*x)/2]))

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1534 vs. \(2 (390 ) = 780\).
time = 3.23, size = 1535, normalized size = 3.66

method	result	size
risch	\(\text {Expression too large to display}\)	\(1535\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 940 vs. \(2 (387) = 774\).
time = 0.47, size = 940, normalized size = 2.24 \begin {gather*} -{\left (\frac {2 \, {\left (e^{\left (-d x - c\right )} - i \, e^{\left (-2 \, d x - 2 \, c\right )} + 2 i\right )}}{{\left (a e^{\left (-d x - c\right )} - i \, a e^{\left (-2 \, d x - 2 \, c\right )} - a e^{\left (-3 \, d x - 3 \, c\right )} + i \, a\right )} d} - \frac {i \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{a d} + \frac {i \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{a d}\right )} e^{3} - \frac {12 \, f x e^{2}}{a d} - \frac {2 \, {\left (-2 i \, f^{3} x^{3} - 6 i \, f^{2} x^{2} e - 6 i \, f x e^{2} - {\left (-i \, f^{3} x^{3} e^{\left (2 \, c\right )} - 3 i \, f^{2} x^{2} e^{\left (2 \, c + 1\right )} - 3 i \, f x e^{\left (2 \, c + 2\right )}\right )} e^{\left (2 \, d x\right )} + {\left (f^{3} x^{3} e^{c} + 3 \, f^{2} x^{2} e^{\left (c + 1\right )} + 3 \, f x e^{\left (c + 2\right )}\right )} e^{\left (d x\right )}\right )}}{a d e^{\left (3 \, d x + 3 \, c\right )} - i \, a d e^{\left (2 \, d x + 2 \, c\right )} - a d e^{\left (d x + c\right )} + i \, a d} + \frac {12 \, {\left (d x \log \left (i \, e^{\left (d x + c\right )} + 1\right ) + {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right )\right )} f^{2} e}{a d^{3}} + \frac {3 \, f e^{2} \log \left (e^{\left (d x + c\right )} + 1\right )}{a d^{2}} + \frac {6 \, f e^{2} \log \left (e^{\left (d x + c\right )} - i\right )}{a d^{2}} + \frac {3 \, f e^{2} \log \left (e^{\left (d x + c\right )} - 1\right )}{a d^{2}} + \frac {i \, {\left (d^{3} x^{3} \log \left (e^{\left (d x + c\right )} + 1\right ) + 3 \, d^{2} x^{2} {\rm Li}_2\left (-e^{\left (d x + c\right )}\right ) - 6 \, d x {\rm Li}_{3}(-e^{\left (d x + c\right )}) + 6 \, {\rm Li}_{4}(-e^{\left (d x + c\right )})\right )} f^{3}}{a d^{4}} - \frac {i \, {\left (d^{3} x^{3} \log \left (-e^{\left (d x + c\right )} + 1\right ) + 3 \, d^{2} x^{2} {\rm Li}_2\left (e^{\left (d x + c\right )}\right ) - 6 \, d x {\rm Li}_{3}(e^{\left (d x + c\right )}) + 6 \, {\rm Li}_{4}(e^{\left (d x + c\right )})\right )} f^{3}}{a d^{4}} + \frac {6 \, {\left (d^{2} x^{2} \log \left (i \, e^{\left (d x + c\right )} + 1\right ) + 2 \, d x {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right ) - 2 \, {\rm Li}_{3}(-i \, e^{\left (d x + c\right )})\right )} f^{3}}{a d^{4}} - \frac {3 \, {\left (-i \, d f e^{2} - 2 \, f^{2} e\right )} {\left (d x \log \left (e^{\left (d x + c\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (d x + c\right )}\right )\right )}}{a d^{3}} + \frac {3 \, {\left (-i \, d f e^{2} + 2 \, f^{2} e\right )} {\left (d x \log \left (-e^{\left (d x + c\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (d x + c\right )}\right )\right )}}{a d^{3}} + \frac {3 \, {\left (d^{2} x^{2} \log \left (-e^{\left (d x + c\right )} + 1\right ) + 2 \, d x {\rm Li}_2\left (e^{\left (d x + c\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (d x + c\right )})\right )} {\left (-i \, d f^{2} e + f^{3}\right )}}{a d^{4}} - \frac {3 \, {\left (d^{2} x^{2} \log \left (e^{\left (d x + c\right )} + 1\right ) + 2 \, d x {\rm Li}_2\left (-e^{\left (d x + c\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (d x + c\right )})\right )} {\left (-i \, d f^{2} e - f^{3}\right )}}{a d^{4}} + \frac {i \, d^{4} f^{3} x^{4} - 4 \, {\left (-i \, d f^{2} e + f^{3}\right )} d^{3} x^{3} - 6 \, {\left (-i \, d^{2} f e^{2} + 2 \, d f^{2} e\right )} d^{2} x^{2}}{4 \, a d^{4}} - \frac {i \, d^{4} f^{3} x^{4} - 4 \, {\left (-i \, d f^{2} e - f^{3}\right )} d^{3} x^{3} - 6 \, {\left (-i \, d^{2} f e^{2} - 2 \, d f^{2} e\right )} d^{2} x^{2}}{4 \, a d^{4}} - \frac {2 \, {\left (d^{3} f^{3} x^{3} + 3 \, d^{3} f^{2} x^{2} e\right )}}{a d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2568 vs. \(2 (387) = 774\).
time = 0.45, size = 2568, normalized size = 6.13 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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