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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.87, antiderivative size = 546, normalized size of antiderivative = 1.00, number of steps used = 40, number of rules used = 13, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {5694, 4271, 4267, 2317, 2438, 2611, 6744, 2320, 6724, 4269, 3797, 2221, 3399} \begin {gather*} -\frac {3 f^3 \text {Li}_2\left (-e^{c+d x}\right )}{a d^4}+\frac {3 f^3 \text {Li}_2\left (e^{c+d x}\right )}{a d^4}+\frac {12 i f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{a d^4}+\frac {3 i f^3 \text {Li}_3\left (e^{2 (c+d x)}\right )}{2 a d^4}+\frac {9 f^3 \text {Li}_4\left (-e^{c+d x}\right )}{a d^4}-\frac {9 f^3 \text {Li}_4\left (e^{c+d x}\right )}{a d^4}-\frac {12 i f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}-\frac {3 i f^2 (e+f x) \text {Li}_2\left (e^{2 (c+d x)}\right )}{a d^3}-\frac {9 f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac {9 f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac {6 f^2 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^3}+\frac {9 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{2 a d^2}-\frac {9 f (e+f x)^2 \text {Li}_2\left (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 {3 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {i (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{a d}+\frac {i (e+f x)^3 \coth (c+d x)}{a d}-\frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 a d}+\frac {2 i (e+f x)^3}{a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

________________________________________________________________________________________

Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2394\) vs. \(2(546)=1092\).
time = 52.33, size = 2394, normalized size = 4.38 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-3*e^3*Log[Tanh[(c + d*x)/2]])/(2*a*d) + (3*e*f^2*Log[Tanh[(c + d*x)/2]])/(a*d^3) - (9*e^2*f*(-(c*Log[Tanh[(c
+ d*x)/2]]) - I*((I*c + I*d*x)*(Log[1 - E^(I*(I*c + I*d*x))] - Log[1 + E^(I*(I*c + I*d*x))]) + I*(PolyLog[2,
-E^(I*(I*c + I*d*x))] - PolyLog[2, E^(I*(I*c + I*d*x))]))))/(2*a*d^2) + (3*f^3*(-(c*Log[Tanh[(c + d*x)/2]]) -
I*((I*c + I*d*x)*(Log[1 - E^(I*(I*c + I*d*x))] - Log[1 + E^(I*(I*c + I*d*x))]) + I*(PolyLog[2, -E^(I*(I*c + I*
d*x))] - PolyLog[2, E^(I*(I*c + I*d*x))]))))/(a*d^4) - (2*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)) + ((I/4)*f^3*Csch[c]*(2*d^2*x^2*(2*d*E^
(2*c)*x - 3*(-1 + E^(2*c))*Log[1 - E^(2*(c + d*x))]) - 6*d*(-1 + E^(2*c))*x*PolyLog[2, E^(2*(c + d*x))] + 3*(-
1 + E^(2*c))*PolyLog[3, E^(2*(c + d*x))]))/(a*d^4*E^c) + (9*e*f^2*(d^2*x^2*ArcTanh[Cosh[c + d*x] + Sinh[c + d*
x]] + d*x*PolyLog[2, -Cosh[c + d*x] - Sinh[c + d*x]] - d*x*PolyLog[2, Cosh[c + d*x] + Sinh[c + d*x]] - PolyLog
[3, -Cosh[c + d*x] - Sinh[c + d*x]] + PolyLog[3, Cosh[c + d*x] + Sinh[c + d*x]]))/(a*d^3) - (3*f^3*(d^3*x^3*Lo
g[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*PolyLog[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) + ((3*I)*e^2*f*Csch[c]*(-(d*x*Cosh[c]) + Log[Cosh[d*x]*Sinh[c] + Cosh[c]*
Sinh[d*x]]*Sinh[c]))/(a*d^2*(-Cosh[c]^2 + Sinh[c]^2)) + (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*Co
sh[c - (d*x)/2] + (5*I)*d*f^3*x^3*Cosh[c - (d*x)/2] - I*d*e^3*Cosh[c + (d*x)/2] - (3*I)*d*e^2*f*x*Cosh[c + (d*
x)/2] - (3*I)*d*e*f^2*x^2*Cosh[c + (d*x)/2] - I*d*f^3*x^3*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*Co
sh[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*Co
sh[3*c + (3*d*x)/2] - (4*I)*d*e^3*Cosh[c + (5*d*x)/2] - (12*I)*d*e^2*f*x*Cosh[c + (5*d*x)/2] - (12*I)*d*e*f^2*
x^2*Cosh[c + (5*d*x)/2] - (4*I)*d*f^3*x^3*Cosh[c + (5*d*x)/2] + (2*I)*d*e^3*Cosh[3*c + (5*d*x)/2] + (6*I)*d*e^
2*f*x*Cosh[3*c + (5*d*x)/2] + (6*I)*d*e*f^2*x^2*Cosh[3*c + (5*d*x)/2] + (2*I)*d*f^3*x^3*Cosh[3*c + (5*d*x)/2]
- d*e^3*Sinh[(d*x)/2] - 3*d*e^2*f*x*Sinh[(d*x)/2] - 3*d*e*f^2*x^2*Sinh[(d*x)/2] - d*f^3*x^3*Sinh[(d*x)/2] - d*
e^3*Sinh[(3*d*x)/2] - 3*d*e^2*f*x*Sinh[(3*d*x)/2] - 3*d*e*f^2*x^2*Sinh[(3*d*x)/2] - d*f^3*x^3*Sinh[(3*d*x)/2]
+ (3*I)*e^2*f*Sinh[c - (d*x)/2] + (6*I)*e*f^2*x*Sinh[c - (d*x)/2] + (3*I)*f^3*x^2*Sinh[c - (d*x)/2] + (3*I)*e^
2*f*Sinh[c + (d*x)/2] + (6*I)*e*f^2*x*Sinh[c + (d*x)/2] + (3*I)*f^3*x^2*Sinh[c + (d*x)/2] - 3*d*e^3*Sinh[2*c +
(d*x)/2] - 9*d*e^2*f*x*Sinh[2*c + (d*x)/2] - 9*d*e*f^2*x^2*Sinh[2*c + (d*x)/2] - 3*d*f^3*x^3*Sinh[2*c + (d*x)
/2] + (3*I)*e^2*f*Sinh[c + (3*d*x)/2] + (6*I)*e*f^2*x*Sinh[c + (3*d*x)/2] + (3*I)*f^3*x^2*Sinh[c + (3*d*x)/2]
- d*e^3*Sinh[2*c + (3*d*x)/2] - 3*d*e^2*f*x*Sinh[2*c + (3*d*x)/2] - 3*d*e*f^2*x^2*Sinh[2*c + (3*d*x)/2] - d*f^
3*x^3*Sinh[2*c + (3*d*x)/2] - (3*I)*e^2*f*Sinh[3*c + (3*d*x)/2] - (6*I)*e*f^2*x*Sinh[3*c + (3*d*x)/2] - (3*I)*
f^3*x^2*Sinh[3*c + (3*d*x)/2] + 2*d*e^3*Sinh[2*c + (5*d*x)/2] + 6*d*e^2*f*x*Sinh[2*c + (5*d*x)/2] + 6*d*e*f^2*
x^2*Sinh[2*c + (5*d*x)/2] + 2*d*f^3*x^3*Sinh[2*c + (5*d*x)/2]))/(8*a*d^2*(Cosh[c/2] + I*Sinh[c/2])*(Cosh[c/2 +
(d*x)/2] + I*Sinh[c/2 + (d*x)/2])) + ((3*I)*e*f^2*Csch[c]*Sech[c]*((d^2*x^2)/E^ArcTanh[Tanh[c]] - (I*(-(d*x*(
-Pi + (2*I)*ArcTanh[Tanh[c]])) - Pi*Log[1 + E^(2*d*x)] - 2*(I*d*x + I*ArcTanh[Tanh[c]])*Log[1 - E^((2*I)*(I*d*
x + I*ArcTanh[Tanh[c]]))] + Pi*Log[Cosh[d*x]] + (2*I)*ArcTanh[Tanh[c]]*Log[I*Sinh[d*x + ArcTanh[Tanh[c]]]] + I
*PolyLog[2, E^((2*I)*(I*d*x + I*ArcTanh[Tanh[c]]))])*Tanh[c])/Sqrt[1 - Tanh[c]^2]))/(a*d^3*Sqrt[Sech[c]^2*(Cos
h[c]^2 - Sinh[c]^2)])

________________________________________________________________________________________

Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2057 vs. \(2 (504 ) = 1008\).
time = 3.44, size = 2058, normalized size = 3.77

method	result	size
risch	\(\text {Expression too large to display}\)	\(2058\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1353 vs. \(2 (500) = 1000\).
time = 0.60, size = 1353, normalized size = 2.48 \begin {gather*} -\frac {1}{2} \, {\left (\frac {2 \, {\left (-i \, e^{\left (-d x - c\right )} - 5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 i \, e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4\right )}}{{\left (a e^{\left (-d x - c\right )} - 2 i \, a e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, a e^{\left (-3 \, d x - 3 \, c\right )} + i \, a e^{\left (-4 \, d x - 4 \, c\right )} + a e^{\left (-5 \, d x - 5 \, c\right )} + i \, a\right )} d} - \frac {3 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{a d} + \frac {3 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{a d}\right )} e^{3} + \frac {6 i \, f x e^{2}}{a d} - \frac {4 \, d f^{3} x^{3} + 12 \, d f^{2} x^{2} e + 12 \, d f x e^{2} + 3 \, {\left (d f^{3} x^{3} e^{\left (4 \, c\right )} + {\left (f^{3} e^{\left (4 \, c\right )} + 3 \, d f^{2} e^{\left (4 \, c + 1\right )}\right )} x^{2} + {\left (3 \, d f e^{\left (4 \, c + 2\right )} + 2 \, f^{2} e^{\left (4 \, c + 1\right )}\right )} x + f e^{\left (4 \, c + 2\right )}\right )} e^{\left (4 \, d x\right )} - 3 \, {\left (i \, d f^{3} x^{3} e^{\left (3 \, c\right )} + {\left (i \, f^{3} e^{\left (3 \, c\right )} + 3 i \, d f^{2} e^{\left (3 \, c + 1\right )}\right )} x^{2} + {\left (3 i \, d f e^{\left (3 \, c + 2\right )} + 2 i \, f^{2} e^{\left (3 \, c + 1\right )}\right )} x + i \, f e^{\left (3 \, c + 2\right )}\right )} e^{\left (3 \, d x\right )} - {\left (5 \, d f^{3} x^{3} e^{\left (2 \, c\right )} + 3 \, {\left (f^{3} e^{\left (2 \, c\right )} + 5 \, d f^{2} e^{\left (2 \, c + 1\right )}\right )} x^{2} + 3 \, {\left (5 \, d f e^{\left (2 \, c + 2\right )} + 2 \, f^{2} e^{\left (2 \, c + 1\right )}\right )} x + 3 \, f e^{\left (2 \, c + 2\right )}\right )} e^{\left (2 \, d x\right )} + {\left (i \, d f^{3} x^{3} e^{c} - 3 \, {\left (-i \, d f^{2} e^{\left (c + 1\right )} - i \, f^{3} e^{c}\right )} x^{2} - 3 \, {\left (-i \, d f e^{\left (c + 2\right )} - 2 i \, f^{2} e^{\left (c + 1\right )}\right )} x + 3 i \, f e^{\left (c + 2\right )}\right )} e^{\left (d x\right )}}{a d^{2} e^{\left (5 \, d x + 5 \, c\right )} - i \, a d^{2} e^{\left (4 \, d x + 4 \, c\right )} - 2 \, a d^{2} e^{\left (3 \, d x + 3 \, c\right )} + 2 i \, a d^{2} e^{\left (2 \, d x + 2 \, c\right )} + a d^{2} e^{\left (d x + c\right )} - i \, a d^{2}} - \frac {12 i \, {\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 {6 i \, f e^{2} \log \left (i \, e^{\left (d x + c\right )} + 1\right )}{a d^{2}} + \frac {3 \, {\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}}{2 \, a d^{4}} - \frac {3 \, {\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}}{2 \, a d^{4}} - \frac {6 i \, {\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} + f^{2} e\right )} x}{a d^{2}} - \frac {3 \, {\left (-i \, d f e^{2} - f^{2} e\right )} x}{a d^{2}} + \frac {3 \, {\left (-i \, d f e^{2} - f^{2} e\right )} \log \left (e^{\left (d x + c\right )} + 1\right )}{a d^{3}} + \frac {3 \, {\left (-i \, d f e^{2} + f^{2} e\right )} \log \left (e^{\left (d x + c\right )} - 1\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 (3 \, d f^{2} e + 2 i \, f^{3}\right )}}{2 \, 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 (3 \, d f^{2} e - 2 i \, f^{3}\right )}}{2 \, a d^{4}} + \frac {3 \, {\left (3 \, d^{2} f e^{2} - 4 i \, d f^{2} e - 2 \, f^{3}\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 )}}{2 \, a d^{4}} - \frac {3 \, {\left (3 \, d^{2} f e^{2} + 4 i \, d f^{2} e - 2 \, f^{3}\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 )}}{2 \, a d^{4}} + \frac {3 \, d^{4} f^{3} x^{4} + 4 \, {\left (3 \, d f^{2} e + 2 i \, f^{3}\right )} d^{3} x^{3} + 6 \, {\left (3 \, d^{2} f e^{2} + 4 i \, d f^{2} e - 2 \, f^{3}\right )} d^{2} x^{2}}{8 \, a d^{4}} - \frac {3 \, d^{4} f^{3} x^{4} + 4 \, {\left (3 \, d f^{2} e - 2 i \, f^{3}\right )} d^{3} x^{3} + 6 \, {\left (3 \, d^{2} f e^{2} - 4 i \, d f^{2} e - 2 \, f^{3}\right )} d^{2} x^{2}}{8 \, a d^{4}} - \frac {2 \, {\left (-i \, d^{3} f^{3} x^{3} - 3 i \, 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)^3/(a+I*a*sinh(d*x+c)),x, algorithm="maxima")

[Out]

-1/2*(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*lo
g(e^(-d*x - c) + 1)/(a*d) + 3*log(e^(-d*x - c) - 1)/(a*d))*e^3 + 6*I*f*x*e^2/(a*d) - (4*d*f^3*x^3 + 12*d*f^2*x
^2*e + 12*d*f*x*e^2 + 3*(d*f^3*x^3*e^(4*c) + (f^3*e^(4*c) + 3*d*f^2*e^(4*c + 1))*x^2 + (3*d*f*e^(4*c + 2) + 2*
f^2*e^(4*c + 1))*x + f*e^(4*c + 2))*e^(4*d*x) - 3*(I*d*f^3*x^3*e^(3*c) + (I*f^3*e^(3*c) + 3*I*d*f^2*e^(3*c + 1
))*x^2 + (3*I*d*f*e^(3*c + 2) + 2*I*f^2*e^(3*c + 1))*x + I*f*e^(3*c + 2))*e^(3*d*x) - (5*d*f^3*x^3*e^(2*c) + 3
*(f^3*e^(2*c) + 5*d*f^2*e^(2*c + 1))*x^2 + 3*(5*d*f*e^(2*c + 2) + 2*f^2*e^(2*c + 1))*x + 3*f*e^(2*c + 2))*e^(2
*d*x) + (I*d*f^3*x^3*e^c - 3*(-I*d*f^2*e^(c + 1) - I*f^3*e^c)*x^2 - 3*(-I*d*f*e^(c + 2) - 2*I*f^2*e^(c + 1))*x
+ 3*I*f*e^(c + 2))*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)
))*f^2*e/(a*d^3) - 6*I*f*e^2*log(I*e^(d*x + c) + 1)/(a*d^2) + 3/2*(d^3*x^3*log(e^(d*x + c) + 1) + 3*d^2*x^2*di
log(-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*lo
g(-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*f*e^2 + f^2*e)*x/(a*d^2) - 3*(-I*d*f*e^2 - f^2*e)*x/(a*d^2) + 3*(-I*d*f*e^2 - f^2*e)
*log(e^(d*x + c) + 1)/(a*d^3) + 3*(-I*d*f*e^2 + f^2*e)*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*f^2*e + 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*f^2*e - 2*I*f^3)/(a*d^4)
+ 3/2*(3*d^2*f*e^2 - 4*I*d*f^2*e - 2*f^3)*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))/(a*d^4) - 3/2*(3*d
^2*f*e^2 + 4*I*d*f^2*e - 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*f^2*e + 2*I*f^3)*d^3*x^3 + 6*(3*d^2*f*e^2 + 4*I*d*f^2*e - 2*f^3)*d^2*x^2)/(a*d^4) - 1/8*(3*d^4*f^3*x^
4 + 4*(3*d*f^2*e - 2*I*f^3)*d^3*x^3 + 6*(3*d^2*f*e^2 - 4*I*d*f^2*e - 2*f^3)*d^2*x^2)/(a*d^4) - 2*(-I*d^3*f^3*x
^3 - 3*I*d^3*f^2*x^2*e)/(a*d^4)

________________________________________________________________________________________

Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 4261 vs. \(2 (500) = 1000\).
time = 0.43, size = 4261, normalized size = 7.80 \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)^3/(a+I*a*sinh(d*x+c)),x, algorithm="fricas")

[Out]

1/2*(8*c^3*f^3 - 24*c^2*d*f^2*e + 24*c*d^2*f*e^2 - 8*d^3*e^3 - 24*(d*f^3*x + d*f^2*e + (I*d*f^3*x + I*d*f^2*e)
*e^(5*d*x + 5*c) + (d*f^3*x + d*f^2*e)*e^(4*d*x + 4*c) + 2*(-I*d*f^3*x - I*d*f^2*e)*e^(3*d*x + 3*c) - 2*(d*f^3
*x + d*f^2*e)*e^(2*d*x + 2*c) + (I*d*f^3*x + I*d*f^2*e)*e^(d*x + c))*dilog(-I*e^(d*x + c)) - 3*(3*I*d^2*f^3*x^
2 + 4*d*f^3*x + 3*I*d^2*f*e^2 - 2*I*f^3 + 2*(3*I*d^2*f^2*x + 2*d*f^2)*e - (3*d^2*f^3*x^2 - 4*I*d*f^3*x + 3*d^2
*f*e^2 - 2*f^3 + 2*(3*d^2*f^2*x - 2*I*d*f^2)*e)*e^(5*d*x + 5*c) + (3*I*d^2*f^3*x^2 + 4*d*f^3*x + 3*I*d^2*f*e^2
- 2*I*f^3 + 2*(3*I*d^2*f^2*x + 2*d*f^2)*e)*e^(4*d*x + 4*c) + 2*(3*d^2*f^3*x^2 - 4*I*d*f^3*x + 3*d^2*f*e^2 - 2
*f^3 + 2*(3*d^2*f^2*x - 2*I*d*f^2)*e)*e^(3*d*x + 3*c) + 2*(-3*I*d^2*f^3*x^2 - 4*d*f^3*x - 3*I*d^2*f*e^2 + 2*I*
f^3 + 2*(-3*I*d^2*f^2*x - 2*d*f^2)*e)*e^(2*d*x + 2*c) - (3*d^2*f^3*x^2 - 4*I*d*f^3*x + 3*d^2*f*e^2 - 2*f^3 + 2
*(3*d^2*f^2*x - 2*I*d*f^2)*e)*e^(d*x + c))*dilog(-e^(d*x + c)) - 3*(-3*I*d^2*f^3*x^2 + 4*d*f^3*x - 3*I*d^2*f*e
^2 + 2*I*f^3 + 2*(-3*I*d^2*f^2*x + 2*d*f^2)*e + (3*d^2*f^3*x^2 + 4*I*d*f^3*x + 3*d^2*f*e^2 - 2*f^3 + 2*(3*d^2*
f^2*x + 2*I*d*f^2)*e)*e^(5*d*x + 5*c) + (-3*I*d^2*f^3*x^2 + 4*d*f^3*x - 3*I*d^2*f*e^2 + 2*I*f^3 + 2*(-3*I*d^2*
f^2*x + 2*d*f^2)*e)*e^(4*d*x + 4*c) - 2*(3*d^2*f^3*x^2 + 4*I*d*f^3*x + 3*d^2*f*e^2 - 2*f^3 + 2*(3*d^2*f^2*x +
2*I*d*f^2)*e)*e^(3*d*x + 3*c) + 2*(3*I*d^2*f^3*x^2 - 4*d*f^3*x + 3*I*d^2*f*e^2 - 2*I*f^3 + 2*(3*I*d^2*f^2*x -
2*d*f^2)*e)*e^(2*d*x + 2*c) + (3*d^2*f^3*x^2 + 4*I*d*f^3*x + 3*d^2*f*e^2 - 2*f^3 + 2*(3*d^2*f^2*x + 2*I*d*f^2)
*e)*e^(d*x + c))*dilog(e^(d*x + c)) - 8*(-I*d^3*f^3*x^3 - I*c^3*f^3 + 3*(-I*d^3*f*x - I*c*d^2*f)*e^2 + 3*(-I*d
^3*f^2*x^2 + I*c^2*d*f^2)*e)*e^(5*d*x + 5*c) + 2*(d^3*f^3*x^3 - 3*d^2*f^3*x^2 + 4*c^3*f^3 - 3*d^3*e^3 + 3*(d^3
*f*x + (4*c - 1)*d^2*f)*e^2 + 3*(d^3*f^2*x^2 - 4*c^2*d*f^2 - 2*d^2*f^2*x)*e)*e^(4*d*x + 4*c) - 2*(5*I*d^3*f^3*
x^3 - 3*I*d^2*f^3*x^2 + 8*I*c^3*f^3 - 3*I*d^3*e^3 + 3*(5*I*d^3*f*x + (8*I*c - I)*d^2*f)*e^2 + 3*(5*I*d^3*f^2*x
^2 - 8*I*c^2*d*f^2 - 2*I*d^2*f^2*x)*e)*e^(3*d*x + 3*c) - 2*(3*d^3*f^3*x^3 - 3*d^2*f^3*x^2 + 8*c^3*f^3 - 5*d^3*
e^3 + 3*(3*d^3*f*x + (8*c - 1)*d^2*f)*e^2 + 3*(3*d^3*f^2*x^2 - 8*c^2*d*f^2 - 2*d^2*f^2*x)*e)*e^(2*d*x + 2*c) -
2*(-3*I*d^3*f^3*x^3 + 3*I*d^2*f^3*x^2 - 4*I*c^3*f^3 + I*d^3*e^3 + 3*(-3*I*d^3*f*x + (-4*I*c + I)*d^2*f)*e^2 +
3*(-3*I*d^3*f^2*x^2 + 4*I*c^2*d*f^2 + 2*I*d^2*f^2*x)*e)*e^(d*x + c) - 3*(I*d^3*f^3*x^3 + 2*d^2*f^3*x^2 - 2*I*
d*f^3*x + I*d^3*e^3 + (3*I*d^3*f*x + 2*d^2*f)*e^2 + (3*I*d^3*f^2*x^2 + 4*d^2*f^2*x - 2*I*d*f^2)*e - (d^3*f^3*x
^3 - 2*I*d^2*f^3*x^2 - 2*d*f^3*x + d^3*e^3 + (3*d^3*f*x - 2*I*d^2*f)*e^2 + (3*d^3*f^2*x^2 - 4*I*d^2*f^2*x - 2*
d*f^2)*e)*e^(5*d*x + 5*c) + (I*d^3*f^3*x^3 + 2*d^2*f^3*x^2 - 2*I*d*f^3*x + I*d^3*e^3 + (3*I*d^3*f*x + 2*d^2*f)
*e^2 + (3*I*d^3*f^2*x^2 + 4*d^2*f^2*x - 2*I*d*f^2)*e)*e^(4*d*x + 4*c) + 2*(d^3*f^3*x^3 - 2*I*d^2*f^3*x^2 - 2*d
*f^3*x + d^3*e^3 + (3*d^3*f*x - 2*I*d^2*f)*e^2 + (3*d^3*f^2*x^2 - 4*I*d^2*f^2*x - 2*d*f^2)*e)*e^(3*d*x + 3*c)
+ 2*(-I*d^3*f^3*x^3 - 2*d^2*f^3*x^2 + 2*I*d*f^3*x - I*d^3*e^3 + (-3*I*d^3*f*x - 2*d^2*f)*e^2 + (-3*I*d^3*f^2*x
^2 - 4*d^2*f^2*x + 2*I*d*f^2)*e)*e^(2*d*x + 2*c) - (d^3*f^3*x^3 - 2*I*d^2*f^3*x^2 - 2*d*f^3*x + d^3*e^3 + (3*d
^3*f*x - 2*I*d^2*f)*e^2 + (3*d^3*f^2*x^2 - 4*I*d^2*f^2*x - 2*d*f^2)*e)*e^(d*x + c))*log(e^(d*x + c) + 1) - 12*
(c^2*f^3 - 2*c*d*f^2*e + d^2*f*e^2 + (I*c^2*f^3 - 2*I*c*d*f^2*e + I*d^2*f*e^2)*e^(5*d*x + 5*c) + (c^2*f^3 - 2*
c*d*f^2*e + d^2*f*e^2)*e^(4*d*x + 4*c) + 2*(-I*c^2*f^3 + 2*I*c*d*f^2*e - I*d^2*f*e^2)*e^(3*d*x + 3*c) - 2*(c^2
*f^3 - 2*c*d*f^2*e + d^2*f*e^2)*e^(2*d*x + 2*c) + (I*c^2*f^3 - 2*I*c*d*f^2*e + I*d^2*f*e^2)*e^(d*x + c))*log(e
^(d*x + c) - I) - 3*((3*I*c + 2)*d^2*f*e^2 + (-3*I*c^2 - 4*c + 2*I)*d*f^2*e + (I*c^3 + 2*c^2 - 2*I*c)*f^3 - I*
d^3*e^3 - ((3*c - 2*I)*d^2*f*e^2 - (3*c^2 - 4*I*c - 2)*d*f^2*e + (c^3 - 2*I*c^2 - 2*c)*f^3 - d^3*e^3)*e^(5*d*x
+ 5*c) + ((3*I*c + 2)*d^2*f*e^2 + (-3*I*c^2 - 4*c + 2*I)*d*f^2*e + (I*c^3 + 2*c^2 - 2*I*c)*f^3 - I*d^3*e^3)*e
^(4*d*x + 4*c) + 2*((3*c - 2*I)*d^2*f*e^2 - (3*c^2 - 4*I*c - 2)*d*f^2*e + (c^3 - 2*I*c^2 - 2*c)*f^3 - d^3*e^3)
*e^(3*d*x + 3*c) + 2*((-3*I*c - 2)*d^2*f*e^2 + (3*I*c^2 + 4*c - 2*I)*d*f^2*e + (-I*c^3 - 2*c^2 + 2*I*c)*f^3 +
I*d^3*e^3)*e^(2*d*x + 2*c) - ((3*c - 2*I)*d^2*f*e^2 - (3*c^2 - 4*I*c - 2)*d*f^2*e + (c^3 - 2*I*c^2 - 2*c)*f^3
- d^3*e^3)*e^(d*x + c))*log(e^(d*x + c) - 1) - 12*(d^2*f^3*x^2 - c^2*f^3 + 2*(d^2*f^2*x + c*d*f^2)*e + (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^(5*d*x + 5*c) + (d^2*f^3*x^2 - c^2*f^3 + 2*(d^2*f^2*x +
c*d*f^2)*e)*e^(4*d*x + 4*c) + 2*(-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^(3*d*x + 3*c)
- 2*(d^2*f^3*x^2 - c^2*f^3 + 2*(d^2*f^2*x + c*d*f^2)*e)*e^(2*d*x + 2*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^(d*x + c))*log(I*e^(d*x + c) + 1) - 3*(-I*d^3*f^3*x^3 + 2*d^2*f^3*x^2 + 2*I*d*f^3*x
+ (-I*c^3 - 2*c^2 + 2*I*c)*f^3 + 3*(-I*d^3*f*x - I*c*d^2*f)*e^2 + (-3*I*d^3*f^2*x^2 + 4*d^2*f^2*x + (3*I*c^2
+ 4*c)*d*f^2)*e + (d^3*f^3*x^3 + 2*I*d^2*f^3*x^...

________________________________________________________________________________________

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)**3/(a+I*a*sinh(d*x+c)),x)

[Out]

Timed out

________________________________________________________________________________________

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)^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]
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 )}^3\,\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)^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)

________________________________________________________________________________________