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

Optimal. Leaf size=393 \[ \frac{12 i f^2 (e+f x) \text{PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac{12 i f^3 \text{PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}+\frac{6 f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac{3 i f^2 (e+f x) \sinh (c+d x) \cosh (c+d x)}{4 a d^3}+\frac{6 i f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac{3 i f (e+f x)^2 \sinh ^2(c+d x)}{4 a d^2}-\frac{3 f (e+f x)^2 \sinh (c+d x)}{a d^2}+\frac{3 i f^3 \sinh ^2(c+d x)}{8 a d^4}-\frac{6 f^3 \sinh (c+d x)}{a d^4}+\frac{(e+f x)^3 \cosh (c+d x)}{a d}-\frac{i (e+f x)^3 \tanh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right )}{a d}-\frac{i (e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 a d}+\frac{3 i e f^2 x}{4 a d^2}+\frac{3 i f^3 x^2}{8 a d^2}-\frac{i (e+f x)^3}{a d}+\frac{3 i (e+f x)^4}{8 a f} \]

[Out]

(((3*I)/4)*e*f^2*x)/(a*d^2) + (((3*I)/8)*f^3*x^2)/(a*d^2) - (I*(e + f*x)^3)/(a*d) + (((3*I)/8)*(e + f*x)^4)/(a
*f) + (6*f^2*(e + f*x)*Cosh[c + d*x])/(a*d^3) + ((e + f*x)^3*Cosh[c + d*x])/(a*d) + ((6*I)*f*(e + f*x)^2*Log[1
+ I*E^(c + d*x)])/(a*d^2) + ((12*I)*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^3) - ((12*I)*f^3*PolyLog
[3, (-I)*E^(c + d*x)])/(a*d^4) - (6*f^3*Sinh[c + d*x])/(a*d^4) - (3*f*(e + f*x)^2*Sinh[c + d*x])/(a*d^2) - (((
3*I)/4)*f^2*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(a*d^3) - ((I/2)*(e + f*x)^3*Cosh[c + d*x]*Sinh[c + d*x])/(
a*d) + (((3*I)/8)*f^3*Sinh[c + d*x]^2)/(a*d^4) + (((3*I)/4)*f*(e + f*x)^2*Sinh[c + d*x]^2)/(a*d^2) - (I*(e + f
*x)^3*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(a*d)

________________________________________________________________________________________

Rubi [A]  time = 0.702415, antiderivative size = 393, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 13, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.419, Rules used = {5557, 3311, 32, 3310, 3296, 2637, 3318, 4184, 3716, 2190, 2531, 2282, 6589} \[ \frac{12 i f^2 (e+f x) \text{PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac{12 i f^3 \text{PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}+\frac{6 f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac{3 i f^2 (e+f x) \sinh (c+d x) \cosh (c+d x)}{4 a d^3}+\frac{6 i f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac{3 i f (e+f x)^2 \sinh ^2(c+d x)}{4 a d^2}-\frac{3 f (e+f x)^2 \sinh (c+d x)}{a d^2}+\frac{3 i f^3 \sinh ^2(c+d x)}{8 a d^4}-\frac{6 f^3 \sinh (c+d x)}{a d^4}+\frac{(e+f x)^3 \cosh (c+d x)}{a d}-\frac{i (e+f x)^3 \tanh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right )}{a d}-\frac{i (e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 a d}+\frac{3 i e f^2 x}{4 a d^2}+\frac{3 i f^3 x^2}{8 a d^2}-\frac{i (e+f x)^3}{a d}+\frac{3 i (e+f x)^4}{8 a f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((3*I)/4)*e*f^2*x)/(a*d^2) + (((3*I)/8)*f^3*x^2)/(a*d^2) - (I*(e + f*x)^3)/(a*d) + (((3*I)/8)*(e + f*x)^4)/(a
*f) + (6*f^2*(e + f*x)*Cosh[c + d*x])/(a*d^3) + ((e + f*x)^3*Cosh[c + d*x])/(a*d) + ((6*I)*f*(e + f*x)^2*Log[1
+ I*E^(c + d*x)])/(a*d^2) + ((12*I)*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^3) - ((12*I)*f^3*PolyLog
[3, (-I)*E^(c + d*x)])/(a*d^4) - (6*f^3*Sinh[c + d*x])/(a*d^4) - (3*f*(e + f*x)^2*Sinh[c + d*x])/(a*d^2) - (((
3*I)/4)*f^2*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(a*d^3) - ((I/2)*(e + f*x)^3*Cosh[c + d*x]*Sinh[c + d*x])/(
a*d) + (((3*I)/8)*f^3*Sinh[c + d*x]^2)/(a*d^4) + (((3*I)/4)*f*(e + f*x)^2*Sinh[c + d*x]^2)/(a*d^2) - (I*(e + f
*x)^3*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(a*d)

Rule 5557

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[1/b, Int[(e + f*x)^m*Sinh[c + d*x]^(n - 1), x], x] - Dist[a/b, Int[((e + f*x)^m*Sinh[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 3311

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 3310

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

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3318

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

Rule 4184

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp[((c + d*x)^m*Cot[e + f*x])/f, x]
+ Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3716

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

Rule 2190

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

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin{align*} \int \frac{(e+f x)^3 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx &=i \int \frac{(e+f x)^3 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx-\frac{i \int (e+f x)^3 \sinh ^2(c+d x) \, dx}{a}\\ &=-\frac{i (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac{3 i f (e+f x)^2 \sinh ^2(c+d x)}{4 a d^2}+\frac{i \int (e+f x)^3 \, dx}{2 a}+\frac{\int (e+f x)^3 \sinh (c+d x) \, dx}{a}-\frac{\left (3 i f^2\right ) \int (e+f x) \sinh ^2(c+d x) \, dx}{2 a d^2}-\int \frac{(e+f x)^3 \sinh (c+d x)}{a+i a \sinh (c+d x)} \, dx\\ &=\frac{i (e+f x)^4}{8 a f}+\frac{(e+f x)^3 \cosh (c+d x)}{a d}-\frac{3 i f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac{i (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac{3 i f^3 \sinh ^2(c+d x)}{8 a d^4}+\frac{3 i f (e+f x)^2 \sinh ^2(c+d x)}{4 a d^2}-i \int \frac{(e+f x)^3}{a+i a \sinh (c+d x)} \, dx+\frac{i \int (e+f x)^3 \, dx}{a}-\frac{(3 f) \int (e+f x)^2 \cosh (c+d x) \, dx}{a d}+\frac{\left (3 i f^2\right ) \int (e+f x) \, dx}{4 a d^2}\\ &=\frac{3 i e f^2 x}{4 a d^2}+\frac{3 i f^3 x^2}{8 a d^2}+\frac{3 i (e+f x)^4}{8 a f}+\frac{(e+f x)^3 \cosh (c+d x)}{a d}-\frac{3 f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac{3 i f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac{i (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac{3 i f^3 \sinh ^2(c+d x)}{8 a d^4}+\frac{3 i f (e+f x)^2 \sinh ^2(c+d x)}{4 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{\left (6 f^2\right ) \int (e+f x) \sinh (c+d x) \, dx}{a d^2}\\ &=\frac{3 i e f^2 x}{4 a d^2}+\frac{3 i f^3 x^2}{8 a d^2}+\frac{3 i (e+f x)^4}{8 a f}+\frac{6 f^2 (e+f x) \cosh (c+d x)}{a d^3}+\frac{(e+f x)^3 \cosh (c+d x)}{a d}-\frac{3 f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac{3 i f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac{i (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac{3 i f^3 \sinh ^2(c+d x)}{8 a d^4}+\frac{3 i f (e+f x)^2 \sinh ^2(c+d x)}{4 a d^2}-\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 f^3\right ) \int \cosh (c+d x) \, dx}{a d^3}\\ &=\frac{3 i e f^2 x}{4 a d^2}+\frac{3 i f^3 x^2}{8 a d^2}-\frac{i (e+f x)^3}{a d}+\frac{3 i (e+f x)^4}{8 a f}+\frac{6 f^2 (e+f x) \cosh (c+d x)}{a d^3}+\frac{(e+f x)^3 \cosh (c+d x)}{a d}-\frac{6 f^3 \sinh (c+d x)}{a d^4}-\frac{3 f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac{3 i f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac{i (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac{3 i f^3 \sinh ^2(c+d x)}{8 a d^4}+\frac{3 i f (e+f x)^2 \sinh ^2(c+d x)}{4 a d^2}-\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{3 i e f^2 x}{4 a d^2}+\frac{3 i f^3 x^2}{8 a d^2}-\frac{i (e+f x)^3}{a d}+\frac{3 i (e+f x)^4}{8 a f}+\frac{6 f^2 (e+f x) \cosh (c+d x)}{a d^3}+\frac{(e+f x)^3 \cosh (c+d x)}{a d}+\frac{6 i f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac{6 f^3 \sinh (c+d x)}{a d^4}-\frac{3 f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac{3 i f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac{i (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac{3 i f^3 \sinh ^2(c+d x)}{8 a d^4}+\frac{3 i f (e+f x)^2 \sinh ^2(c+d x)}{4 a d^2}-\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{3 i e f^2 x}{4 a d^2}+\frac{3 i f^3 x^2}{8 a d^2}-\frac{i (e+f x)^3}{a d}+\frac{3 i (e+f x)^4}{8 a f}+\frac{6 f^2 (e+f x) \cosh (c+d x)}{a d^3}+\frac{(e+f x)^3 \cosh (c+d x)}{a d}+\frac{6 i f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{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{6 f^3 \sinh (c+d x)}{a d^4}-\frac{3 f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac{3 i f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac{i (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac{3 i f^3 \sinh ^2(c+d x)}{8 a d^4}+\frac{3 i f (e+f x)^2 \sinh ^2(c+d x)}{4 a d^2}-\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{3 i e f^2 x}{4 a d^2}+\frac{3 i f^3 x^2}{8 a d^2}-\frac{i (e+f x)^3}{a d}+\frac{3 i (e+f x)^4}{8 a f}+\frac{6 f^2 (e+f x) \cosh (c+d x)}{a d^3}+\frac{(e+f x)^3 \cosh (c+d x)}{a d}+\frac{6 i f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{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{6 f^3 \sinh (c+d x)}{a d^4}-\frac{3 f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac{3 i f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac{i (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac{3 i f^3 \sinh ^2(c+d x)}{8 a d^4}+\frac{3 i f (e+f x)^2 \sinh ^2(c+d x)}{4 a d^2}-\frac{i (e+f x)^3 \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{a d}-\frac{\left (12 i f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{2 \left (\frac{c}{2}+\frac{d x}{2}\right )}\right )}{a d^4}\\ &=\frac{3 i e f^2 x}{4 a d^2}+\frac{3 i f^3 x^2}{8 a d^2}-\frac{i (e+f x)^3}{a d}+\frac{3 i (e+f x)^4}{8 a f}+\frac{6 f^2 (e+f x) \cosh (c+d x)}{a d^3}+\frac{(e+f x)^3 \cosh (c+d x)}{a d}+\frac{6 i f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{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{12 i f^3 \text{Li}_3\left (-i e^{c+d x}\right )}{a d^4}-\frac{6 f^3 \sinh (c+d x)}{a d^4}-\frac{3 f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac{3 i f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac{i (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac{3 i f^3 \sinh ^2(c+d x)}{8 a d^4}+\frac{3 i f (e+f x)^2 \sinh ^2(c+d x)}{4 a d^2}-\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 [A]  time = 7.31088, size = 376, normalized size = 0.96 \[ \frac{-\frac{192 i f^2 \left (d (e+f x) \text{PolyLog}\left (2,i e^{-c-d x}\right )+f \text{PolyLog}\left (3,i e^{-c-d x}\right )\right )}{d^4}-\frac{6 i f^2 (e+f x) \sinh (2 (c+d x))}{d^3}+\frac{96 f^2 (e+f x) \cosh (c+d x)}{d^3}+\frac{96 i f (e+f x)^2 \log \left (1-i e^{-c-d x}\right )}{d^2}-\frac{48 f (e+f x)^2 \sinh (c+d x)}{d^2}+\frac{6 i f (e+f x)^2 \cosh (2 (c+d x))}{d^2}-\frac{96 f^3 \sinh (c+d x)}{d^4}+\frac{3 i f^3 \cosh (2 (c+d x))}{d^4}+\frac{32 (e+f x)^3}{\left (e^c-i\right ) d}-\frac{4 i (e+f x)^3 \sinh (2 (c+d x))}{d}+\frac{16 (e+f x)^3 \cosh (c+d x)}{d}-\frac{32 i \sinh \left (\frac{d x}{2}\right ) (e+f x)^3}{d \left (\cosh \left (\frac{c}{2}\right )+i \sinh \left (\frac{c}{2}\right )\right ) \left (\cosh \left (\frac{1}{2} (c+d x)\right )+i \sinh \left (\frac{1}{2} (c+d x)\right )\right )}+36 i e^2 f x^2+24 i e^3 x+24 i e f^2 x^3+6 i f^3 x^4}{16 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((24*I)*e^3*x + (36*I)*e^2*f*x^2 + (24*I)*e*f^2*x^3 + (6*I)*f^3*x^4 + (32*(e + f*x)^3)/(d*(-I + E^c)) + (96*f^
2*(e + f*x)*Cosh[c + d*x])/d^3 + (16*(e + f*x)^3*Cosh[c + d*x])/d + ((3*I)*f^3*Cosh[2*(c + d*x)])/d^4 + ((6*I)
*f*(e + f*x)^2*Cosh[2*(c + d*x)])/d^2 + ((96*I)*f*(e + f*x)^2*Log[1 - I*E^(-c - d*x)])/d^2 - ((192*I)*f^2*(d*(
e + f*x)*PolyLog[2, I*E^(-c - d*x)] + f*PolyLog[3, I*E^(-c - d*x)]))/d^4 - ((32*I)*(e + f*x)^3*Sinh[(d*x)/2])/
(d*(Cosh[c/2] + I*Sinh[c/2])*(Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2])) - (96*f^3*Sinh[c + d*x])/d^4 - (48*f*(
e + f*x)^2*Sinh[c + d*x])/d^2 - ((6*I)*f^2*(e + f*x)*Sinh[2*(c + d*x)])/d^3 - ((4*I)*(e + f*x)^3*Sinh[2*(c + d
*x)])/d)/(16*a)

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Maple [B]  time = 0.155, size = 928, normalized size = 2.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(f^3*x^3+3*e*f^2*x^2+3*e^2*f*x+e^3)/d/a/(exp(d*x+c)-I)+6*I/d^2/a*f^3*ln(1+I*exp(d*x+c))*x^2-6*I/d^2/a*ln(exp
(d*x+c))*e^2*f-6*I/d/a*e*f^2*x^2+6*I/d^2/a*ln(exp(d*x+c)-I)*e^2*f+6*I/d^3/a*f^3*c^2*x+6*I/d^4/a*f^3*c^2*ln(exp
(d*x+c)-I)+12*I/d^3/a*e*f^2*polylog(2,-I*exp(d*x+c))-6*I/d^4/a*f^3*c^2*ln(exp(d*x+c))-6*I/d^3/a*e*f^2*c^2+12*I
/d^3/a*f^3*polylog(2,-I*exp(d*x+c))*x-6*I/d^4/a*f^3*c^2*ln(1+I*exp(d*x+c))-12*I/d^2/a*e*f^2*c*x+12*I/d^3/a*e*f
^2*ln(1+I*exp(d*x+c))*c+12*I/d^2/a*e*f^2*ln(1+I*exp(d*x+c))*x+12*I/d^3/a*e*f^2*c*ln(exp(d*x+c))-12*I*f^3*polyl
og(3,-I*exp(d*x+c))/a/d^4-2*I/d/a*f^3*x^3+4*I/d^4/a*f^3*c^3-1/32*I*(4*d^3*f^3*x^3+12*d^3*e*f^2*x^2+12*d^3*e^2*
f*x-6*d^2*f^3*x^2+4*d^3*e^3-12*d^2*e*f^2*x-6*d^2*e^2*f+6*d*f^3*x+6*d*e*f^2-3*f^3)/a/d^4*exp(2*d*x+2*c)-12*I/d^
3/a*e*f^2*c*ln(exp(d*x+c)-I)+3/8*I/a*x^4*f^3+3/2*I/a*e^3*x+1/32*I*(4*d^3*f^3*x^3+12*d^3*e*f^2*x^2+12*d^3*e^2*f
*x+6*d^2*f^3*x^2+4*d^3*e^3+12*d^2*e*f^2*x+6*d^2*e^2*f+6*d*f^3*x+6*d*e*f^2+3*f^3)/a/d^4*exp(-2*d*x-2*c)+1/2*(d^
3*f^3*x^3+3*d^3*e*f^2*x^2+3*d^3*e^2*f*x+3*d^2*f^3*x^2+d^3*e^3+6*d^2*e*f^2*x+3*d^2*e^2*f+6*d*f^3*x+6*d*e*f^2+6*
f^3)/a/d^4*exp(-d*x-c)+1/2*(d^3*f^3*x^3+3*d^3*e*f^2*x^2+3*d^3*e^2*f*x-3*d^2*f^3*x^2+d^3*e^3-6*d^2*e*f^2*x-3*d^
2*e^2*f+6*d*f^3*x+6*d*e*f^2-6*f^3)/a/d^4*exp(d*x+c)+3/2*I/a*e*f^2*x^3+9/4*I/a*e^2*f*x^2

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [C]  time = 2.78172, size = 2449, normalized size = 6.23 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*d^3*f^3*x^3 + 4*d^3*e^3 + 6*d^2*e^2*f + 6*d*e*f^2 + 3*f^3 + 6*(2*d^3*e*f^2 + d^2*f^3)*x^2 + 6*(2*d^3*e^2*f
+ 2*d^2*e*f^2 + d*f^3)*x + ((384*I*d*f^3*x + 384*I*d*e*f^2)*e^(3*d*x + 3*c) + 384*(d*f^3*x + d*e*f^2)*e^(2*d*x
+ 2*c))*dilog(-I*e^(d*x + c)) + (-4*I*d^3*f^3*x^3 - 4*I*d^3*e^3 + 6*I*d^2*e^2*f - 6*I*d*e*f^2 + 3*I*f^3 + (-1
2*I*d^3*e*f^2 + 6*I*d^2*f^3)*x^2 + (-12*I*d^3*e^2*f + 12*I*d^2*e*f^2 - 6*I*d*f^3)*x)*e^(5*d*x + 5*c) + 3*(4*d^
3*f^3*x^3 + 4*d^3*e^3 - 14*d^2*e^2*f + 30*d*e*f^2 - 31*f^3 + 2*(6*d^3*e*f^2 - 7*d^2*f^3)*x^2 + 2*(6*d^3*e^2*f
- 14*d^2*e*f^2 + 15*d*f^3)*x)*e^(4*d*x + 4*c) + (12*I*d^4*f^3*x^4 - 16*I*d^3*e^3 + (-192*I*c + 48*I)*d^2*e^2*f
+ (192*I*c^2 - 96*I)*d*e*f^2 + (-64*I*c^3 + 96*I)*f^3 + (48*I*d^4*e*f^2 - 80*I*d^3*f^3)*x^3 + (72*I*d^4*e^2*f
- 240*I*d^3*e*f^2 + 48*I*d^2*f^3)*x^2 + (48*I*d^4*e^3 - 240*I*d^3*e^2*f + 96*I*d^2*e*f^2 - 96*I*d*f^3)*x)*e^(
3*d*x + 3*c) + 4*(3*d^4*f^3*x^4 + 20*d^3*e^3 - 12*(4*c - 1)*d^2*e^2*f + 24*(2*c^2 + 1)*d*e*f^2 - 8*(2*c^3 - 3)
*f^3 + 4*(3*d^4*e*f^2 + d^3*f^3)*x^3 + 6*(3*d^4*e^2*f + 2*d^3*e*f^2 + 2*d^2*f^3)*x^2 + 12*(d^4*e^3 + d^3*e^2*f
+ 2*d^2*e*f^2 + 2*d*f^3)*x)*e^(2*d*x + 2*c) + (-12*I*d^3*f^3*x^3 - 12*I*d^3*e^3 - 42*I*d^2*e^2*f - 90*I*d*e*f
^2 - 93*I*f^3 + (-36*I*d^3*e*f^2 - 42*I*d^2*f^3)*x^2 + (-36*I*d^3*e^2*f - 84*I*d^2*e*f^2 - 90*I*d*f^3)*x)*e^(d
*x + c) + ((192*I*d^2*e^2*f - 384*I*c*d*e*f^2 + 192*I*c^2*f^3)*e^(3*d*x + 3*c) + 192*(d^2*e^2*f - 2*c*d*e*f^2
+ c^2*f^3)*e^(2*d*x + 2*c))*log(e^(d*x + c) - I) + ((192*I*d^2*f^3*x^2 + 384*I*d^2*e*f^2*x + 384*I*c*d*e*f^2 -
192*I*c^2*f^3)*e^(3*d*x + 3*c) + 192*(d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3)*e^(2*d*x + 2*c))*l
og(I*e^(d*x + c) + 1) + (-384*I*f^3*e^(3*d*x + 3*c) - 384*f^3*e^(2*d*x + 2*c))*polylog(3, -I*e^(d*x + c)))/(32
*a*d^4*e^(3*d*x + 3*c) - 32*I*a*d^4*e^(2*d*x + 2*c))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (f x + e\right )}^{3} \sinh \left (d x + c\right )^{3}}{i \, a \sinh \left (d x + c\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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