\(\int \frac {(e+f x)^3 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx\) [199]

Optimal result

Integrand size = 31, antiderivative size = 393 \[ \int \frac {(e+f x)^3 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {3 i e f^2 x}{4 a d^2}+\frac {3 i f^3 x^2}{8 a d^2}-\frac {i (e+f x)^3}{a d}+\frac {3 i (e+f x)^4}{8 a f}+\frac {6 f^2 (e+f x) \cosh (c+d x)}{a d^3}+\frac {(e+f x)^3 \cosh (c+d x)}{a d}+\frac {6 i f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {12 i f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}-\frac {6 f^3 \sinh (c+d x)}{a d^4}-\frac {3 f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {3 i f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {3 i f^3 \sinh ^2(c+d x)}{8 a d^4}+\frac {3 i f (e+f x)^2 \sinh ^2(c+d x)}{4 a d^2}-\frac {i (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \]

[Out]

-12*I*f^3*polylog(3,-I*exp(d*x+c))/a/d^4-I*(f*x+e)^3*tanh(1/2*c+1/4*I*Pi+1/2*d*x)/a/d+3/4*I*f*(f*x+e)^2*sinh(d
*x+c)^2/a/d^2+3/8*I*(f*x+e)^4/a/f+6*f^2*(f*x+e)*cosh(d*x+c)/a/d^3+(f*x+e)^3*cosh(d*x+c)/a/d-I*(f*x+e)^3/a/d+6*
I*f*(f*x+e)^2*ln(1+I*exp(d*x+c))/a/d^2+3/8*I*f^3*sinh(d*x+c)^2/a/d^4-6*f^3*sinh(d*x+c)/a/d^4-3*f*(f*x+e)^2*sin
h(d*x+c)/a/d^2+12*I*f^2*(f*x+e)*polylog(2,-I*exp(d*x+c))/a/d^3+3/8*I*f^3*x^2/a/d^2-1/2*I*(f*x+e)^3*cosh(d*x+c)
*sinh(d*x+c)/a/d+3/4*I*e*f^2*x/a/d^2-3/4*I*f^2*(f*x+e)*cosh(d*x+c)*sinh(d*x+c)/a/d^3

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {5676, 3392, 32, 3391, 3377, 2717, 3399, 4269, 3797, 2221, 2611, 2320, 6724} \[ \int \frac {(e+f x)^3 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {12 i f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}+\frac {3 i f^3 \sinh ^2(c+d x)}{8 a d^4}-\frac {6 f^3 \sinh (c+d x)}{a d^4}+\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}+\frac {6 f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac {3 i f^2 (e+f x) \sinh (c+d x) \cosh (c+d x)}{4 a d^3}+\frac {6 i f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {3 i f (e+f x)^2 \sinh ^2(c+d x)}{4 a d^2}-\frac {3 f (e+f x)^2 \sinh (c+d x)}{a d^2}+\frac {(e+f x)^3 \cosh (c+d x)}{a d}-\frac {i (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{a d}-\frac {i (e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 a d}+\frac {3 i e f^2 x}{4 a d^2}+\frac {3 i f^3 x^2}{8 a d^2}-\frac {i (e+f x)^3}{a d}+\frac {3 i (e+f x)^4}{8 a f} \]

[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 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 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 2717

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

Rule 3377

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 3391

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 3392

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 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 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 5676

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 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]

Rubi steps \begin{align*} \text {integral}& = 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) \operatorname {PolyLog}\left (2,-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 \operatorname {PolyLog}\left (2,-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) \operatorname {PolyLog}\left (2,-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 ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )}\right )}{a d^4} \\ & = \frac {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) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {12 i f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}-\frac {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] (verified)

Time = 4.70 (sec) , antiderivative size = 376, normalized size of antiderivative = 0.96 \[ \int \frac {(e+f x)^3 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {24 i e^3 x+36 i e^2 f x^2+24 i e f^2 x^3+6 i f^3 x^4+\frac {32 (e+f x)^3}{d \left (-i+e^c\right )}+\frac {96 f^2 (e+f x) \cosh (c+d x)}{d^3}+\frac {16 (e+f x)^3 \cosh (c+d x)}{d}+\frac {3 i f^3 \cosh (2 (c+d x))}{d^4}+\frac {6 i f (e+f x)^2 \cosh (2 (c+d x))}{d^2}+\frac {96 i f (e+f x)^2 \log \left (1-i e^{-c-d x}\right )}{d^2}-\frac {192 i f^2 \left (d (e+f x) \operatorname {PolyLog}\left (2,i e^{-c-d x}\right )+f \operatorname {PolyLog}\left (3,i e^{-c-d x}\right )\right )}{d^4}-\frac {32 i (e+f x)^3 \sinh \left (\frac {d x}{2}\right )}{d \left (\cosh \left (\frac {c}{2}\right )+i \sinh \left (\frac {c}{2}\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {96 f^3 \sinh (c+d x)}{d^4}-\frac {48 f (e+f x)^2 \sinh (c+d x)}{d^2}-\frac {6 i f^2 (e+f x) \sinh (2 (c+d x))}{d^3}-\frac {4 i (e+f x)^3 \sinh (2 (c+d x))}{d}}{16 a} \]

[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)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1005 vs. \(2 (354 ) = 708\).

Time = 2.53 (sec) , antiderivative size = 1006, normalized size of antiderivative = 2.56

method	result	size
risch	\(-\frac {6 e^{2} f \arctan \left ({\mathrm e}^{d x +c}\right )}{a \,d^{2}}-\frac {6 c^{2} f^{3} \arctan \left ({\mathrm e}^{d x +c}\right )}{a \,d^{4}}+\frac {3 i f^{3} x^{4}}{8 a}+\frac {3 i e^{4}}{8 a f}+\frac {3 i e^{3} x}{2 a}+\frac {12 i c \,f^{2} e \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{3}}-\frac {6 i c \,f^{2} e \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{a \,d^{3}}-\frac {12 i f^{2} e c x}{a \,d^{2}}+\frac {\left (d^{3} x^{3} f^{3}+3 d^{3} e \,f^{2} x^{2}+3 d^{3} e^{2} f x +3 d^{2} f^{3} x^{2}+e^{3} d^{3}+6 d^{2} e \,f^{2} x +3 d^{2} e^{2} f +6 d \,f^{3} x +6 e \,f^{2} d +6 f^{3}\right ) {\mathrm e}^{-d x -c}}{2 d^{4} a}+\frac {12 c \,f^{2} e \arctan \left ({\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {12 i f^{2} e \ln \left (1+i {\mathrm e}^{d x +c}\right ) x}{a \,d^{2}}+\frac {12 i f^{2} e \ln \left (1+i {\mathrm e}^{d x +c}\right ) c}{a \,d^{3}}+\frac {3 i c^{2} f^{3} \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{a \,d^{4}}-\frac {6 i c^{2} f^{3} \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{4}}-\frac {6 i e^{2} f \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{2}}+\frac {12 i f^{2} e \operatorname {polylog}\left (2, -i {\mathrm e}^{d x +c}\right )}{a \,d^{3}}-\frac {6 i f^{2} e \,c^{2}}{a \,d^{3}}-\frac {6 i f^{3} \ln \left (1+i {\mathrm e}^{d x +c}\right ) c^{2}}{a \,d^{4}}+\frac {12 i f^{3} \operatorname {polylog}\left (2, -i {\mathrm e}^{d x +c}\right ) x}{a \,d^{3}}+\frac {3 i e^{2} f \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{a \,d^{2}}+\frac {6 i f^{3} \ln \left (1+i {\mathrm e}^{d x +c}\right ) x^{2}}{a \,d^{2}}-\frac {12 i f^{3} \operatorname {polylog}\left (3, -i {\mathrm e}^{d x +c}\right )}{a \,d^{4}}+\frac {i \left (4 d^{3} x^{3} f^{3}+12 d^{3} e \,f^{2} x^{2}+12 d^{3} e^{2} f x +6 d^{2} f^{3} x^{2}+4 e^{3} d^{3}+12 d^{2} e \,f^{2} x +6 d^{2} e^{2} f +6 d \,f^{3} x +6 e \,f^{2} d +3 f^{3}\right ) {\mathrm e}^{-2 d x -2 c}}{32 d^{4} a}-\frac {i \left (4 d^{3} x^{3} f^{3}+12 d^{3} e \,f^{2} x^{2}+12 d^{3} e^{2} f x -6 d^{2} f^{3} x^{2}+4 e^{3} d^{3}-12 d^{2} e \,f^{2} x -6 d^{2} e^{2} f +6 d \,f^{3} x +6 e \,f^{2} d -3 f^{3}\right ) {\mathrm e}^{2 d x +2 c}}{32 d^{4} a}+\frac {4 i f^{3} c^{3}}{a \,d^{4}}+\frac {6 i f^{3} x \,c^{2}}{a \,d^{3}}-\frac {6 i f^{2} e \,x^{2}}{a d}+\frac {3 i f^{2} e \,x^{3}}{2 a}-\frac {2 i f^{3} x^{3}}{a d}+\frac {\left (d^{3} x^{3} f^{3}+3 d^{3} e \,f^{2} x^{2}+3 d^{3} e^{2} f x -3 d^{2} f^{3} x^{2}+e^{3} d^{3}-6 d^{2} e \,f^{2} x -3 d^{2} e^{2} f +6 d \,f^{3} x +6 e \,f^{2} d -6 f^{3}\right ) {\mathrm e}^{d x +c}}{2 d^{4} a}+\frac {9 i f \,e^{2} x^{2}}{4 a}+\frac {2 f^{3} x^{3}+6 e \,f^{2} x^{2}+6 e^{2} f x +2 e^{3}}{d a \left ({\mathrm e}^{d x +c}-i\right )}\)	\(1006\)

[In]

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

[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/a/d^2*e^2*f*arctan(exp(d*x+c))-6/a/d^4*c^2*f^3*arct
an(exp(d*x+c))+12*I/a/d^3*e*f^2*ln(1+I*exp(d*x+c))*c-6*I/a/d^3*c*e*f^2*ln(1+exp(2*d*x+2*c))+12*I/a/d^3*c*e*f^2
*ln(exp(d*x+c))-12*I/a/d^2*e*f^2*c*x+12*I/a/d^2*e*f^2*ln(1+I*exp(d*x+c))*x+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)/d^4/a*exp(-d*x-c)+12/a/
d^3*c*f^2*e*arctan(exp(d*x+c))-12*I*f^3*polylog(3,-I*exp(d*x+c))/a/d^4+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)/d^4/a*exp(-2*d*x-
2*c)-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)/d^4/a*exp(2*d*x+2*c)+4*I/a/d^4*f^3*c^3+3/8*I/a*f^3*x^4+3/8*I/a/f*e^4-6*I/a/d*e*f^2*
x^2+6*I/a/d^3*f^3*x*c^2+6*I/a/d^2*f^3*ln(1+I*exp(d*x+c))*x^2+3*I/a/d^4*c^2*f^3*ln(1+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
)/d^4/a*exp(d*x+c)+3/2*I/a*e^3*x+3/2*I/a*f^2*e*x^3+9/4*I/a*f*e^2*x^2-6*I/a/d^3*e*f^2*c^2-6*I/a/d^4*f^3*ln(1+I*
exp(d*x+c))*c^2-6*I/a/d^4*c^2*f^3*ln(exp(d*x+c))-6*I/a/d^2*e^2*f*ln(exp(d*x+c))+12*I/a/d^3*e*f^2*polylog(2,-I*
exp(d*x+c))+12*I/a/d^3*f^3*polylog(2,-I*exp(d*x+c))*x+3*I/a/d^2*e^2*f*ln(1+exp(2*d*x+2*c))-2*I/a/d*f^3*x^3

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1044 vs. \(2 (337) = 674\).

Time = 0.31 (sec) , antiderivative size = 1044, normalized size of antiderivative = 2.66 \[ \int \frac {(e+f x)^3 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/32*(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 - I*d*e*f^2)*e^(3*d*x + 3*c) - (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 - 6*(2*
I*d^3*e*f^2 - I*d^2*f^3)*x^2 - 6*(2*I*d^3*e^2*f - 2*I*d^2*e*f^2 + 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) - 4*(-3*I*d^4*f^3*x^4 + 4*I*d^3*e^3 + 12*(4*I*c - I)*d^2*e^2*f + 24*(-2
*I*c^2 + I)*d*e*f^2 + 8*(2*I*c^3 - 3*I)*f^3 + 4*(-3*I*d^4*e*f^2 + 5*I*d^3*f^3)*x^3 + 6*(-3*I*d^4*e^2*f + 10*I*
d^3*e*f^2 - 2*I*d^2*f^3)*x^2 + 12*(-I*d^4*e^3 + 5*I*d^3*e^2*f - 2*I*d^2*e*f^2 + 2*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) - 3*(4*I*d^3*f^3*x^3 + 4*I*d^3*e^3 + 14*I*d^2*e^2*f + 30*I*d*e*f^2 + 31*I*f^3
+ 2*(6*I*d^3*e*f^2 + 7*I*d^2*f^3)*x^2 + 2*(6*I*d^3*e^2*f + 14*I*d^2*e*f^2 + 15*I*d*f^3)*x)*e^(d*x + c) - 192*(
(-I*d^2*e^2*f + 2*I*c*d*e*f^2 - I*c^2*f^3)*e^(3*d*x + 3*c) - (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 - 2*I*d^2*e*f^2*x - 2*I*c*d*e*f^2 + I*c^2*f^3)*e^(3*d*x + 3*c)
- (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))*log(I*e^(d*x + c) + 1) - 384*(I*f^3*
e^(3*d*x + 3*c) + f^3*e^(2*d*x + 2*c))*polylog(3, -I*e^(d*x + c)))/(a*d^4*e^(3*d*x + 3*c) - I*a*d^4*e^(2*d*x +
2*c))

Sympy [F]

\[ \int \frac {(e+f x)^3 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {2 e^{3} + 6 e^{2} f x + 6 e f^{2} x^{2} + 2 f^{3} x^{3}}{a d e^{c} e^{d x} - i a d} - \frac {i \left (\int \left (- \frac {i d e^{3}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {i d f^{3} x^{3}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {d e^{3} e^{c} e^{d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {4 d e^{3} e^{3 c} e^{3 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \frac {d e^{3} e^{5 c} e^{5 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \left (- \frac {3 i d e f^{2} x^{2}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {3 i d e^{2} f x}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \frac {4 i d e^{3} e^{2 c} e^{2 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \frac {i d e^{3} e^{4 c} e^{4 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \left (- \frac {24 i e^{2} f e^{2 c} e^{2 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {24 i f^{3} x^{2} e^{2 c} e^{2 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {d f^{3} x^{3} e^{c} e^{d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {4 d f^{3} x^{3} e^{3 c} e^{3 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \frac {d f^{3} x^{3} e^{5 c} e^{5 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \frac {4 i d f^{3} x^{3} e^{2 c} e^{2 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \frac {i d f^{3} x^{3} e^{4 c} e^{4 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \left (- \frac {48 i e f^{2} x e^{2 c} e^{2 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {3 d e f^{2} x^{2} e^{c} e^{d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {12 d e f^{2} x^{2} e^{3 c} e^{3 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \frac {3 d e f^{2} x^{2} e^{5 c} e^{5 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \left (- \frac {3 d e^{2} f x e^{c} e^{d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {12 d e^{2} f x e^{3 c} e^{3 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \frac {3 d e^{2} f x e^{5 c} e^{5 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \frac {12 i d e f^{2} x^{2} e^{2 c} e^{2 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \frac {3 i d e f^{2} x^{2} e^{4 c} e^{4 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \frac {12 i d e^{2} f x e^{2 c} e^{2 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \frac {3 i d e^{2} f x e^{4 c} e^{4 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx\right ) e^{- 2 c}}{4 a d} \]

[In]

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

[Out]

(2*e**3 + 6*e**2*f*x + 6*e*f**2*x**2 + 2*f**3*x**3)/(a*d*exp(c)*exp(d*x) - I*a*d) - I*(Integral(-I*d*e**3/(exp
(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(-I*d*f**3*x**3/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral
(-d*e**3*exp(c)*exp(d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(-4*d*e**3*exp(3*c)*exp(3*d*x)/(exp(
c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(d*e**3*exp(5*c)*exp(5*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x)
+ Integral(-3*I*d*e*f**2*x**2/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(-3*I*d*e**2*f*x/(exp(c)*exp(3
*d*x) - I*exp(2*d*x)), x) + Integral(4*I*d*e**3*exp(2*c)*exp(2*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + I
ntegral(I*d*e**3*exp(4*c)*exp(4*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(-24*I*e**2*f*exp(2*c)*e
xp(2*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(-24*I*f**3*x**2*exp(2*c)*exp(2*d*x)/(exp(c)*exp(3*
d*x) - I*exp(2*d*x)), x) + Integral(-d*f**3*x**3*exp(c)*exp(d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Inte
gral(-4*d*f**3*x**3*exp(3*c)*exp(3*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(d*f**3*x**3*exp(5*c)
*exp(5*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(4*I*d*f**3*x**3*exp(2*c)*exp(2*d*x)/(exp(c)*exp(
3*d*x) - I*exp(2*d*x)), x) + Integral(I*d*f**3*x**3*exp(4*c)*exp(4*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x)
+ Integral(-48*I*e*f**2*x*exp(2*c)*exp(2*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(-3*d*e*f**2*x
**2*exp(c)*exp(d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(-12*d*e*f**2*x**2*exp(3*c)*exp(3*d*x)/(e
xp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(3*d*e*f**2*x**2*exp(5*c)*exp(5*d*x)/(exp(c)*exp(3*d*x) - I*exp
(2*d*x)), x) + Integral(-3*d*e**2*f*x*exp(c)*exp(d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(-12*d*
e**2*f*x*exp(3*c)*exp(3*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(3*d*e**2*f*x*exp(5*c)*exp(5*d*x
)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(12*I*d*e*f**2*x**2*exp(2*c)*exp(2*d*x)/(exp(c)*exp(3*d*x)
- I*exp(2*d*x)), x) + Integral(3*I*d*e*f**2*x**2*exp(4*c)*exp(4*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) +
Integral(12*I*d*e**2*f*x*exp(2*c)*exp(2*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(3*I*d*e**2*f*x*
exp(4*c)*exp(4*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x))*exp(-2*c)/(4*a*d)

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e+f x)^3 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Exception raised: RuntimeError} \]

[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 >> ECL says: THROW: The catch RAT-ERR is undefined.

Giac [F]

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

[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)

Mupad [F(-1)]

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

[In]

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

[Out]

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