\(\int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx\) [193]

Optimal result

Integrand size = 31, antiderivative size = 241 \[ \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {(e+f x)^3}{a d}+\frac {(e+f x)^4}{4 a f}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {6 f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {12 f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {12 f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}+\frac {6 i f^3 \sinh (c+d x)}{a d^4}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {5676, 3377, 2717, 32, 3399, 4269, 3797, 2221, 2611, 2320, 6724} \[ \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {12 f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}+\frac {6 i f^3 \sinh (c+d x)}{a d^4}+\frac {12 f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}+\frac {6 f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{a d}-\frac {(e+f x)^3}{a d}+\frac {(e+f x)^4}{4 a f} \]

[In]

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

[Out]

-((e + f*x)^3/(a*d)) + (e + f*x)^4/(4*a*f) - ((6*I)*f^2*(e + f*x)*Cosh[c + d*x])/(a*d^3) - (I*(e + f*x)^3*Cosh
[c + d*x])/(a*d) + (6*f*(e + f*x)^2*Log[1 + I*E^(c + d*x)])/(a*d^2) + (12*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c +
d*x)])/(a*d^3) - (12*f^3*PolyLog[3, (-I)*E^(c + d*x)])/(a*d^4) + ((6*I)*f^3*Sinh[c + d*x])/(a*d^4) + ((3*I)*f
*(e + f*x)^2*Sinh[c + d*x])/(a*d^2) - ((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 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 (c+d x)}{a+i a \sinh (c+d x)} \, dx-\frac {i \int (e+f x)^3 \sinh (c+d x) \, dx}{a} \\ & = -\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {\int (e+f x)^3 \, dx}{a}+\frac {(3 i f) \int (e+f x)^2 \cosh (c+d x) \, dx}{a d}-\int \frac {(e+f x)^3}{a+i a \sinh (c+d x)} \, dx \\ & = \frac {(e+f x)^4}{4 a f}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {\int (e+f x)^3 \csc ^2\left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {i d x}{2}\right ) \, dx}{2 a}-\frac {\left (6 i f^2\right ) \int (e+f x) \sinh (c+d x) \, dx}{a d^2} \\ & = \frac {(e+f x)^4}{4 a f}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(3 f) \int (e+f x)^2 \coth \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{a d}+\frac {\left (6 i f^3\right ) \int \cosh (c+d x) \, dx}{a d^3} \\ & = -\frac {(e+f x)^3}{a d}+\frac {(e+f x)^4}{4 a f}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {6 i f^3 \sinh (c+d x)}{a d^4}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(6 i f) \int \frac {e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )} (e+f x)^2}{1+i e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )}} \, dx}{a d} \\ & = -\frac {(e+f x)^3}{a d}+\frac {(e+f x)^4}{4 a f}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {6 f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {6 i f^3 \sinh (c+d x)}{a d^4}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {\left (12 f^2\right ) \int (e+f x) \log \left (1+i e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )}\right ) \, dx}{a d^2} \\ & = -\frac {(e+f x)^3}{a d}+\frac {(e+f x)^4}{4 a f}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {6 f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {12 f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}+\frac {6 i f^3 \sinh (c+d x)}{a d^4}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {\left (12 f^3\right ) \int \operatorname {PolyLog}\left (2,-i e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )}\right ) \, dx}{a d^3} \\ & = -\frac {(e+f x)^3}{a d}+\frac {(e+f x)^4}{4 a f}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {6 f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {12 f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}+\frac {6 i f^3 \sinh (c+d x)}{a d^4}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {\left (12 f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )}\right )}{a d^4} \\ & = -\frac {(e+f x)^3}{a d}+\frac {(e+f x)^4}{4 a f}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {6 f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {12 f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {12 f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}+\frac {6 i f^3 \sinh (c+d x)}{a d^4}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2619\) vs. \(2(241)=482\).

Time = 3.58 (sec) , antiderivative size = 2619, normalized size of antiderivative = 10.87 \[ \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

((-10*I)*d^3*e^3*E^c*Cosh[(d*x)/2] - 2*d^3*e^3*E^(2*c)*Cosh[(d*x)/2] - (6*I)*d^2*e^2*E^c*f*Cosh[(d*x)/2] + 6*d
^2*e^2*E^(2*c)*f*Cosh[(d*x)/2] - (12*I)*d*e*E^c*f^2*Cosh[(d*x)/2] - 12*d*e*E^(2*c)*f^2*Cosh[(d*x)/2] - (12*I)*
E^c*f^3*Cosh[(d*x)/2] + 12*E^(2*c)*f^3*Cosh[(d*x)/2] - (4*I)*d^4*e^3*E^c*x*Cosh[(d*x)/2] + 4*d^4*e^3*E^(2*c)*x
*Cosh[(d*x)/2] - (30*I)*d^3*e^2*E^c*f*x*Cosh[(d*x)/2] - 6*d^3*e^2*E^(2*c)*f*x*Cosh[(d*x)/2] - (12*I)*d^2*e*E^c
*f^2*x*Cosh[(d*x)/2] + 12*d^2*e*E^(2*c)*f^2*x*Cosh[(d*x)/2] - (12*I)*d*E^c*f^3*x*Cosh[(d*x)/2] - 12*d*E^(2*c)*
f^3*x*Cosh[(d*x)/2] - (6*I)*d^4*e^2*E^c*f*x^2*Cosh[(d*x)/2] + 6*d^4*e^2*E^(2*c)*f*x^2*Cosh[(d*x)/2] - (30*I)*d
^3*e*E^c*f^2*x^2*Cosh[(d*x)/2] - 6*d^3*e*E^(2*c)*f^2*x^2*Cosh[(d*x)/2] - (6*I)*d^2*E^c*f^3*x^2*Cosh[(d*x)/2] +
6*d^2*E^(2*c)*f^3*x^2*Cosh[(d*x)/2] - (4*I)*d^4*e*E^c*f^2*x^3*Cosh[(d*x)/2] + 4*d^4*e*E^(2*c)*f^2*x^3*Cosh[(d
*x)/2] - (10*I)*d^3*E^c*f^3*x^3*Cosh[(d*x)/2] - 2*d^3*E^(2*c)*f^3*x^3*Cosh[(d*x)/2] - I*d^4*E^c*f^3*x^4*Cosh[(
d*x)/2] + d^4*E^(2*c)*f^3*x^4*Cosh[(d*x)/2] - 2*d^3*e^3*Cosh[(3*d*x)/2] - (2*I)*d^3*e^3*E^(3*c)*Cosh[(3*d*x)/2
] - 6*d^2*e^2*f*Cosh[(3*d*x)/2] + (6*I)*d^2*e^2*E^(3*c)*f*Cosh[(3*d*x)/2] - 12*d*e*f^2*Cosh[(3*d*x)/2] - (12*I
)*d*e*E^(3*c)*f^2*Cosh[(3*d*x)/2] - 12*f^3*Cosh[(3*d*x)/2] + (12*I)*E^(3*c)*f^3*Cosh[(3*d*x)/2] - 6*d^3*e^2*f*
x*Cosh[(3*d*x)/2] - (6*I)*d^3*e^2*E^(3*c)*f*x*Cosh[(3*d*x)/2] - 12*d^2*e*f^2*x*Cosh[(3*d*x)/2] + (12*I)*d^2*e*
E^(3*c)*f^2*x*Cosh[(3*d*x)/2] - 12*d*f^3*x*Cosh[(3*d*x)/2] - (12*I)*d*E^(3*c)*f^3*x*Cosh[(3*d*x)/2] - 6*d^3*e*
f^2*x^2*Cosh[(3*d*x)/2] - (6*I)*d^3*e*E^(3*c)*f^2*x^2*Cosh[(3*d*x)/2] - 6*d^2*f^3*x^2*Cosh[(3*d*x)/2] + (6*I)*
d^2*E^(3*c)*f^3*x^2*Cosh[(3*d*x)/2] - 2*d^3*f^3*x^3*Cosh[(3*d*x)/2] - (2*I)*d^3*E^(3*c)*f^3*x^3*Cosh[(3*d*x)/2
] - (24*I)*d^2*e^2*E^c*f*Cosh[(d*x)/2]*Log[1 - I*E^(-c - d*x)] + 24*d^2*e^2*E^(2*c)*f*Cosh[(d*x)/2]*Log[1 - I*
E^(-c - d*x)] - (48*I)*d^2*e*E^c*f^2*x*Cosh[(d*x)/2]*Log[1 - I*E^(-c - d*x)] + 48*d^2*e*E^(2*c)*f^2*x*Cosh[(d*
x)/2]*Log[1 - I*E^(-c - d*x)] - (24*I)*d^2*E^c*f^3*x^2*Cosh[(d*x)/2]*Log[1 - I*E^(-c - d*x)] + 24*d^2*E^(2*c)*
f^3*x^2*Cosh[(d*x)/2]*Log[1 - I*E^(-c - d*x)] + (10*I)*d^3*e^3*E^c*Sinh[(d*x)/2] - 2*d^3*e^3*E^(2*c)*Sinh[(d*x
)/2] + (6*I)*d^2*e^2*E^c*f*Sinh[(d*x)/2] + 6*d^2*e^2*E^(2*c)*f*Sinh[(d*x)/2] + (12*I)*d*e*E^c*f^2*Sinh[(d*x)/2
] - 12*d*e*E^(2*c)*f^2*Sinh[(d*x)/2] + (12*I)*E^c*f^3*Sinh[(d*x)/2] + 12*E^(2*c)*f^3*Sinh[(d*x)/2] + (4*I)*d^4
*e^3*E^c*x*Sinh[(d*x)/2] + 4*d^4*e^3*E^(2*c)*x*Sinh[(d*x)/2] + (30*I)*d^3*e^2*E^c*f*x*Sinh[(d*x)/2] - 6*d^3*e^
2*E^(2*c)*f*x*Sinh[(d*x)/2] + (12*I)*d^2*e*E^c*f^2*x*Sinh[(d*x)/2] + 12*d^2*e*E^(2*c)*f^2*x*Sinh[(d*x)/2] + (1
2*I)*d*E^c*f^3*x*Sinh[(d*x)/2] - 12*d*E^(2*c)*f^3*x*Sinh[(d*x)/2] + (6*I)*d^4*e^2*E^c*f*x^2*Sinh[(d*x)/2] + 6*
d^4*e^2*E^(2*c)*f*x^2*Sinh[(d*x)/2] + (30*I)*d^3*e*E^c*f^2*x^2*Sinh[(d*x)/2] - 6*d^3*e*E^(2*c)*f^2*x^2*Sinh[(d
*x)/2] + (6*I)*d^2*E^c*f^3*x^2*Sinh[(d*x)/2] + 6*d^2*E^(2*c)*f^3*x^2*Sinh[(d*x)/2] + (4*I)*d^4*e*E^c*f^2*x^3*S
inh[(d*x)/2] + 4*d^4*e*E^(2*c)*f^2*x^3*Sinh[(d*x)/2] + (10*I)*d^3*E^c*f^3*x^3*Sinh[(d*x)/2] - 2*d^3*E^(2*c)*f^
3*x^3*Sinh[(d*x)/2] + I*d^4*E^c*f^3*x^4*Sinh[(d*x)/2] + d^4*E^(2*c)*f^3*x^4*Sinh[(d*x)/2] + (24*I)*d^2*e^2*E^c
*f*Log[1 - I*E^(-c - d*x)]*Sinh[(d*x)/2] + 24*d^2*e^2*E^(2*c)*f*Log[1 - I*E^(-c - d*x)]*Sinh[(d*x)/2] + (48*I)
*d^2*e*E^c*f^2*x*Log[1 - I*E^(-c - d*x)]*Sinh[(d*x)/2] + 48*d^2*e*E^(2*c)*f^2*x*Log[1 - I*E^(-c - d*x)]*Sinh[(
d*x)/2] + (24*I)*d^2*E^c*f^3*x^2*Log[1 - I*E^(-c - d*x)]*Sinh[(d*x)/2] + 24*d^2*E^(2*c)*f^3*x^2*Log[1 - I*E^(-
c - d*x)]*Sinh[(d*x)/2] - 48*d*E^c*f^2*(e + f*x)*PolyLog[2, I*E^(-c - d*x)]*((-I + E^c)*Cosh[(d*x)/2] + (I + E
^c)*Sinh[(d*x)/2]) - 48*E^c*f^3*PolyLog[3, I*E^(-c - d*x)]*((-I + E^c)*Cosh[(d*x)/2] + (I + E^c)*Sinh[(d*x)/2]
) + 2*d^3*e^3*Sinh[(3*d*x)/2] - (2*I)*d^3*e^3*E^(3*c)*Sinh[(3*d*x)/2] + 6*d^2*e^2*f*Sinh[(3*d*x)/2] + (6*I)*d^
2*e^2*E^(3*c)*f*Sinh[(3*d*x)/2] + 12*d*e*f^2*Sinh[(3*d*x)/2] - (12*I)*d*e*E^(3*c)*f^2*Sinh[(3*d*x)/2] + 12*f^3
*Sinh[(3*d*x)/2] + (12*I)*E^(3*c)*f^3*Sinh[(3*d*x)/2] + 6*d^3*e^2*f*x*Sinh[(3*d*x)/2] - (6*I)*d^3*e^2*E^(3*c)*
f*x*Sinh[(3*d*x)/2] + 12*d^2*e*f^2*x*Sinh[(3*d*x)/2] + (12*I)*d^2*e*E^(3*c)*f^2*x*Sinh[(3*d*x)/2] + 12*d*f^3*x
*Sinh[(3*d*x)/2] - (12*I)*d*E^(3*c)*f^3*x*Sinh[(3*d*x)/2] + 6*d^3*e*f^2*x^2*Sinh[(3*d*x)/2] - (6*I)*d^3*e*E^(3
*c)*f^2*x^2*Sinh[(3*d*x)/2] + 6*d^2*f^3*x^2*Sinh[(3*d*x)/2] + (6*I)*d^2*E^(3*c)*f^3*x^2*Sinh[(3*d*x)/2] + 2*d^
3*f^3*x^3*Sinh[(3*d*x)/2] - (2*I)*d^3*E^(3*c)*f^3*x^3*Sinh[(3*d*x)/2])/(4*a*d^4*E^c*((-I + E^c)*Cosh[(d*x)/2]
+ (I + E^c)*Sinh[(d*x)/2]))

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 698 vs. \(2 (222 ) = 444\).

Time = 2.90 (sec) , antiderivative size = 699, normalized size of antiderivative = 2.90

method	result	size
risch	\(\frac {f^{3} x^{4}}{4 a}+\frac {e^{4}}{4 a f}+\frac {12 f^{2} e \ln \left (1+i {\mathrm e}^{d x +c}\right ) x}{a \,d^{2}}-\frac {12 f^{2} e c \ln \left ({\mathrm e}^{d x +c}-i\right )}{a \,d^{3}}+\frac {12 f^{2} e \ln \left (1+i {\mathrm e}^{d x +c}\right ) c}{a \,d^{3}}-\frac {6 f^{2} e \,c^{2}}{a \,d^{3}}-\frac {6 f \ln \left ({\mathrm e}^{d x +c}\right ) e^{2}}{a \,d^{2}}-\frac {6 f^{2} e \,x^{2}}{a d}-\frac {6 f^{3} c^{2} \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{4}}+\frac {6 f^{3} x \,c^{2}}{a \,d^{3}}+\frac {f^{2} e \,x^{3}}{a}+\frac {3 f \,e^{2} x^{2}}{2 a}+\frac {e^{3} x}{a}+\frac {4 f^{3} c^{3}}{a \,d^{4}}-\frac {2 f^{3} x^{3}}{a d}-\frac {12 f^{2} e c x}{a \,d^{2}}+\frac {12 f^{2} e c \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{3}}-\frac {12 f^{3} \operatorname {polylog}\left (3, -i {\mathrm e}^{d x +c}\right )}{a \,d^{4}}-\frac {i \left (d^{3} x^{3} f^{3}+3 d^{3} e \,f^{2} x^{2}+3 d^{3} e^{2} f x -3 d^{2} f^{3} x^{2}+e^{3} d^{3}-6 d^{2} e \,f^{2} x -3 d^{2} e^{2} f +6 d \,f^{3} x +6 e \,f^{2} d -6 f^{3}\right ) {\mathrm e}^{d x +c}}{2 d^{4} a}-\frac {2 i \left (f^{3} x^{3}+3 e \,f^{2} x^{2}+3 e^{2} f x +e^{3}\right )}{d a \left ({\mathrm e}^{d x +c}-i\right )}+\frac {6 f^{3} \ln \left (1+i {\mathrm e}^{d x +c}\right ) x^{2}}{a \,d^{2}}+\frac {12 f^{2} e \operatorname {polylog}\left (2, -i {\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {6 f \ln \left ({\mathrm e}^{d x +c}-i\right ) e^{2}}{a \,d^{2}}+\frac {6 f^{3} c^{2} \ln \left ({\mathrm e}^{d x +c}-i\right )}{a \,d^{4}}+\frac {12 f^{3} \operatorname {polylog}\left (2, -i {\mathrm e}^{d x +c}\right ) x}{a \,d^{3}}-\frac {6 f^{3} \ln \left (1+i {\mathrm e}^{d x +c}\right ) c^{2}}{a \,d^{4}}-\frac {i \left (d^{3} x^{3} f^{3}+3 d^{3} e \,f^{2} x^{2}+3 d^{3} e^{2} f x +3 d^{2} f^{3} x^{2}+e^{3} d^{3}+6 d^{2} e \,f^{2} x +3 d^{2} e^{2} f +6 d \,f^{3} x +6 e \,f^{2} d +6 f^{3}\right ) {\mathrm e}^{-d x -c}}{2 d^{4} a}\)	\(699\)

[In]

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

[Out]

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

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. 823 vs. \(2 (213) = 426\).

Time = 0.26 (sec) , antiderivative size = 823, normalized size of antiderivative = 3.41 \[ \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {2 \, d^{3} f^{3} x^{3} + 2 \, d^{3} e^{3} + 6 \, d^{2} e^{2} f + 12 \, d e f^{2} + 12 \, f^{3} + 6 \, {\left (d^{3} e f^{2} + d^{2} f^{3}\right )} x^{2} + 6 \, {\left (d^{3} e^{2} f + 2 \, d^{2} e f^{2} + 2 \, d f^{3}\right )} x - 48 \, {\left ({\left (d f^{3} x + d e f^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )} - {\left (i \, d f^{3} x + i \, d e f^{2}\right )} e^{\left (d x + c\right )}\right )} {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right ) + 2 \, {\left (i \, d^{3} f^{3} x^{3} + i \, d^{3} e^{3} - 3 i \, d^{2} e^{2} f + 6 i \, d e f^{2} - 6 i \, f^{3} + 3 \, {\left (i \, d^{3} e f^{2} - i \, d^{2} f^{3}\right )} x^{2} + 3 \, {\left (i \, d^{3} e^{2} f - 2 i \, d^{2} e f^{2} + 2 i \, d f^{3}\right )} x\right )} e^{\left (3 \, d x + 3 \, c\right )} - {\left (d^{4} f^{3} x^{4} - 2 \, d^{3} e^{3} - 6 \, {\left (4 \, c - 1\right )} d^{2} e^{2} f + 12 \, {\left (2 \, c^{2} - 1\right )} d e f^{2} - 4 \, {\left (2 \, c^{3} - 3\right )} f^{3} + 2 \, {\left (2 \, d^{4} e f^{2} - 5 \, d^{3} f^{3}\right )} x^{3} + 6 \, {\left (d^{4} e^{2} f - 5 \, d^{3} e f^{2} + d^{2} f^{3}\right )} x^{2} + 2 \, {\left (2 \, d^{4} e^{3} - 15 \, d^{3} e^{2} f + 6 \, d^{2} e f^{2} - 6 \, d f^{3}\right )} x\right )} e^{\left (2 \, d x + 2 \, c\right )} - {\left (-i \, d^{4} f^{3} x^{4} - 10 i \, d^{3} e^{3} - 6 \, {\left (-4 i \, c + i\right )} d^{2} e^{2} f - 12 \, {\left (2 i \, c^{2} + i\right )} d e f^{2} - 4 \, {\left (-2 i \, c^{3} + 3 i\right )} f^{3} - 2 \, {\left (2 i \, d^{4} e f^{2} + i \, d^{3} f^{3}\right )} x^{3} - 6 \, {\left (i \, d^{4} e^{2} f + i \, d^{3} e f^{2} + i \, d^{2} f^{3}\right )} x^{2} - 2 \, {\left (2 i \, d^{4} e^{3} + 3 i \, d^{3} e^{2} f + 6 i \, d^{2} e f^{2} + 6 i \, d f^{3}\right )} x\right )} e^{\left (d x + c\right )} - 24 \, {\left ({\left (d^{2} e^{2} f - 2 \, c d e f^{2} + c^{2} f^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )} - {\left (i \, d^{2} e^{2} f - 2 i \, c d e f^{2} + i \, c^{2} f^{3}\right )} e^{\left (d x + c\right )}\right )} \log \left (e^{\left (d x + c\right )} - i\right ) - 24 \, {\left ({\left (d^{2} f^{3} x^{2} + 2 \, d^{2} e f^{2} x + 2 \, c d e f^{2} - c^{2} f^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )} - {\left (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}\right )} e^{\left (d x + c\right )}\right )} \log \left (i \, e^{\left (d x + c\right )} + 1\right ) + 48 \, {\left (f^{3} e^{\left (2 \, d x + 2 \, c\right )} - i \, f^{3} e^{\left (d x + c\right )}\right )} {\rm polylog}\left (3, -i \, e^{\left (d x + c\right )}\right )}{4 \, {\left (a d^{4} e^{\left (2 \, d x + 2 \, c\right )} - i \, a d^{4} e^{\left (d x + c\right )}\right )}} \]

[In]

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

[Out]

-1/4*(2*d^3*f^3*x^3 + 2*d^3*e^3 + 6*d^2*e^2*f + 12*d*e*f^2 + 12*f^3 + 6*(d^3*e*f^2 + d^2*f^3)*x^2 + 6*(d^3*e^2
*f + 2*d^2*e*f^2 + 2*d*f^3)*x - 48*((d*f^3*x + d*e*f^2)*e^(2*d*x + 2*c) - (I*d*f^3*x + I*d*e*f^2)*e^(d*x + c))
*dilog(-I*e^(d*x + c)) + 2*(I*d^3*f^3*x^3 + I*d^3*e^3 - 3*I*d^2*e^2*f + 6*I*d*e*f^2 - 6*I*f^3 + 3*(I*d^3*e*f^2
- I*d^2*f^3)*x^2 + 3*(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) - (d^4*f^3*x^4 - 2*d^3*e^3
- 6*(4*c - 1)*d^2*e^2*f + 12*(2*c^2 - 1)*d*e*f^2 - 4*(2*c^3 - 3)*f^3 + 2*(2*d^4*e*f^2 - 5*d^3*f^3)*x^3 + 6*(d^
4*e^2*f - 5*d^3*e*f^2 + d^2*f^3)*x^2 + 2*(2*d^4*e^3 - 15*d^3*e^2*f + 6*d^2*e*f^2 - 6*d*f^3)*x)*e^(2*d*x + 2*c)
- (-I*d^4*f^3*x^4 - 10*I*d^3*e^3 - 6*(-4*I*c + I)*d^2*e^2*f - 12*(2*I*c^2 + I)*d*e*f^2 - 4*(-2*I*c^3 + 3*I)*f
^3 - 2*(2*I*d^4*e*f^2 + I*d^3*f^3)*x^3 - 6*(I*d^4*e^2*f + I*d^3*e*f^2 + I*d^2*f^3)*x^2 - 2*(2*I*d^4*e^3 + 3*I*
d^3*e^2*f + 6*I*d^2*e*f^2 + 6*I*d*f^3)*x)*e^(d*x + c) - 24*((d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*e^(2*d*x + 2*c
) - (I*d^2*e^2*f - 2*I*c*d*e*f^2 + I*c^2*f^3)*e^(d*x + c))*log(e^(d*x + c) - I) - 24*((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) - (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^(d*x + c))*log(I*e^(d*x + c) + 1) + 48*(f^3*e^(2*d*x + 2*c) - I*f^3*e^(d*x + c))*polylog(3, -I*e^(d*x + c)))
/(a*d^4*e^(2*d*x + 2*c) - I*a*d^4*e^(d*x + c))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 671 vs. \(2 (213) = 426\).

Time = 0.38 (sec) , antiderivative size = 671, normalized size of antiderivative = 2.78 \[ \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {3}{2} \, e^{2} f {\left (\frac {2 \, x e^{\left (d x + c\right )}}{a d e^{\left (d x + c\right )} - i \, a d} + \frac {i \, d^{2} x^{2} e^{c} + i \, d x e^{c} - {\left (-i \, d x e^{\left (3 \, c\right )} + i \, e^{\left (3 \, c\right )}\right )} e^{\left (2 \, d x\right )} - {\left (d^{2} x^{2} e^{\left (2 \, c\right )} - 3 \, d x e^{\left (2 \, c\right )} + e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} + {\left (d x + 1\right )} e^{\left (-d x\right )} + i \, e^{c}}{a d^{2} e^{\left (d x + 2 \, c\right )} - i \, a d^{2} e^{c}} - \frac {4 \, \log \left ({\left (e^{\left (d x + c\right )} - i\right )} e^{\left (-c\right )}\right )}{a d^{2}}\right )} + \frac {1}{2} \, e^{3} {\left (\frac {2 \, {\left (d x + c\right )}}{a d} + \frac {-5 i \, e^{\left (-d x - c\right )} + 1}{{\left (i \, a e^{\left (-d x - c\right )} + a e^{\left (-2 \, d x - 2 \, c\right )}\right )} d} - \frac {i \, e^{\left (-d x - c\right )}}{a d}\right )} + \frac {-i \, d^{4} f^{3} x^{4} - 12 i \, d e f^{2} + 2 \, {\left (-2 i \, d^{4} e f^{2} - 5 i \, d^{3} f^{3}\right )} x^{3} - 12 i \, f^{3} + 6 \, {\left (-5 i \, d^{3} e f^{2} - i \, d^{2} f^{3}\right )} x^{2} + 12 \, {\left (-i \, d^{2} e f^{2} - i \, d f^{3}\right )} x + 2 \, {\left (-i \, d^{3} f^{3} x^{3} e^{\left (2 \, c\right )} + 3 \, {\left (-i \, d^{3} e f^{2} + i \, d^{2} f^{3}\right )} x^{2} e^{\left (2 \, c\right )} + 6 \, {\left (i \, d^{2} e f^{2} - i \, d f^{3}\right )} x e^{\left (2 \, c\right )} + 6 \, {\left (-i \, d e f^{2} + i \, f^{3}\right )} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} + {\left (d^{4} f^{3} x^{4} e^{c} + 2 \, {\left (2 \, d^{4} e f^{2} - d^{3} f^{3}\right )} x^{3} e^{c} - 6 \, {\left (d^{3} e f^{2} - d^{2} f^{3}\right )} x^{2} e^{c} + 12 \, {\left (d^{2} e f^{2} - d f^{3}\right )} x e^{c} - 12 \, {\left (d e f^{2} - f^{3}\right )} e^{c}\right )} e^{\left (d x\right )}}{4 \, {\left (a d^{4} e^{\left (d x + c\right )} - i \, a d^{4}\right )}} + \frac {12 \, {\left (d x \log \left (i \, e^{\left (d x + c\right )} + 1\right ) + {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right )\right )} e f^{2}}{a d^{3}} + \frac {6 \, {\left (d^{2} x^{2} \log \left (i \, e^{\left (d x + c\right )} + 1\right ) + 2 \, d x {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right ) - 2 \, {\rm Li}_{3}(-i \, e^{\left (d x + c\right )})\right )} f^{3}}{a d^{4}} - \frac {2 \, {\left (d^{3} f^{3} x^{3} + 3 \, d^{3} e f^{2} x^{2}\right )}}{a d^{4}} \]

[In]

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

[Out]

-3/2*e^2*f*(2*x*e^(d*x + c)/(a*d*e^(d*x + c) - I*a*d) + (I*d^2*x^2*e^c + I*d*x*e^c - (-I*d*x*e^(3*c) + I*e^(3*
c))*e^(2*d*x) - (d^2*x^2*e^(2*c) - 3*d*x*e^(2*c) + e^(2*c))*e^(d*x) + (d*x + 1)*e^(-d*x) + I*e^c)/(a*d^2*e^(d*
x + 2*c) - I*a*d^2*e^c) - 4*log((e^(d*x + c) - I)*e^(-c))/(a*d^2)) + 1/2*e^3*(2*(d*x + c)/(a*d) + (-5*I*e^(-d*
x - c) + 1)/((I*a*e^(-d*x - c) + a*e^(-2*d*x - 2*c))*d) - I*e^(-d*x - c)/(a*d)) + 1/4*(-I*d^4*f^3*x^4 - 12*I*d
*e*f^2 + 2*(-2*I*d^4*e*f^2 - 5*I*d^3*f^3)*x^3 - 12*I*f^3 + 6*(-5*I*d^3*e*f^2 - I*d^2*f^3)*x^2 + 12*(-I*d^2*e*f
^2 - I*d*f^3)*x + 2*(-I*d^3*f^3*x^3*e^(2*c) + 3*(-I*d^3*e*f^2 + I*d^2*f^3)*x^2*e^(2*c) + 6*(I*d^2*e*f^2 - I*d*
f^3)*x*e^(2*c) + 6*(-I*d*e*f^2 + I*f^3)*e^(2*c))*e^(2*d*x) + (d^4*f^3*x^4*e^c + 2*(2*d^4*e*f^2 - d^3*f^3)*x^3*
e^c - 6*(d^3*e*f^2 - d^2*f^3)*x^2*e^c + 12*(d^2*e*f^2 - d*f^3)*x*e^c - 12*(d*e*f^2 - f^3)*e^c)*e^(d*x))/(a*d^4
*e^(d*x + c) - I*a*d^4) + 12*(d*x*log(I*e^(d*x + c) + 1) + dilog(-I*e^(d*x + c)))*e*f^2/(a*d^3) + 6*(d^2*x^2*l
og(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) - 2*(d^3*f^3*x
^3 + 3*d^3*e*f^2*x^2)/(a*d^4)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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