3.141 \(\int \frac{x^3}{(a+i a \sinh (e+f x))^{3/2}} \, dx\)

Optimal. Leaf size=807 \[ \frac{\tanh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) x^3}{2 a f \sqrt{i \sinh (e+f x) a+a}}+\frac{i \tanh ^{-1}\left (e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right ) \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) x^3}{a f \sqrt{i \sinh (e+f x) a+a}}+\frac{3 i \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (2,-e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right ) x^2}{a f^2 \sqrt{i \sinh (e+f x) a+a}}-\frac{3 i \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (2,e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right ) x^2}{a f^2 \sqrt{i \sinh (e+f x) a+a}}+\frac{3 x^2}{a f^2 \sqrt{i \sinh (e+f x) a+a}}-\frac{24 i \tanh ^{-1}\left (e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right ) \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) x}{a f^3 \sqrt{i \sinh (e+f x) a+a}}-\frac{12 i \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (3,-e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right ) x}{a f^3 \sqrt{i \sinh (e+f x) a+a}}+\frac{12 i \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (3,e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right ) x}{a f^3 \sqrt{i \sinh (e+f x) a+a}}-\frac{24 i \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (2,-e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right )}{a f^4 \sqrt{i \sinh (e+f x) a+a}}+\frac{24 i \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (2,e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right )}{a f^4 \sqrt{i \sinh (e+f x) a+a}}+\frac{24 i \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (4,-e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right )}{a f^4 \sqrt{i \sinh (e+f x) a+a}}-\frac{24 i \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (4,e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right )}{a f^4 \sqrt{i \sinh (e+f x) a+a}} \]

[Out]

(3*x^2)/(a*f^2*Sqrt[a + I*a*Sinh[e + f*x]]) - ((24*I)*x*ArcTanh[E^((2*e - I*Pi)/4 + (f*x)/2)]*Cosh[e/2 + (I/4)
*Pi + (f*x)/2])/(a*f^3*Sqrt[a + I*a*Sinh[e + f*x]]) + (I*x^3*ArcTanh[E^((2*e - I*Pi)/4 + (f*x)/2)]*Cosh[e/2 +
(I/4)*Pi + (f*x)/2])/(a*f*Sqrt[a + I*a*Sinh[e + f*x]]) - ((24*I)*Cosh[e/2 + (I/4)*Pi + (f*x)/2]*PolyLog[2, -E^
((2*e - I*Pi)/4 + (f*x)/2)])/(a*f^4*Sqrt[a + I*a*Sinh[e + f*x]]) + ((3*I)*x^2*Cosh[e/2 + (I/4)*Pi + (f*x)/2]*P
olyLog[2, -E^((2*e - I*Pi)/4 + (f*x)/2)])/(a*f^2*Sqrt[a + I*a*Sinh[e + f*x]]) + ((24*I)*Cosh[e/2 + (I/4)*Pi +
(f*x)/2]*PolyLog[2, E^((2*e - I*Pi)/4 + (f*x)/2)])/(a*f^4*Sqrt[a + I*a*Sinh[e + f*x]]) - ((3*I)*x^2*Cosh[e/2 +
(I/4)*Pi + (f*x)/2]*PolyLog[2, E^((2*e - I*Pi)/4 + (f*x)/2)])/(a*f^2*Sqrt[a + I*a*Sinh[e + f*x]]) - ((12*I)*x
*Cosh[e/2 + (I/4)*Pi + (f*x)/2]*PolyLog[3, -E^((2*e - I*Pi)/4 + (f*x)/2)])/(a*f^3*Sqrt[a + I*a*Sinh[e + f*x]])
+ ((12*I)*x*Cosh[e/2 + (I/4)*Pi + (f*x)/2]*PolyLog[3, E^((2*e - I*Pi)/4 + (f*x)/2)])/(a*f^3*Sqrt[a + I*a*Sinh
[e + f*x]]) + ((24*I)*Cosh[e/2 + (I/4)*Pi + (f*x)/2]*PolyLog[4, -E^((2*e - I*Pi)/4 + (f*x)/2)])/(a*f^4*Sqrt[a
+ I*a*Sinh[e + f*x]]) - ((24*I)*Cosh[e/2 + (I/4)*Pi + (f*x)/2]*PolyLog[4, E^((2*e - I*Pi)/4 + (f*x)/2)])/(a*f^
4*Sqrt[a + I*a*Sinh[e + f*x]]) + (x^3*Tanh[e/2 + (I/4)*Pi + (f*x)/2])/(2*a*f*Sqrt[a + I*a*Sinh[e + f*x]])

________________________________________________________________________________________

Rubi [A]  time = 0.437724, antiderivative size = 807, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {3319, 4186, 4182, 2279, 2391, 2531, 6609, 2282, 6589} \[ \frac{\tanh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) x^3}{2 a f \sqrt{i \sinh (e+f x) a+a}}+\frac{i \tanh ^{-1}\left (e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right ) \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) x^3}{a f \sqrt{i \sinh (e+f x) a+a}}+\frac{3 i \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (2,-e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right ) x^2}{a f^2 \sqrt{i \sinh (e+f x) a+a}}-\frac{3 i \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (2,e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right ) x^2}{a f^2 \sqrt{i \sinh (e+f x) a+a}}+\frac{3 x^2}{a f^2 \sqrt{i \sinh (e+f x) a+a}}-\frac{24 i \tanh ^{-1}\left (e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right ) \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) x}{a f^3 \sqrt{i \sinh (e+f x) a+a}}-\frac{12 i \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (3,-e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right ) x}{a f^3 \sqrt{i \sinh (e+f x) a+a}}+\frac{12 i \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (3,e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right ) x}{a f^3 \sqrt{i \sinh (e+f x) a+a}}-\frac{24 i \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (2,-e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right )}{a f^4 \sqrt{i \sinh (e+f x) a+a}}+\frac{24 i \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (2,e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right )}{a f^4 \sqrt{i \sinh (e+f x) a+a}}+\frac{24 i \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (4,-e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right )}{a f^4 \sqrt{i \sinh (e+f x) a+a}}-\frac{24 i \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (4,e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right )}{a f^4 \sqrt{i \sinh (e+f x) a+a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x^2)/(a*f^2*Sqrt[a + I*a*Sinh[e + f*x]]) - ((24*I)*x*ArcTanh[E^((2*e - I*Pi)/4 + (f*x)/2)]*Cosh[e/2 + (I/4)
*Pi + (f*x)/2])/(a*f^3*Sqrt[a + I*a*Sinh[e + f*x]]) + (I*x^3*ArcTanh[E^((2*e - I*Pi)/4 + (f*x)/2)]*Cosh[e/2 +
(I/4)*Pi + (f*x)/2])/(a*f*Sqrt[a + I*a*Sinh[e + f*x]]) - ((24*I)*Cosh[e/2 + (I/4)*Pi + (f*x)/2]*PolyLog[2, -E^
((2*e - I*Pi)/4 + (f*x)/2)])/(a*f^4*Sqrt[a + I*a*Sinh[e + f*x]]) + ((3*I)*x^2*Cosh[e/2 + (I/4)*Pi + (f*x)/2]*P
olyLog[2, -E^((2*e - I*Pi)/4 + (f*x)/2)])/(a*f^2*Sqrt[a + I*a*Sinh[e + f*x]]) + ((24*I)*Cosh[e/2 + (I/4)*Pi +
(f*x)/2]*PolyLog[2, E^((2*e - I*Pi)/4 + (f*x)/2)])/(a*f^4*Sqrt[a + I*a*Sinh[e + f*x]]) - ((3*I)*x^2*Cosh[e/2 +
(I/4)*Pi + (f*x)/2]*PolyLog[2, E^((2*e - I*Pi)/4 + (f*x)/2)])/(a*f^2*Sqrt[a + I*a*Sinh[e + f*x]]) - ((12*I)*x
*Cosh[e/2 + (I/4)*Pi + (f*x)/2]*PolyLog[3, -E^((2*e - I*Pi)/4 + (f*x)/2)])/(a*f^3*Sqrt[a + I*a*Sinh[e + f*x]])
+ ((12*I)*x*Cosh[e/2 + (I/4)*Pi + (f*x)/2]*PolyLog[3, E^((2*e - I*Pi)/4 + (f*x)/2)])/(a*f^3*Sqrt[a + I*a*Sinh
[e + f*x]]) + ((24*I)*Cosh[e/2 + (I/4)*Pi + (f*x)/2]*PolyLog[4, -E^((2*e - I*Pi)/4 + (f*x)/2)])/(a*f^4*Sqrt[a
+ I*a*Sinh[e + f*x]]) - ((24*I)*Cosh[e/2 + (I/4)*Pi + (f*x)/2]*PolyLog[4, E^((2*e - I*Pi)/4 + (f*x)/2)])/(a*f^
4*Sqrt[a + I*a*Sinh[e + f*x]]) + (x^3*Tanh[e/2 + (I/4)*Pi + (f*x)/2])/(2*a*f*Sqrt[a + I*a*Sinh[e + f*x]])

Rule 3319

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[((2*a)^IntPart[n
]*(a + b*Sin[e + f*x])^FracPart[n])/Sin[e/2 + (a*Pi)/(4*b) + (f*x)/2]^(2*FracPart[n]), Int[(c + d*x)^m*Sin[e/2
+ (a*Pi)/(4*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
+ 1/2] && (GtQ[n, 0] || IGtQ[m, 0])

Rule 4186

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]) /; FreeQ[
{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rule 4182

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 2279

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 2391

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 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 6609

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]

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{x^3}{(a+i a \sinh (e+f x))^{3/2}} \, dx &=-\frac{\sinh \left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right ) \int x^3 \text{csch}^3\left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right ) \, dx}{2 a \sqrt{a+i a \sinh (e+f x)}}\\ &=\frac{3 x^2}{a f^2 \sqrt{a+i a \sinh (e+f x)}}+\frac{x^3 \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{2 a f \sqrt{a+i a \sinh (e+f x)}}+\frac{\sinh \left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right ) \int x^3 \text{csch}\left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right ) \, dx}{4 a \sqrt{a+i a \sinh (e+f x)}}-\frac{\left (6 \sinh \left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right )\right ) \int x \text{csch}\left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right ) \, dx}{a f^2 \sqrt{a+i a \sinh (e+f x)}}\\ &=\frac{3 x^2}{a f^2 \sqrt{a+i a \sinh (e+f x)}}-\frac{24 i x \tanh ^{-1}\left (e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right ) \cosh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{a f^3 \sqrt{a+i a \sinh (e+f x)}}+\frac{i x^3 \tanh ^{-1}\left (e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right ) \cosh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{a f \sqrt{a+i a \sinh (e+f x)}}+\frac{x^3 \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{2 a f \sqrt{a+i a \sinh (e+f x)}}+\frac{\left (12 \sinh \left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right )\right ) \int \log \left (1-e^{-i \left (\frac{i e}{2}+\frac{\pi }{4}\right )+\frac{f x}{2}}\right ) \, dx}{a f^3 \sqrt{a+i a \sinh (e+f x)}}-\frac{\left (12 \sinh \left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right )\right ) \int \log \left (1+e^{-i \left (\frac{i e}{2}+\frac{\pi }{4}\right )+\frac{f x}{2}}\right ) \, dx}{a f^3 \sqrt{a+i a \sinh (e+f x)}}-\frac{\left (3 \sinh \left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right )\right ) \int x^2 \log \left (1-e^{-i \left (\frac{i e}{2}+\frac{\pi }{4}\right )+\frac{f x}{2}}\right ) \, dx}{2 a f \sqrt{a+i a \sinh (e+f x)}}+\frac{\left (3 \sinh \left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right )\right ) \int x^2 \log \left (1+e^{-i \left (\frac{i e}{2}+\frac{\pi }{4}\right )+\frac{f x}{2}}\right ) \, dx}{2 a f \sqrt{a+i a \sinh (e+f x)}}\\ &=\frac{3 x^2}{a f^2 \sqrt{a+i a \sinh (e+f x)}}-\frac{24 i x \tanh ^{-1}\left (e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right ) \cosh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{a f^3 \sqrt{a+i a \sinh (e+f x)}}+\frac{i x^3 \tanh ^{-1}\left (e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right ) \cosh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{a f \sqrt{a+i a \sinh (e+f x)}}+\frac{3 i x^2 \cosh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \text{Li}_2\left (-e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right )}{a f^2 \sqrt{a+i a \sinh (e+f x)}}-\frac{3 i x^2 \cosh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \text{Li}_2\left (e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right )}{a f^2 \sqrt{a+i a \sinh (e+f x)}}+\frac{x^3 \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{2 a f \sqrt{a+i a \sinh (e+f x)}}+\frac{\left (24 \sinh \left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right )\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{-i \left (\frac{i e}{2}+\frac{\pi }{4}\right )+\frac{f x}{2}}\right )}{a f^4 \sqrt{a+i a \sinh (e+f x)}}-\frac{\left (24 \sinh \left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right )\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{-i \left (\frac{i e}{2}+\frac{\pi }{4}\right )+\frac{f x}{2}}\right )}{a f^4 \sqrt{a+i a \sinh (e+f x)}}+\frac{\left (6 \sinh \left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right )\right ) \int x \text{Li}_2\left (-e^{-i \left (\frac{i e}{2}+\frac{\pi }{4}\right )+\frac{f x}{2}}\right ) \, dx}{a f^2 \sqrt{a+i a \sinh (e+f x)}}-\frac{\left (6 \sinh \left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right )\right ) \int x \text{Li}_2\left (e^{-i \left (\frac{i e}{2}+\frac{\pi }{4}\right )+\frac{f x}{2}}\right ) \, dx}{a f^2 \sqrt{a+i a \sinh (e+f x)}}\\ &=\frac{3 x^2}{a f^2 \sqrt{a+i a \sinh (e+f x)}}-\frac{24 i x \tanh ^{-1}\left (e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right ) \cosh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{a f^3 \sqrt{a+i a \sinh (e+f x)}}+\frac{i x^3 \tanh ^{-1}\left (e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right ) \cosh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{a f \sqrt{a+i a \sinh (e+f x)}}-\frac{24 i \cosh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \text{Li}_2\left (-e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right )}{a f^4 \sqrt{a+i a \sinh (e+f x)}}+\frac{3 i x^2 \cosh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \text{Li}_2\left (-e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right )}{a f^2 \sqrt{a+i a \sinh (e+f x)}}+\frac{24 i \cosh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \text{Li}_2\left (e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right )}{a f^4 \sqrt{a+i a \sinh (e+f x)}}-\frac{3 i x^2 \cosh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \text{Li}_2\left (e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right )}{a f^2 \sqrt{a+i a \sinh (e+f x)}}-\frac{12 i x \cosh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \text{Li}_3\left (-e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right )}{a f^3 \sqrt{a+i a \sinh (e+f x)}}+\frac{12 i x \cosh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \text{Li}_3\left (e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right )}{a f^3 \sqrt{a+i a \sinh (e+f x)}}+\frac{x^3 \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{2 a f \sqrt{a+i a \sinh (e+f x)}}-\frac{\left (12 \sinh \left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right )\right ) \int \text{Li}_3\left (-e^{-i \left (\frac{i e}{2}+\frac{\pi }{4}\right )+\frac{f x}{2}}\right ) \, dx}{a f^3 \sqrt{a+i a \sinh (e+f x)}}+\frac{\left (12 \sinh \left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right )\right ) \int \text{Li}_3\left (e^{-i \left (\frac{i e}{2}+\frac{\pi }{4}\right )+\frac{f x}{2}}\right ) \, dx}{a f^3 \sqrt{a+i a \sinh (e+f x)}}\\ &=\frac{3 x^2}{a f^2 \sqrt{a+i a \sinh (e+f x)}}-\frac{24 i x \tanh ^{-1}\left (e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right ) \cosh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{a f^3 \sqrt{a+i a \sinh (e+f x)}}+\frac{i x^3 \tanh ^{-1}\left (e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right ) \cosh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{a f \sqrt{a+i a \sinh (e+f x)}}-\frac{24 i \cosh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \text{Li}_2\left (-e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right )}{a f^4 \sqrt{a+i a \sinh (e+f x)}}+\frac{3 i x^2 \cosh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \text{Li}_2\left (-e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right )}{a f^2 \sqrt{a+i a \sinh (e+f x)}}+\frac{24 i \cosh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \text{Li}_2\left (e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right )}{a f^4 \sqrt{a+i a \sinh (e+f x)}}-\frac{3 i x^2 \cosh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \text{Li}_2\left (e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right )}{a f^2 \sqrt{a+i a \sinh (e+f x)}}-\frac{12 i x \cosh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \text{Li}_3\left (-e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right )}{a f^3 \sqrt{a+i a \sinh (e+f x)}}+\frac{12 i x \cosh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \text{Li}_3\left (e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right )}{a f^3 \sqrt{a+i a \sinh (e+f x)}}+\frac{x^3 \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{2 a f \sqrt{a+i a \sinh (e+f x)}}-\frac{\left (24 \sinh \left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right )\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{-i \left (\frac{i e}{2}+\frac{\pi }{4}\right )+\frac{f x}{2}}\right )}{a f^4 \sqrt{a+i a \sinh (e+f x)}}+\frac{\left (24 \sinh \left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right )\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{-i \left (\frac{i e}{2}+\frac{\pi }{4}\right )+\frac{f x}{2}}\right )}{a f^4 \sqrt{a+i a \sinh (e+f x)}}\\ &=\frac{3 x^2}{a f^2 \sqrt{a+i a \sinh (e+f x)}}-\frac{24 i x \tanh ^{-1}\left (e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right ) \cosh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{a f^3 \sqrt{a+i a \sinh (e+f x)}}+\frac{i x^3 \tanh ^{-1}\left (e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right ) \cosh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{a f \sqrt{a+i a \sinh (e+f x)}}-\frac{24 i \cosh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \text{Li}_2\left (-e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right )}{a f^4 \sqrt{a+i a \sinh (e+f x)}}+\frac{3 i x^2 \cosh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \text{Li}_2\left (-e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right )}{a f^2 \sqrt{a+i a \sinh (e+f x)}}+\frac{24 i \cosh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \text{Li}_2\left (e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right )}{a f^4 \sqrt{a+i a \sinh (e+f x)}}-\frac{3 i x^2 \cosh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \text{Li}_2\left (e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right )}{a f^2 \sqrt{a+i a \sinh (e+f x)}}-\frac{12 i x \cosh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \text{Li}_3\left (-e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right )}{a f^3 \sqrt{a+i a \sinh (e+f x)}}+\frac{12 i x \cosh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \text{Li}_3\left (e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right )}{a f^3 \sqrt{a+i a \sinh (e+f x)}}+\frac{24 i \cosh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \text{Li}_4\left (-e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right )}{a f^4 \sqrt{a+i a \sinh (e+f x)}}-\frac{24 i \cosh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \text{Li}_4\left (e^{\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}}\right )}{a f^4 \sqrt{a+i a \sinh (e+f x)}}+\frac{x^3 \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{2 a f \sqrt{a+i a \sinh (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 3.23932, size = 546, normalized size = 0.68 \[ \frac{\left (\cosh \left (\frac{1}{2} (e+f x)\right )+i \sinh \left (\frac{1}{2} (e+f x)\right )\right ) \left (\left (\frac{1}{2}-\frac{i}{2}\right ) (-1)^{3/4} \left (\cosh \left (\frac{1}{2} (e+f x)\right )+i \sinh \left (\frac{1}{2} (e+f x)\right )\right )^2 \left (-6 \left (f^2 x^2-8\right ) \text{PolyLog}\left (2,-(-1)^{3/4} e^{\frac{1}{2} (e+f x)}\right )+6 \left (f^2 x^2-8\right ) \text{PolyLog}\left (2,(-1)^{3/4} e^{\frac{1}{2} (e+f x)}\right )+24 f x \text{PolyLog}\left (3,-(-1)^{3/4} e^{\frac{1}{2} (e+f x)}\right )-24 f x \text{PolyLog}\left (3,(-1)^{3/4} e^{\frac{1}{2} (e+f x)}\right )-48 \text{PolyLog}\left (4,-(-1)^{3/4} e^{\frac{1}{2} (e+f x)}\right )+48 \text{PolyLog}\left (4,(-1)^{3/4} e^{\frac{1}{2} (e+f x)}\right )+e^3 \log \left (1-(-1)^{3/4} e^{\frac{1}{2} (e+f x)}\right )-e^3 \log \left ((-1)^{3/4} e^{\frac{1}{2} (e+f x)}+1\right )+2 e^3 \tanh ^{-1}\left ((-1)^{3/4} e^{\frac{1}{2} (e+f x)}\right )+f^3 x^3 \log \left (1-(-1)^{3/4} e^{\frac{1}{2} (e+f x)}\right )-f^3 x^3 \log \left ((-1)^{3/4} e^{\frac{1}{2} (e+f x)}+1\right )-24 e \log \left (1-(-1)^{3/4} e^{\frac{1}{2} (e+f x)}\right )+24 e \log \left ((-1)^{3/4} e^{\frac{1}{2} (e+f x)}+1\right )-24 f x \log \left (1-(-1)^{3/4} e^{\frac{1}{2} (e+f x)}\right )+24 f x \log \left ((-1)^{3/4} e^{\frac{1}{2} (e+f x)}+1\right )-48 e \tanh ^{-1}\left ((-1)^{3/4} e^{\frac{1}{2} (e+f x)}\right )\right )+2 f^3 x^3 \sinh \left (\frac{1}{2} (e+f x)\right )+f^2 x^2 (6+i f x) \left (\cosh \left (\frac{1}{2} (e+f x)\right )+i \sinh \left (\frac{1}{2} (e+f x)\right )\right )\right )}{2 f^4 (a+i a \sinh (e+f x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((Cosh[(e + f*x)/2] + I*Sinh[(e + f*x)/2])*(f^2*x^2*(6 + I*f*x)*(Cosh[(e + f*x)/2] + I*Sinh[(e + f*x)/2]) + (1
/2 - I/2)*(-1)^(3/4)*(-48*e*ArcTanh[(-1)^(3/4)*E^((e + f*x)/2)] + 2*e^3*ArcTanh[(-1)^(3/4)*E^((e + f*x)/2)] -
24*e*Log[1 - (-1)^(3/4)*E^((e + f*x)/2)] + e^3*Log[1 - (-1)^(3/4)*E^((e + f*x)/2)] - 24*f*x*Log[1 - (-1)^(3/4)
*E^((e + f*x)/2)] + f^3*x^3*Log[1 - (-1)^(3/4)*E^((e + f*x)/2)] + 24*e*Log[1 + (-1)^(3/4)*E^((e + f*x)/2)] - e
^3*Log[1 + (-1)^(3/4)*E^((e + f*x)/2)] + 24*f*x*Log[1 + (-1)^(3/4)*E^((e + f*x)/2)] - f^3*x^3*Log[1 + (-1)^(3/
4)*E^((e + f*x)/2)] - 6*(-8 + f^2*x^2)*PolyLog[2, -((-1)^(3/4)*E^((e + f*x)/2))] + 6*(-8 + f^2*x^2)*PolyLog[2,
(-1)^(3/4)*E^((e + f*x)/2)] + 24*f*x*PolyLog[3, -((-1)^(3/4)*E^((e + f*x)/2))] - 24*f*x*PolyLog[3, (-1)^(3/4)
*E^((e + f*x)/2)] - 48*PolyLog[4, -((-1)^(3/4)*E^((e + f*x)/2))] + 48*PolyLog[4, (-1)^(3/4)*E^((e + f*x)/2)])*
(Cosh[(e + f*x)/2] + I*Sinh[(e + f*x)/2])^2 + 2*f^3*x^3*Sinh[(e + f*x)/2]))/(2*f^4*(a + I*a*Sinh[e + f*x])^(3/
2))

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Maple [F]  time = 0.047, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3} \left ( a+ia\sinh \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a+I*a*sinh(f*x+e))^(3/2),x)

[Out]

int(x^3/(a+I*a*sinh(f*x+e))^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{{\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\sqrt{\frac{1}{2}}{\left ({\left (-i \, f x^{3} - 6 i \, x^{2}\right )} e^{\left (2 \, f x + 2 \, e\right )} +{\left (f x^{3} - 6 \, x^{2}\right )} e^{\left (f x + e\right )}\right )} \sqrt{i \, a e^{\left (2 \, f x + 2 \, e\right )} + 2 \, a e^{\left (f x + e\right )} - i \, a} e^{\left (-\frac{1}{2} \, f x - \frac{1}{2} \, e\right )} +{\left (a^{2} f^{2} e^{\left (3 \, f x + 3 \, e\right )} - 3 i \, a^{2} f^{2} e^{\left (2 \, f x + 2 \, e\right )} - 3 \, a^{2} f^{2} e^{\left (f x + e\right )} + i \, a^{2} f^{2}\right )}{\rm integral}\left (\frac{\sqrt{\frac{1}{2}}{\left (-i \, f^{2} x^{3} + 24 i \, x\right )} \sqrt{i \, a e^{\left (2 \, f x + 2 \, e\right )} + 2 \, a e^{\left (f x + e\right )} - i \, a} e^{\left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}}{2 \, a^{2} f^{2} e^{\left (2 \, f x + 2 \, e\right )} - 4 i \, a^{2} f^{2} e^{\left (f x + e\right )} - 2 \, a^{2} f^{2}}, x\right )}{a^{2} f^{2} e^{\left (3 \, f x + 3 \, e\right )} - 3 i \, a^{2} f^{2} e^{\left (2 \, f x + 2 \, e\right )} - 3 \, a^{2} f^{2} e^{\left (f x + e\right )} + i \, a^{2} f^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sqrt(1/2)*((-I*f*x^3 - 6*I*x^2)*e^(2*f*x + 2*e) + (f*x^3 - 6*x^2)*e^(f*x + e))*sqrt(I*a*e^(2*f*x + 2*e) + 2*a
*e^(f*x + e) - I*a)*e^(-1/2*f*x - 1/2*e) + (a^2*f^2*e^(3*f*x + 3*e) - 3*I*a^2*f^2*e^(2*f*x + 2*e) - 3*a^2*f^2*
e^(f*x + e) + I*a^2*f^2)*integral(sqrt(1/2)*(-I*f^2*x^3 + 24*I*x)*sqrt(I*a*e^(2*f*x + 2*e) + 2*a*e^(f*x + e) -
I*a)*e^(1/2*f*x + 1/2*e)/(2*a^2*f^2*e^(2*f*x + 2*e) - 4*I*a^2*f^2*e^(f*x + e) - 2*a^2*f^2), x))/(a^2*f^2*e^(3
*f*x + 3*e) - 3*I*a^2*f^2*e^(2*f*x + 2*e) - 3*a^2*f^2*e^(f*x + e) + I*a^2*f^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\left (a \left (i \sinh{\left (e + f x \right )} + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3/(a*(I*sinh(e + f*x) + 1))**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{{\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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