3.146 \(\int \frac{x^3}{(a+i a \sinh (c+d x))^{5/2}} \, dx\)

Optimal. Leaf size=1016 \[ \text{result too large to display} \]

[Out]

-(1/(a^2*d^4*Sqrt[a + I*a*Sinh[c + d*x]])) + (9*x^2)/(8*a^2*d^2*Sqrt[a + I*a*Sinh[c + d*x]]) - ((10*I)*x*ArcTa
nh[E^((2*c - I*Pi)/4 + (d*x)/2)]*Cosh[c/2 + (I/4)*Pi + (d*x)/2])/(a^2*d^3*Sqrt[a + I*a*Sinh[c + d*x]]) + (((3*
I)/8)*x^3*ArcTanh[E^((2*c - I*Pi)/4 + (d*x)/2)]*Cosh[c/2 + (I/4)*Pi + (d*x)/2])/(a^2*d*Sqrt[a + I*a*Sinh[c + d
*x]]) - ((10*I)*Cosh[c/2 + (I/4)*Pi + (d*x)/2]*PolyLog[2, -E^((2*c - I*Pi)/4 + (d*x)/2)])/(a^2*d^4*Sqrt[a + I*
a*Sinh[c + d*x]]) + (((9*I)/8)*x^2*Cosh[c/2 + (I/4)*Pi + (d*x)/2]*PolyLog[2, -E^((2*c - I*Pi)/4 + (d*x)/2)])/(
a^2*d^2*Sqrt[a + I*a*Sinh[c + d*x]]) + ((10*I)*Cosh[c/2 + (I/4)*Pi + (d*x)/2]*PolyLog[2, E^((2*c - I*Pi)/4 + (
d*x)/2)])/(a^2*d^4*Sqrt[a + I*a*Sinh[c + d*x]]) - (((9*I)/8)*x^2*Cosh[c/2 + (I/4)*Pi + (d*x)/2]*PolyLog[2, E^(
(2*c - I*Pi)/4 + (d*x)/2)])/(a^2*d^2*Sqrt[a + I*a*Sinh[c + d*x]]) - (((9*I)/2)*x*Cosh[c/2 + (I/4)*Pi + (d*x)/2
]*PolyLog[3, -E^((2*c - I*Pi)/4 + (d*x)/2)])/(a^2*d^3*Sqrt[a + I*a*Sinh[c + d*x]]) + (((9*I)/2)*x*Cosh[c/2 + (
I/4)*Pi + (d*x)/2]*PolyLog[3, E^((2*c - I*Pi)/4 + (d*x)/2)])/(a^2*d^3*Sqrt[a + I*a*Sinh[c + d*x]]) + ((9*I)*Co
sh[c/2 + (I/4)*Pi + (d*x)/2]*PolyLog[4, -E^((2*c - I*Pi)/4 + (d*x)/2)])/(a^2*d^4*Sqrt[a + I*a*Sinh[c + d*x]])
- ((9*I)*Cosh[c/2 + (I/4)*Pi + (d*x)/2]*PolyLog[4, E^((2*c - I*Pi)/4 + (d*x)/2)])/(a^2*d^4*Sqrt[a + I*a*Sinh[c
+ d*x]]) + (x^2*Sech[c/2 + (I/4)*Pi + (d*x)/2]^2)/(4*a^2*d^2*Sqrt[a + I*a*Sinh[c + d*x]]) - (x*Tanh[c/2 + (I/
4)*Pi + (d*x)/2])/(2*a^2*d^3*Sqrt[a + I*a*Sinh[c + d*x]]) + (3*x^3*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(16*a^2*d*S
qrt[a + I*a*Sinh[c + d*x]]) + (x^3*Sech[c/2 + (I/4)*Pi + (d*x)/2]^2*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(8*a^2*d*S
qrt[a + I*a*Sinh[c + d*x]])

________________________________________________________________________________________

Rubi [A]  time = 0.676576, antiderivative size = 1016, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {3319, 4186, 4185, 4182, 2279, 2391, 2531, 6609, 2282, 6589} \[ \frac{\text{sech}^2\left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \tanh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) x^3}{8 a^2 d \sqrt{i \sinh (c+d x) a+a}}+\frac{3 \tanh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) x^3}{16 a^2 d \sqrt{i \sinh (c+d x) a+a}}+\frac{3 i \tanh ^{-1}\left (e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right ) \cosh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) x^3}{8 a^2 d \sqrt{i \sinh (c+d x) a+a}}+\frac{\text{sech}^2\left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) x^2}{4 a^2 d^2 \sqrt{i \sinh (c+d x) a+a}}+\frac{9 i \cosh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (2,-e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right ) x^2}{8 a^2 d^2 \sqrt{i \sinh (c+d x) a+a}}-\frac{9 i \cosh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (2,e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right ) x^2}{8 a^2 d^2 \sqrt{i \sinh (c+d x) a+a}}+\frac{9 x^2}{8 a^2 d^2 \sqrt{i \sinh (c+d x) a+a}}-\frac{\tanh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) x}{2 a^2 d^3 \sqrt{i \sinh (c+d x) a+a}}-\frac{10 i \tanh ^{-1}\left (e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right ) \cosh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) x}{a^2 d^3 \sqrt{i \sinh (c+d x) a+a}}-\frac{9 i \cosh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (3,-e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right ) x}{2 a^2 d^3 \sqrt{i \sinh (c+d x) a+a}}+\frac{9 i \cosh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (3,e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right ) x}{2 a^2 d^3 \sqrt{i \sinh (c+d x) a+a}}-\frac{10 i \cosh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (2,-e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right )}{a^2 d^4 \sqrt{i \sinh (c+d x) a+a}}+\frac{10 i \cosh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (2,e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right )}{a^2 d^4 \sqrt{i \sinh (c+d x) a+a}}+\frac{9 i \cosh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (4,-e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right )}{a^2 d^4 \sqrt{i \sinh (c+d x) a+a}}-\frac{9 i \cosh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (4,e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right )}{a^2 d^4 \sqrt{i \sinh (c+d x) a+a}}-\frac{1}{a^2 d^4 \sqrt{i \sinh (c+d x) a+a}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + I*a*Sinh[c + d*x])^(5/2),x]

[Out]

-(1/(a^2*d^4*Sqrt[a + I*a*Sinh[c + d*x]])) + (9*x^2)/(8*a^2*d^2*Sqrt[a + I*a*Sinh[c + d*x]]) - ((10*I)*x*ArcTa
nh[E^((2*c - I*Pi)/4 + (d*x)/2)]*Cosh[c/2 + (I/4)*Pi + (d*x)/2])/(a^2*d^3*Sqrt[a + I*a*Sinh[c + d*x]]) + (((3*
I)/8)*x^3*ArcTanh[E^((2*c - I*Pi)/4 + (d*x)/2)]*Cosh[c/2 + (I/4)*Pi + (d*x)/2])/(a^2*d*Sqrt[a + I*a*Sinh[c + d
*x]]) - ((10*I)*Cosh[c/2 + (I/4)*Pi + (d*x)/2]*PolyLog[2, -E^((2*c - I*Pi)/4 + (d*x)/2)])/(a^2*d^4*Sqrt[a + I*
a*Sinh[c + d*x]]) + (((9*I)/8)*x^2*Cosh[c/2 + (I/4)*Pi + (d*x)/2]*PolyLog[2, -E^((2*c - I*Pi)/4 + (d*x)/2)])/(
a^2*d^2*Sqrt[a + I*a*Sinh[c + d*x]]) + ((10*I)*Cosh[c/2 + (I/4)*Pi + (d*x)/2]*PolyLog[2, E^((2*c - I*Pi)/4 + (
d*x)/2)])/(a^2*d^4*Sqrt[a + I*a*Sinh[c + d*x]]) - (((9*I)/8)*x^2*Cosh[c/2 + (I/4)*Pi + (d*x)/2]*PolyLog[2, E^(
(2*c - I*Pi)/4 + (d*x)/2)])/(a^2*d^2*Sqrt[a + I*a*Sinh[c + d*x]]) - (((9*I)/2)*x*Cosh[c/2 + (I/4)*Pi + (d*x)/2
]*PolyLog[3, -E^((2*c - I*Pi)/4 + (d*x)/2)])/(a^2*d^3*Sqrt[a + I*a*Sinh[c + d*x]]) + (((9*I)/2)*x*Cosh[c/2 + (
I/4)*Pi + (d*x)/2]*PolyLog[3, E^((2*c - I*Pi)/4 + (d*x)/2)])/(a^2*d^3*Sqrt[a + I*a*Sinh[c + d*x]]) + ((9*I)*Co
sh[c/2 + (I/4)*Pi + (d*x)/2]*PolyLog[4, -E^((2*c - I*Pi)/4 + (d*x)/2)])/(a^2*d^4*Sqrt[a + I*a*Sinh[c + d*x]])
- ((9*I)*Cosh[c/2 + (I/4)*Pi + (d*x)/2]*PolyLog[4, E^((2*c - I*Pi)/4 + (d*x)/2)])/(a^2*d^4*Sqrt[a + I*a*Sinh[c
+ d*x]]) + (x^2*Sech[c/2 + (I/4)*Pi + (d*x)/2]^2)/(4*a^2*d^2*Sqrt[a + I*a*Sinh[c + d*x]]) - (x*Tanh[c/2 + (I/
4)*Pi + (d*x)/2])/(2*a^2*d^3*Sqrt[a + I*a*Sinh[c + d*x]]) + (3*x^3*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(16*a^2*d*S
qrt[a + I*a*Sinh[c + d*x]]) + (x^3*Sech[c/2 + (I/4)*Pi + (d*x)/2]^2*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(8*a^2*d*S
qrt[a + I*a*Sinh[c + d*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 4185

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> -Simp[(b^2*(c + d*x)*Cot[e + f*x]*
(b*Csc[e + f*x])^(n - 2))/(f*(n - 1)), x] + (Dist[(b^2*(n - 2))/(n - 1), Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2
), x], x] - Simp[(b^2*d*(b*Csc[e + f*x])^(n - 2))/(f^2*(n - 1)*(n - 2)), x]) /; FreeQ[{b, c, d, e, f}, x] && G
tQ[n, 1] && NeQ[n, 2]

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 (c+d x))^{5/2}} \, dx &=\frac{\sinh \left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \int x^3 \text{csch}^5\left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \, dx}{4 a^2 \sqrt{a+i a \sinh (c+d x)}}\\ &=\frac{x^2 \text{sech}^2\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{4 a^2 d^2 \sqrt{a+i a \sinh (c+d x)}}+\frac{x^3 \text{sech}^2\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{8 a^2 d \sqrt{a+i a \sinh (c+d x)}}-\frac{\left (3 \sinh \left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right )\right ) \int x^3 \text{csch}^3\left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \, dx}{16 a^2 \sqrt{a+i a \sinh (c+d x)}}+\frac{\sinh \left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \int x \text{csch}^3\left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \, dx}{2 a^2 d^2 \sqrt{a+i a \sinh (c+d x)}}\\ &=-\frac{1}{a^2 d^4 \sqrt{a+i a \sinh (c+d x)}}+\frac{9 x^2}{8 a^2 d^2 \sqrt{a+i a \sinh (c+d x)}}+\frac{x^2 \text{sech}^2\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{4 a^2 d^2 \sqrt{a+i a \sinh (c+d x)}}-\frac{x \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{2 a^2 d^3 \sqrt{a+i a \sinh (c+d x)}}+\frac{3 x^3 \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{16 a^2 d \sqrt{a+i a \sinh (c+d x)}}+\frac{x^3 \text{sech}^2\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{8 a^2 d \sqrt{a+i a \sinh (c+d x)}}+\frac{\left (3 \sinh \left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right )\right ) \int x^3 \text{csch}\left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \, dx}{32 a^2 \sqrt{a+i a \sinh (c+d x)}}-\frac{\sinh \left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \int x \text{csch}\left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \, dx}{4 a^2 d^2 \sqrt{a+i a \sinh (c+d x)}}-\frac{\left (9 \sinh \left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right )\right ) \int x \text{csch}\left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \, dx}{4 a^2 d^2 \sqrt{a+i a \sinh (c+d x)}}\\ &=-\frac{1}{a^2 d^4 \sqrt{a+i a \sinh (c+d x)}}+\frac{9 x^2}{8 a^2 d^2 \sqrt{a+i a \sinh (c+d x)}}-\frac{10 i x \tanh ^{-1}\left (e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right ) \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{a^2 d^3 \sqrt{a+i a \sinh (c+d x)}}+\frac{3 i x^3 \tanh ^{-1}\left (e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right ) \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{8 a^2 d \sqrt{a+i a \sinh (c+d x)}}+\frac{x^2 \text{sech}^2\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{4 a^2 d^2 \sqrt{a+i a \sinh (c+d x)}}-\frac{x \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{2 a^2 d^3 \sqrt{a+i a \sinh (c+d x)}}+\frac{3 x^3 \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{16 a^2 d \sqrt{a+i a \sinh (c+d x)}}+\frac{x^3 \text{sech}^2\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{8 a^2 d \sqrt{a+i a \sinh (c+d x)}}+\frac{\sinh \left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \int \log \left (1-e^{-i \left (\frac{i c}{2}+\frac{\pi }{4}\right )+\frac{d x}{2}}\right ) \, dx}{2 a^2 d^3 \sqrt{a+i a \sinh (c+d x)}}-\frac{\sinh \left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \int \log \left (1+e^{-i \left (\frac{i c}{2}+\frac{\pi }{4}\right )+\frac{d x}{2}}\right ) \, dx}{2 a^2 d^3 \sqrt{a+i a \sinh (c+d x)}}+\frac{\left (9 \sinh \left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right )\right ) \int \log \left (1-e^{-i \left (\frac{i c}{2}+\frac{\pi }{4}\right )+\frac{d x}{2}}\right ) \, dx}{2 a^2 d^3 \sqrt{a+i a \sinh (c+d x)}}-\frac{\left (9 \sinh \left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right )\right ) \int \log \left (1+e^{-i \left (\frac{i c}{2}+\frac{\pi }{4}\right )+\frac{d x}{2}}\right ) \, dx}{2 a^2 d^3 \sqrt{a+i a \sinh (c+d x)}}-\frac{\left (9 \sinh \left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right )\right ) \int x^2 \log \left (1-e^{-i \left (\frac{i c}{2}+\frac{\pi }{4}\right )+\frac{d x}{2}}\right ) \, dx}{16 a^2 d \sqrt{a+i a \sinh (c+d x)}}+\frac{\left (9 \sinh \left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right )\right ) \int x^2 \log \left (1+e^{-i \left (\frac{i c}{2}+\frac{\pi }{4}\right )+\frac{d x}{2}}\right ) \, dx}{16 a^2 d \sqrt{a+i a \sinh (c+d x)}}\\ &=-\frac{1}{a^2 d^4 \sqrt{a+i a \sinh (c+d x)}}+\frac{9 x^2}{8 a^2 d^2 \sqrt{a+i a \sinh (c+d x)}}-\frac{10 i x \tanh ^{-1}\left (e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right ) \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{a^2 d^3 \sqrt{a+i a \sinh (c+d x)}}+\frac{3 i x^3 \tanh ^{-1}\left (e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right ) \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{8 a^2 d \sqrt{a+i a \sinh (c+d x)}}+\frac{9 i x^2 \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \text{Li}_2\left (-e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right )}{8 a^2 d^2 \sqrt{a+i a \sinh (c+d x)}}-\frac{9 i x^2 \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \text{Li}_2\left (e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right )}{8 a^2 d^2 \sqrt{a+i a \sinh (c+d x)}}+\frac{x^2 \text{sech}^2\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{4 a^2 d^2 \sqrt{a+i a \sinh (c+d x)}}-\frac{x \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{2 a^2 d^3 \sqrt{a+i a \sinh (c+d x)}}+\frac{3 x^3 \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{16 a^2 d \sqrt{a+i a \sinh (c+d x)}}+\frac{x^3 \text{sech}^2\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{8 a^2 d \sqrt{a+i a \sinh (c+d x)}}+\frac{\sinh \left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{-i \left (\frac{i c}{2}+\frac{\pi }{4}\right )+\frac{d x}{2}}\right )}{a^2 d^4 \sqrt{a+i a \sinh (c+d x)}}-\frac{\sinh \left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{-i \left (\frac{i c}{2}+\frac{\pi }{4}\right )+\frac{d x}{2}}\right )}{a^2 d^4 \sqrt{a+i a \sinh (c+d x)}}+\frac{\left (9 \sinh \left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right )\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{-i \left (\frac{i c}{2}+\frac{\pi }{4}\right )+\frac{d x}{2}}\right )}{a^2 d^4 \sqrt{a+i a \sinh (c+d x)}}-\frac{\left (9 \sinh \left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right )\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{-i \left (\frac{i c}{2}+\frac{\pi }{4}\right )+\frac{d x}{2}}\right )}{a^2 d^4 \sqrt{a+i a \sinh (c+d x)}}+\frac{\left (9 \sinh \left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right )\right ) \int x \text{Li}_2\left (-e^{-i \left (\frac{i c}{2}+\frac{\pi }{4}\right )+\frac{d x}{2}}\right ) \, dx}{4 a^2 d^2 \sqrt{a+i a \sinh (c+d x)}}-\frac{\left (9 \sinh \left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right )\right ) \int x \text{Li}_2\left (e^{-i \left (\frac{i c}{2}+\frac{\pi }{4}\right )+\frac{d x}{2}}\right ) \, dx}{4 a^2 d^2 \sqrt{a+i a \sinh (c+d x)}}\\ &=-\frac{1}{a^2 d^4 \sqrt{a+i a \sinh (c+d x)}}+\frac{9 x^2}{8 a^2 d^2 \sqrt{a+i a \sinh (c+d x)}}-\frac{10 i x \tanh ^{-1}\left (e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right ) \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{a^2 d^3 \sqrt{a+i a \sinh (c+d x)}}+\frac{3 i x^3 \tanh ^{-1}\left (e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right ) \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{8 a^2 d \sqrt{a+i a \sinh (c+d x)}}-\frac{10 i \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \text{Li}_2\left (-e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right )}{a^2 d^4 \sqrt{a+i a \sinh (c+d x)}}+\frac{9 i x^2 \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \text{Li}_2\left (-e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right )}{8 a^2 d^2 \sqrt{a+i a \sinh (c+d x)}}+\frac{10 i \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \text{Li}_2\left (e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right )}{a^2 d^4 \sqrt{a+i a \sinh (c+d x)}}-\frac{9 i x^2 \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \text{Li}_2\left (e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right )}{8 a^2 d^2 \sqrt{a+i a \sinh (c+d x)}}-\frac{9 i x \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \text{Li}_3\left (-e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right )}{2 a^2 d^3 \sqrt{a+i a \sinh (c+d x)}}+\frac{9 i x \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \text{Li}_3\left (e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right )}{2 a^2 d^3 \sqrt{a+i a \sinh (c+d x)}}+\frac{x^2 \text{sech}^2\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{4 a^2 d^2 \sqrt{a+i a \sinh (c+d x)}}-\frac{x \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{2 a^2 d^3 \sqrt{a+i a \sinh (c+d x)}}+\frac{3 x^3 \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{16 a^2 d \sqrt{a+i a \sinh (c+d x)}}+\frac{x^3 \text{sech}^2\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{8 a^2 d \sqrt{a+i a \sinh (c+d x)}}-\frac{\left (9 \sinh \left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right )\right ) \int \text{Li}_3\left (-e^{-i \left (\frac{i c}{2}+\frac{\pi }{4}\right )+\frac{d x}{2}}\right ) \, dx}{2 a^2 d^3 \sqrt{a+i a \sinh (c+d x)}}+\frac{\left (9 \sinh \left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right )\right ) \int \text{Li}_3\left (e^{-i \left (\frac{i c}{2}+\frac{\pi }{4}\right )+\frac{d x}{2}}\right ) \, dx}{2 a^2 d^3 \sqrt{a+i a \sinh (c+d x)}}\\ &=-\frac{1}{a^2 d^4 \sqrt{a+i a \sinh (c+d x)}}+\frac{9 x^2}{8 a^2 d^2 \sqrt{a+i a \sinh (c+d x)}}-\frac{10 i x \tanh ^{-1}\left (e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right ) \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{a^2 d^3 \sqrt{a+i a \sinh (c+d x)}}+\frac{3 i x^3 \tanh ^{-1}\left (e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right ) \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{8 a^2 d \sqrt{a+i a \sinh (c+d x)}}-\frac{10 i \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \text{Li}_2\left (-e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right )}{a^2 d^4 \sqrt{a+i a \sinh (c+d x)}}+\frac{9 i x^2 \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \text{Li}_2\left (-e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right )}{8 a^2 d^2 \sqrt{a+i a \sinh (c+d x)}}+\frac{10 i \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \text{Li}_2\left (e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right )}{a^2 d^4 \sqrt{a+i a \sinh (c+d x)}}-\frac{9 i x^2 \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \text{Li}_2\left (e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right )}{8 a^2 d^2 \sqrt{a+i a \sinh (c+d x)}}-\frac{9 i x \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \text{Li}_3\left (-e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right )}{2 a^2 d^3 \sqrt{a+i a \sinh (c+d x)}}+\frac{9 i x \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \text{Li}_3\left (e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right )}{2 a^2 d^3 \sqrt{a+i a \sinh (c+d x)}}+\frac{x^2 \text{sech}^2\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{4 a^2 d^2 \sqrt{a+i a \sinh (c+d x)}}-\frac{x \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{2 a^2 d^3 \sqrt{a+i a \sinh (c+d x)}}+\frac{3 x^3 \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{16 a^2 d \sqrt{a+i a \sinh (c+d x)}}+\frac{x^3 \text{sech}^2\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{8 a^2 d \sqrt{a+i a \sinh (c+d x)}}-\frac{\left (9 \sinh \left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right )\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{-i \left (\frac{i c}{2}+\frac{\pi }{4}\right )+\frac{d x}{2}}\right )}{a^2 d^4 \sqrt{a+i a \sinh (c+d x)}}+\frac{\left (9 \sinh \left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right )\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{-i \left (\frac{i c}{2}+\frac{\pi }{4}\right )+\frac{d x}{2}}\right )}{a^2 d^4 \sqrt{a+i a \sinh (c+d x)}}\\ &=-\frac{1}{a^2 d^4 \sqrt{a+i a \sinh (c+d x)}}+\frac{9 x^2}{8 a^2 d^2 \sqrt{a+i a \sinh (c+d x)}}-\frac{10 i x \tanh ^{-1}\left (e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right ) \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{a^2 d^3 \sqrt{a+i a \sinh (c+d x)}}+\frac{3 i x^3 \tanh ^{-1}\left (e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right ) \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{8 a^2 d \sqrt{a+i a \sinh (c+d x)}}-\frac{10 i \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \text{Li}_2\left (-e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right )}{a^2 d^4 \sqrt{a+i a \sinh (c+d x)}}+\frac{9 i x^2 \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \text{Li}_2\left (-e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right )}{8 a^2 d^2 \sqrt{a+i a \sinh (c+d x)}}+\frac{10 i \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \text{Li}_2\left (e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right )}{a^2 d^4 \sqrt{a+i a \sinh (c+d x)}}-\frac{9 i x^2 \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \text{Li}_2\left (e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right )}{8 a^2 d^2 \sqrt{a+i a \sinh (c+d x)}}-\frac{9 i x \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \text{Li}_3\left (-e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right )}{2 a^2 d^3 \sqrt{a+i a \sinh (c+d x)}}+\frac{9 i x \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \text{Li}_3\left (e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right )}{2 a^2 d^3 \sqrt{a+i a \sinh (c+d x)}}+\frac{9 i \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \text{Li}_4\left (-e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right )}{a^2 d^4 \sqrt{a+i a \sinh (c+d x)}}-\frac{9 i \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \text{Li}_4\left (e^{\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}}\right )}{a^2 d^4 \sqrt{a+i a \sinh (c+d x)}}+\frac{x^2 \text{sech}^2\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{4 a^2 d^2 \sqrt{a+i a \sinh (c+d x)}}-\frac{x \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{2 a^2 d^3 \sqrt{a+i a \sinh (c+d x)}}+\frac{3 x^3 \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{16 a^2 d \sqrt{a+i a \sinh (c+d x)}}+\frac{x^3 \text{sech}^2\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{8 a^2 d \sqrt{a+i a \sinh (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 4.53243, size = 1200, normalized size = 1.18 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + I*a*Sinh[c + d*x])^(5/2),x]

[Out]

((Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2])*(-48*Cosh[(c + d*x)/2] + (8*I)*c*Cosh[(c + d*x)/2] + 70*c^2*Cosh[(c
+ d*x)/2] - (11*I)*c^3*Cosh[(c + d*x)/2] - (8*I)*(c + d*x)*Cosh[(c + d*x)/2] - 140*c*(c + d*x)*Cosh[(c + d*x)
/2] + (33*I)*c^2*(c + d*x)*Cosh[(c + d*x)/2] + 70*(c + d*x)^2*Cosh[(c + d*x)/2] - (33*I)*c*(c + d*x)^2*Cosh[(c
+ d*x)/2] + (11*I)*(c + d*x)^3*Cosh[(c + d*x)/2] + 16*Cosh[(3*(c + d*x))/2] + (8*I)*c*Cosh[(3*(c + d*x))/2] -
18*c^2*Cosh[(3*(c + d*x))/2] - (3*I)*c^3*Cosh[(3*(c + d*x))/2] - (8*I)*(c + d*x)*Cosh[(3*(c + d*x))/2] + 36*c
*(c + d*x)*Cosh[(3*(c + d*x))/2] + (9*I)*c^2*(c + d*x)*Cosh[(3*(c + d*x))/2] - 18*(c + d*x)^2*Cosh[(3*(c + d*x
))/2] - (9*I)*c*(c + d*x)^2*Cosh[(3*(c + d*x))/2] + (3*I)*(c + d*x)^3*Cosh[(3*(c + d*x))/2] + (1 - I)*(-1)^(3/
4)*(-160*c*ArcTanh[(-1)^(3/4)*E^((c + d*x)/2)] + 6*c^3*ArcTanh[(-1)^(3/4)*E^((c + d*x)/2)] - 80*c*Log[1 - (-1)
^(3/4)*E^((c + d*x)/2)] + 3*c^3*Log[1 - (-1)^(3/4)*E^((c + d*x)/2)] - 80*d*x*Log[1 - (-1)^(3/4)*E^((c + d*x)/2
)] + 3*d^3*x^3*Log[1 - (-1)^(3/4)*E^((c + d*x)/2)] + 80*c*Log[1 + (-1)^(3/4)*E^((c + d*x)/2)] - 3*c^3*Log[1 +
(-1)^(3/4)*E^((c + d*x)/2)] + 80*d*x*Log[1 + (-1)^(3/4)*E^((c + d*x)/2)] - 3*d^3*x^3*Log[1 + (-1)^(3/4)*E^((c
+ d*x)/2)] - 2*(-80 + 9*d^2*x^2)*PolyLog[2, -((-1)^(3/4)*E^((c + d*x)/2))] + 2*(-80 + 9*d^2*x^2)*PolyLog[2, (-
1)^(3/4)*E^((c + d*x)/2)] + 72*d*x*PolyLog[3, -((-1)^(3/4)*E^((c + d*x)/2))] - 72*d*x*PolyLog[3, (-1)^(3/4)*E^
((c + d*x)/2)] - 144*PolyLog[4, -((-1)^(3/4)*E^((c + d*x)/2))] + 144*PolyLog[4, (-1)^(3/4)*E^((c + d*x)/2)])*(
Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2])^4 - (48*I)*Sinh[(c + d*x)/2] + 8*c*Sinh[(c + d*x)/2] + (70*I)*c^2*Sin
h[(c + d*x)/2] - 11*c^3*Sinh[(c + d*x)/2] - 8*(c + d*x)*Sinh[(c + d*x)/2] - (140*I)*c*(c + d*x)*Sinh[(c + d*x)
/2] + 33*c^2*(c + d*x)*Sinh[(c + d*x)/2] + (70*I)*(c + d*x)^2*Sinh[(c + d*x)/2] - 33*c*(c + d*x)^2*Sinh[(c + d
*x)/2] + 11*(c + d*x)^3*Sinh[(c + d*x)/2] - (16*I)*Sinh[(3*(c + d*x))/2] - 8*c*Sinh[(3*(c + d*x))/2] + (18*I)*
c^2*Sinh[(3*(c + d*x))/2] + 3*c^3*Sinh[(3*(c + d*x))/2] + 8*(c + d*x)*Sinh[(3*(c + d*x))/2] - (36*I)*c*(c + d*
x)*Sinh[(3*(c + d*x))/2] - 9*c^2*(c + d*x)*Sinh[(3*(c + d*x))/2] + (18*I)*(c + d*x)^2*Sinh[(3*(c + d*x))/2] +
9*c*(c + d*x)^2*Sinh[(3*(c + d*x))/2] - 3*(c + d*x)^3*Sinh[(3*(c + d*x))/2]))/(32*d^4*(a + I*a*Sinh[c + d*x])^
(5/2))

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3} \left ( a+ia\sinh \left ( dx+c \right ) \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a+I*a*sinh(d*x+c))^(5/2),x)

[Out]

int(x^3/(a+I*a*sinh(d*x+c))^(5/2),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 (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+I*a*sinh(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate(x^3/(I*a*sinh(d*x + c) + a)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\sqrt{\frac{1}{2}}{\left ({\left (-3 i \, d^{3} x^{3} - 18 i \, d^{2} x^{2} + 8 i \, d x + 16 i\right )} e^{\left (4 \, d x + 4 \, c\right )} -{\left (11 \, d^{3} x^{3} + 70 \, d^{2} x^{2} - 8 \, d x - 48\right )} e^{\left (3 \, d x + 3 \, c\right )} +{\left (-11 i \, d^{3} x^{3} + 70 i \, d^{2} x^{2} + 8 i \, d x - 48 i\right )} e^{\left (2 \, d x + 2 \, c\right )} -{\left (3 \, d^{3} x^{3} - 18 \, d^{2} x^{2} - 8 \, d x + 16\right )} e^{\left (d x + c\right )}\right )} \sqrt{i \, a e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a e^{\left (d x + c\right )} - i \, a} e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )} +{\left (8 \, a^{3} d^{4} e^{\left (5 \, d x + 5 \, c\right )} - 40 i \, a^{3} d^{4} e^{\left (4 \, d x + 4 \, c\right )} - 80 \, a^{3} d^{4} e^{\left (3 \, d x + 3 \, c\right )} + 80 i \, a^{3} d^{4} e^{\left (2 \, d x + 2 \, c\right )} + 40 \, a^{3} d^{4} e^{\left (d x + c\right )} - 8 i \, a^{3} d^{4}\right )}{\rm integral}\left (\frac{\sqrt{\frac{1}{2}}{\left (-3 i \, d^{2} x^{3} + 80 i \, x\right )} \sqrt{i \, a e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a e^{\left (d x + c\right )} - i \, a} e^{\left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}}{16 \, a^{3} d^{2} e^{\left (2 \, d x + 2 \, c\right )} - 32 i \, a^{3} d^{2} e^{\left (d x + c\right )} - 16 \, a^{3} d^{2}}, x\right )}{8 \, a^{3} d^{4} e^{\left (5 \, d x + 5 \, c\right )} - 40 i \, a^{3} d^{4} e^{\left (4 \, d x + 4 \, c\right )} - 80 \, a^{3} d^{4} e^{\left (3 \, d x + 3 \, c\right )} + 80 i \, a^{3} d^{4} e^{\left (2 \, d x + 2 \, c\right )} + 40 \, a^{3} d^{4} e^{\left (d x + c\right )} - 8 i \, a^{3} d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+I*a*sinh(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

(sqrt(1/2)*((-3*I*d^3*x^3 - 18*I*d^2*x^2 + 8*I*d*x + 16*I)*e^(4*d*x + 4*c) - (11*d^3*x^3 + 70*d^2*x^2 - 8*d*x
- 48)*e^(3*d*x + 3*c) + (-11*I*d^3*x^3 + 70*I*d^2*x^2 + 8*I*d*x - 48*I)*e^(2*d*x + 2*c) - (3*d^3*x^3 - 18*d^2*
x^2 - 8*d*x + 16)*e^(d*x + c))*sqrt(I*a*e^(2*d*x + 2*c) + 2*a*e^(d*x + c) - I*a)*e^(-1/2*d*x - 1/2*c) + (8*a^3
*d^4*e^(5*d*x + 5*c) - 40*I*a^3*d^4*e^(4*d*x + 4*c) - 80*a^3*d^4*e^(3*d*x + 3*c) + 80*I*a^3*d^4*e^(2*d*x + 2*c
) + 40*a^3*d^4*e^(d*x + c) - 8*I*a^3*d^4)*integral(sqrt(1/2)*(-3*I*d^2*x^3 + 80*I*x)*sqrt(I*a*e^(2*d*x + 2*c)
+ 2*a*e^(d*x + c) - I*a)*e^(1/2*d*x + 1/2*c)/(16*a^3*d^2*e^(2*d*x + 2*c) - 32*I*a^3*d^2*e^(d*x + c) - 16*a^3*d
^2), x))/(8*a^3*d^4*e^(5*d*x + 5*c) - 40*I*a^3*d^4*e^(4*d*x + 4*c) - 80*a^3*d^4*e^(3*d*x + 3*c) + 80*I*a^3*d^4
*e^(2*d*x + 2*c) + 40*a^3*d^4*e^(d*x + c) - 8*I*a^3*d^4)

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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(x**3/(a+I*a*sinh(d*x+c))**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{{\left (i \, a \sinh \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+I*a*sinh(d*x+c))^(5/2),x, algorithm="giac")

[Out]

integrate(x^3/(I*a*sinh(d*x + c) + a)^(5/2), x)