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

Optimal. Leaf size=638 \[ -\frac{128 a^2 x^2 \sqrt{a+i a \sinh (c+d x)}}{5 d^2}-\frac{48 a^2 x^2 \cosh ^4\left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}}{25 d^2}-\frac{64 a^2 x^2 \cosh ^2\left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}}{15 d^2}-\frac{265216 a^2 \sqrt{a+i a \sinh (c+d x)}}{1125 d^4}+\frac{132608 a^2 x \tanh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}}{1125 d^3}-\frac{384 a^2 \cosh ^4\left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}}{625 d^4}+\frac{192 a^2 x \sinh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \cosh ^3\left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}}{125 d^3}-\frac{17408 a^2 \cosh ^2\left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}}{3375 d^4}+\frac{8704 a^2 x \sinh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \cosh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}}{1125 d^3}+\frac{64 a^2 x^3 \tanh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}}{15 d}+\frac{8 a^2 x^3 \sinh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \cosh ^3\left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}}{5 d}+\frac{32 a^2 x^3 \sinh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \cosh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}}{15 d} \]

[Out]

(-265216*a^2*Sqrt[a + I*a*Sinh[c + d*x]])/(1125*d^4) - (128*a^2*x^2*Sqrt[a + I*a*Sinh[c + d*x]])/(5*d^2) - (17
408*a^2*Cosh[c/2 + (I/4)*Pi + (d*x)/2]^2*Sqrt[a + I*a*Sinh[c + d*x]])/(3375*d^4) - (64*a^2*x^2*Cosh[c/2 + (I/4
)*Pi + (d*x)/2]^2*Sqrt[a + I*a*Sinh[c + d*x]])/(15*d^2) - (384*a^2*Cosh[c/2 + (I/4)*Pi + (d*x)/2]^4*Sqrt[a + I
*a*Sinh[c + d*x]])/(625*d^4) - (48*a^2*x^2*Cosh[c/2 + (I/4)*Pi + (d*x)/2]^4*Sqrt[a + I*a*Sinh[c + d*x]])/(25*d
^2) + (8704*a^2*x*Cosh[c/2 + (I/4)*Pi + (d*x)/2]*Sinh[c/2 + (I/4)*Pi + (d*x)/2]*Sqrt[a + I*a*Sinh[c + d*x]])/(
1125*d^3) + (32*a^2*x^3*Cosh[c/2 + (I/4)*Pi + (d*x)/2]*Sinh[c/2 + (I/4)*Pi + (d*x)/2]*Sqrt[a + I*a*Sinh[c + d*
x]])/(15*d) + (192*a^2*x*Cosh[c/2 + (I/4)*Pi + (d*x)/2]^3*Sinh[c/2 + (I/4)*Pi + (d*x)/2]*Sqrt[a + I*a*Sinh[c +
d*x]])/(125*d^3) + (8*a^2*x^3*Cosh[c/2 + (I/4)*Pi + (d*x)/2]^3*Sinh[c/2 + (I/4)*Pi + (d*x)/2]*Sqrt[a + I*a*Si
nh[c + d*x]])/(5*d) + (132608*a^2*x*Sqrt[a + I*a*Sinh[c + d*x]]*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(1125*d^3) + (
64*a^2*x^3*Sqrt[a + I*a*Sinh[c + d*x]]*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(15*d)

________________________________________________________________________________________

Rubi [A]  time = 0.639757, antiderivative size = 638, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3319, 3311, 3296, 2638, 3310} \[ -\frac{128 a^2 x^2 \sqrt{a+i a \sinh (c+d x)}}{5 d^2}-\frac{48 a^2 x^2 \cosh ^4\left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}}{25 d^2}-\frac{64 a^2 x^2 \cosh ^2\left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}}{15 d^2}-\frac{265216 a^2 \sqrt{a+i a \sinh (c+d x)}}{1125 d^4}+\frac{132608 a^2 x \tanh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}}{1125 d^3}-\frac{384 a^2 \cosh ^4\left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}}{625 d^4}+\frac{192 a^2 x \sinh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \cosh ^3\left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}}{125 d^3}-\frac{17408 a^2 \cosh ^2\left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}}{3375 d^4}+\frac{8704 a^2 x \sinh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \cosh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}}{1125 d^3}+\frac{64 a^2 x^3 \tanh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}}{15 d}+\frac{8 a^2 x^3 \sinh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \cosh ^3\left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}}{5 d}+\frac{32 a^2 x^3 \sinh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \cosh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}}{15 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-265216*a^2*Sqrt[a + I*a*Sinh[c + d*x]])/(1125*d^4) - (128*a^2*x^2*Sqrt[a + I*a*Sinh[c + d*x]])/(5*d^2) - (17
408*a^2*Cosh[c/2 + (I/4)*Pi + (d*x)/2]^2*Sqrt[a + I*a*Sinh[c + d*x]])/(3375*d^4) - (64*a^2*x^2*Cosh[c/2 + (I/4
)*Pi + (d*x)/2]^2*Sqrt[a + I*a*Sinh[c + d*x]])/(15*d^2) - (384*a^2*Cosh[c/2 + (I/4)*Pi + (d*x)/2]^4*Sqrt[a + I
*a*Sinh[c + d*x]])/(625*d^4) - (48*a^2*x^2*Cosh[c/2 + (I/4)*Pi + (d*x)/2]^4*Sqrt[a + I*a*Sinh[c + d*x]])/(25*d
^2) + (8704*a^2*x*Cosh[c/2 + (I/4)*Pi + (d*x)/2]*Sinh[c/2 + (I/4)*Pi + (d*x)/2]*Sqrt[a + I*a*Sinh[c + d*x]])/(
1125*d^3) + (32*a^2*x^3*Cosh[c/2 + (I/4)*Pi + (d*x)/2]*Sinh[c/2 + (I/4)*Pi + (d*x)/2]*Sqrt[a + I*a*Sinh[c + d*
x]])/(15*d) + (192*a^2*x*Cosh[c/2 + (I/4)*Pi + (d*x)/2]^3*Sinh[c/2 + (I/4)*Pi + (d*x)/2]*Sqrt[a + I*a*Sinh[c +
d*x]])/(125*d^3) + (8*a^2*x^3*Cosh[c/2 + (I/4)*Pi + (d*x)/2]^3*Sinh[c/2 + (I/4)*Pi + (d*x)/2]*Sqrt[a + I*a*Si
nh[c + d*x]])/(5*d) + (132608*a^2*x*Sqrt[a + I*a*Sinh[c + d*x]]*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(1125*d^3) + (
64*a^2*x^3*Sqrt[a + I*a*Sinh[c + d*x]]*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(15*d)

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 3311

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

Rule 3296

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

Rule 2638

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

Rule 3310

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

Rubi steps

\begin{align*} \int x^3 (a+i a \sinh (c+d x))^{5/2} \, dx &=\left (4 a^2 \text{csch}\left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}\right ) \int x^3 \sinh ^5\left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \, dx\\ &=-\frac{48 a^2 x^2 \cosh ^4\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{25 d^2}+\frac{8 a^2 x^3 \cosh ^3\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sinh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{5 d}-\frac{1}{5} \left (16 a^2 \text{csch}\left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}\right ) \int x^3 \sinh ^3\left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \, dx+\frac{\left (96 a^2 \text{csch}\left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}\right ) \int x \sinh ^5\left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \, dx}{25 d^2}\\ &=-\frac{64 a^2 x^2 \cosh ^2\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{15 d^2}-\frac{384 a^2 \cosh ^4\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{625 d^4}-\frac{48 a^2 x^2 \cosh ^4\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{25 d^2}+\frac{32 a^2 x^3 \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sinh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{15 d}+\frac{192 a^2 x \cosh ^3\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sinh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{125 d^3}+\frac{8 a^2 x^3 \cosh ^3\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sinh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{5 d}+\frac{1}{15} \left (32 a^2 \text{csch}\left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}\right ) \int x^3 \sinh \left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \, dx-\frac{\left (384 a^2 \text{csch}\left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}\right ) \int x \sinh ^3\left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \, dx}{125 d^2}-\frac{\left (128 a^2 \text{csch}\left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}\right ) \int x \sinh ^3\left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \, dx}{15 d^2}\\ &=-\frac{17408 a^2 \cosh ^2\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{3375 d^4}-\frac{64 a^2 x^2 \cosh ^2\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{15 d^2}-\frac{384 a^2 \cosh ^4\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{625 d^4}-\frac{48 a^2 x^2 \cosh ^4\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{25 d^2}+\frac{8704 a^2 x \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sinh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{1125 d^3}+\frac{32 a^2 x^3 \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sinh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{15 d}+\frac{192 a^2 x \cosh ^3\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sinh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{125 d^3}+\frac{8 a^2 x^3 \cosh ^3\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sinh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{5 d}+\frac{64 a^2 x^3 \sqrt{a+i a \sinh (c+d x)} \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{15 d}+\frac{\left (256 a^2 \text{csch}\left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}\right ) \int x \sinh \left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \, dx}{125 d^2}+\frac{\left (256 a^2 \text{csch}\left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}\right ) \int x \sinh \left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \, dx}{45 d^2}-\frac{\left (64 a^2 \text{csch}\left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}\right ) \int x^2 \cosh \left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \, dx}{5 d}\\ &=-\frac{128 a^2 x^2 \sqrt{a+i a \sinh (c+d x)}}{5 d^2}-\frac{17408 a^2 \cosh ^2\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{3375 d^4}-\frac{64 a^2 x^2 \cosh ^2\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{15 d^2}-\frac{384 a^2 \cosh ^4\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{625 d^4}-\frac{48 a^2 x^2 \cosh ^4\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{25 d^2}+\frac{8704 a^2 x \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sinh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{1125 d^3}+\frac{32 a^2 x^3 \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sinh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{15 d}+\frac{192 a^2 x \cosh ^3\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sinh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{125 d^3}+\frac{8 a^2 x^3 \cosh ^3\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sinh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{5 d}+\frac{17408 a^2 x \sqrt{a+i a \sinh (c+d x)} \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{1125 d^3}+\frac{64 a^2 x^3 \sqrt{a+i a \sinh (c+d x)} \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{15 d}-\frac{\left (512 a^2 \text{csch}\left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}\right ) \int \cosh \left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \, dx}{125 d^3}-\frac{\left (512 a^2 \text{csch}\left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}\right ) \int \cosh \left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \, dx}{45 d^3}-\frac{\left (256 i a^2 \text{csch}\left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}\right ) \int x \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \, dx}{5 d^2}\\ &=-\frac{34816 a^2 \sqrt{a+i a \sinh (c+d x)}}{1125 d^4}-\frac{128 a^2 x^2 \sqrt{a+i a \sinh (c+d x)}}{5 d^2}-\frac{17408 a^2 \cosh ^2\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{3375 d^4}-\frac{64 a^2 x^2 \cosh ^2\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{15 d^2}-\frac{384 a^2 \cosh ^4\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{625 d^4}-\frac{48 a^2 x^2 \cosh ^4\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{25 d^2}+\frac{8704 a^2 x \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sinh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{1125 d^3}+\frac{32 a^2 x^3 \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sinh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{15 d}+\frac{192 a^2 x \cosh ^3\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sinh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{125 d^3}+\frac{8 a^2 x^3 \cosh ^3\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sinh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{5 d}+\frac{132608 a^2 x \sqrt{a+i a \sinh (c+d x)} \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{1125 d^3}+\frac{64 a^2 x^3 \sqrt{a+i a \sinh (c+d x)} \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{15 d}-\frac{\left (512 a^2 \text{csch}\left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}\right ) \int \cosh \left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \, dx}{5 d^3}\\ &=-\frac{265216 a^2 \sqrt{a+i a \sinh (c+d x)}}{1125 d^4}-\frac{128 a^2 x^2 \sqrt{a+i a \sinh (c+d x)}}{5 d^2}-\frac{17408 a^2 \cosh ^2\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{3375 d^4}-\frac{64 a^2 x^2 \cosh ^2\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{15 d^2}-\frac{384 a^2 \cosh ^4\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{625 d^4}-\frac{48 a^2 x^2 \cosh ^4\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{25 d^2}+\frac{8704 a^2 x \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sinh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{1125 d^3}+\frac{32 a^2 x^3 \cosh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sinh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{15 d}+\frac{192 a^2 x \cosh ^3\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sinh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{125 d^3}+\frac{8 a^2 x^3 \cosh ^3\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sinh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)}}{5 d}+\frac{132608 a^2 x \sqrt{a+i a \sinh (c+d x)} \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{1125 d^3}+\frac{64 a^2 x^3 \sqrt{a+i a \sinh (c+d x)} \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{15 d}\\ \end{align*}

Mathematica [B]  time = 7.43947, size = 2918, normalized size = 4.57 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(((-1/135000 - I/135000)*Cosh[5*(c/2 + (d*x)/2)])/d^3 + ((1/135000 + I/135000)*Sinh[5*(c/2 + (d*x)/2)])/d^3
)*(1296*I - (3240*I)*c + (4050*I)*c^2 - (3375*I)*c^3 + (6480*I)*(c/2 + (d*x)/2) - (16200*I)*c*(c/2 + (d*x)/2)
+ (20250*I)*c^2*(c/2 + (d*x)/2) + (16200*I)*(c/2 + (d*x)/2)^2 - (40500*I)*c*(c/2 + (d*x)/2)^2 + (27000*I)*(c/2
+ (d*x)/2)^3 - 50000*Cosh[2*(c/2 + (d*x)/2)] + 75000*c*Cosh[2*(c/2 + (d*x)/2)] - 56250*c^2*Cosh[2*(c/2 + (d*x
)/2)] + 28125*c^3*Cosh[2*(c/2 + (d*x)/2)] - 150000*(c/2 + (d*x)/2)*Cosh[2*(c/2 + (d*x)/2)] + 225000*c*(c/2 + (
d*x)/2)*Cosh[2*(c/2 + (d*x)/2)] - 168750*c^2*(c/2 + (d*x)/2)*Cosh[2*(c/2 + (d*x)/2)] - 225000*(c/2 + (d*x)/2)^
2*Cosh[2*(c/2 + (d*x)/2)] + 337500*c*(c/2 + (d*x)/2)^2*Cosh[2*(c/2 + (d*x)/2)] - 225000*(c/2 + (d*x)/2)^3*Cosh
[2*(c/2 + (d*x)/2)] - (8100000*I)*Cosh[4*(c/2 + (d*x)/2)] + (4050000*I)*c*Cosh[4*(c/2 + (d*x)/2)] - (1012500*I
)*c^2*Cosh[4*(c/2 + (d*x)/2)] + (168750*I)*c^3*Cosh[4*(c/2 + (d*x)/2)] - (8100000*I)*(c/2 + (d*x)/2)*Cosh[4*(c
/2 + (d*x)/2)] + (4050000*I)*c*(c/2 + (d*x)/2)*Cosh[4*(c/2 + (d*x)/2)] - (1012500*I)*c^2*(c/2 + (d*x)/2)*Cosh[
4*(c/2 + (d*x)/2)] - (4050000*I)*(c/2 + (d*x)/2)^2*Cosh[4*(c/2 + (d*x)/2)] + (2025000*I)*c*(c/2 + (d*x)/2)^2*C
osh[4*(c/2 + (d*x)/2)] - (1350000*I)*(c/2 + (d*x)/2)^3*Cosh[4*(c/2 + (d*x)/2)] + 8100000*Cosh[6*(c/2 + (d*x)/2
)] + 4050000*c*Cosh[6*(c/2 + (d*x)/2)] + 1012500*c^2*Cosh[6*(c/2 + (d*x)/2)] + 168750*c^3*Cosh[6*(c/2 + (d*x)/
2)] - 8100000*(c/2 + (d*x)/2)*Cosh[6*(c/2 + (d*x)/2)] - 4050000*c*(c/2 + (d*x)/2)*Cosh[6*(c/2 + (d*x)/2)] - 10
12500*c^2*(c/2 + (d*x)/2)*Cosh[6*(c/2 + (d*x)/2)] + 4050000*(c/2 + (d*x)/2)^2*Cosh[6*(c/2 + (d*x)/2)] + 202500
0*c*(c/2 + (d*x)/2)^2*Cosh[6*(c/2 + (d*x)/2)] - 1350000*(c/2 + (d*x)/2)^3*Cosh[6*(c/2 + (d*x)/2)] + (50000*I)*
Cosh[8*(c/2 + (d*x)/2)] + (75000*I)*c*Cosh[8*(c/2 + (d*x)/2)] + (56250*I)*c^2*Cosh[8*(c/2 + (d*x)/2)] + (28125
*I)*c^3*Cosh[8*(c/2 + (d*x)/2)] - (150000*I)*(c/2 + (d*x)/2)*Cosh[8*(c/2 + (d*x)/2)] - (225000*I)*c*(c/2 + (d*
x)/2)*Cosh[8*(c/2 + (d*x)/2)] - (168750*I)*c^2*(c/2 + (d*x)/2)*Cosh[8*(c/2 + (d*x)/2)] + (225000*I)*(c/2 + (d*
x)/2)^2*Cosh[8*(c/2 + (d*x)/2)] + (337500*I)*c*(c/2 + (d*x)/2)^2*Cosh[8*(c/2 + (d*x)/2)] - (225000*I)*(c/2 + (
d*x)/2)^3*Cosh[8*(c/2 + (d*x)/2)] - 1296*Cosh[10*(c/2 + (d*x)/2)] - 3240*c*Cosh[10*(c/2 + (d*x)/2)] - 4050*c^2
*Cosh[10*(c/2 + (d*x)/2)] - 3375*c^3*Cosh[10*(c/2 + (d*x)/2)] + 6480*(c/2 + (d*x)/2)*Cosh[10*(c/2 + (d*x)/2)]
+ 16200*c*(c/2 + (d*x)/2)*Cosh[10*(c/2 + (d*x)/2)] + 20250*c^2*(c/2 + (d*x)/2)*Cosh[10*(c/2 + (d*x)/2)] - 1620
0*(c/2 + (d*x)/2)^2*Cosh[10*(c/2 + (d*x)/2)] - 40500*c*(c/2 + (d*x)/2)^2*Cosh[10*(c/2 + (d*x)/2)] + 27000*(c/2
+ (d*x)/2)^3*Cosh[10*(c/2 + (d*x)/2)] - 50000*Sinh[2*(c/2 + (d*x)/2)] + 75000*c*Sinh[2*(c/2 + (d*x)/2)] - 562
50*c^2*Sinh[2*(c/2 + (d*x)/2)] + 28125*c^3*Sinh[2*(c/2 + (d*x)/2)] - 150000*(c/2 + (d*x)/2)*Sinh[2*(c/2 + (d*x
)/2)] + 225000*c*(c/2 + (d*x)/2)*Sinh[2*(c/2 + (d*x)/2)] - 168750*c^2*(c/2 + (d*x)/2)*Sinh[2*(c/2 + (d*x)/2)]
- 225000*(c/2 + (d*x)/2)^2*Sinh[2*(c/2 + (d*x)/2)] + 337500*c*(c/2 + (d*x)/2)^2*Sinh[2*(c/2 + (d*x)/2)] - 2250
00*(c/2 + (d*x)/2)^3*Sinh[2*(c/2 + (d*x)/2)] - (8100000*I)*Sinh[4*(c/2 + (d*x)/2)] + (4050000*I)*c*Sinh[4*(c/2
+ (d*x)/2)] - (1012500*I)*c^2*Sinh[4*(c/2 + (d*x)/2)] + (168750*I)*c^3*Sinh[4*(c/2 + (d*x)/2)] - (8100000*I)*
(c/2 + (d*x)/2)*Sinh[4*(c/2 + (d*x)/2)] + (4050000*I)*c*(c/2 + (d*x)/2)*Sinh[4*(c/2 + (d*x)/2)] - (1012500*I)*
c^2*(c/2 + (d*x)/2)*Sinh[4*(c/2 + (d*x)/2)] - (4050000*I)*(c/2 + (d*x)/2)^2*Sinh[4*(c/2 + (d*x)/2)] + (2025000
*I)*c*(c/2 + (d*x)/2)^2*Sinh[4*(c/2 + (d*x)/2)] - (1350000*I)*(c/2 + (d*x)/2)^3*Sinh[4*(c/2 + (d*x)/2)] + 8100
000*Sinh[6*(c/2 + (d*x)/2)] + 4050000*c*Sinh[6*(c/2 + (d*x)/2)] + 1012500*c^2*Sinh[6*(c/2 + (d*x)/2)] + 168750
*c^3*Sinh[6*(c/2 + (d*x)/2)] - 8100000*(c/2 + (d*x)/2)*Sinh[6*(c/2 + (d*x)/2)] - 4050000*c*(c/2 + (d*x)/2)*Sin
h[6*(c/2 + (d*x)/2)] - 1012500*c^2*(c/2 + (d*x)/2)*Sinh[6*(c/2 + (d*x)/2)] + 4050000*(c/2 + (d*x)/2)^2*Sinh[6*
(c/2 + (d*x)/2)] + 2025000*c*(c/2 + (d*x)/2)^2*Sinh[6*(c/2 + (d*x)/2)] - 1350000*(c/2 + (d*x)/2)^3*Sinh[6*(c/2
+ (d*x)/2)] + (50000*I)*Sinh[8*(c/2 + (d*x)/2)] + (75000*I)*c*Sinh[8*(c/2 + (d*x)/2)] + (56250*I)*c^2*Sinh[8*
(c/2 + (d*x)/2)] + (28125*I)*c^3*Sinh[8*(c/2 + (d*x)/2)] - (150000*I)*(c/2 + (d*x)/2)*Sinh[8*(c/2 + (d*x)/2)]
- (225000*I)*c*(c/2 + (d*x)/2)*Sinh[8*(c/2 + (d*x)/2)] - (168750*I)*c^2*(c/2 + (d*x)/2)*Sinh[8*(c/2 + (d*x)/2)
] + (225000*I)*(c/2 + (d*x)/2)^2*Sinh[8*(c/2 + (d*x)/2)] + (337500*I)*c*(c/2 + (d*x)/2)^2*Sinh[8*(c/2 + (d*x)/
2)] - (225000*I)*(c/2 + (d*x)/2)^3*Sinh[8*(c/2 + (d*x)/2)] - 1296*Sinh[10*(c/2 + (d*x)/2)] - 3240*c*Sinh[10*(c
/2 + (d*x)/2)] - 4050*c^2*Sinh[10*(c/2 + (d*x)/2)] - 3375*c^3*Sinh[10*(c/2 + (d*x)/2)] + 6480*(c/2 + (d*x)/2)*
Sinh[10*(c/2 + (d*x)/2)] + 16200*c*(c/2 + (d*x)/2)*Sinh[10*(c/2 + (d*x)/2)] + 20250*c^2*(c/2 + (d*x)/2)*Sinh[1
0*(c/2 + (d*x)/2)] - 16200*(c/2 + (d*x)/2)^2*Sinh[10*(c/2 + (d*x)/2)] - 40500*c*(c/2 + (d*x)/2)^2*Sinh[10*(c/2
+ (d*x)/2)] + 27000*(c/2 + (d*x)/2)^3*Sinh[10*(c/2 + (d*x)/2)])*(a + I*a*Sinh[c + d*x])^(5/2))/(d*(Cosh[c/2 +
(d*x)/2] + I*Sinh[c/2 + (d*x)/2])^5)

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Maple [F]  time = 0.063, 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{\left (i \, a \sinh \left (d x + c\right ) + a\right )}^{\frac{5}{2}} x^{3}\,{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((I*a*sinh(d*x + c) + a)^(5/2)*x^3, x)

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

Exception raised: UnboundLocalError

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