3.2.30 \(\int x^3 (a+i a \sinh (c+d x))^{5/2} \, dx\) [130]

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

[Out]

-265216/1125*a^2*(a+I*a*sinh(d*x+c))^(1/2)/d^4-128/5*a^2*x^2*(a+I*a*sinh(d*x+c))^(1/2)/d^2-17408/3375*a^2*cosh
(1/2*c+1/4*I*Pi+1/2*d*x)^2*(a+I*a*sinh(d*x+c))^(1/2)/d^4-64/15*a^2*x^2*cosh(1/2*c+1/4*I*Pi+1/2*d*x)^2*(a+I*a*s
inh(d*x+c))^(1/2)/d^2-384/625*a^2*cosh(1/2*c+1/4*I*Pi+1/2*d*x)^4*(a+I*a*sinh(d*x+c))^(1/2)/d^4-48/25*a^2*x^2*c
osh(1/2*c+1/4*I*Pi+1/2*d*x)^4*(a+I*a*sinh(d*x+c))^(1/2)/d^2+8704/1125*a^2*x*cosh(1/2*c+1/4*I*Pi+1/2*d*x)*sinh(
1/2*c+1/4*I*Pi+1/2*d*x)*(a+I*a*sinh(d*x+c))^(1/2)/d^3+32/15*a^2*x^3*cosh(1/2*c+1/4*I*Pi+1/2*d*x)*sinh(1/2*c+1/
4*I*Pi+1/2*d*x)*(a+I*a*sinh(d*x+c))^(1/2)/d+192/125*a^2*x*cosh(1/2*c+1/4*I*Pi+1/2*d*x)^3*sinh(1/2*c+1/4*I*Pi+1
/2*d*x)*(a+I*a*sinh(d*x+c))^(1/2)/d^3+8/5*a^2*x^3*cosh(1/2*c+1/4*I*Pi+1/2*d*x)^3*sinh(1/2*c+1/4*I*Pi+1/2*d*x)*
(a+I*a*sinh(d*x+c))^(1/2)/d+132608/1125*a^2*x*(a+I*a*sinh(d*x+c))^(1/2)*tanh(1/2*c+1/4*I*Pi+1/2*d*x)/d^3+64/15
*a^2*x^3*(a+I*a*sinh(d*x+c))^(1/2)*tanh(1/2*c+1/4*I*Pi+1/2*d*x)/d

________________________________________________________________________________________

Rubi [A]
time = 0.44, antiderivative size = 638, normalized size of antiderivative = 1.00, 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 = {3400, 3392, 3377, 2718, 3391} \begin {gather*} -\frac {265216 a^2 \sqrt {a+i a \sinh (c+d x)}}{1125 d^4}-\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 {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 {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 {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 {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 {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 {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} \end {gather*}

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 2718

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

Rule 3377

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

Rule 3391

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

Rule 3392

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

Rule 3400

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])

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*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2918\) vs. \(2(638)=1276\).
time = 6.32, size = 2918, normalized size = 4.57 \begin {gather*} \text {Result too large to show} \end {gather*}

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*...

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Maple [F]
time = 0.38, size = 0, normalized size = 0.00 \[\int x^{3} \left (a +i a \sinh \left (d x +c \right )\right )^{\frac {5}{2}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 3878 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^3\,{\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + a*sinh(c + d*x)*1i)^(5/2),x)

[Out]

int(x^3*(a + a*sinh(c + d*x)*1i)^(5/2), x)

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