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

Optimal. Leaf size=536 \[ -\frac{25}{32} i a^2 d^2 \sinh \left (\frac{5 c}{2}-\frac{i \pi }{4}\right ) \text{Chi}\left (\frac{5 d x}{2}\right ) \text{sech}\left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}+\frac{5}{16} i a^2 d^2 \sinh \left (\frac{1}{4} (2 c-i \pi )\right ) \text{Chi}\left (\frac{d x}{2}\right ) \text{sech}\left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}+\frac{45}{32} i a^2 d^2 \sinh \left (\frac{1}{4} (6 c+i \pi )\right ) \text{Chi}\left (\frac{3 d x}{2}\right ) \text{sech}\left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}+\frac{5}{16} i a^2 d^2 \cosh \left (\frac{1}{4} (2 c-i \pi )\right ) \text{Shi}\left (\frac{d x}{2}\right ) \text{sech}\left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}+\frac{45}{32} i a^2 d^2 \cosh \left (\frac{1}{4} (6 c+i \pi )\right ) \text{Shi}\left (\frac{3 d x}{2}\right ) \text{sech}\left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}-\frac{25}{32} i a^2 d^2 \cosh \left (\frac{5 c}{2}-\frac{i \pi }{4}\right ) \text{Shi}\left (\frac{5 d x}{2}\right ) \text{sech}\left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}-\frac{2 a^2 \cosh ^4\left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}}{x^2}-\frac{5 a^2 d \sinh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \cosh ^3\left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}}{x} \]

[Out]

(-2*a^2*Cosh[c/2 + (I/4)*Pi + (d*x)/2]^4*Sqrt[a + I*a*Sinh[c + d*x]])/x^2 - ((25*I)/32)*a^2*d^2*CoshIntegral[(
5*d*x)/2]*Sech[c/2 + (I/4)*Pi + (d*x)/2]*Sinh[(5*c)/2 - (I/4)*Pi]*Sqrt[a + I*a*Sinh[c + d*x]] + ((5*I)/16)*a^2
*d^2*CoshIntegral[(d*x)/2]*Sech[c/2 + (I/4)*Pi + (d*x)/2]*Sinh[(2*c - I*Pi)/4]*Sqrt[a + I*a*Sinh[c + d*x]] + (
(45*I)/32)*a^2*d^2*CoshIntegral[(3*d*x)/2]*Sech[c/2 + (I/4)*Pi + (d*x)/2]*Sinh[(6*c + I*Pi)/4]*Sqrt[a + I*a*Si
nh[c + d*x]] - (5*a^2*d*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]])/x + ((5*I)/16)*a^2*d^2*Cosh[(2*c - I*Pi)/4]*Sech[c/2 + (I/4)*Pi + (d*x)/2]*Sqrt[a + I*a*Sinh[c + d*x]]*
SinhIntegral[(d*x)/2] + ((45*I)/32)*a^2*d^2*Cosh[(6*c + I*Pi)/4]*Sech[c/2 + (I/4)*Pi + (d*x)/2]*Sqrt[a + I*a*S
inh[c + d*x]]*SinhIntegral[(3*d*x)/2] - ((25*I)/32)*a^2*d^2*Cosh[(5*c)/2 - (I/4)*Pi]*Sech[c/2 + (I/4)*Pi + (d*
x)/2]*Sqrt[a + I*a*Sinh[c + d*x]]*SinhIntegral[(5*d*x)/2]

________________________________________________________________________________________

Rubi [A]  time = 0.638273, antiderivative size = 536, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3319, 3314, 3312, 3303, 3298, 3301} \[ -\frac{25}{32} i a^2 d^2 \sinh \left (\frac{5 c}{2}-\frac{i \pi }{4}\right ) \text{Chi}\left (\frac{5 d x}{2}\right ) \text{sech}\left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}+\frac{5}{16} i a^2 d^2 \sinh \left (\frac{1}{4} (2 c-i \pi )\right ) \text{Chi}\left (\frac{d x}{2}\right ) \text{sech}\left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}+\frac{45}{32} i a^2 d^2 \sinh \left (\frac{1}{4} (6 c+i \pi )\right ) \text{Chi}\left (\frac{3 d x}{2}\right ) \text{sech}\left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}+\frac{5}{16} i a^2 d^2 \cosh \left (\frac{1}{4} (2 c-i \pi )\right ) \text{Shi}\left (\frac{d x}{2}\right ) \text{sech}\left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}+\frac{45}{32} i a^2 d^2 \cosh \left (\frac{1}{4} (6 c+i \pi )\right ) \text{Shi}\left (\frac{3 d x}{2}\right ) \text{sech}\left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}-\frac{25}{32} i a^2 d^2 \cosh \left (\frac{5 c}{2}-\frac{i \pi }{4}\right ) \text{Shi}\left (\frac{5 d x}{2}\right ) \text{sech}\left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}-\frac{2 a^2 \cosh ^4\left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}}{x^2}-\frac{5 a^2 d \sinh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \cosh ^3\left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^2*Cosh[c/2 + (I/4)*Pi + (d*x)/2]^4*Sqrt[a + I*a*Sinh[c + d*x]])/x^2 - ((25*I)/32)*a^2*d^2*CoshIntegral[(
5*d*x)/2]*Sech[c/2 + (I/4)*Pi + (d*x)/2]*Sinh[(5*c)/2 - (I/4)*Pi]*Sqrt[a + I*a*Sinh[c + d*x]] + ((5*I)/16)*a^2
*d^2*CoshIntegral[(d*x)/2]*Sech[c/2 + (I/4)*Pi + (d*x)/2]*Sinh[(2*c - I*Pi)/4]*Sqrt[a + I*a*Sinh[c + d*x]] + (
(45*I)/32)*a^2*d^2*CoshIntegral[(3*d*x)/2]*Sech[c/2 + (I/4)*Pi + (d*x)/2]*Sinh[(6*c + I*Pi)/4]*Sqrt[a + I*a*Si
nh[c + d*x]] - (5*a^2*d*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]])/x + ((5*I)/16)*a^2*d^2*Cosh[(2*c - I*Pi)/4]*Sech[c/2 + (I/4)*Pi + (d*x)/2]*Sqrt[a + I*a*Sinh[c + d*x]]*
SinhIntegral[(d*x)/2] + ((45*I)/32)*a^2*d^2*Cosh[(6*c + I*Pi)/4]*Sech[c/2 + (I/4)*Pi + (d*x)/2]*Sqrt[a + I*a*S
inh[c + d*x]]*SinhIntegral[(3*d*x)/2] - ((25*I)/32)*a^2*d^2*Cosh[(5*c)/2 - (I/4)*Pi]*Sech[c/2 + (I/4)*Pi + (d*
x)/2]*Sqrt[a + I*a*Sinh[c + d*x]]*SinhIntegral[(5*d*x)/2]

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 3314

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

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 3303

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

Rule 3298

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(I*SinhIntegral[(c*f*fz)
/d + f*fz*x])/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3301

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[(c*f*fz)/d
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rubi steps

\begin{align*} \int \frac{(a+i a \sinh (c+d x))^{5/2}}{x^3} \, 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 \frac{\sinh ^5\left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right )}{x^3} \, dx\\ &=-\frac{2 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)}}{x^2}-\frac{5 a^2 d \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)}}{x}+\left (10 a^2 d^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 \frac{\sinh ^3\left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right )}{x} \, dx+\frac{1}{2} \left (25 a^2 d^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 \frac{\sinh ^5\left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right )}{x} \, dx\\ &=-\frac{2 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)}}{x^2}-\frac{5 a^2 d \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)}}{x}+\left (10 i a^2 d^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 \left (\frac{3 i \sinh \left (\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}\right )}{4 x}+\frac{i \sinh \left (\frac{1}{4} (6 c+i \pi )+\frac{3 d x}{2}\right )}{4 x}\right ) \, dx-\frac{1}{2} \left (25 i a^2 d^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 \left (\frac{5 i \sinh \left (\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}\right )}{8 x}+\frac{5 i \sinh \left (\frac{1}{4} (6 c+i \pi )+\frac{3 d x}{2}\right )}{16 x}-\frac{i \sinh \left (\frac{1}{4} (10 c-i \pi )+\frac{5 d x}{2}\right )}{16 x}\right ) \, dx\\ &=-\frac{2 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)}}{x^2}-\frac{5 a^2 d \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)}}{x}-\frac{1}{32} \left (25 a^2 d^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 \frac{\sinh \left (\frac{1}{4} (10 c-i \pi )+\frac{5 d x}{2}\right )}{x} \, dx-\frac{1}{2} \left (5 a^2 d^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 \frac{\sinh \left (\frac{1}{4} (6 c+i \pi )+\frac{3 d x}{2}\right )}{x} \, dx+\frac{1}{32} \left (125 a^2 d^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 \frac{\sinh \left (\frac{1}{4} (6 c+i \pi )+\frac{3 d x}{2}\right )}{x} \, dx-\frac{1}{2} \left (15 a^2 d^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 \frac{\sinh \left (\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}\right )}{x} \, dx+\frac{1}{16} \left (125 a^2 d^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 \frac{\sinh \left (\frac{1}{4} (2 c-i \pi )+\frac{d x}{2}\right )}{x} \, dx\\ &=-\frac{2 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)}}{x^2}-\frac{5 a^2 d \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)}}{x}-\frac{1}{32} \left (25 a^2 d^2 \cosh \left (\frac{5 c}{2}-\frac{i \pi }{4}\right ) \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 \frac{\sinh \left (\frac{5 d x}{2}\right )}{x} \, dx-\frac{1}{2} \left (15 a^2 d^2 \cosh \left (\frac{1}{4} (2 c-i \pi )\right ) \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 \frac{\sinh \left (\frac{d x}{2}\right )}{x} \, dx+\frac{1}{16} \left (125 a^2 d^2 \cosh \left (\frac{1}{4} (2 c-i \pi )\right ) \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 \frac{\sinh \left (\frac{d x}{2}\right )}{x} \, dx-\frac{1}{2} \left (5 a^2 d^2 \cosh \left (\frac{1}{4} (6 c+i \pi )\right ) \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 \frac{\sinh \left (\frac{3 d x}{2}\right )}{x} \, dx+\frac{1}{32} \left (125 a^2 d^2 \cosh \left (\frac{1}{4} (6 c+i \pi )\right ) \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 \frac{\sinh \left (\frac{3 d x}{2}\right )}{x} \, dx-\frac{1}{32} \left (25 a^2 d^2 \text{csch}\left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \sinh \left (\frac{5 c}{2}-\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}\right ) \int \frac{\cosh \left (\frac{5 d x}{2}\right )}{x} \, dx-\frac{1}{2} \left (15 a^2 d^2 \text{csch}\left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \sinh \left (\frac{1}{4} (2 c-i \pi )\right ) \sqrt{a+i a \sinh (c+d x)}\right ) \int \frac{\cosh \left (\frac{d x}{2}\right )}{x} \, dx+\frac{1}{16} \left (125 a^2 d^2 \text{csch}\left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \sinh \left (\frac{1}{4} (2 c-i \pi )\right ) \sqrt{a+i a \sinh (c+d x)}\right ) \int \frac{\cosh \left (\frac{d x}{2}\right )}{x} \, dx-\frac{1}{2} \left (5 a^2 d^2 \text{csch}\left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \sinh \left (\frac{1}{4} (6 c+i \pi )\right ) \sqrt{a+i a \sinh (c+d x)}\right ) \int \frac{\cosh \left (\frac{3 d x}{2}\right )}{x} \, dx+\frac{1}{32} \left (125 a^2 d^2 \text{csch}\left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \sinh \left (\frac{1}{4} (6 c+i \pi )\right ) \sqrt{a+i a \sinh (c+d x)}\right ) \int \frac{\cosh \left (\frac{3 d x}{2}\right )}{x} \, dx\\ &=-\frac{2 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)}}{x^2}-\frac{25}{32} i a^2 d^2 \text{Chi}\left (\frac{5 d x}{2}\right ) \text{sech}\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sinh \left (\frac{5 c}{2}-\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (c+d x)}+\frac{5}{16} i a^2 d^2 \text{Chi}\left (\frac{d x}{2}\right ) \text{sech}\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sinh \left (\frac{1}{4} (2 c-i \pi )\right ) \sqrt{a+i a \sinh (c+d x)}+\frac{45}{32} i a^2 d^2 \text{Chi}\left (\frac{3 d x}{2}\right ) \text{sech}\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sinh \left (\frac{1}{4} (6 c+i \pi )\right ) \sqrt{a+i a \sinh (c+d x)}-\frac{5 a^2 d \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)}}{x}+\frac{5}{16} i a^2 d^2 \cosh \left (\frac{1}{4} (2 c-i \pi )\right ) \text{sech}\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)} \text{Shi}\left (\frac{d x}{2}\right )+\frac{45}{32} i a^2 d^2 \cosh \left (\frac{1}{4} (6 c+i \pi )\right ) \text{sech}\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)} \text{Shi}\left (\frac{3 d x}{2}\right )-\frac{25}{32} i a^2 d^2 \cosh \left (\frac{5 c}{2}-\frac{i \pi }{4}\right ) \text{sech}\left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right ) \sqrt{a+i a \sinh (c+d x)} \text{Shi}\left (\frac{5 d x}{2}\right )\\ \end{align*}

Mathematica [B]  time = 7.68558, size = 4751, normalized size = 8.86 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((1/128 + I/128)*Cosh[5*(c/2 + (d*x)/2)] - (1/128 + I/128)*Sinh[5*(c/2 + (d*x)/2)])*(a + I*a*Sinh[c + d*x])
^(5/2)*((-4*I)*d^3 - (10*I)*c*d^3 + (20*I)*d^3*(c/2 + (d*x)/2) + 20*d^3*Cosh[2*(c/2 + (d*x)/2)] + 30*c*d^3*Cos
h[2*(c/2 + (d*x)/2)] - 60*d^3*(c/2 + (d*x)/2)*Cosh[2*(c/2 + (d*x)/2)] + (40*I)*d^3*Cosh[4*(c/2 + (d*x)/2)] + (
20*I)*c*d^3*Cosh[4*(c/2 + (d*x)/2)] - (40*I)*d^3*(c/2 + (d*x)/2)*Cosh[4*(c/2 + (d*x)/2)] - 40*d^3*Cosh[6*(c/2
+ (d*x)/2)] + 20*c*d^3*Cosh[6*(c/2 + (d*x)/2)] - 40*d^3*(c/2 + (d*x)/2)*Cosh[6*(c/2 + (d*x)/2)] - (20*I)*d^3*C
osh[8*(c/2 + (d*x)/2)] + (30*I)*c*d^3*Cosh[8*(c/2 + (d*x)/2)] - (60*I)*d^3*(c/2 + (d*x)/2)*Cosh[8*(c/2 + (d*x)
/2)] + 4*d^3*Cosh[10*(c/2 + (d*x)/2)] - 10*c*d^3*Cosh[10*(c/2 + (d*x)/2)] + 20*d^3*(c/2 + (d*x)/2)*Cosh[10*(c/
2 + (d*x)/2)] - (10*I)*c^2*d^3*Cosh[c/2 - 5*(c/2 + (d*x)/2)]*CoshIntegral[(d*x)/2] + (40*I)*c*d^3*(c/2 + (d*x)
/2)*Cosh[c/2 - 5*(c/2 + (d*x)/2)]*CoshIntegral[(d*x)/2] - (40*I)*d^3*(c/2 + (d*x)/2)^2*Cosh[c/2 - 5*(c/2 + (d*
x)/2)]*CoshIntegral[(d*x)/2] + 10*c^2*d^3*Cosh[c/2 + 5*(c/2 + (d*x)/2)]*CoshIntegral[(d*x)/2] - 40*c*d^3*(c/2
+ (d*x)/2)*Cosh[c/2 + 5*(c/2 + (d*x)/2)]*CoshIntegral[(d*x)/2] + 40*d^3*(c/2 + (d*x)/2)^2*Cosh[c/2 + 5*(c/2 +
(d*x)/2)]*CoshIntegral[(d*x)/2] - 45*c^2*d^3*Cosh[(3*c)/2 - 5*(c/2 + (d*x)/2)]*CoshIntegral[(-3*c)/2 + 3*(c/2
+ (d*x)/2)] + 180*c*d^3*(c/2 + (d*x)/2)*Cosh[(3*c)/2 - 5*(c/2 + (d*x)/2)]*CoshIntegral[(-3*c)/2 + 3*(c/2 + (d*
x)/2)] - 180*d^3*(c/2 + (d*x)/2)^2*Cosh[(3*c)/2 - 5*(c/2 + (d*x)/2)]*CoshIntegral[(-3*c)/2 + 3*(c/2 + (d*x)/2)
] + (45*I)*c^2*d^3*Cosh[(3*c)/2 + 5*(c/2 + (d*x)/2)]*CoshIntegral[(-3*c)/2 + 3*(c/2 + (d*x)/2)] - (180*I)*c*d^
3*(c/2 + (d*x)/2)*Cosh[(3*c)/2 + 5*(c/2 + (d*x)/2)]*CoshIntegral[(-3*c)/2 + 3*(c/2 + (d*x)/2)] + (180*I)*d^3*(
c/2 + (d*x)/2)^2*Cosh[(3*c)/2 + 5*(c/2 + (d*x)/2)]*CoshIntegral[(-3*c)/2 + 3*(c/2 + (d*x)/2)] + (25*I)*c^2*d^3
*Cosh[(5*c)/2 - 5*(c/2 + (d*x)/2)]*CoshIntegral[(-5*c)/2 + 5*(c/2 + (d*x)/2)] - (100*I)*c*d^3*(c/2 + (d*x)/2)*
Cosh[(5*c)/2 - 5*(c/2 + (d*x)/2)]*CoshIntegral[(-5*c)/2 + 5*(c/2 + (d*x)/2)] + (100*I)*d^3*(c/2 + (d*x)/2)^2*C
osh[(5*c)/2 - 5*(c/2 + (d*x)/2)]*CoshIntegral[(-5*c)/2 + 5*(c/2 + (d*x)/2)] - 25*c^2*d^3*Cosh[(5*c)/2 + 5*(c/2
+ (d*x)/2)]*CoshIntegral[(-5*c)/2 + 5*(c/2 + (d*x)/2)] + 100*c*d^3*(c/2 + (d*x)/2)*Cosh[(5*c)/2 + 5*(c/2 + (d
*x)/2)]*CoshIntegral[(-5*c)/2 + 5*(c/2 + (d*x)/2)] - 100*d^3*(c/2 + (d*x)/2)^2*Cosh[(5*c)/2 + 5*(c/2 + (d*x)/2
)]*CoshIntegral[(-5*c)/2 + 5*(c/2 + (d*x)/2)] + 20*d^3*Sinh[2*(c/2 + (d*x)/2)] + 30*c*d^3*Sinh[2*(c/2 + (d*x)/
2)] - 60*d^3*(c/2 + (d*x)/2)*Sinh[2*(c/2 + (d*x)/2)] + (40*I)*d^3*Sinh[4*(c/2 + (d*x)/2)] + (20*I)*c*d^3*Sinh[
4*(c/2 + (d*x)/2)] - (40*I)*d^3*(c/2 + (d*x)/2)*Sinh[4*(c/2 + (d*x)/2)] - 40*d^3*Sinh[6*(c/2 + (d*x)/2)] + 20*
c*d^3*Sinh[6*(c/2 + (d*x)/2)] - 40*d^3*(c/2 + (d*x)/2)*Sinh[6*(c/2 + (d*x)/2)] - (20*I)*d^3*Sinh[8*(c/2 + (d*x
)/2)] + (30*I)*c*d^3*Sinh[8*(c/2 + (d*x)/2)] - (60*I)*d^3*(c/2 + (d*x)/2)*Sinh[8*(c/2 + (d*x)/2)] + 4*d^3*Sinh
[10*(c/2 + (d*x)/2)] - 10*c*d^3*Sinh[10*(c/2 + (d*x)/2)] + 20*d^3*(c/2 + (d*x)/2)*Sinh[10*(c/2 + (d*x)/2)] + (
10*I)*c^2*d^3*CoshIntegral[(d*x)/2]*Sinh[c/2 - 5*(c/2 + (d*x)/2)] - (40*I)*c*d^3*(c/2 + (d*x)/2)*CoshIntegral[
(d*x)/2]*Sinh[c/2 - 5*(c/2 + (d*x)/2)] + (40*I)*d^3*(c/2 + (d*x)/2)^2*CoshIntegral[(d*x)/2]*Sinh[c/2 - 5*(c/2
+ (d*x)/2)] + 45*c^2*d^3*CoshIntegral[(-3*c)/2 + 3*(c/2 + (d*x)/2)]*Sinh[(3*c)/2 - 5*(c/2 + (d*x)/2)] - 180*c*
d^3*(c/2 + (d*x)/2)*CoshIntegral[(-3*c)/2 + 3*(c/2 + (d*x)/2)]*Sinh[(3*c)/2 - 5*(c/2 + (d*x)/2)] + 180*d^3*(c/
2 + (d*x)/2)^2*CoshIntegral[(-3*c)/2 + 3*(c/2 + (d*x)/2)]*Sinh[(3*c)/2 - 5*(c/2 + (d*x)/2)] - (25*I)*c^2*d^3*C
oshIntegral[(-5*c)/2 + 5*(c/2 + (d*x)/2)]*Sinh[(5*c)/2 - 5*(c/2 + (d*x)/2)] + (100*I)*c*d^3*(c/2 + (d*x)/2)*Co
shIntegral[(-5*c)/2 + 5*(c/2 + (d*x)/2)]*Sinh[(5*c)/2 - 5*(c/2 + (d*x)/2)] - (100*I)*d^3*(c/2 + (d*x)/2)^2*Cos
hIntegral[(-5*c)/2 + 5*(c/2 + (d*x)/2)]*Sinh[(5*c)/2 - 5*(c/2 + (d*x)/2)] + 10*c^2*d^3*CoshIntegral[(d*x)/2]*S
inh[c/2 + 5*(c/2 + (d*x)/2)] - 40*c*d^3*(c/2 + (d*x)/2)*CoshIntegral[(d*x)/2]*Sinh[c/2 + 5*(c/2 + (d*x)/2)] +
40*d^3*(c/2 + (d*x)/2)^2*CoshIntegral[(d*x)/2]*Sinh[c/2 + 5*(c/2 + (d*x)/2)] + (45*I)*c^2*d^3*CoshIntegral[(-3
*c)/2 + 3*(c/2 + (d*x)/2)]*Sinh[(3*c)/2 + 5*(c/2 + (d*x)/2)] - (180*I)*c*d^3*(c/2 + (d*x)/2)*CoshIntegral[(-3*
c)/2 + 3*(c/2 + (d*x)/2)]*Sinh[(3*c)/2 + 5*(c/2 + (d*x)/2)] + (180*I)*d^3*(c/2 + (d*x)/2)^2*CoshIntegral[(-3*c
)/2 + 3*(c/2 + (d*x)/2)]*Sinh[(3*c)/2 + 5*(c/2 + (d*x)/2)] - 25*c^2*d^3*CoshIntegral[(-5*c)/2 + 5*(c/2 + (d*x)
/2)]*Sinh[(5*c)/2 + 5*(c/2 + (d*x)/2)] + 100*c*d^3*(c/2 + (d*x)/2)*CoshIntegral[(-5*c)/2 + 5*(c/2 + (d*x)/2)]*
Sinh[(5*c)/2 + 5*(c/2 + (d*x)/2)] - 100*d^3*(c/2 + (d*x)/2)^2*CoshIntegral[(-5*c)/2 + 5*(c/2 + (d*x)/2)]*Sinh[
(5*c)/2 + 5*(c/2 + (d*x)/2)] + (10*I)*c^2*d^3*Cosh[c/2 - 5*(c/2 + (d*x)/2)]*SinhIntegral[(d*x)/2] - (40*I)*c*d
^3*(c/2 + (d*x)/2)*Cosh[c/2 - 5*(c/2 + (d*x)/2)]*SinhIntegral[(d*x)/2] + (40*I)*d^3*(c/2 + (d*x)/2)^2*Cosh[c/2
- 5*(c/2 + (d*x)/2)]*SinhIntegral[(d*x)/2] + 10*c^2*d^3*Cosh[c/2 + 5*(c/2 + (d*x)/2)]*SinhIntegral[(d*x)/2] -
40*c*d^3*(c/2 + (d*x)/2)*Cosh[c/2 + 5*(c/2 + (d*x)/2)]*SinhIntegral[(d*x)/2] + 40*d^3*(c/2 + (d*x)/2)^2*Cosh[
c/2 + 5*(c/2 + (d*x)/2)]*SinhIntegral[(d*x)/2] - (10*I)*c^2*d^3*Sinh[c/2 - 5*(c/2 + (d*x)/2)]*SinhIntegral[(d*
x)/2] + (40*I)*c*d^3*(c/2 + (d*x)/2)*Sinh[c/2 - 5*(c/2 + (d*x)/2)]*SinhIntegral[(d*x)/2] - (40*I)*d^3*(c/2 + (
d*x)/2)^2*Sinh[c/2 - 5*(c/2 + (d*x)/2)]*SinhIntegral[(d*x)/2] + 10*c^2*d^3*Sinh[c/2 + 5*(c/2 + (d*x)/2)]*SinhI
ntegral[(d*x)/2] - 40*c*d^3*(c/2 + (d*x)/2)*Sinh[c/2 + 5*(c/2 + (d*x)/2)]*SinhIntegral[(d*x)/2] + 40*d^3*(c/2
+ (d*x)/2)^2*Sinh[c/2 + 5*(c/2 + (d*x)/2)]*SinhIntegral[(d*x)/2] + (25*I)*c^2*d^3*Cosh[(5*c)/2 - 5*(c/2 + (d*x
)/2)]*SinhIntegral[(5*c)/2 - 5*(c/2 + (d*x)/2)] - (100*I)*c*d^3*(c/2 + (d*x)/2)*Cosh[(5*c)/2 - 5*(c/2 + (d*x)/
2)]*SinhIntegral[(5*c)/2 - 5*(c/2 + (d*x)/2)] + (100*I)*d^3*(c/2 + (d*x)/2)^2*Cosh[(5*c)/2 - 5*(c/2 + (d*x)/2)
]*SinhIntegral[(5*c)/2 - 5*(c/2 + (d*x)/2)] + 25*c^2*d^3*Cosh[(5*c)/2 + 5*(c/2 + (d*x)/2)]*SinhIntegral[(5*c)/
2 - 5*(c/2 + (d*x)/2)] - 100*c*d^3*(c/2 + (d*x)/2)*Cosh[(5*c)/2 + 5*(c/2 + (d*x)/2)]*SinhIntegral[(5*c)/2 - 5*
(c/2 + (d*x)/2)] + 100*d^3*(c/2 + (d*x)/2)^2*Cosh[(5*c)/2 + 5*(c/2 + (d*x)/2)]*SinhIntegral[(5*c)/2 - 5*(c/2 +
(d*x)/2)] - (25*I)*c^2*d^3*Sinh[(5*c)/2 - 5*(c/2 + (d*x)/2)]*SinhIntegral[(5*c)/2 - 5*(c/2 + (d*x)/2)] + (100
*I)*c*d^3*(c/2 + (d*x)/2)*Sinh[(5*c)/2 - 5*(c/2 + (d*x)/2)]*SinhIntegral[(5*c)/2 - 5*(c/2 + (d*x)/2)] - (100*I
)*d^3*(c/2 + (d*x)/2)^2*Sinh[(5*c)/2 - 5*(c/2 + (d*x)/2)]*SinhIntegral[(5*c)/2 - 5*(c/2 + (d*x)/2)] + 25*c^2*d
^3*Sinh[(5*c)/2 + 5*(c/2 + (d*x)/2)]*SinhIntegral[(5*c)/2 - 5*(c/2 + (d*x)/2)] - 100*c*d^3*(c/2 + (d*x)/2)*Sin
h[(5*c)/2 + 5*(c/2 + (d*x)/2)]*SinhIntegral[(5*c)/2 - 5*(c/2 + (d*x)/2)] + 100*d^3*(c/2 + (d*x)/2)^2*Sinh[(5*c
)/2 + 5*(c/2 + (d*x)/2)]*SinhIntegral[(5*c)/2 - 5*(c/2 + (d*x)/2)] - 45*c^2*d^3*Cosh[(3*c)/2 - 5*(c/2 + (d*x)/
2)]*SinhIntegral[(3*c)/2 - 3*(c/2 + (d*x)/2)] + 180*c*d^3*(c/2 + (d*x)/2)*Cosh[(3*c)/2 - 5*(c/2 + (d*x)/2)]*Si
nhIntegral[(3*c)/2 - 3*(c/2 + (d*x)/2)] - 180*d^3*(c/2 + (d*x)/2)^2*Cosh[(3*c)/2 - 5*(c/2 + (d*x)/2)]*SinhInte
gral[(3*c)/2 - 3*(c/2 + (d*x)/2)] - (45*I)*c^2*d^3*Cosh[(3*c)/2 + 5*(c/2 + (d*x)/2)]*SinhIntegral[(3*c)/2 - 3*
(c/2 + (d*x)/2)] + (180*I)*c*d^3*(c/2 + (d*x)/2)*Cosh[(3*c)/2 + 5*(c/2 + (d*x)/2)]*SinhIntegral[(3*c)/2 - 3*(c
/2 + (d*x)/2)] - (180*I)*d^3*(c/2 + (d*x)/2)^2*Cosh[(3*c)/2 + 5*(c/2 + (d*x)/2)]*SinhIntegral[(3*c)/2 - 3*(c/2
+ (d*x)/2)] + 45*c^2*d^3*Sinh[(3*c)/2 - 5*(c/2 + (d*x)/2)]*SinhIntegral[(3*c)/2 - 3*(c/2 + (d*x)/2)] - 180*c*
d^3*(c/2 + (d*x)/2)*Sinh[(3*c)/2 - 5*(c/2 + (d*x)/2)]*SinhIntegral[(3*c)/2 - 3*(c/2 + (d*x)/2)] + 180*d^3*(c/2
+ (d*x)/2)^2*Sinh[(3*c)/2 - 5*(c/2 + (d*x)/2)]*SinhIntegral[(3*c)/2 - 3*(c/2 + (d*x)/2)] - (45*I)*c^2*d^3*Sin
h[(3*c)/2 + 5*(c/2 + (d*x)/2)]*SinhIntegral[(3*c)/2 - 3*(c/2 + (d*x)/2)] + (180*I)*c*d^3*(c/2 + (d*x)/2)*Sinh[
(3*c)/2 + 5*(c/2 + (d*x)/2)]*SinhIntegral[(3*c)/2 - 3*(c/2 + (d*x)/2)] - (180*I)*d^3*(c/2 + (d*x)/2)^2*Sinh[(3
*c)/2 + 5*(c/2 + (d*x)/2)]*SinhIntegral[(3*c)/2 - 3*(c/2 + (d*x)/2)]))/(d*(-c + 2*(c/2 + (d*x)/2))^2*(Cosh[c/2
+ (d*x)/2] + I*Sinh[c/2 + (d*x)/2])^5)

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Maple [F]  time = 0.044, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{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((a+I*a*sinh(d*x+c))^(5/2)/x^3,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\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((a+I*a*sinh(d*x+c))^(5/2)/x^3,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((a+I*a*sinh(d*x+c))^(5/2)/x^3,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((a+I*a*sinh(d*x+c))**(5/2)/x**3,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\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((a+I*a*sinh(d*x+c))^(5/2)/x^3,x, algorithm="giac")

[Out]

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