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

Optimal. Leaf size=302 \[ -\frac{3}{4} a f \sinh \left (\frac{1}{4} (6 e-i \pi )\right ) \text{Chi}\left (\frac{3 f x}{2}\right ) \text{sech}\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}+\frac{3}{4} a f \sinh \left (\frac{1}{4} (2 e+i \pi )\right ) \text{Chi}\left (\frac{f x}{2}\right ) \text{sech}\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}+\frac{3}{4} a f \cosh \left (\frac{1}{4} (2 e+i \pi )\right ) \text{Shi}\left (\frac{f x}{2}\right ) \text{sech}\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}-\frac{3}{4} a f \cosh \left (\frac{1}{4} (6 e-i \pi )\right ) \text{Shi}\left (\frac{3 f x}{2}\right ) \text{sech}\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}-\frac{2 a \cosh ^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}}{x} \]

[Out]

(-2*a*Cosh[e/2 + (I/4)*Pi + (f*x)/2]^2*Sqrt[a + I*a*Sinh[e + f*x]])/x - (3*a*f*CoshIntegral[(3*f*x)/2]*Sech[e/
2 + (I/4)*Pi + (f*x)/2]*Sinh[(6*e - I*Pi)/4]*Sqrt[a + I*a*Sinh[e + f*x]])/4 + (3*a*f*CoshIntegral[(f*x)/2]*Sec
h[e/2 + (I/4)*Pi + (f*x)/2]*Sinh[(2*e + I*Pi)/4]*Sqrt[a + I*a*Sinh[e + f*x]])/4 + (3*a*f*Cosh[(2*e + I*Pi)/4]*
Sech[e/2 + (I/4)*Pi + (f*x)/2]*Sqrt[a + I*a*Sinh[e + f*x]]*SinhIntegral[(f*x)/2])/4 - (3*a*f*Cosh[(6*e - I*Pi)
/4]*Sech[e/2 + (I/4)*Pi + (f*x)/2]*Sqrt[a + I*a*Sinh[e + f*x]]*SinhIntegral[(3*f*x)/2])/4

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Rubi [A]  time = 0.289703, antiderivative size = 302, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3319, 3313, 3303, 3298, 3301} \[ -\frac{3}{4} a f \sinh \left (\frac{1}{4} (6 e-i \pi )\right ) \text{Chi}\left (\frac{3 f x}{2}\right ) \text{sech}\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}+\frac{3}{4} a f \sinh \left (\frac{1}{4} (2 e+i \pi )\right ) \text{Chi}\left (\frac{f x}{2}\right ) \text{sech}\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}+\frac{3}{4} a f \cosh \left (\frac{1}{4} (2 e+i \pi )\right ) \text{Shi}\left (\frac{f x}{2}\right ) \text{sech}\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}-\frac{3}{4} a f \cosh \left (\frac{1}{4} (6 e-i \pi )\right ) \text{Shi}\left (\frac{3 f x}{2}\right ) \text{sech}\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}-\frac{2 a \cosh ^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*Cosh[e/2 + (I/4)*Pi + (f*x)/2]^2*Sqrt[a + I*a*Sinh[e + f*x]])/x - (3*a*f*CoshIntegral[(3*f*x)/2]*Sech[e/
2 + (I/4)*Pi + (f*x)/2]*Sinh[(6*e - I*Pi)/4]*Sqrt[a + I*a*Sinh[e + f*x]])/4 + (3*a*f*CoshIntegral[(f*x)/2]*Sec
h[e/2 + (I/4)*Pi + (f*x)/2]*Sinh[(2*e + I*Pi)/4]*Sqrt[a + I*a*Sinh[e + f*x]])/4 + (3*a*f*Cosh[(2*e + I*Pi)/4]*
Sech[e/2 + (I/4)*Pi + (f*x)/2]*Sqrt[a + I*a*Sinh[e + f*x]]*SinhIntegral[(f*x)/2])/4 - (3*a*f*Cosh[(6*e - I*Pi)
/4]*Sech[e/2 + (I/4)*Pi + (f*x)/2]*Sqrt[a + I*a*Sinh[e + f*x]]*SinhIntegral[(3*f*x)/2])/4

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 3313

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x]^
n)/(d*(m + 1)), x] - Dist[(f*n)/(d*(m + 1)), Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]
^(n - 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] && 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 (e+f x))^{3/2}}{x^2} \, dx &=-\left (\left (2 a \text{csch}\left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right ) \sqrt{a+i a \sinh (e+f x)}\right ) \int \frac{\sinh ^3\left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right )}{x^2} \, dx\right )\\ &=-\frac{2 a \cosh ^2\left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \sqrt{a+i a \sinh (e+f x)}}{x}+\left (3 a f \text{csch}\left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right ) \sqrt{a+i a \sinh (e+f x)}\right ) \int \left (\frac{\cosh \left (\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}\right )}{4 x}+\frac{\cosh \left (\frac{1}{4} (6 e+i \pi )+\frac{3 f x}{2}\right )}{4 x}\right ) \, dx\\ &=-\frac{2 a \cosh ^2\left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \sqrt{a+i a \sinh (e+f x)}}{x}+\frac{1}{4} \left (3 a f \text{csch}\left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right ) \sqrt{a+i a \sinh (e+f x)}\right ) \int \frac{\cosh \left (\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}\right )}{x} \, dx+\frac{1}{4} \left (3 a f \text{csch}\left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right ) \sqrt{a+i a \sinh (e+f x)}\right ) \int \frac{\cosh \left (\frac{1}{4} (6 e+i \pi )+\frac{3 f x}{2}\right )}{x} \, dx\\ &=-\frac{2 a \cosh ^2\left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \sqrt{a+i a \sinh (e+f x)}}{x}+\frac{1}{4} \left (3 i a f \cosh \left (\frac{1}{4} (6 e-i \pi )\right ) \text{csch}\left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right ) \sqrt{a+i a \sinh (e+f x)}\right ) \int \frac{\sinh \left (\frac{3 f x}{2}\right )}{x} \, dx-\frac{1}{4} \left (3 i a f \cosh \left (\frac{1}{4} (2 e+i \pi )\right ) \text{csch}\left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right ) \sqrt{a+i a \sinh (e+f x)}\right ) \int \frac{\sinh \left (\frac{f x}{2}\right )}{x} \, dx+\frac{1}{4} \left (3 i a f \text{csch}\left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right ) \sinh \left (\frac{1}{4} (6 e-i \pi )\right ) \sqrt{a+i a \sinh (e+f x)}\right ) \int \frac{\cosh \left (\frac{3 f x}{2}\right )}{x} \, dx-\frac{1}{4} \left (3 i a f \text{csch}\left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right ) \sinh \left (\frac{1}{4} (2 e+i \pi )\right ) \sqrt{a+i a \sinh (e+f x)}\right ) \int \frac{\cosh \left (\frac{f x}{2}\right )}{x} \, dx\\ &=-\frac{2 a \cosh ^2\left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \sqrt{a+i a \sinh (e+f x)}}{x}-\frac{3}{4} a f \text{Chi}\left (\frac{3 f x}{2}\right ) \text{sech}\left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \sinh \left (\frac{1}{4} (6 e-i \pi )\right ) \sqrt{a+i a \sinh (e+f x)}+\frac{3}{4} a f \text{Chi}\left (\frac{f x}{2}\right ) \text{sech}\left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \sinh \left (\frac{1}{4} (2 e+i \pi )\right ) \sqrt{a+i a \sinh (e+f x)}+\frac{3}{4} a f \cosh \left (\frac{1}{4} (2 e+i \pi )\right ) \text{sech}\left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \sqrt{a+i a \sinh (e+f x)} \text{Shi}\left (\frac{f x}{2}\right )-\frac{3}{4} a f \cosh \left (\frac{1}{4} (6 e-i \pi )\right ) \text{sech}\left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \sqrt{a+i a \sinh (e+f x)} \text{Shi}\left (\frac{3 f x}{2}\right )\\ \end{align*}

Mathematica [A]  time = 0.973868, size = 243, normalized size = 0.8 \[ \frac{a (\sinh (e+f x)-i) \sqrt{a+i a \sinh (e+f x)} \left (-3 f x \text{Chi}\left (\frac{f x}{2}\right ) \left (\cosh \left (\frac{e}{2}\right )-i \sinh \left (\frac{e}{2}\right )\right )-3 f x \text{Chi}\left (\frac{3 f x}{2}\right ) \left (\cosh \left (\frac{3 e}{2}\right )+i \sinh \left (\frac{3 e}{2}\right )\right )-3 f x \sinh \left (\frac{e}{2}\right ) \text{Shi}\left (\frac{f x}{2}\right )-3 f x \sinh \left (\frac{3 e}{2}\right ) \text{Shi}\left (\frac{3 f x}{2}\right )+3 i f x \cosh \left (\frac{e}{2}\right ) \text{Shi}\left (\frac{f x}{2}\right )-3 i f x \cosh \left (\frac{3 e}{2}\right ) \text{Shi}\left (\frac{3 f x}{2}\right )+6 \sinh \left (\frac{1}{2} (e+f x)\right )+2 \sinh \left (\frac{3}{2} (e+f x)\right )-6 i \cosh \left (\frac{1}{2} (e+f x)\right )+2 i \cosh \left (\frac{3}{2} (e+f x)\right )\right )}{4 x \left (\cosh \left (\frac{1}{2} (e+f x)\right )+i \sinh \left (\frac{1}{2} (e+f x)\right )\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(-I + Sinh[e + f*x])*Sqrt[a + I*a*Sinh[e + f*x]]*((-6*I)*Cosh[(e + f*x)/2] + (2*I)*Cosh[(3*(e + f*x))/2] -
3*f*x*CoshIntegral[(f*x)/2]*(Cosh[e/2] - I*Sinh[e/2]) - 3*f*x*CoshIntegral[(3*f*x)/2]*(Cosh[(3*e)/2] + I*Sinh[
(3*e)/2]) + 6*Sinh[(e + f*x)/2] + 2*Sinh[(3*(e + f*x))/2] + (3*I)*f*x*Cosh[e/2]*SinhIntegral[(f*x)/2] - 3*f*x*
Sinh[e/2]*SinhIntegral[(f*x)/2] - (3*I)*f*x*Cosh[(3*e)/2]*SinhIntegral[(3*f*x)/2] - 3*f*x*Sinh[(3*e)/2]*SinhIn
tegral[(3*f*x)/2]))/(4*x*(Cosh[(e + f*x)/2] + I*Sinh[(e + f*x)/2])^3)

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( a+ia\sinh \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((I*a*sinh(f*x + e) + a)^(3/2)/x^2, 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(f*x+e))^(3/2)/x^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((a+I*a*sinh(f*x+e))**(3/2)/x**2,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 (f x + e\right ) + a\right )}^{\frac{3}{2}}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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