3.124 \(\int \frac{\sqrt{a+i a \sinh (e+f x)}}{x^3} \, dx\)

Optimal. Leaf size=204 \[ \frac{1}{8} i f^2 \sinh \left (\frac{1}{4} (2 e-i \pi )\right ) \text{Chi}\left (\frac{f x}{2}\right ) \text{sech}\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}+\frac{1}{8} i f^2 \cosh \left (\frac{1}{4} (2 e-i \pi )\right ) \text{Shi}\left (\frac{f x}{2}\right ) \text{sech}\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}-\frac{\sqrt{a+i a \sinh (e+f x)}}{2 x^2}-\frac{f \tanh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}}{4 x} \]

[Out]

-Sqrt[a + I*a*Sinh[e + f*x]]/(2*x^2) + (I/8)*f^2*CoshIntegral[(f*x)/2]*Sech[e/2 + (I/4)*Pi + (f*x)/2]*Sinh[(2*
e - I*Pi)/4]*Sqrt[a + I*a*Sinh[e + f*x]] + (I/8)*f^2*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] - (f*Sqrt[a + I*a*Sinh[e + f*x]]*Tanh[e/2 + (I/4)*Pi + (f*x)/2])/
(4*x)

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Rubi [A]  time = 0.198161, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3319, 3297, 3303, 3298, 3301} \[ \frac{1}{8} i f^2 \sinh \left (\frac{1}{4} (2 e-i \pi )\right ) \text{Chi}\left (\frac{f x}{2}\right ) \text{sech}\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}+\frac{1}{8} i f^2 \cosh \left (\frac{1}{4} (2 e-i \pi )\right ) \text{Shi}\left (\frac{f x}{2}\right ) \text{sech}\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}-\frac{\sqrt{a+i a \sinh (e+f x)}}{2 x^2}-\frac{f \tanh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}}{4 x} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + I*a*Sinh[e + f*x]]/x^3,x]

[Out]

-Sqrt[a + I*a*Sinh[e + f*x]]/(2*x^2) + (I/8)*f^2*CoshIntegral[(f*x)/2]*Sech[e/2 + (I/4)*Pi + (f*x)/2]*Sinh[(2*
e - I*Pi)/4]*Sqrt[a + I*a*Sinh[e + f*x]] + (I/8)*f^2*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] - (f*Sqrt[a + I*a*Sinh[e + f*x]]*Tanh[e/2 + (I/4)*Pi + (f*x)/2])/
(4*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 3297

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x])/(d*(
m + 1)), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && 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{\sqrt{a+i a \sinh (e+f x)}}{x^3} \, dx &=\left (\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{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right )}{x^3} \, dx\\ &=-\frac{\sqrt{a+i a \sinh (e+f x)}}{2 x^2}+\frac{1}{4} \left (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{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right )}{x^2} \, dx\\ &=-\frac{\sqrt{a+i a \sinh (e+f x)}}{2 x^2}-\frac{f \sqrt{a+i a \sinh (e+f x)} \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{4 x}-\frac{1}{8} \left (i f^2 \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{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{x} \, dx\\ &=-\frac{\sqrt{a+i a \sinh (e+f x)}}{2 x^2}-\frac{f \sqrt{a+i a \sinh (e+f x)} \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{4 x}+\frac{1}{8} \left (f^2 \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}{8} \left (f^2 \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{\sqrt{a+i a \sinh (e+f x)}}{2 x^2}+\frac{1}{8} i f^2 \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{1}{8} i f^2 \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{f \sqrt{a+i a \sinh (e+f x)} \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{4 x}\\ \end{align*}

Mathematica [A]  time = 0.339556, size = 170, normalized size = 0.83 \[ \frac{\sqrt{a+i a \sinh (e+f x)} \left (f^2 x^2 \text{Chi}\left (\frac{f x}{2}\right ) \left (\cosh \left (\frac{e}{2}\right )+i \sinh \left (\frac{e}{2}\right )\right )+f^2 x^2 \left (\sinh \left (\frac{e}{2}\right )+i \cosh \left (\frac{e}{2}\right )\right ) \text{Shi}\left (\frac{f x}{2}\right )-2 f x \sinh \left (\frac{1}{2} (e+f x)\right )-4 i \sinh \left (\frac{1}{2} (e+f x)\right )-2 i f x \cosh \left (\frac{1}{2} (e+f x)\right )-4 \cosh \left (\frac{1}{2} (e+f x)\right )\right )}{8 x^2 \left (\cosh \left (\frac{1}{2} (e+f x)\right )+i \sinh \left (\frac{1}{2} (e+f x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + I*a*Sinh[e + f*x]]/x^3,x]

[Out]

(Sqrt[a + I*a*Sinh[e + f*x]]*(-4*Cosh[(e + f*x)/2] - (2*I)*f*x*Cosh[(e + f*x)/2] + f^2*x^2*CoshIntegral[(f*x)/
2]*(Cosh[e/2] + I*Sinh[e/2]) - (4*I)*Sinh[(e + f*x)/2] - 2*f*x*Sinh[(e + f*x)/2] + f^2*x^2*(I*Cosh[e/2] + Sinh
[e/2])*SinhIntegral[(f*x)/2]))/(8*x^2*(Cosh[(e + f*x)/2] + I*Sinh[(e + f*x)/2]))

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}}\sqrt{a+ia\sinh \left ( fx+e \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+I*a*sinh(f*x+e))^(1/2)/x^3,x)

[Out]

int((a+I*a*sinh(f*x+e))^(1/2)/x^3,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*sinh(f*x+e))^(1/2)/x^3,x, algorithm="maxima")

[Out]

integrate(sqrt(I*a*sinh(f*x + e) + a)/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(f*x+e))^(1/2)/x^3,x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a \left (i \sinh{\left (e + f x \right )} + 1\right )}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*sinh(f*x+e))**(1/2)/x**3,x)

[Out]

Integral(sqrt(a*(I*sinh(e + f*x) + 1))/x**3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*sinh(f*x+e))^(1/2)/x^3,x, algorithm="giac")

[Out]

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