Optimal. Leaf size=261 \[ \frac{3}{2} i a \sinh \left (\frac{1}{4} (2 e-i \pi )\right ) \text{Chi}\left (\frac{f x}{2}\right ) \text{sech}\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}+\frac{1}{2} i a \sinh \left (\frac{1}{4} (6 e+i \pi )\right ) \text{Chi}\left (\frac{3 f x}{2}\right ) \text{sech}\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}+\frac{3}{2} i a \cosh \left (\frac{1}{4} (2 e-i \pi )\right ) \text{Shi}\left (\frac{f x}{2}\right ) \text{sech}\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}+\frac{1}{2} i a \cosh \left (\frac{1}{4} (6 e+i \pi )\right ) \text{Shi}\left (\frac{3 f x}{2}\right ) \text{sech}\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)} \]
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Rubi [A] time = 0.273379, antiderivative size = 261, 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, 3312, 3303, 3298, 3301} \[ \frac{3}{2} i a \sinh \left (\frac{1}{4} (2 e-i \pi )\right ) \text{Chi}\left (\frac{f x}{2}\right ) \text{sech}\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}+\frac{1}{2} i a \sinh \left (\frac{1}{4} (6 e+i \pi )\right ) \text{Chi}\left (\frac{3 f x}{2}\right ) \text{sech}\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}+\frac{3}{2} i a \cosh \left (\frac{1}{4} (2 e-i \pi )\right ) \text{Shi}\left (\frac{f x}{2}\right ) \text{sech}\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}+\frac{1}{2} i a \cosh \left (\frac{1}{4} (6 e+i \pi )\right ) \text{Shi}\left (\frac{3 f x}{2}\right ) \text{sech}\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)} \]
Antiderivative was successfully verified.
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Rule 3319
Rule 3312
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{(a+i a \sinh (e+f x))^{3/2}}{x} \, 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} \, dx\right )\\ &=-\left (\left (2 i 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 \left (\frac{3 i \sinh \left (\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}\right )}{4 x}+\frac{i \sinh \left (\frac{1}{4} (6 e+i \pi )+\frac{3 f x}{2}\right )}{4 x}\right ) \, dx\right )\\ &=\frac{1}{2} \left (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 \left (\frac{1}{4} (6 e+i \pi )+\frac{3 f x}{2}\right )}{x} \, dx+\frac{1}{2} \left (3 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 \left (\frac{1}{4} (2 e-i \pi )+\frac{f x}{2}\right )}{x} \, dx\\ &=\frac{1}{2} \left (3 a \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}{2} \left (a \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}{2} \left (3 a \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{1}{2} \left (a \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{3}{2} i a \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}{2} i a \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}{2} i a \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{1}{2} i a \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 = 0.784664, size = 146, normalized size = 0.56 \[ \frac{a \sqrt{a+i a \sinh (e+f x)} \left (3 \text{Chi}\left (\frac{f x}{2}\right ) \left (\cosh \left (\frac{e}{2}\right )+i \sinh \left (\frac{e}{2}\right )\right )-\text{Chi}\left (\frac{3 f x}{2}\right ) \left (\cosh \left (\frac{3 e}{2}\right )-i \sinh \left (\frac{3 e}{2}\right )\right )+\left (\sinh \left (\frac{e}{2}\right )+i \cosh \left (\frac{e}{2}\right )\right ) \left (3 \text{Shi}\left (\frac{f x}{2}\right )+(1+2 i \sinh (e)) \text{Shi}\left (\frac{3 f x}{2}\right )\right )\right )}{2 \left (\cosh \left (\frac{1}{2} (e+f x)\right )+i \sinh \left (\frac{1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.045, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x} \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.
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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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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