Optimal. Leaf size=149 \[ \frac{1}{2} f \sinh \left (\frac{1}{4} (2 e+i \pi )\right ) \text{Chi}\left (\frac{f x}{2}\right ) \text{sech}\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}+\frac{1}{2} f \cosh \left (\frac{1}{4} (2 e+i \pi )\right ) \text{Shi}\left (\frac{f x}{2}\right ) \text{sech}\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}-\frac{\sqrt{a+i a \sinh (e+f x)}}{x} \]
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Rubi [A] time = 0.175202, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 5, 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} \[ \frac{1}{2} f \sinh \left (\frac{1}{4} (2 e+i \pi )\right ) \text{Chi}\left (\frac{f x}{2}\right ) \text{sech}\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}+\frac{1}{2} f \cosh \left (\frac{1}{4} (2 e+i \pi )\right ) \text{Shi}\left (\frac{f x}{2}\right ) \text{sech}\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}-\frac{\sqrt{a+i a \sinh (e+f x)}}{x} \]
Antiderivative was successfully verified.
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Rule 3319
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{\sqrt{a+i a \sinh (e+f x)}}{x^2} \, 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^2} \, dx\\ &=-\frac{\sqrt{a+i a \sinh (e+f x)}}{x}+\frac{1}{2} \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} \, dx\\ &=-\frac{\sqrt{a+i a \sinh (e+f x)}}{x}-\frac{1}{2} \left (i 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}{2} \left (i 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{\sqrt{a+i a \sinh (e+f x)}}{x}+\frac{1}{2} 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{1}{2} 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 )\\ \end{align*}
Mathematica [A] time = 0.25353, size = 133, normalized size = 0.89 \[ \frac{\sqrt{a+i a \sinh (e+f x)} \left (f x \text{Chi}\left (\frac{f x}{2}\right ) \left (\sinh \left (\frac{e}{2}\right )+i \cosh \left (\frac{e}{2}\right )\right )+f x \left (\cosh \left (\frac{e}{2}\right )+i \sinh \left (\frac{e}{2}\right )\right ) \text{Shi}\left (\frac{f x}{2}\right )-2 \left (\cosh \left (\frac{1}{2} (e+f x)\right )+i \sinh \left (\frac{1}{2} (e+f x)\right )\right )\right )}{2 x \left (\cosh \left (\frac{1}{2} (e+f x)\right )+i \sinh \left (\frac{1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.062, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}\sqrt{a+ia\sinh \left ( fx+e \right ) }}\, 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{\sqrt{i \, a \sinh \left (f x + e\right ) + a}}{x^{2}}\,{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a \left (i \sinh{\left (e + f x \right )} + 1\right )}}{x^{2}}\, dx \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{\sqrt{i \, a \sinh \left (f x + e\right ) + a}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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