Optimal. Leaf size=60 \[ -\frac{2 \sinh (x) \cosh (x)}{\sqrt{a \sinh ^3(x)}}+\frac{2 i \sinh ^2(x) E\left (\left .\frac{\pi }{4}-\frac{i x}{2}\right |2\right )}{\sqrt{i \sinh (x)} \sqrt{a \sinh ^3(x)}} \]
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Rubi [A] time = 0.0298034, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {3207, 2636, 2640, 2639} \[ -\frac{2 \sinh (x) \cosh (x)}{\sqrt{a \sinh ^3(x)}}+\frac{2 i \sinh ^2(x) E\left (\left .\frac{\pi }{4}-\frac{i x}{2}\right |2\right )}{\sqrt{i \sinh (x)} \sqrt{a \sinh ^3(x)}} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 2636
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a \sinh ^3(x)}} \, dx &=\frac{\sinh ^{\frac{3}{2}}(x) \int \frac{1}{\sinh ^{\frac{3}{2}}(x)} \, dx}{\sqrt{a \sinh ^3(x)}}\\ &=-\frac{2 \cosh (x) \sinh (x)}{\sqrt{a \sinh ^3(x)}}+\frac{\sinh ^{\frac{3}{2}}(x) \int \sqrt{\sinh (x)} \, dx}{\sqrt{a \sinh ^3(x)}}\\ &=-\frac{2 \cosh (x) \sinh (x)}{\sqrt{a \sinh ^3(x)}}+\frac{\sinh ^2(x) \int \sqrt{i \sinh (x)} \, dx}{\sqrt{i \sinh (x)} \sqrt{a \sinh ^3(x)}}\\ &=-\frac{2 \cosh (x) \sinh (x)}{\sqrt{a \sinh ^3(x)}}+\frac{2 i E\left (\left .\frac{\pi }{4}-\frac{i x}{2}\right |2\right ) \sinh ^2(x)}{\sqrt{i \sinh (x)} \sqrt{a \sinh ^3(x)}}\\ \end{align*}
Mathematica [A] time = 0.0297168, size = 42, normalized size = 0.7 \[ -\frac{2 \sinh (x) \left (\cosh (x)-\sqrt{i \sinh (x)} E\left (\left .\frac{1}{4} (\pi -2 i x)\right |2\right )\right )}{\sqrt{a \sinh ^3(x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.065, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt{a \left ( \sinh \left ( x \right ) \right ) ^{3}}}}\, 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{1}{\sqrt{a \sinh \left (x\right )^{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a \sinh \left (x\right )^{3}}}{a \sinh \left (x\right )^{3}}, x\right ) \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{1}{\sqrt{a \sinh ^{3}{\left (x \right )}}}\, 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{1}{\sqrt{a \sinh \left (x\right )^{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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