Optimal. Leaf size=46 \[ \frac{2 \sinh (x)}{a \sqrt{a \cosh (x)}}+\frac{2 i E\left (\left .\frac{i x}{2}\right |2\right ) \sqrt{a \cosh (x)}}{a^2 \sqrt{\cosh (x)}} \]
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Rubi [A] time = 0.0253687, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {2636, 2640, 2639} \[ \frac{2 \sinh (x)}{a \sqrt{a \cosh (x)}}+\frac{2 i E\left (\left .\frac{i x}{2}\right |2\right ) \sqrt{a \cosh (x)}}{a^2 \sqrt{\cosh (x)}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{(a \cosh (x))^{3/2}} \, dx &=\frac{2 \sinh (x)}{a \sqrt{a \cosh (x)}}-\frac{\int \sqrt{a \cosh (x)} \, dx}{a^2}\\ &=\frac{2 \sinh (x)}{a \sqrt{a \cosh (x)}}-\frac{\sqrt{a \cosh (x)} \int \sqrt{\cosh (x)} \, dx}{a^2 \sqrt{\cosh (x)}}\\ &=\frac{2 i \sqrt{a \cosh (x)} E\left (\left .\frac{i x}{2}\right |2\right )}{a^2 \sqrt{\cosh (x)}}+\frac{2 \sinh (x)}{a \sqrt{a \cosh (x)}}\\ \end{align*}
Mathematica [A] time = 0.0216983, size = 34, normalized size = 0.74 \[ \frac{2 \cosh (x) \left (\sinh (x)+i \sqrt{\cosh (x)} E\left (\left .\frac{i x}{2}\right |2\right )\right )}{(a \cosh (x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.052, size = 159, normalized size = 3.5 \begin{align*} -{\frac{1}{a}\sqrt{2\, \left ( \sinh \left ( x/2 \right ) \right ) ^{4}a+ \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{2}a} \left ( \sqrt{2}\sqrt{-2\, \left ( \sinh \left ( x/2 \right ) \right ) ^{2}-1}\sqrt{- \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{2}}{\it EllipticF} \left ( \cosh \left ({\frac{x}{2}} \right ) \sqrt{2},{\frac{\sqrt{2}}{2}} \right ) -2\,\sqrt{2}\sqrt{-2\, \left ( \sinh \left ( x/2 \right ) \right ) ^{2}-1}\sqrt{- \left ( \sinh \left ( x/2 \right ) \right ) ^{2}}{\it EllipticE} \left ( \cosh \left ( x/2 \right ) \sqrt{2},1/2\,\sqrt{2} \right ) -4\, \left ( \sinh \left ( x/2 \right ) \right ) ^{2}\cosh \left ( x/2 \right ) \right ){\frac{1}{\sqrt{a \left ( 2\, \left ( \sinh \left ( x/2 \right ) \right ) ^{4}+ \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{2} \right ) }}} \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{a \left ( 2\, \left ( \cosh \left ( x/2 \right ) \right ) ^{2}-1 \right ) }}}} \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}{\left (a \cosh \left (x\right )\right )^{\frac{3}{2}}}\,{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 \cosh \left (x\right )}}{a^{2} \cosh \left (x\right )^{2}}, x\right ) \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{1}{\left (a \cosh \left (x\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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