Optimal. Leaf size=50 \[ \frac{2 \sinh (x)}{3 a (a \cosh (x))^{3/2}}-\frac{2 i \sqrt{\cosh (x)} \text{EllipticF}\left (\frac{i x}{2},2\right )}{3 a^2 \sqrt{a \cosh (x)}} \]
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Rubi [A] time = 0.0267797, antiderivative size = 50, 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, 2642, 2641} \[ \frac{2 \sinh (x)}{3 a (a \cosh (x))^{3/2}}-\frac{2 i \sqrt{\cosh (x)} F\left (\left .\frac{i x}{2}\right |2\right )}{3 a^2 \sqrt{a \cosh (x)}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{(a \cosh (x))^{5/2}} \, dx &=\frac{2 \sinh (x)}{3 a (a \cosh (x))^{3/2}}+\frac{\int \frac{1}{\sqrt{a \cosh (x)}} \, dx}{3 a^2}\\ &=\frac{2 \sinh (x)}{3 a (a \cosh (x))^{3/2}}+\frac{\sqrt{\cosh (x)} \int \frac{1}{\sqrt{\cosh (x)}} \, dx}{3 a^2 \sqrt{a \cosh (x)}}\\ &=-\frac{2 i \sqrt{\cosh (x)} F\left (\left .\frac{i x}{2}\right |2\right )}{3 a^2 \sqrt{a \cosh (x)}}+\frac{2 \sinh (x)}{3 a (a \cosh (x))^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0347288, size = 56, normalized size = 1.12 \[ \frac{2 \left (\sqrt{\sinh (2 x)+\cosh (2 x)+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\cosh (2 x)-\sinh (2 x)\right )+\tanh (x)\right )}{3 a^2 \sqrt{a \cosh (x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.052, size = 177, normalized size = 3.5 \begin{align*}{\frac{1}{3\,{a}^{2}} \left ( 2\,\sqrt{2}\sqrt{-2\, \left ( \sinh \left ( x/2 \right ) \right ) ^{2}-1}\sqrt{- \left ( \sinh \left ( x/2 \right ) \right ) ^{2}}{\it EllipticF} \left ( \cosh \left ( x/2 \right ) \sqrt{2},1/2\,\sqrt{2} \right ) \left ( \sinh \left ( x/2 \right ) \right ) ^{2}+\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 ) +4\, \left ( \sinh \left ( x/2 \right ) \right ) ^{2}\cosh \left ( x/2 \right ) \right ) \sqrt{a \left ( 2\, \left ( \cosh \left ( x/2 \right ) \right ) ^{2}-1 \right ) \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{2}}{\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 ( 2\, \left ( \cosh \left ( x/2 \right ) \right ) ^{2}-1 \right ) ^{-1} \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{5}{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^{3} \cosh \left (x\right )^{3}}, 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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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