Optimal. Leaf size=48 \[ \frac{2}{5} a \sinh (x) (a \cosh (x))^{3/2}-\frac{6 i a^2 E\left (\left .\frac{i x}{2}\right |2\right ) \sqrt{a \cosh (x)}}{5 \sqrt{\cosh (x)}} \]
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Rubi [A] time = 0.0252784, antiderivative size = 48, 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 = {2635, 2640, 2639} \[ \frac{2}{5} a \sinh (x) (a \cosh (x))^{3/2}-\frac{6 i a^2 E\left (\left .\frac{i x}{2}\right |2\right ) \sqrt{a \cosh (x)}}{5 \sqrt{\cosh (x)}} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int (a \cosh (x))^{5/2} \, dx &=\frac{2}{5} a (a \cosh (x))^{3/2} \sinh (x)+\frac{1}{5} \left (3 a^2\right ) \int \sqrt{a \cosh (x)} \, dx\\ &=\frac{2}{5} a (a \cosh (x))^{3/2} \sinh (x)+\frac{\left (3 a^2 \sqrt{a \cosh (x)}\right ) \int \sqrt{\cosh (x)} \, dx}{5 \sqrt{\cosh (x)}}\\ &=-\frac{6 i a^2 \sqrt{a \cosh (x)} E\left (\left .\frac{i x}{2}\right |2\right )}{5 \sqrt{\cosh (x)}}+\frac{2}{5} a (a \cosh (x))^{3/2} \sinh (x)\\ \end{align*}
Mathematica [A] time = 0.0413884, size = 41, normalized size = 0.85 \[ \frac{2 (a \cosh (x))^{5/2} \left (\sinh (x) \cosh ^{\frac{3}{2}}(x)-3 i E\left (\left .\frac{i x}{2}\right |2\right )\right )}{5 \cosh ^{\frac{5}{2}}(x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.043, size = 184, normalized size = 3.8 \begin{align*}{\frac{{a}^{3}}{5}\sqrt{a \left ( 2\, \left ( \cosh \left ( x/2 \right ) \right ) ^{2}-1 \right ) \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{2}} \left ( 16\,\cosh \left ( x/2 \right ) \left ( \sinh \left ( x/2 \right ) \right ) ^{6}+16\, \left ( \sinh \left ( x/2 \right ) \right ) ^{4}\cosh \left ( x/2 \right ) +3\,\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 ) -6\,\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 \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 (\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 \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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