Optimal. Leaf size=48 \[ \frac{2}{3} a \sinh (x) \sqrt{a \cosh (x)}-\frac{2 i a^2 \sqrt{\cosh (x)} \text{EllipticF}\left (\frac{i x}{2},2\right )}{3 \sqrt{a \cosh (x)}} \]
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Rubi [A] time = 0.0243887, 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, 2642, 2641} \[ \frac{2}{3} a \sinh (x) \sqrt{a \cosh (x)}-\frac{2 i a^2 \sqrt{\cosh (x)} F\left (\left .\frac{i x}{2}\right |2\right )}{3 \sqrt{a \cosh (x)}} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int (a \cosh (x))^{3/2} \, dx &=\frac{2}{3} a \sqrt{a \cosh (x)} \sinh (x)+\frac{1}{3} a^2 \int \frac{1}{\sqrt{a \cosh (x)}} \, dx\\ &=\frac{2}{3} a \sqrt{a \cosh (x)} \sinh (x)+\frac{\left (a^2 \sqrt{\cosh (x)}\right ) \int \frac{1}{\sqrt{\cosh (x)}} \, dx}{3 \sqrt{a \cosh (x)}}\\ &=-\frac{2 i a^2 \sqrt{\cosh (x)} F\left (\left .\frac{i x}{2}\right |2\right )}{3 \sqrt{a \cosh (x)}}+\frac{2}{3} a \sqrt{a \cosh (x)} \sinh (x)\\ \end{align*}
Mathematica [C] time = 0.0553999, size = 57, normalized size = 1.19 \[ \frac{2}{3} (a \cosh (x))^{3/2} \left (\text{sech}^2(x) \sqrt{\sinh (2 x)+\cosh (2 x)+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\cosh (2 x)-\sinh (2 x)\right )+\tanh (x)\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.041, size = 130, normalized size = 2.7 \begin{align*}{\frac{{a}^{2}}{3}\sqrt{a \left ( 2\, \left ( \cosh \left ( x/2 \right ) \right ) ^{2}-1 \right ) \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{2}} \left ( 8\, \left ( \sinh \left ( x/2 \right ) \right ) ^{4}\cosh \left ( x/2 \right ) +\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 ){\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{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 (\sqrt{a \cosh \left (x\right )} a \cosh \left (x\right ), 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{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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