Optimal. Leaf size=48 \[ \frac{2}{3} \tanh (x) \sqrt{a \cosh ^3(x)}-\frac{2 i \text{EllipticF}\left (\frac{i x}{2},2\right ) \sqrt{a \cosh ^3(x)}}{3 \cosh ^{\frac{3}{2}}(x)} \]
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Rubi [A] time = 0.0240773, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3207, 2635, 2641} \[ \frac{2}{3} \tanh (x) \sqrt{a \cosh ^3(x)}-\frac{2 i F\left (\left .\frac{i x}{2}\right |2\right ) \sqrt{a \cosh ^3(x)}}{3 \cosh ^{\frac{3}{2}}(x)} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 2635
Rule 2641
Rubi steps
\begin{align*} \int \sqrt{a \cosh ^3(x)} \, dx &=\frac{\sqrt{a \cosh ^3(x)} \int \cosh ^{\frac{3}{2}}(x) \, dx}{\cosh ^{\frac{3}{2}}(x)}\\ &=\frac{2}{3} \sqrt{a \cosh ^3(x)} \tanh (x)+\frac{\sqrt{a \cosh ^3(x)} \int \frac{1}{\sqrt{\cosh (x)}} \, dx}{3 \cosh ^{\frac{3}{2}}(x)}\\ &=-\frac{2 i \sqrt{a \cosh ^3(x)} F\left (\left .\frac{i x}{2}\right |2\right )}{3 \cosh ^{\frac{3}{2}}(x)}+\frac{2}{3} \sqrt{a \cosh ^3(x)} \tanh (x)\\ \end{align*}
Mathematica [C] time = 0.0513766, size = 59, normalized size = 1.23 \[ \frac{2}{3} \sqrt{a \cosh ^3(x)} \left (\text{sech}^2(x) \sqrt{\sinh (2 x)+\cosh (2 x)+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\cosh (2 x)-\sinh (2 x)\right )+\tanh (x)\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.05, size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \left ( \cosh \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 \sqrt{a \cosh \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 (\sqrt{a \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 \sqrt{a \cosh \left (x\right )^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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