Optimal. Leaf size=46 \[ \frac{2 \sinh (x) \cosh (x)}{\sqrt{a \cosh ^3(x)}}+\frac{2 i \cosh ^{\frac{3}{2}}(x) E\left (\left .\frac{i x}{2}\right |2\right )}{\sqrt{a \cosh ^3(x)}} \]
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Rubi [A] time = 0.0238207, antiderivative size = 46, 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, 2636, 2639} \[ \frac{2 \sinh (x) \cosh (x)}{\sqrt{a \cosh ^3(x)}}+\frac{2 i \cosh ^{\frac{3}{2}}(x) E\left (\left .\frac{i x}{2}\right |2\right )}{\sqrt{a \cosh ^3(x)}} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 2636
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a \cosh ^3(x)}} \, dx &=\frac{\cosh ^{\frac{3}{2}}(x) \int \frac{1}{\cosh ^{\frac{3}{2}}(x)} \, dx}{\sqrt{a \cosh ^3(x)}}\\ &=\frac{2 \cosh (x) \sinh (x)}{\sqrt{a \cosh ^3(x)}}-\frac{\cosh ^{\frac{3}{2}}(x) \int \sqrt{\cosh (x)} \, dx}{\sqrt{a \cosh ^3(x)}}\\ &=\frac{2 i \cosh ^{\frac{3}{2}}(x) E\left (\left .\frac{i x}{2}\right |2\right )}{\sqrt{a \cosh ^3(x)}}+\frac{2 \cosh (x) \sinh (x)}{\sqrt{a \cosh ^3(x)}}\\ \end{align*}
Mathematica [A] time = 0.021119, size = 36, normalized size = 0.78 \[ \frac{2 \cosh (x) \left (\sinh (x)+i \sqrt{\cosh (x)} E\left (\left .\frac{i x}{2}\right |2\right )\right )}{\sqrt{a \cosh ^3(x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\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 \frac{1}{\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 (\frac{\sqrt{a \cosh \left (x\right )^{3}}}{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 \frac{1}{\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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