Optimal. Leaf size=121 \[ \frac{154 \sinh (x) \cosh (x)}{195 a^2 \sqrt{a \cosh ^3(x)}}+\frac{154 \tanh (x)}{585 a^2 \sqrt{a \cosh ^3(x)}}+\frac{154 i \cosh ^{\frac{3}{2}}(x) E\left (\left .\frac{i x}{2}\right |2\right )}{195 a^2 \sqrt{a \cosh ^3(x)}}+\frac{2 \tanh (x) \text{sech}^4(x)}{13 a^2 \sqrt{a \cosh ^3(x)}}+\frac{22 \tanh (x) \text{sech}^2(x)}{117 a^2 \sqrt{a \cosh ^3(x)}} \]
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Rubi [A] time = 0.051845, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3207, 2636, 2639} \[ \frac{154 \sinh (x) \cosh (x)}{195 a^2 \sqrt{a \cosh ^3(x)}}+\frac{154 \tanh (x)}{585 a^2 \sqrt{a \cosh ^3(x)}}+\frac{154 i \cosh ^{\frac{3}{2}}(x) E\left (\left .\frac{i x}{2}\right |2\right )}{195 a^2 \sqrt{a \cosh ^3(x)}}+\frac{2 \tanh (x) \text{sech}^4(x)}{13 a^2 \sqrt{a \cosh ^3(x)}}+\frac{22 \tanh (x) \text{sech}^2(x)}{117 a^2 \sqrt{a \cosh ^3(x)}} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 2636
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{\left (a \cosh ^3(x)\right )^{5/2}} \, dx &=\frac{\cosh ^{\frac{3}{2}}(x) \int \frac{1}{\cosh ^{\frac{15}{2}}(x)} \, dx}{a^2 \sqrt{a \cosh ^3(x)}}\\ &=\frac{2 \text{sech}^4(x) \tanh (x)}{13 a^2 \sqrt{a \cosh ^3(x)}}+\frac{\left (11 \cosh ^{\frac{3}{2}}(x)\right ) \int \frac{1}{\cosh ^{\frac{11}{2}}(x)} \, dx}{13 a^2 \sqrt{a \cosh ^3(x)}}\\ &=\frac{22 \text{sech}^2(x) \tanh (x)}{117 a^2 \sqrt{a \cosh ^3(x)}}+\frac{2 \text{sech}^4(x) \tanh (x)}{13 a^2 \sqrt{a \cosh ^3(x)}}+\frac{\left (77 \cosh ^{\frac{3}{2}}(x)\right ) \int \frac{1}{\cosh ^{\frac{7}{2}}(x)} \, dx}{117 a^2 \sqrt{a \cosh ^3(x)}}\\ &=\frac{154 \tanh (x)}{585 a^2 \sqrt{a \cosh ^3(x)}}+\frac{22 \text{sech}^2(x) \tanh (x)}{117 a^2 \sqrt{a \cosh ^3(x)}}+\frac{2 \text{sech}^4(x) \tanh (x)}{13 a^2 \sqrt{a \cosh ^3(x)}}+\frac{\left (77 \cosh ^{\frac{3}{2}}(x)\right ) \int \frac{1}{\cosh ^{\frac{3}{2}}(x)} \, dx}{195 a^2 \sqrt{a \cosh ^3(x)}}\\ &=\frac{154 \cosh (x) \sinh (x)}{195 a^2 \sqrt{a \cosh ^3(x)}}+\frac{154 \tanh (x)}{585 a^2 \sqrt{a \cosh ^3(x)}}+\frac{22 \text{sech}^2(x) \tanh (x)}{117 a^2 \sqrt{a \cosh ^3(x)}}+\frac{2 \text{sech}^4(x) \tanh (x)}{13 a^2 \sqrt{a \cosh ^3(x)}}-\frac{\left (77 \cosh ^{\frac{3}{2}}(x)\right ) \int \sqrt{\cosh (x)} \, dx}{195 a^2 \sqrt{a \cosh ^3(x)}}\\ &=\frac{154 i \cosh ^{\frac{3}{2}}(x) E\left (\left .\frac{i x}{2}\right |2\right )}{195 a^2 \sqrt{a \cosh ^3(x)}}+\frac{154 \cosh (x) \sinh (x)}{195 a^2 \sqrt{a \cosh ^3(x)}}+\frac{154 \tanh (x)}{585 a^2 \sqrt{a \cosh ^3(x)}}+\frac{22 \text{sech}^2(x) \tanh (x)}{117 a^2 \sqrt{a \cosh ^3(x)}}+\frac{2 \text{sech}^4(x) \tanh (x)}{13 a^2 \sqrt{a \cosh ^3(x)}}\\ \end{align*}
Mathematica [A] time = 0.100225, size = 61, normalized size = 0.5 \[ \frac{462 i \cosh ^{\frac{3}{2}}(x) E\left (\left .\frac{i x}{2}\right |2\right )+462 \sinh (x) \cosh (x)+2 \tanh (x) \left (45 \text{sech}^4(x)+55 \text{sech}^2(x)+77\right )}{585 a^2 \sqrt{a \cosh ^3(x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.052, size = 0, normalized size = 0. \begin{align*} \int \left ( a \left ( \cosh \left ( x \right ) \right ) ^{3} \right ) ^{-{\frac{5}{2}}}\, 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}{\left (a \cosh \left (x\right )^{3}\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 )^{3}}}{a^{3} \cosh \left (x\right )^{9}}, 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 )^{3}\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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