Optimal. Leaf size=75 \[ -\frac{10 i \cosh ^{\frac{3}{2}}(x) \text{EllipticF}\left (\frac{i x}{2},2\right )}{21 a \sqrt{a \cosh ^3(x)}}+\frac{10 \sinh (x)}{21 a \sqrt{a \cosh ^3(x)}}+\frac{2 \tanh (x) \text{sech}(x)}{7 a \sqrt{a \cosh ^3(x)}} \]
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Rubi [A] time = 0.0342725, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3207, 2636, 2641} \[ \frac{10 \sinh (x)}{21 a \sqrt{a \cosh ^3(x)}}-\frac{10 i \cosh ^{\frac{3}{2}}(x) F\left (\left .\frac{i x}{2}\right |2\right )}{21 a \sqrt{a \cosh ^3(x)}}+\frac{2 \tanh (x) \text{sech}(x)}{7 a \sqrt{a \cosh ^3(x)}} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 2636
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{\left (a \cosh ^3(x)\right )^{3/2}} \, dx &=\frac{\cosh ^{\frac{3}{2}}(x) \int \frac{1}{\cosh ^{\frac{9}{2}}(x)} \, dx}{a \sqrt{a \cosh ^3(x)}}\\ &=\frac{2 \text{sech}(x) \tanh (x)}{7 a \sqrt{a \cosh ^3(x)}}+\frac{\left (5 \cosh ^{\frac{3}{2}}(x)\right ) \int \frac{1}{\cosh ^{\frac{5}{2}}(x)} \, dx}{7 a \sqrt{a \cosh ^3(x)}}\\ &=\frac{10 \sinh (x)}{21 a \sqrt{a \cosh ^3(x)}}+\frac{2 \text{sech}(x) \tanh (x)}{7 a \sqrt{a \cosh ^3(x)}}+\frac{\left (5 \cosh ^{\frac{3}{2}}(x)\right ) \int \frac{1}{\sqrt{\cosh (x)}} \, dx}{21 a \sqrt{a \cosh ^3(x)}}\\ &=-\frac{10 i \cosh ^{\frac{3}{2}}(x) F\left (\left .\frac{i x}{2}\right |2\right )}{21 a \sqrt{a \cosh ^3(x)}}+\frac{10 \sinh (x)}{21 a \sqrt{a \cosh ^3(x)}}+\frac{2 \text{sech}(x) \tanh (x)}{7 a \sqrt{a \cosh ^3(x)}}\\ \end{align*}
Mathematica [A] time = 0.0588545, size = 48, normalized size = 0.64 \[ \frac{2 \cosh ^2(x) \left (-5 i \cosh ^{\frac{5}{2}}(x) \text{EllipticF}\left (\frac{i x}{2},2\right )+3 \tanh (x)+5 \sinh (x) \cosh (x)\right )}{21 \left (a \cosh ^3(x)\right )^{3/2}} \]
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{3}{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{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 (\frac{\sqrt{a \cosh \left (x\right )^{3}}}{a^{2} \cosh \left (x\right )^{6}}, 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{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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