Optimal. Leaf size=67 \[ \frac{6 \sinh (x)}{5 a^3 \sqrt{a \cosh (x)}}+\frac{6 i E\left (\left .\frac{i x}{2}\right |2\right ) \sqrt{a \cosh (x)}}{5 a^4 \sqrt{\cosh (x)}}+\frac{2 \sinh (x)}{5 a (a \cosh (x))^{5/2}} \]
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Rubi [A] time = 0.0387202, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {2636, 2640, 2639} \[ \frac{6 \sinh (x)}{5 a^3 \sqrt{a \cosh (x)}}+\frac{6 i E\left (\left .\frac{i x}{2}\right |2\right ) \sqrt{a \cosh (x)}}{5 a^4 \sqrt{\cosh (x)}}+\frac{2 \sinh (x)}{5 a (a \cosh (x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{(a \cosh (x))^{7/2}} \, dx &=\frac{2 \sinh (x)}{5 a (a \cosh (x))^{5/2}}+\frac{3 \int \frac{1}{(a \cosh (x))^{3/2}} \, dx}{5 a^2}\\ &=\frac{2 \sinh (x)}{5 a (a \cosh (x))^{5/2}}+\frac{6 \sinh (x)}{5 a^3 \sqrt{a \cosh (x)}}-\frac{3 \int \sqrt{a \cosh (x)} \, dx}{5 a^4}\\ &=\frac{2 \sinh (x)}{5 a (a \cosh (x))^{5/2}}+\frac{6 \sinh (x)}{5 a^3 \sqrt{a \cosh (x)}}-\frac{\left (3 \sqrt{a \cosh (x)}\right ) \int \sqrt{\cosh (x)} \, dx}{5 a^4 \sqrt{\cosh (x)}}\\ &=\frac{6 i \sqrt{a \cosh (x)} E\left (\left .\frac{i x}{2}\right |2\right )}{5 a^4 \sqrt{\cosh (x)}}+\frac{2 \sinh (x)}{5 a (a \cosh (x))^{5/2}}+\frac{6 \sinh (x)}{5 a^3 \sqrt{a \cosh (x)}}\\ \end{align*}
Mathematica [A] time = 0.0446776, size = 43, normalized size = 0.64 \[ \frac{2 \left (\tanh (x)+3 i \cosh ^{\frac{3}{2}}(x) E\left (\left .\frac{i x}{2}\right |2\right )+3 \sinh (x) \cosh (x)\right )}{5 a^2 (a \cosh (x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.082, size = 254, normalized size = 3.8 \begin{align*} 2\,{\frac{\sqrt{a \left ( 2\, \left ( \cosh \left ( x/2 \right ) \right ) ^{2}-1 \right ) \left ( \sinh \left ( x/2 \right ) \right ) ^{2}}}{{a}^{3}\sinh \left ( x/2 \right ) \sqrt{a \left ( 2\, \left ( \cosh \left ( x/2 \right ) \right ) ^{2}-1 \right ) }} \left ( 1/20\,{\frac{\cosh \left ( x/2 \right ) \sqrt{a \left ( 2\, \left ( \sinh \left ( x/2 \right ) \right ) ^{4}+ \left ( \sinh \left ( x/2 \right ) \right ) ^{2} \right ) }}{a \left ( \left ( \cosh \left ( x/2 \right ) \right ) ^{2}-1/2 \right ) ^{3}}}+6/5\,{\frac{ \left ( \sinh \left ( x/2 \right ) \right ) ^{2}\cosh \left ( x/2 \right ) }{\sqrt{a \left ( 2\, \left ( \cosh \left ( x/2 \right ) \right ) ^{2}-1 \right ) \left ( \sinh \left ( x/2 \right ) \right ) ^{2}}}}+3/10\,{\frac{\sqrt{2}\sqrt{-2\, \left ( \cosh \left ( x/2 \right ) \right ) ^{2}+1}\sqrt{- \left ( \sinh \left ( x/2 \right ) \right ) ^{2}}{\it EllipticF} \left ( \cosh \left ( x/2 \right ) \sqrt{2},1/2\,\sqrt{2} \right ) }{\sqrt{a \left ( 2\, \left ( \sinh \left ( x/2 \right ) \right ) ^{4}+ \left ( \sinh \left ( x/2 \right ) \right ) ^{2} \right ) }}}-3/5\,{\frac{\sqrt{2}\sqrt{-2\, \left ( \cosh \left ( x/2 \right ) \right ) ^{2}+1}\sqrt{- \left ( \sinh \left ( x/2 \right ) \right ) ^{2}} \left ({\it EllipticF} \left ( \cosh \left ( x/2 \right ) \sqrt{2},1/2\,\sqrt{2} \right ) -{\it EllipticE} \left ( \cosh \left ( x/2 \right ) \sqrt{2},1/2\,\sqrt{2} \right ) \right ) }{\sqrt{a \left ( 2\, \left ( \sinh \left ( x/2 \right ) \right ) ^{4}+ \left ( \sinh \left ( x/2 \right ) \right ) ^{2} \right ) }}} \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 \frac{1}{\left (a \cosh \left (x\right )\right )^{\frac{7}{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 )}}{a^{4} \cosh \left (x\right )^{4}}, 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 )\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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