Optimal. Leaf size=46 \[ \frac {2 i \cosh ^{\frac {3}{2}}(x) E\left (\left .\frac {i x}{2}\right |2\right )}{\sqrt {a \cosh ^3(x)}}+\frac {2 \cosh (x) \sinh (x)}{\sqrt {a \cosh ^3(x)}} \]
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Rubi [A]
time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3286, 2716,
2719} \begin {gather*} \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)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2716
Rule 2719
Rule 3286
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*}
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Mathematica [A]
time = 0.02, size = 36, normalized size = 0.78 \begin {gather*} \frac {2 \cosh (x) \left (i \sqrt {\cosh (x)} E\left (\left .\frac {i x}{2}\right |2\right )+\sinh (x)\right )}{\sqrt {a \cosh ^3(x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {a \left (\cosh ^{3}\left (x \right )\right )}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.11, size = 93, normalized size = 2.02 \begin {gather*} \frac {2 \, {\left ({\left (\sqrt {2} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} \sinh \left (x\right )^{2} + \sqrt {2}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) + 2 \, \sqrt {a \cosh \left (x\right )} {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )}\right )}}{a \cosh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) \sinh \left (x\right ) + a \sinh \left (x\right )^{2} + a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a \cosh ^{3}{\left (x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\sqrt {a\,{\mathrm {cosh}\left (x\right )}^3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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