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 = 0.02, 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 = {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 2636
Rule 2639
Rule 3207
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 \[ \frac {2 \cosh (x) \left (\sinh (x)+i \sqrt {\cosh (x)} E\left (\left .\frac {i x}{2}\right |2\right )\right )}{\sqrt {a \cosh ^3(x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.00, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \cosh \relax (x)^{3}}}{a \cosh \relax (x)^{3}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {a \cosh \relax (x)^{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.18, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \left (\cosh ^{3}\relax (x )\right )}}\, dx \]
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 \[ \int \frac {1}{\sqrt {a \cosh \relax (x)^{3}}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {a\,{\mathrm {cosh}\relax (x)}^3}} \,d x \]
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 \[ \int \frac {1}{\sqrt {a \cosh ^{3}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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