Optimal. Leaf size=46 \[ \frac {2 \sinh (x)}{a \sqrt {a \cosh (x)}}+\frac {2 i E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \cosh (x)}}{a^2 \sqrt {\cosh (x)}} \]
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Rubi [A] time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2636, 2640, 2639} \[ \frac {2 \sinh (x)}{a \sqrt {a \cosh (x)}}+\frac {2 i E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \cosh (x)}}{a^2 \sqrt {\cosh (x)}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rule 2640
Rubi steps
\begin {align*} \int \frac {1}{(a \cosh (x))^{3/2}} \, dx &=\frac {2 \sinh (x)}{a \sqrt {a \cosh (x)}}-\frac {\int \sqrt {a \cosh (x)} \, dx}{a^2}\\ &=\frac {2 \sinh (x)}{a \sqrt {a \cosh (x)}}-\frac {\sqrt {a \cosh (x)} \int \sqrt {\cosh (x)} \, dx}{a^2 \sqrt {\cosh (x)}}\\ &=\frac {2 i \sqrt {a \cosh (x)} E\left (\left .\frac {i x}{2}\right |2\right )}{a^2 \sqrt {\cosh (x)}}+\frac {2 \sinh (x)}{a \sqrt {a \cosh (x)}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 34, normalized size = 0.74 \[ \frac {2 \cosh (x) \left (\sinh (x)+i \sqrt {\cosh (x)} E\left (\left .\frac {i x}{2}\right |2\right )\right )}{(a \cosh (x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \cosh \relax (x)}}{a^{2} \cosh \relax (x)^{2}}, x\right ) \]
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 \[ \int \frac {1}{\left (a \cosh \relax (x)\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.37, size = 159, normalized size = 3.46 \[ -\frac {\sqrt {2 a \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+a \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \left (\sqrt {2}\, \sqrt {-2 \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )-1}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \EllipticF \left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )-2 \sqrt {2}\, \sqrt {-2 \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )-1}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \EllipticE \left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )-4 \left (\sinh ^{2}\left (\frac {x}{2}\right )\right ) \cosh \left (\frac {x}{2}\right )\right )}{a \sqrt {a \left (2 \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \sinh \left (\frac {x}{2}\right ) \sqrt {a \left (2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )-1\right )}} \]
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 \[ \int \frac {1}{\left (a \cosh \relax (x)\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (a\,\mathrm {cosh}\relax (x)\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (a \cosh {\relax (x )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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