Optimal. Leaf size=50 \[ \frac {2 \sinh (x)}{3 a (a \cosh (x))^{3/2}}-\frac {2 i \sqrt {\cosh (x)} F\left (\left .\frac {i x}{2}\right |2\right )}{3 a^2 \sqrt {a \cosh (x)}} \]
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Rubi [A] time = 0.03, antiderivative size = 50, 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, 2642, 2641} \[ \frac {2 \sinh (x)}{3 a (a \cosh (x))^{3/2}}-\frac {2 i \sqrt {\cosh (x)} F\left (\left .\frac {i x}{2}\right |2\right )}{3 a^2 \sqrt {a \cosh (x)}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2641
Rule 2642
Rubi steps
\begin {align*} \int \frac {1}{(a \cosh (x))^{5/2}} \, dx &=\frac {2 \sinh (x)}{3 a (a \cosh (x))^{3/2}}+\frac {\int \frac {1}{\sqrt {a \cosh (x)}} \, dx}{3 a^2}\\ &=\frac {2 \sinh (x)}{3 a (a \cosh (x))^{3/2}}+\frac {\sqrt {\cosh (x)} \int \frac {1}{\sqrt {\cosh (x)}} \, dx}{3 a^2 \sqrt {a \cosh (x)}}\\ &=-\frac {2 i \sqrt {\cosh (x)} F\left (\left .\frac {i x}{2}\right |2\right )}{3 a^2 \sqrt {a \cosh (x)}}+\frac {2 \sinh (x)}{3 a (a \cosh (x))^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 56, normalized size = 1.12 \[ \frac {2 \left (\sqrt {\sinh (2 x)+\cosh (2 x)+1} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\cosh (2 x)-\sinh (2 x)\right )+\tanh (x)\right )}{3 a^2 \sqrt {a \cosh (x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \cosh \relax (x)}}{a^{3} \cosh \relax (x)^{3}}, x\right ) \]
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 \[ \int \frac {1}{\left (a \cosh \relax (x)\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.34, size = 177, normalized size = 3.54 \[ \frac {\left (2 \sqrt {-2 \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )-1}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \EllipticF \left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )+\sqrt {2}\, \sqrt {-2 \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )-1}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \EllipticF \left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )+4 \left (\sinh ^{2}\left (\frac {x}{2}\right )\right ) \cosh \left (\frac {x}{2}\right )\right ) \sqrt {a \left (2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}}{3 a^{2} \sqrt {a \left (2 \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \left (2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )-1\right ) \sinh \left (\frac {x}{2}\right ) \sqrt {a \left (2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )-1\right )}} \]
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 \[ \int \frac {1}{\left (a \cosh \relax (x)\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (a\,\mathrm {cosh}\relax (x)\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\left (a \cosh {\relax (x )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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