Optimal. Leaf size=67 \[ \frac {6 i E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \cosh (x)}}{5 a^4 \sqrt {\cosh (x)}}+\frac {6 \sinh (x)}{5 a^3 \sqrt {a \cosh (x)}}+\frac {2 \sinh (x)}{5 a (a \cosh (x))^{5/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 67, normalized size of antiderivative = 1.00, 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 2639
Rule 2640
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*}
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Mathematica [A] time = 0.05, 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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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \cosh \relax (x)}}{a^{4} \cosh \relax (x)^{4}}, x\right ) \]
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 \[ \int \frac {1}{\left (a \cosh \relax (x)\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.60, size = 254, normalized size = 3.79 \[ \frac {2 \sqrt {a \left (2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \left (\frac {\cosh \left (\frac {x}{2}\right ) \sqrt {a \left (2 \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+\sinh ^{2}\left (\frac {x}{2}\right )\right )}}{20 a \left (\cosh ^{2}\left (\frac {x}{2}\right )-\frac {1}{2}\right )^{3}}+\frac {6 \left (\sinh ^{2}\left (\frac {x}{2}\right )\right ) \cosh \left (\frac {x}{2}\right )}{5 \sqrt {a \left (2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}}+\frac {3 \sqrt {2}\, \sqrt {-2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )+1}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \EllipticF \left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )}{10 \sqrt {a \left (2 \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+\sinh ^{2}\left (\frac {x}{2}\right )\right )}}-\frac {3 \sqrt {2}\, \sqrt {-2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )+1}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \left (\EllipticF \left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )-\EllipticE \left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )\right )}{5 \sqrt {a \left (2 \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+\sinh ^{2}\left (\frac {x}{2}\right )\right )}}\right )}{a^{3} \sinh \left (\frac {x}{2}\right ) \sqrt {a \left (2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )-1\right )}} \]
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 \[ \int \frac {1}{\left (a \cosh \relax (x)\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a\,\mathrm {cosh}\relax (x)\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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