Optimal. Leaf size=75 \[ \frac {10 \sinh (x)}{21 a \sqrt {a \cosh ^3(x)}}-\frac {10 i \cosh ^{\frac {3}{2}}(x) F\left (\left .\frac {i x}{2}\right |2\right )}{21 a \sqrt {a \cosh ^3(x)}}+\frac {2 \tanh (x) \text {sech}(x)}{7 a \sqrt {a \cosh ^3(x)}} \]
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Rubi [A] time = 0.03, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3207, 2636, 2641} \[ \frac {10 \sinh (x)}{21 a \sqrt {a \cosh ^3(x)}}-\frac {10 i \cosh ^{\frac {3}{2}}(x) F\left (\left .\frac {i x}{2}\right |2\right )}{21 a \sqrt {a \cosh ^3(x)}}+\frac {2 \tanh (x) \text {sech}(x)}{7 a \sqrt {a \cosh ^3(x)}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2641
Rule 3207
Rubi steps
\begin {align*} \int \frac {1}{\left (a \cosh ^3(x)\right )^{3/2}} \, dx &=\frac {\cosh ^{\frac {3}{2}}(x) \int \frac {1}{\cosh ^{\frac {9}{2}}(x)} \, dx}{a \sqrt {a \cosh ^3(x)}}\\ &=\frac {2 \text {sech}(x) \tanh (x)}{7 a \sqrt {a \cosh ^3(x)}}+\frac {\left (5 \cosh ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\cosh ^{\frac {5}{2}}(x)} \, dx}{7 a \sqrt {a \cosh ^3(x)}}\\ &=\frac {10 \sinh (x)}{21 a \sqrt {a \cosh ^3(x)}}+\frac {2 \text {sech}(x) \tanh (x)}{7 a \sqrt {a \cosh ^3(x)}}+\frac {\left (5 \cosh ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\sqrt {\cosh (x)}} \, dx}{21 a \sqrt {a \cosh ^3(x)}}\\ &=-\frac {10 i \cosh ^{\frac {3}{2}}(x) F\left (\left .\frac {i x}{2}\right |2\right )}{21 a \sqrt {a \cosh ^3(x)}}+\frac {10 \sinh (x)}{21 a \sqrt {a \cosh ^3(x)}}+\frac {2 \text {sech}(x) \tanh (x)}{7 a \sqrt {a \cosh ^3(x)}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 48, normalized size = 0.64 \[ \frac {2 \cosh ^2(x) \left (3 \tanh (x)-5 i \cosh ^{\frac {5}{2}}(x) F\left (\left .\frac {i x}{2}\right |2\right )+5 \sinh (x) \cosh (x)\right )}{21 \left (a \cosh ^3(x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \cosh \relax (x)^{3}}}{a^{2} \cosh \relax (x)^{6}}, x\right ) \]
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 \[ \int \frac {1}{\left (a \cosh \relax (x)^{3}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.17, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \left (\cosh ^{3}\relax (x )\right )\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {1}{\left (a \cosh \relax (x)^{3}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a\,{\mathrm {cosh}\relax (x)}^3\right )}^{3/2}} \,d x \]
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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