Optimal. Leaf size=42 \[ \frac {\tanh (x)}{2 a \sqrt {a \cosh ^2(x)}}+\frac {\cosh (x) \tan ^{-1}(\sinh (x))}{2 a \sqrt {a \cosh ^2(x)}} \]
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Rubi [A] time = 0.02, antiderivative size = 42, 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 = {3204, 3207, 3770} \[ \frac {\tanh (x)}{2 a \sqrt {a \cosh ^2(x)}}+\frac {\cosh (x) \tan ^{-1}(\sinh (x))}{2 a \sqrt {a \cosh ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3204
Rule 3207
Rule 3770
Rubi steps
\begin {align*} \int \frac {1}{\left (a \cosh ^2(x)\right )^{3/2}} \, dx &=\frac {\tanh (x)}{2 a \sqrt {a \cosh ^2(x)}}+\frac {\int \frac {1}{\sqrt {a \cosh ^2(x)}} \, dx}{2 a}\\ &=\frac {\tanh (x)}{2 a \sqrt {a \cosh ^2(x)}}+\frac {\cosh (x) \int \text {sech}(x) \, dx}{2 a \sqrt {a \cosh ^2(x)}}\\ &=\frac {\tan ^{-1}(\sinh (x)) \cosh (x)}{2 a \sqrt {a \cosh ^2(x)}}+\frac {\tanh (x)}{2 a \sqrt {a \cosh ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 31, normalized size = 0.74 \[ \frac {\tanh (x)+2 \cosh (x) \tan ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right )}{2 a \sqrt {a \cosh ^2(x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.26, size = 299, normalized size = 7.12 \[ \frac {{\left (3 \, \cosh \relax (x) e^{x} \sinh \relax (x)^{2} + e^{x} \sinh \relax (x)^{3} + {\left (3 \, \cosh \relax (x)^{2} - 1\right )} e^{x} \sinh \relax (x) + {\left (4 \, \cosh \relax (x) e^{x} \sinh \relax (x)^{3} + e^{x} \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} + 1\right )} e^{x} \sinh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} + \cosh \relax (x)\right )} e^{x} \sinh \relax (x) + {\left (\cosh \relax (x)^{4} + 2 \, \cosh \relax (x)^{2} + 1\right )} e^{x}\right )} \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right ) + {\left (\cosh \relax (x)^{3} - \cosh \relax (x)\right )} e^{x}\right )} \sqrt {a e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} + a} e^{\left (-x\right )}}{a^{2} \cosh \relax (x)^{4} + {\left (a^{2} e^{\left (2 \, x\right )} + a^{2}\right )} \sinh \relax (x)^{4} + 2 \, a^{2} \cosh \relax (x)^{2} + 4 \, {\left (a^{2} \cosh \relax (x) e^{\left (2 \, x\right )} + a^{2} \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 2 \, {\left (3 \, a^{2} \cosh \relax (x)^{2} + a^{2} + {\left (3 \, a^{2} \cosh \relax (x)^{2} + a^{2}\right )} e^{\left (2 \, x\right )}\right )} \sinh \relax (x)^{2} + a^{2} + {\left (a^{2} \cosh \relax (x)^{4} + 2 \, a^{2} \cosh \relax (x)^{2} + a^{2}\right )} e^{\left (2 \, x\right )} + 4 \, {\left (a^{2} \cosh \relax (x)^{3} + a^{2} \cosh \relax (x) + {\left (a^{2} \cosh \relax (x)^{3} + a^{2} \cosh \relax (x)\right )} e^{\left (2 \, x\right )}\right )} \sinh \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 56, normalized size = 1.33 \[ \frac {\frac {\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )}{\sqrt {a}} - \frac {4 \, {\left (e^{\left (-x\right )} - e^{x}\right )}}{{\left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )} \sqrt {a}}}{4 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.30, size = 82, normalized size = 1.95 \[ \frac {\sqrt {a \left (\sinh ^{2}\relax (x )\right )}\, \left (-\ln \left (\frac {2 \sqrt {-a}\, \sqrt {a \left (\sinh ^{2}\relax (x )\right )}-2 a}{\cosh \relax (x )}\right ) a \left (\cosh ^{2}\relax (x )\right )+\sqrt {-a}\, \sqrt {a \left (\sinh ^{2}\relax (x )\right )}\right )}{2 a^{2} \cosh \relax (x ) \sqrt {-a}\, \sinh \relax (x ) \sqrt {a \left (\cosh ^{2}\relax (x )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 41, normalized size = 0.98 \[ \frac {e^{\left (3 \, x\right )} - e^{x}}{a^{\frac {3}{2}} e^{\left (4 \, x\right )} + 2 \, a^{\frac {3}{2}} e^{\left (2 \, x\right )} + a^{\frac {3}{2}}} + \frac {\arctan \left (e^{x}\right )}{a^{\frac {3}{2}}} \]
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 \[ \int \frac {1}{{\left (a\,{\mathrm {cosh}\relax (x)}^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (a \cosh ^{2}{\relax (x )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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