Optimal. Leaf size=42 \[ \frac {\sinh (x) \tanh ^{-1}(\cosh (x))}{2 a \sqrt {a \sinh ^2(x)}}-\frac {\coth (x)}{2 a \sqrt {a \sinh ^2(x)}} \]
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Rubi [A] time = 0.03, 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} \[ \frac {\sinh (x) \tanh ^{-1}(\cosh (x))}{2 a \sqrt {a \sinh ^2(x)}}-\frac {\coth (x)}{2 a \sqrt {a \sinh ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3204
Rule 3207
Rule 3770
Rubi steps
\begin {align*} \int \frac {1}{\left (a \sinh ^2(x)\right )^{3/2}} \, dx &=-\frac {\coth (x)}{2 a \sqrt {a \sinh ^2(x)}}-\frac {\int \frac {1}{\sqrt {a \sinh ^2(x)}} \, dx}{2 a}\\ &=-\frac {\coth (x)}{2 a \sqrt {a \sinh ^2(x)}}-\frac {\sinh (x) \int \text {csch}(x) \, dx}{2 a \sqrt {a \sinh ^2(x)}}\\ &=-\frac {\coth (x)}{2 a \sqrt {a \sinh ^2(x)}}+\frac {\tanh ^{-1}(\cosh (x)) \sinh (x)}{2 a \sqrt {a \sinh ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 44, normalized size = 1.05 \[ -\frac {\sinh ^3(x) \left (\text {csch}^2\left (\frac {x}{2}\right )+\text {sech}^2\left (\frac {x}{2}\right )+4 \log \left (\tanh \left (\frac {x}{2}\right )\right )\right )}{8 \left (a \sinh ^2(x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.59, size = 327, normalized size = 7.79 \[ \frac {{\left (6 \, \cosh \relax (x) e^{x} \sinh \relax (x)^{2} + 2 \, e^{x} \sinh \relax (x)^{3} + 2 \, {\left (3 \, \cosh \relax (x)^{2} + 1\right )} e^{x} \sinh \relax (x) + 2 \, {\left (\cosh \relax (x)^{3} + \cosh \relax (x)\right )} e^{x} - {\left (4 \, \cosh \relax (x) e^{x} \sinh \relax (x)^{3} + e^{x} \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} - 1\right )} e^{x} \sinh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} - \cosh \relax (x)\right )} e^{x} \sinh \relax (x) + {\left (\cosh \relax (x)^{4} - 2 \, \cosh \relax (x)^{2} + 1\right )} e^{x}\right )} \log \left (\frac {\cosh \relax (x) + \sinh \relax (x) + 1}{\cosh \relax (x) + \sinh \relax (x) - 1}\right )\right )} \sqrt {a e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + a} e^{\left (-x\right )}}{2 \, {\left (a^{2} \cosh \relax (x)^{4} - {\left (a^{2} e^{\left (2 \, x\right )} - a^{2}\right )} \sinh \relax (x)^{4} - 2 \, a^{2} \cosh \relax (x)^{2} - 4 \, {\left (a^{2} \cosh \relax (x) e^{\left (2 \, x\right )} - a^{2} \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 2 \, {\left (3 \, a^{2} \cosh \relax (x)^{2} - a^{2} - {\left (3 \, a^{2} \cosh \relax (x)^{2} - a^{2}\right )} e^{\left (2 \, x\right )}\right )} \sinh \relax (x)^{2} + a^{2} - {\left (a^{2} \cosh \relax (x)^{4} - 2 \, a^{2} \cosh \relax (x)^{2} + a^{2}\right )} e^{\left (2 \, x\right )} + 4 \, {\left (a^{2} \cosh \relax (x)^{3} - a^{2} \cosh \relax (x) - {\left (a^{2} \cosh \relax (x)^{3} - a^{2} \cosh \relax (x)\right )} e^{\left (2 \, x\right )}\right )} \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 37, normalized size = 0.88 \[ -\frac {e^{\left (-x\right )} + e^{x}}{{\left ({\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4\right )} a^{\frac {3}{2}} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 71, normalized size = 1.69 \[ -\frac {\sqrt {a \left (\cosh ^{2}\relax (x )\right )}\, \left (-\ln \left (\frac {2 \sqrt {a}\, \sqrt {a \left (\cosh ^{2}\relax (x )\right )}+2 a}{\sinh \relax (x )}\right ) a \left (\sinh ^{2}\relax (x )\right )+\sqrt {a}\, \sqrt {a \left (\cosh ^{2}\relax (x )\right )}\right )}{2 a^{\frac {5}{2}} \sinh \relax (x ) \cosh \relax (x ) \sqrt {a \left (\sinh ^{2}\relax (x )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 62, normalized size = 1.48 \[ -\frac {e^{\left (-x\right )} + e^{\left (-3 \, x\right )}}{2 \, a^{\frac {3}{2}} e^{\left (-2 \, x\right )} - a^{\frac {3}{2}} e^{\left (-4 \, x\right )} - a^{\frac {3}{2}}} - \frac {\log \left (e^{\left (-x\right )} + 1\right )}{2 \, a^{\frac {3}{2}}} + \frac {\log \left (e^{\left (-x\right )} - 1\right )}{2 \, a^{\frac {3}{2}}} \]
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 \[ \int \frac {1}{{\left (a\,{\mathrm {sinh}\relax (x)}^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (a \sinh ^{2}{\relax (x )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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