Optimal. Leaf size=61 \[ \frac{3 \coth (x)}{8 a^2 \sqrt{a \sinh ^2(x)}}-\frac{3 \sinh (x) \tanh ^{-1}(\cosh (x))}{8 a^2 \sqrt{a \sinh ^2(x)}}-\frac{\coth (x)}{4 a \left (a \sinh ^2(x)\right )^{3/2}} \]
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Rubi [A] time = 0.0369936, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3204, 3207, 3770} \[ \frac{3 \coth (x)}{8 a^2 \sqrt{a \sinh ^2(x)}}-\frac{3 \sinh (x) \tanh ^{-1}(\cosh (x))}{8 a^2 \sqrt{a \sinh ^2(x)}}-\frac{\coth (x)}{4 a \left (a \sinh ^2(x)\right )^{3/2}} \]
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 )^{5/2}} \, dx &=-\frac{\coth (x)}{4 a \left (a \sinh ^2(x)\right )^{3/2}}-\frac{3 \int \frac{1}{\left (a \sinh ^2(x)\right )^{3/2}} \, dx}{4 a}\\ &=-\frac{\coth (x)}{4 a \left (a \sinh ^2(x)\right )^{3/2}}+\frac{3 \coth (x)}{8 a^2 \sqrt{a \sinh ^2(x)}}+\frac{3 \int \frac{1}{\sqrt{a \sinh ^2(x)}} \, dx}{8 a^2}\\ &=-\frac{\coth (x)}{4 a \left (a \sinh ^2(x)\right )^{3/2}}+\frac{3 \coth (x)}{8 a^2 \sqrt{a \sinh ^2(x)}}+\frac{(3 \sinh (x)) \int \text{csch}(x) \, dx}{8 a^2 \sqrt{a \sinh ^2(x)}}\\ &=-\frac{\coth (x)}{4 a \left (a \sinh ^2(x)\right )^{3/2}}+\frac{3 \coth (x)}{8 a^2 \sqrt{a \sinh ^2(x)}}-\frac{3 \tanh ^{-1}(\cosh (x)) \sinh (x)}{8 a^2 \sqrt{a \sinh ^2(x)}}\\ \end{align*}
Mathematica [A] time = 0.100122, size = 67, normalized size = 1.1 \[ -\frac{\text{csch}(x) \sqrt{a \sinh ^2(x)} \left (\text{csch}^4\left (\frac{x}{2}\right )-6 \text{csch}^2\left (\frac{x}{2}\right )-\text{sech}^4\left (\frac{x}{2}\right )-6 \text{sech}^2\left (\frac{x}{2}\right )-24 \log \left (\tanh \left (\frac{x}{2}\right )\right )\right )}{64 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 89, normalized size = 1.5 \begin{align*}{\frac{1}{8\, \left ( \sinh \left ( x \right ) \right ) ^{3}\cosh \left ( x \right ) }\sqrt{a \left ( \cosh \left ( x \right ) \right ) ^{2}} \left ( -3\,\ln \left ( 2\,{\frac{\sqrt{a}\sqrt{a \left ( \cosh \left ( x \right ) \right ) ^{2}}+a}{\sinh \left ( x \right ) }} \right ) a \left ( \sinh \left ( x \right ) \right ) ^{4}+3\,\sqrt{a \left ( \cosh \left ( x \right ) \right ) ^{2}} \left ( \sinh \left ( x \right ) \right ) ^{2}\sqrt{a}-2\,\sqrt{a}\sqrt{a \left ( \cosh \left ( x \right ) \right ) ^{2}} \right ){a}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{a \left ( \sinh \left ( x \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.82704, size = 130, normalized size = 2.13 \begin{align*} \frac{3 \, e^{\left (-x\right )} - 11 \, e^{\left (-3 \, x\right )} - 11 \, e^{\left (-5 \, x\right )} + 3 \, e^{\left (-7 \, x\right )}}{4 \,{\left (4 \, a^{\frac{5}{2}} e^{\left (-2 \, x\right )} - 6 \, a^{\frac{5}{2}} e^{\left (-4 \, x\right )} + 4 \, a^{\frac{5}{2}} e^{\left (-6 \, x\right )} - a^{\frac{5}{2}} e^{\left (-8 \, x\right )} - a^{\frac{5}{2}}\right )}} + \frac{3 \, \log \left (e^{\left (-x\right )} + 1\right )}{8 \, a^{\frac{5}{2}}} - \frac{3 \, \log \left (e^{\left (-x\right )} - 1\right )}{8 \, a^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.96298, size = 2514, normalized size = 41.21 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \sinh ^{2}{\left (x \right )}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.36389, size = 149, normalized size = 2.44 \begin{align*} -\frac{3 \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right )}{16 \, a^{\frac{5}{2}} \mathrm{sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right )} + \frac{3 \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right )}{16 \, a^{\frac{5}{2}} \mathrm{sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right )} + \frac{3 \, \sqrt{a}{\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 20 \, \sqrt{a}{\left (e^{\left (-x\right )} + e^{x}\right )}}{4 \,{\left ({\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4\right )}^{2} a^{3} \mathrm{sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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