Optimal. Leaf size=36 \[ \frac{\coth (x)}{3 \left (a \text{csch}^2(x)\right )^{3/2}}-\frac{2 \coth (x)}{3 a \sqrt{a \text{csch}^2(x)}} \]
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Rubi [A] time = 0.0216984, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4122, 192, 191} \[ \frac{\coth (x)}{3 \left (a \text{csch}^2(x)\right )^{3/2}}-\frac{2 \coth (x)}{3 a \sqrt{a \text{csch}^2(x)}} \]
Antiderivative was successfully verified.
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Rule 4122
Rule 192
Rule 191
Rubi steps
\begin{align*} \int \frac{1}{\left (a \text{csch}^2(x)\right )^{3/2}} \, dx &=-\left (a \operatorname{Subst}\left (\int \frac{1}{\left (-a+a x^2\right )^{5/2}} \, dx,x,\coth (x)\right )\right )\\ &=\frac{\coth (x)}{3 \left (a \text{csch}^2(x)\right )^{3/2}}+\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{\left (-a+a x^2\right )^{3/2}} \, dx,x,\coth (x)\right )\\ &=\frac{\coth (x)}{3 \left (a \text{csch}^2(x)\right )^{3/2}}-\frac{2 \coth (x)}{3 a \sqrt{a \text{csch}^2(x)}}\\ \end{align*}
Mathematica [A] time = 0.0232281, size = 27, normalized size = 0.75 \[ \frac{(\cosh (3 x)-9 \cosh (x)) \text{csch}^3(x)}{12 \left (a \text{csch}^2(x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.049, size = 130, normalized size = 3.6 \begin{align*}{\frac{{{\rm e}^{4\,x}}}{24\,a \left ({{\rm e}^{2\,x}}-1 \right ) }{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}-{\frac{3\,{{\rm e}^{2\,x}}}{8\,a \left ({{\rm e}^{2\,x}}-1 \right ) }{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}-{\frac{3}{8\,a \left ({{\rm e}^{2\,x}}-1 \right ) }{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}+{\frac{{{\rm e}^{-2\,x}}}{24\,a \left ({{\rm e}^{2\,x}}-1 \right ) }{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.70034, size = 47, normalized size = 1.31 \begin{align*} -\frac{e^{\left (3 \, x\right )}}{24 \, a^{\frac{3}{2}}} + \frac{3 \, e^{\left (-x\right )}}{8 \, a^{\frac{3}{2}}} - \frac{e^{\left (-3 \, x\right )}}{24 \, a^{\frac{3}{2}}} + \frac{3 \, e^{x}}{8 \, a^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.84244, size = 856, normalized size = 23.78 \begin{align*} \frac{{\left ({\left (e^{\left (2 \, x\right )} - 1\right )} \sinh \left (x\right )^{6} - \cosh \left (x\right )^{6} + 6 \,{\left (\cosh \left (x\right ) e^{\left (2 \, x\right )} - \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} - 3 \,{\left (5 \, \cosh \left (x\right )^{2} -{\left (5 \, \cosh \left (x\right )^{2} - 3\right )} e^{\left (2 \, x\right )} - 3\right )} \sinh \left (x\right )^{4} + 9 \, \cosh \left (x\right )^{4} - 4 \,{\left (5 \, \cosh \left (x\right )^{3} -{\left (5 \, \cosh \left (x\right )^{3} - 9 \, \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} - 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} - 3 \,{\left (5 \, \cosh \left (x\right )^{4} - 18 \, \cosh \left (x\right )^{2} -{\left (5 \, \cosh \left (x\right )^{4} - 18 \, \cosh \left (x\right )^{2} - 3\right )} e^{\left (2 \, x\right )} - 3\right )} \sinh \left (x\right )^{2} + 9 \, \cosh \left (x\right )^{2} +{\left (\cosh \left (x\right )^{6} - 9 \, \cosh \left (x\right )^{4} - 9 \, \cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, x\right )} - 6 \,{\left (\cosh \left (x\right )^{5} - 6 \, \cosh \left (x\right )^{3} -{\left (\cosh \left (x\right )^{5} - 6 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - 1\right )} \sqrt{\frac{a}{e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1}} e^{x}}{24 \,{\left (a^{2} \cosh \left (x\right )^{3} e^{x} + 3 \, a^{2} \cosh \left (x\right )^{2} e^{x} \sinh \left (x\right ) + 3 \, a^{2} \cosh \left (x\right ) e^{x} \sinh \left (x\right )^{2} + a^{2} e^{x} \sinh \left (x\right )^{3}\right )}} \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 \operatorname{csch}^{2}{\left (x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17792, size = 73, normalized size = 2.03 \begin{align*} -\frac{\frac{{\left (9 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-3 \, x\right )}}{\mathrm{sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right )} - \frac{e^{\left (3 \, x\right )} - 9 \, e^{x}}{\mathrm{sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right )}}{24 \, a^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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