Optimal. Leaf size=34 \[ \frac{1}{2} \coth (x) \sqrt{-\coth ^2(x)}-\tanh (x) \sqrt{-\coth ^2(x)} \log (\sinh (x)) \]
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Rubi [A] time = 0.0373094, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4121, 3658, 3473, 3475} \[ \frac{1}{2} \coth (x) \sqrt{-\coth ^2(x)}-\tanh (x) \sqrt{-\coth ^2(x)} \log (\sinh (x)) \]
Antiderivative was successfully verified.
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Rule 4121
Rule 3658
Rule 3473
Rule 3475
Rubi steps
\begin{align*} \int \left (-1-\text{csch}^2(x)\right )^{3/2} \, dx &=\int \left (-\coth ^2(x)\right )^{3/2} \, dx\\ &=-\left (\left (\sqrt{-\coth ^2(x)} \tanh (x)\right ) \int \coth ^3(x) \, dx\right )\\ &=\frac{1}{2} \coth (x) \sqrt{-\coth ^2(x)}-\left (\sqrt{-\coth ^2(x)} \tanh (x)\right ) \int \coth (x) \, dx\\ &=\frac{1}{2} \coth (x) \sqrt{-\coth ^2(x)}-\sqrt{-\coth ^2(x)} \log (\sinh (x)) \tanh (x)\\ \end{align*}
Mathematica [A] time = 0.0156181, size = 26, normalized size = 0.76 \[ \frac{1}{2} \tanh (x) \sqrt{-\coth ^2(x)} \left (\text{csch}^2(x)-2 \log (\sinh (x))\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.098, size = 123, normalized size = 3.6 \begin{align*}{\frac{ \left ({{\rm e}^{2\,x}}-1 \right ) x}{{{\rm e}^{2\,x}}+1}\sqrt{-{\frac{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}+2\,{\frac{{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) \left ({{\rm e}^{2\,x}}-1 \right ) }\sqrt{-{\frac{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}-{\frac{ \left ({{\rm e}^{2\,x}}-1 \right ) \ln \left ({{\rm e}^{2\,x}}-1 \right ) }{{{\rm e}^{2\,x}}+1}\sqrt{-{\frac{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{ \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 [C] time = 1.51645, size = 59, normalized size = 1.74 \begin{align*} i \, x + \frac{2 i \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} + i \, \log \left (e^{\left (-x\right )} + 1\right ) + i \, \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.11553, size = 165, normalized size = 4.85 \begin{align*} \frac{i \, x e^{\left (4 \, x\right )} +{\left (-2 i \, x + 2 i\right )} e^{\left (2 \, x\right )} +{\left (-i \, e^{\left (4 \, x\right )} + 2 i \, e^{\left (2 \, x\right )} - i\right )} \log \left (e^{\left (2 \, x\right )} - 1\right ) + i \, x}{e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1} \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 \left (- \operatorname{csch}^{2}{\left (x \right )} - 1\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.16153, size = 113, normalized size = 3.32 \begin{align*} -i \, x \mathrm{sgn}\left (-e^{\left (4 \, x\right )} + 1\right ) + i \, \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \mathrm{sgn}\left (-e^{\left (4 \, x\right )} + 1\right ) - \frac{i \,{\left (3 \, e^{\left (4 \, x\right )} \mathrm{sgn}\left (-e^{\left (4 \, x\right )} + 1\right ) - 2 \, e^{\left (2 \, x\right )} \mathrm{sgn}\left (-e^{\left (4 \, x\right )} + 1\right ) + 3 \, \mathrm{sgn}\left (-e^{\left (4 \, x\right )} + 1\right )\right )}}{2 \,{\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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