Optimal. Leaf size=3 \[ \sin ^{-1}(\coth (x)) \]
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Rubi [A] time = 0.0081699, antiderivative size = 3, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4122, 216} \[ \sin ^{-1}(\coth (x)) \]
Antiderivative was successfully verified.
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Rule 4122
Rule 216
Rubi steps
\begin{align*} \int \sqrt{-\text{csch}^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,\coth (x)\right )\\ &=\sin ^{-1}(\coth (x))\\ \end{align*}
Mathematica [B] time = 0.0048837, size = 20, normalized size = 6.67 \[ \sinh (x) \sqrt{-\text{csch}^2(x)} \log \left (\tanh \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.052, size = 67, normalized size = 22.3 \begin{align*} -{{\rm e}^{-x}} \left ({{\rm e}^{2\,x}}-1 \right ) \sqrt{-{\frac{{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}\ln \left ({{\rm e}^{x}}+1 \right ) +{{\rm e}^{-x}} \left ({{\rm e}^{2\,x}}-1 \right ) \sqrt{-{\frac{{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}\ln \left ({{\rm e}^{x}}-1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.74671, size = 26, normalized size = 8.67 \begin{align*} 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 = 1.50787, size = 46, normalized size = 15.33 \begin{align*} -i \, \log \left (e^{x} + 1\right ) + i \, \log \left (e^{x} - 1\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 \sqrt{- \operatorname{csch}^{2}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.15374, size = 36, normalized size = 12. \begin{align*}{\left (i \, \log \left (e^{x} + 1\right ) - i \, \log \left ({\left | e^{x} - 1 \right |}\right )\right )} \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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