Optimal. Leaf size=27 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} \coth (x)}{\sqrt{-\coth ^2(x)-1}}\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.0204859, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3661, 377, 203} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} \coth (x)}{\sqrt{-\coth ^2(x)-1}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 3661
Rule 377
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-1-\coth ^2(x)}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{-1-x^2} \left (1-x^2\right )} \, dx,x,\coth (x)\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{1+2 x^2} \, dx,x,\frac{\coth (x)}{\sqrt{-1-\coth ^2(x)}}\right )\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{2} \coth (x)}{\sqrt{-1-\coth ^2(x)}}\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0349675, size = 46, normalized size = 1.7 \[ \frac{\sqrt{\cosh (2 x)} \text{csch}(x) \log \left (\sqrt{2} \cosh (x)+\sqrt{\cosh (2 x)}\right )}{\sqrt{2} \sqrt{-\coth ^2(x)-1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.046, size = 66, normalized size = 2.4 \begin{align*}{\frac{\sqrt{2}}{4}\arctan \left ({\frac{ \left ( 2\,{\rm coth} \left (x\right )-2 \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{- \left ( 1+{\rm coth} \left (x\right ) \right ) ^{2}+2\,{\rm coth} \left (x\right )}}}} \right ) }-{\frac{\sqrt{2}}{4}\arctan \left ({\frac{ \left ( -2-2\,{\rm coth} \left (x\right ) \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{- \left ({\rm coth} \left (x\right )-1 \right ) ^{2}-2\,{\rm coth} \left (x\right )}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-\coth \left (x\right )^{2} - 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.03047, size = 551, normalized size = 20.41 \begin{align*} \frac{1}{8} i \, \sqrt{2} \log \left (\frac{1}{2} \,{\left (i \, \sqrt{2} \sqrt{-2 \, e^{\left (4 \, x\right )} - 2} + 2 \, e^{\left (2 \, x\right )} - 2\right )} e^{\left (-2 \, x\right )}\right ) - \frac{1}{8} i \, \sqrt{2} \log \left (\frac{1}{2} \,{\left (-i \, \sqrt{2} \sqrt{-2 \, e^{\left (4 \, x\right )} - 2} + 2 \, e^{\left (2 \, x\right )} - 2\right )} e^{\left (-2 \, x\right )}\right ) - \frac{1}{8} i \, \sqrt{2} \log \left ({\left (\sqrt{-2 \, e^{\left (4 \, x\right )} - 2}{\left (e^{\left (2 \, x\right )} + 2\right )} + i \, \sqrt{2} e^{\left (4 \, x\right )} + i \, \sqrt{2} e^{\left (2 \, x\right )} + 2 i \, \sqrt{2}\right )} e^{\left (-4 \, x\right )}\right ) + \frac{1}{8} i \, \sqrt{2} \log \left ({\left (\sqrt{-2 \, e^{\left (4 \, x\right )} - 2}{\left (e^{\left (2 \, x\right )} + 2\right )} - i \, \sqrt{2} e^{\left (4 \, x\right )} - i \, \sqrt{2} e^{\left (2 \, x\right )} - 2 i \, \sqrt{2}\right )} e^{\left (-4 \, x\right )}\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}{\sqrt{- \coth ^{2}{\left (x \right )} - 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.18379, size = 99, normalized size = 3.67 \begin{align*} -\frac{\sqrt{2}{\left (-i \, \log \left (\sqrt{e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right ) + i \, \log \left (\sqrt{e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right ) + i \, \log \left (-\sqrt{e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1\right )\right )}}{4 \, \mathrm{sgn}\left (-e^{\left (2 \, x\right )} + 1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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