Optimal. Leaf size=27 \[ -\frac{1}{2} \log (\coth (x)) \sqrt{\log ^2(\coth (x))+1}-\frac{1}{2} \sinh ^{-1}(\log (\coth (x))) \]
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Rubi [A] time = 0.17014, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {6696, 195, 215} \[ -\frac{1}{2} \log (\coth (x)) \sqrt{\log ^2(\coth (x))+1}-\frac{1}{2} \sinh ^{-1}(\log (\coth (x))) \]
Antiderivative was successfully verified.
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Rule 6696
Rule 195
Rule 215
Rubi steps
\begin{align*} \int \text{csch}(x) \sqrt{1+\log ^2(\coth (x))} \text{sech}(x) \, dx &=-\operatorname{Subst}\left (\int \sqrt{1+x^2} \, dx,x,\log (\coth (x))\right )\\ &=-\frac{1}{2} \log (\coth (x)) \sqrt{1+\log ^2(\coth (x))}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,\log (\coth (x))\right )\\ &=-\frac{1}{2} \sinh ^{-1}(\log (\coth (x)))-\frac{1}{2} \log (\coth (x)) \sqrt{1+\log ^2(\coth (x))}\\ \end{align*}
Mathematica [A] time = 0.033826, size = 27, normalized size = 1. \[ -\frac{1}{2} \log (\coth (x)) \sqrt{\log ^2(\coth (x))+1}-\frac{1}{2} \sinh ^{-1}(\log (\coth (x))) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.074, size = 22, normalized size = 0.8 \begin{align*} -{\frac{{\it Arcsinh} \left ( \ln \left ({\rm coth} \left (x\right ) \right ) \right ) }{2}}-{\frac{\ln \left ({\rm coth} \left (x\right ) \right ) }{2}\sqrt{1+ \left ( \ln \left ({\rm coth} \left (x\right ) \right ) \right ) ^{2}}} \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 \sqrt{\log \left (\coth \left (x\right )\right )^{2} + 1} \operatorname{csch}\left (x\right ) \operatorname{sech}\left (x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.22986, size = 171, normalized size = 6.33 \begin{align*} -\frac{1}{2} \, \sqrt{\log \left (\frac{\cosh \left (x\right )}{\sinh \left (x\right )}\right )^{2} + 1} \log \left (\frac{\cosh \left (x\right )}{\sinh \left (x\right )}\right ) + \frac{1}{2} \, \log \left (\sqrt{\log \left (\frac{\cosh \left (x\right )}{\sinh \left (x\right )}\right )^{2} + 1} - \log \left (\frac{\cosh \left (x\right )}{\sinh \left (x\right )}\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\log \left (\coth \left (x\right )\right )^{2} + 1} \operatorname{csch}\left (x\right ) \operatorname{sech}\left (x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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