Optimal. Leaf size=14 \[ \tanh (x) \sqrt{\coth ^2(x)} \log (\sinh (x)) \]
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Rubi [A] time = 0.0199036, antiderivative size = 14, 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 = {4121, 3658, 3475} \[ \tanh (x) \sqrt{\coth ^2(x)} \log (\sinh (x)) \]
Antiderivative was successfully verified.
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Rule 4121
Rule 3658
Rule 3475
Rubi steps
\begin{align*} \int \sqrt{1+\text{csch}^2(x)} \, dx &=\int \sqrt{\coth ^2(x)} \, dx\\ &=\left (\sqrt{\coth ^2(x)} \tanh (x)\right ) \int \coth (x) \, dx\\ &=\sqrt{\coth ^2(x)} \log (\sinh (x)) \tanh (x)\\ \end{align*}
Mathematica [A] time = 0.0056012, size = 14, normalized size = 1. \[ \tanh (x) \sqrt{\coth ^2(x)} \log (\sinh (x)) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.118, size = 79, normalized size = 5.6 \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}}}}}+{\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 [A] time = 1.55401, size = 30, normalized size = 2.14 \begin{align*} -x - \log \left (e^{\left (-x\right )} + 1\right ) - \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 [A] time = 2.13975, size = 55, normalized size = 3.93 \begin{align*} -x + \log \left (\frac{2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (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 \sqrt{\operatorname{csch}^{2}{\left (x \right )} + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17434, size = 36, normalized size = 2.57 \begin{align*} -x \mathrm{sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \mathrm{sgn}\left (e^{\left (4 \, x\right )} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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