Optimal. Leaf size=19 \[ \tanh (x) \sqrt{\coth ^2(x)+1}-\sinh ^{-1}(\coth (x)) \]
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Rubi [A] time = 0.0491916, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3663, 277, 215} \[ \tanh (x) \sqrt{\coth ^2(x)+1}-\sinh ^{-1}(\coth (x)) \]
Antiderivative was successfully verified.
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Rule 3663
Rule 277
Rule 215
Rubi steps
\begin{align*} \int \sqrt{1+\coth ^2(x)} \text{sech}^2(x) \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt{1+x^2}}{x^2} \, dx,x,\coth (x)\right )\\ &=\sqrt{1+\coth ^2(x)} \tanh (x)-\operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,\coth (x)\right )\\ &=-\sinh ^{-1}(\coth (x))+\sqrt{1+\coth ^2(x)} \tanh (x)\\ \end{align*}
Mathematica [B] time = 0.208612, size = 51, normalized size = 2.68 \[ \sinh (x) \sqrt{\coth ^2(x)+1} \text{sech}(2 x) \left (\cosh (x)+\sinh (x) \tanh (x)-\sqrt{-\cosh (2 x)} \tan ^{-1}\left (\frac{\cosh (x)}{\sqrt{-\cosh (2 x)}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.232, size = 0, normalized size = 0. \begin{align*} \int \left ({\rm sech} \left (x\right ) \right ) ^{2}\sqrt{1+ \left ({\rm coth} \left (x\right ) \right ) ^{2}}\, dx \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{\coth \left (x\right )^{2} + 1} \operatorname{sech}\left (x\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.06866, size = 786, normalized size = 41.37 \begin{align*} -\frac{{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \log \left (\frac{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 2 \, \sqrt{\frac{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2}}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} + 1}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right ) -{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \log \left (\frac{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 2 \, \sqrt{\frac{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2}}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} + 1}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right ) - 4 \, \sqrt{\frac{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2}}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}}{2 \,{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 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{\coth ^{2}{\left (x \right )} + 1} \operatorname{sech}^{2}{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17636, size = 162, normalized size = 8.53 \begin{align*} \frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} \log \left (\frac{{\left | -2 \, \sqrt{2} + 2 \, \sqrt{e^{\left (4 \, x\right )} + 1} - 2 \, e^{\left (2 \, x\right )} + 2 \right |}}{2 \,{\left (\sqrt{2} + \sqrt{e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right )}}\right ) - \frac{4 \,{\left (\sqrt{e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right )}}{{\left (\sqrt{e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{2} - 2 \, \sqrt{e^{\left (4 \, x\right )} + 1} + 2 \, e^{\left (2 \, x\right )} - 1}\right )} \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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