Optimal. Leaf size=31 \[ \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{2} \coth (x)}{\sqrt{\coth ^2(x)+1}}\right )-\sinh ^{-1}(\coth (x)) \]
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Rubi [A] time = 0.0261764, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3661, 402, 215, 377, 206} \[ \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{2} \coth (x)}{\sqrt{\coth ^2(x)+1}}\right )-\sinh ^{-1}(\coth (x)) \]
Antiderivative was successfully verified.
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Rule 3661
Rule 402
Rule 215
Rule 377
Rule 206
Rubi steps
\begin{align*} \int \sqrt{1+\coth ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{\sqrt{1+x^2}}{1-x^2} \, dx,x,\coth (x)\right )\\ &=2 \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{1+x^2}} \, dx,x,\coth (x)\right )-\operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,\coth (x)\right )\\ &=-\sinh ^{-1}(\coth (x))+2 \operatorname{Subst}\left (\int \frac{1}{1-2 x^2} \, dx,x,\frac{\coth (x)}{\sqrt{1+\coth ^2(x)}}\right )\\ &=-\sinh ^{-1}(\coth (x))+\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{2} \coth (x)}{\sqrt{1+\coth ^2(x)}}\right )\\ \end{align*}
Mathematica [A] time = 0.0722749, size = 60, normalized size = 1.94 \[ \frac{\sinh (x) \sqrt{\coth ^2(x)+1} \left (\sqrt{2} \log \left (\sqrt{2} \cosh (x)+\sqrt{\cosh (2 x)}\right )-\tanh ^{-1}\left (\frac{\cosh (x)}{\sqrt{\cosh (2 x)}}\right )\right )}{\sqrt{\cosh (2 x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.052, size = 97, normalized size = 3.1 \begin{align*}{\frac{1}{2}\sqrt{ \left ( 1+{\rm coth} \left (x\right ) \right ) ^{2}-2\,{\rm coth} \left (x\right )}}-{\it Arcsinh} \left ({\rm coth} \left (x\right ) \right ) -{\frac{\sqrt{2}}{2}{\it Artanh} \left ({\frac{ \left ( 2-2\,{\rm coth} \left (x\right ) \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{ \left ( 1+{\rm coth} \left (x\right ) \right ) ^{2}-2\,{\rm coth} \left (x\right )}}}} \right ) }-{\frac{1}{2}\sqrt{ \left ({\rm coth} \left (x\right )-1 \right ) ^{2}+2\,{\rm coth} \left (x\right )}}+{\frac{\sqrt{2}}{2}{\it Artanh} \left ({\frac{ \left ( 2\,{\rm coth} \left (x\right )+2 \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 \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 [B] time = 2.48234, size = 2319, normalized size = 74.81 \begin{align*} \text{result too large to display} \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}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.24062, size = 161, normalized size = 5.19 \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 ) + \log \left (\sqrt{e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right ) - \log \left (\sqrt{e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right ) - \log \left (-\sqrt{e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1\right )\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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