Optimal. Leaf size=25 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{2} \coth (x)}{\sqrt{\coth ^2(x)+1}}\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.0193541, antiderivative size = 25, 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 = {3661, 377, 206} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{2} \coth (x)}{\sqrt{\coth ^2(x)+1}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 3661
Rule 377
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1+\coth ^2(x)}} \, dx &=\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}{1-2 x^2} \, dx,x,\frac{\coth (x)}{\sqrt{1+\coth ^2(x)}}\right )\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{2} \coth (x)}{\sqrt{1+\coth ^2(x)}}\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0625269, size = 44, normalized size = 1.76 \[ \frac{\sqrt{\cosh (2 x)} \text{csch}(x) \log \left (\sqrt{2} \cosh (x)+\sqrt{\cosh (2 x)}\right )}{\sqrt{2} \sqrt{\coth ^2(x)+1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.046, size = 62, normalized size = 2.5 \begin{align*} -{\frac{\sqrt{2}}{4}{\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{\sqrt{2}}{4}{\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 \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 [B] time = 2.22461, size = 1831, normalized size = 73.24 \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 \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 [B] time = 1.18556, size = 93, normalized size = 3.72 \begin{align*} \frac{\sqrt{2}{\left (\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 )}}{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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