Optimal. Leaf size=3 \[ \sin ^{-1}(\coth (x)) \]
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Rubi [A] time = 0.0183658, antiderivative size = 3, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3657, 4122, 216} \[ \sin ^{-1}(\coth (x)) \]
Antiderivative was successfully verified.
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Rule 3657
Rule 4122
Rule 216
Rubi steps
\begin{align*} \int \sqrt{1-\coth ^2(x)} \, dx &=\int \sqrt{-\text{csch}^2(x)} \, dx\\ &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,\coth (x)\right )\\ &=\sin ^{-1}(\coth (x))\\ \end{align*}
Mathematica [B] time = 0.0064008, size = 20, normalized size = 6.67 \[ \sinh (x) \sqrt{-\text{csch}^2(x)} \log \left (\tanh \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 4, normalized size = 1.3 \begin{align*} \arcsin \left ({\rm coth} \left (x\right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.72328, size = 26, normalized size = 8.67 \begin{align*} i \, \log \left (e^{\left (-x\right )} + 1\right ) - i \, \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 = 2.61366, size = 4, normalized size = 1.33 \begin{align*} 0 \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{1 - \coth ^{2}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.16479, size = 35, normalized size = 11.67 \begin{align*}{\left (i \, \log \left (e^{x} + 1\right ) - i \, \log \left ({\left | e^{x} - 1 \right |}\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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