Optimal. Leaf size=3 \[ \sin ^{-1}(\coth (x)) \]
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Rubi [A] time = 0.02, antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3657, 4122, 216} \[ \sin ^{-1}(\coth (x)) \]
Antiderivative was successfully verified.
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Rule 216
Rule 3657
Rule 4122
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*}
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Mathematica [B] time = 0.01, 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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fricas [A] time = 0.48, size = 1, normalized size = 0.33 \[ 0 \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.14, size = 26, normalized size = 8.67 \[ {\left (i \, \log \left (e^{x} + 1\right ) - i \, \log \left ({\left | e^{x} - 1 \right |}\right )\right )} \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 4, normalized size = 1.33 \[ \arcsin \left (\coth \relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.42, size = 19, normalized size = 6.33 \[ i \, \log \left (e^{\left (-x\right )} + 1\right ) - i \, \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 3, normalized size = 1.00 \[ \mathrm {asin}\left (\mathrm {coth}\relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {1 - \coth ^{2}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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