Optimal. Leaf size=3 \[ \sin ^{-1}(\tanh (x)) \]
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Rubi [A] time = 0.0165158, 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} \[ \sin ^{-1}(\tanh (x)) \]
Antiderivative was successfully verified.
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Rule 3657
Rule 4122
Rule 216
Rubi steps
\begin{align*} \int \sqrt{1-\tanh ^2(x)} \, dx &=\int \sqrt{\text{sech}^2(x)} \, dx\\ &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,\tanh (x)\right )\\ &=\sin ^{-1}(\tanh (x))\\ \end{align*}
Mathematica [B] time = 0.007552, size = 19, normalized size = 6.33 \[ 2 \cosh (x) \sqrt{\text{sech}^2(x)} \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 4, normalized size = 1.3 \begin{align*} \arcsin \left ( \tanh \left ( x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.68692, size = 7, normalized size = 2.33 \begin{align*} 2 \, \arctan \left (e^{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.16123, size = 39, normalized size = 13. \begin{align*} 2 \, \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\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{1 - \tanh ^{2}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16934, size = 7, normalized size = 2.33 \begin{align*} 2 \, \arctan \left (e^{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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