Optimal. Leaf size=16 \[ -\tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{-\text{sech}^2(x)}}\right ) \]
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Rubi [A] time = 0.0205394, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {3657, 4122, 217, 206} \[ -\tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{-\text{sech}^2(x)}}\right ) \]
Antiderivative was successfully verified.
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Rule 3657
Rule 4122
Rule 217
Rule 206
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 )\\ &=-\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\tanh (x)}{\sqrt{-\text{sech}^2(x)}}\right )\\ &=-\tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{-\text{sech}^2(x)}}\right )\\ \end{align*}
Mathematica [A] time = 0.0067079, size = 21, normalized size = 1.31 \[ 2 \cosh (x) \sqrt{-\text{sech}^2(x)} \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 15, normalized size = 0.9 \begin{align*} -\ln \left ( \tanh \left ( x \right ) +\sqrt{-1+ \left ( \tanh \left ( x \right ) \right ) ^{2}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.67764, size = 7, normalized size = 0.44 \begin{align*} 2 i \, \arctan \left (e^{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.2341, size = 4, normalized size = 0.25 \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{\tanh ^{2}{\left (x \right )} - 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.20325, size = 23, normalized size = 1.44 \begin{align*} -\frac{1}{2} \, \log \left (e^{\left (2 \, x\right )} + 1\right ) + \log \left (i \, e^{x} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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