Optimal. Leaf size=16 \[ -\tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {-\text {sech}^2(x)}}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, 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 206
Rule 217
Rule 3657
Rule 4122
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 2.08, size = 1, normalized size = 0.06 \[ 0 \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 1, normalized size = 0.06 \[ 0 \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 15, normalized size = 0.94 \[ -\ln \left (\tanh \relax (x )+\sqrt {-1+\tanh ^{2}\relax (x )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.49, size = 5, normalized size = 0.31 \[ 2 i \, \arctan \left (e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 14, normalized size = 0.88 \[ -\ln \left (\mathrm {tanh}\relax (x)+\sqrt {{\mathrm {tanh}\relax (x)}^2-1}\right ) \]
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 \[ \int \sqrt {\tanh ^{2}{\relax (x )} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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