Optimal. Leaf size=16 \[ \sqrt{-\tanh ^2(x)} \coth (x) \log (\cosh (x)) \]
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Rubi [A] time = 0.0195109, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4121, 3658, 3475} \[ \sqrt{-\tanh ^2(x)} \coth (x) \log (\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 4121
Rule 3658
Rule 3475
Rubi steps
\begin{align*} \int \sqrt{-1+\text{sech}^2(x)} \, dx &=\int \sqrt{-\tanh ^2(x)} \, dx\\ &=\left (\coth (x) \sqrt{-\tanh ^2(x)}\right ) \int \tanh (x) \, dx\\ &=\coth (x) \log (\cosh (x)) \sqrt{-\tanh ^2(x)}\\ \end{align*}
Mathematica [A] time = 0.0064218, size = 16, normalized size = 1. \[ \sqrt{-\tanh ^2(x)} \coth (x) \log (\cosh (x)) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.122, size = 81, normalized size = 5.1 \begin{align*} -{\frac{ \left ({{\rm e}^{2\,x}}+1 \right ) x}{{{\rm e}^{2\,x}}-1}\sqrt{-{\frac{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}}+{\frac{ \left ({{\rm e}^{2\,x}}+1 \right ) \ln \left ({{\rm e}^{2\,x}}+1 \right ) }{{{\rm e}^{2\,x}}-1}\sqrt{-{\frac{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.70793, size = 18, normalized size = 1.12 \begin{align*} -i \, x - i \, \log \left (e^{\left (-2 \, x\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.97561, 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{\operatorname{sech}^{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 = 1.12261, size = 42, normalized size = 2.62 \begin{align*} i \, x \mathrm{sgn}\left (-e^{\left (4 \, x\right )} + 1\right ) - i \, \log \left (e^{\left (2 \, x\right )} + 1\right ) \mathrm{sgn}\left (-e^{\left (4 \, x\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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