Optimal. Leaf size=16 \[ \frac{\tanh (x) \log (\sinh (x))}{\sqrt{-\tanh ^2(x)}} \]
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Rubi [A] time = 0.0219229, 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} \[ \frac{\tanh (x) \log (\sinh (x))}{\sqrt{-\tanh ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 4121
Rule 3658
Rule 3475
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-1+\text{sech}^2(x)}} \, dx &=\int \frac{1}{\sqrt{-\tanh ^2(x)}} \, dx\\ &=\frac{\tanh (x) \int \coth (x) \, dx}{\sqrt{-\tanh ^2(x)}}\\ &=\frac{\log (\sinh (x)) \tanh (x)}{\sqrt{-\tanh ^2(x)}}\\ \end{align*}
Mathematica [A] time = 0.0086223, size = 16, normalized size = 1. \[ \frac{\tanh (x) \log (\sinh (x))}{\sqrt{-\tanh ^2(x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.101, size = 81, normalized size = 5.1 \begin{align*} -{\frac{ \left ({{\rm e}^{2\,x}}-1 \right ) x}{{{\rm e}^{2\,x}}+1}{\frac{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}{\frac{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 = 1.70369, size = 30, normalized size = 1.88 \begin{align*} i \, x + i \, \log \left (e^{\left (-x\right )} + 1\right ) + i \, \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.96996, 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 \frac{1}{\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 = 1.14028, size = 50, normalized size = 3.12 \begin{align*} -\frac{i \, x}{\mathrm{sgn}\left (-e^{\left (4 \, x\right )} + 1\right )} + \frac{i \, \log \left (-i \, e^{\left (2 \, x\right )} + i\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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