Optimal. Leaf size=15 \[ -\frac{\sinh (x) \tanh ^{-1}(\cosh (x))}{\sqrt{\sinh ^2(x)}} \]
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Rubi [A] time = 0.0188273, antiderivative size = 15, 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 = {3176, 3207, 3770} \[ -\frac{\sinh (x) \tanh ^{-1}(\cosh (x))}{\sqrt{\sinh ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3176
Rule 3207
Rule 3770
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-1+\cosh ^2(x)}} \, dx &=\int \frac{1}{\sqrt{\sinh ^2(x)}} \, dx\\ &=\frac{\sinh (x) \int \text{csch}(x) \, dx}{\sqrt{\sinh ^2(x)}}\\ &=-\frac{\tanh ^{-1}(\cosh (x)) \sinh (x)}{\sqrt{\sinh ^2(x)}}\\ \end{align*}
Mathematica [A] time = 0.008162, size = 18, normalized size = 1.2 \[ \frac{\sinh (x) \log \left (\tanh \left (\frac{x}{2}\right )\right )}{\sqrt{\sinh ^2(x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.076, size = 16, normalized size = 1.1 \begin{align*} -{\frac{{\it Artanh} \left ( \cosh \left ( x \right ) \right ) }{\sinh \left ( x \right ) }\sqrt{ \left ( \sinh \left ( x \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.62907, size = 23, normalized size = 1.53 \begin{align*} \log \left (e^{\left (-x\right )} + 1\right ) - \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 = 2.03799, size = 78, normalized size = 5.2 \begin{align*} -\log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\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 \frac{1}{\sqrt{\cosh ^{2}{\left (x \right )} - 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.29688, size = 53, normalized size = 3.53 \begin{align*} -\frac{\log \left (e^{x} + 1\right )}{\mathrm{sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right )} + \frac{\log \left ({\left | e^{x} - 1 \right |}\right )}{\mathrm{sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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