Optimal. Leaf size=17 \[ -\frac{\sinh (x) \tanh ^{-1}(\cosh (x))}{\sqrt{-\sinh ^2(x)}} \]
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Rubi [A] time = 0.0218475, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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 = 0.0072967, size = 20, normalized size = 1.18 \[ \frac{\sinh (x) \log \left (\tanh \left (\frac{x}{2}\right )\right )}{\sqrt{-\sinh ^2(x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.068, size = 34, normalized size = 2. \begin{align*} -{\frac{\sinh \left ( x \right ) }{\cosh \left ( x \right ) }\sqrt{- \left ( \cosh \left ( x \right ) \right ) ^{2}}\arctan \left ({\frac{1}{\sqrt{- \left ( \cosh \left ( x \right ) \right ) ^{2}}}} \right ){\frac{1}{\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 [C] time = 1.66846, size = 26, normalized size = 1.53 \begin{align*} -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 = 2.21145, size = 4, normalized size = 0.24 \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{1 - \cosh ^{2}{\left (x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.21177, size = 54, normalized size = 3.18 \begin{align*} -\frac{i \, \log \left (e^{x} + 1\right )}{\mathrm{sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right )} + \frac{i \, \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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