Optimal. Leaf size=16 \[ \frac{\cosh (x) \tan ^{-1}(\sinh (x))}{\sqrt{-\cosh ^2(x)}} \]
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Rubi [A] time = 0.0237084, antiderivative size = 16, 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} \[ \frac{\cosh (x) \tan ^{-1}(\sinh (x))}{\sqrt{-\cosh ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3176
Rule 3207
Rule 3770
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-1-\sinh ^2(x)}} \, dx &=\int \frac{1}{\sqrt{-\cosh ^2(x)}} \, dx\\ &=\frac{\cosh (x) \int \text{sech}(x) \, dx}{\sqrt{-\cosh ^2(x)}}\\ &=\frac{\tan ^{-1}(\sinh (x)) \cosh (x)}{\sqrt{-\cosh ^2(x)}}\\ \end{align*}
Mathematica [A] time = 0.0079731, size = 21, normalized size = 1.31 \[ \frac{2 \cosh (x) \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )}{\sqrt{-\cosh ^2(x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.04, size = 34, normalized size = 2.1 \begin{align*} -{\frac{\cosh \left ( x \right ) }{\sinh \left ( x \right ) }\sqrt{- \left ( \sinh \left ( x \right ) \right ) ^{2}}{\it Artanh} \left ({\frac{1}{\sqrt{- \left ( \sinh \left ( x \right ) \right ) ^{2}}}} \right ){\frac{1}{\sqrt{- \left ( \cosh \left ( x \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.92463, size = 7, normalized size = 0.44 \begin{align*} -2 i \, \arctan \left (e^{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.77536, size = 39, normalized size = 2.44 \begin{align*} \log \left (e^{x} + i\right ) - \log \left (e^{x} - i\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{- \sinh ^{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 = 1.15006, size = 7, normalized size = 0.44 \begin{align*} -2 i \, \arctan \left (e^{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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