Optimal. Leaf size=16 \[ \frac {\cosh (x) \tan ^{-1}(\sinh (x))}{\sqrt {-\cosh ^2(x)}} \]
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Rubi [A] time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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*}
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Mathematica [A] time = 0.01, 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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fricas [C] time = 1.03, size = 13, normalized size = 0.81 \[ \log \left (e^{x} + i\right ) - \log \left (e^{x} - i\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.14, size = 5, normalized size = 0.31 \[ -2 i \, \arctan \left (e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 34, normalized size = 2.12 \[ -\frac {\cosh \relax (x ) \sqrt {-\left (\sinh ^{2}\relax (x )\right )}\, \arctanh \left (\frac {1}{\sqrt {-\left (\sinh ^{2}\relax (x )\right )}}\right )}{\sinh \relax (x ) \sqrt {-\left (\cosh ^{2}\relax (x )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.59, size = 5, normalized size = 0.31 \[ -2 i \, \arctan \left (e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {1}{\sqrt {-{\mathrm {sinh}\relax (x)}^2-1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- \sinh ^{2}{\relax (x )} - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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