Optimal. Leaf size=17 \[ -\frac {\sinh (x) \tanh ^{-1}(\cosh (x))}{\sqrt {-\sinh ^2(x)}} \]
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Rubi [A] time = 0.02, antiderivative size = 17, 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} \[ -\frac {\sinh (x) \tanh ^{-1}(\cosh (x))}{\sqrt {-\sinh ^2(x)}} \]
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.49, size = 1, normalized size = 0.06 \[ 0 \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.15, size = 40, normalized size = 2.35 \[ -\frac {i \, \log \left (e^{x} + 1\right )}{\mathrm {sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right )} + \frac {i \, \log \left ({\left | e^{x} - 1 \right |}\right )}{\mathrm {sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.14, size = 34, normalized size = 2.00 \[ -\frac {\sinh \relax (x ) \sqrt {-\left (\cosh ^{2}\relax (x )\right )}\, \arctan \left (\frac {1}{\sqrt {-\left (\cosh ^{2}\relax (x )\right )}}\right )}{\cosh \relax (x ) \sqrt {-\left (\sinh ^{2}\relax (x )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.82, size = 19, normalized size = 1.12 \[ -i \, \log \left (e^{\left (-x\right )} + 1\right ) + i \, \log \left (e^{\left (-x\right )} - 1\right ) \]
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 \[ \int \frac {1}{\sqrt {1-{\mathrm {cosh}\relax (x)}^2}} \,d x \]
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 \[ \int \frac {1}{\sqrt {1 - \cosh ^{2}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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