Optimal. Leaf size=15 \[ -\frac {\sinh (x) \tanh ^{-1}(\cosh (x))}{\sqrt {\sinh ^2(x)}} \]
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Rubi [A] time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 = 18, normalized size = 1.20 \[ \frac {\sinh (x) \log \left (\tanh \left (\frac {x}{2}\right )\right )}{\sqrt {\sinh ^2(x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 17, normalized size = 1.13 \[ -\log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 39, normalized size = 2.60 \[ -\frac {\log \left (e^{x} + 1\right )}{\mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right )} + \frac {\log \left ({\left | e^{x} - 1 \right |}\right )}{\mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 16, normalized size = 1.07 \[ -\frac {\sqrt {-\frac {1}{2}+\frac {\cosh \left (2 x \right )}{2}}\, \arctanh \left (\cosh \relax (x )\right )}{\sinh \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 17, normalized size = 1.13 \[ \log \left (e^{\left (-x\right )} + 1\right ) - \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.07 \[ \int \frac {1}{\sqrt {{\mathrm {cosh}\relax (x)}^2-1}} \,d x \]
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 \[ \int \frac {1}{\sqrt {\cosh ^{2}{\relax (x )} - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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