Optimal. Leaf size=11 \[ \sqrt {\sinh ^2(x)} \coth (x) \]
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Rubi [A] time = 0.02, antiderivative size = 11, 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, 2638} \[ \sqrt {\sinh ^2(x)} \coth (x) \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3176
Rule 3207
Rubi steps
\begin {align*} \int \sqrt {-1+\cosh ^2(x)} \, dx &=\int \sqrt {\sinh ^2(x)} \, dx\\ &=\left (\text {csch}(x) \sqrt {\sinh ^2(x)}\right ) \int \sinh (x) \, dx\\ &=\coth (x) \sqrt {\sinh ^2(x)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 11, normalized size = 1.00 \[ \sqrt {\sinh ^2(x)} \coth (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 2, normalized size = 0.18 \[ \cosh \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 31, normalized size = 2.82 \[ \frac {1}{2} \, e^{\left (-x\right )} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right ) + \frac {1}{2} \, e^{x} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 14, normalized size = 1.27 \[ \frac {\sqrt {-\frac {1}{2}+\frac {\cosh \left (2 x \right )}{2}}\, \cosh \relax (x )}{\sinh \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 11, normalized size = 1.00 \[ -\frac {1}{2} \, e^{\left (-x\right )} - \frac {1}{2} \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.95, size = 11, normalized size = 1.00 \[ \mathrm {coth}\relax (x)\,\sqrt {{\mathrm {cosh}\relax (x)}^2-1} \]
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 \[ \int \sqrt {\cosh ^{2}{\relax (x )} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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