Optimal. Leaf size=13 \[ \sqrt{-\sinh ^2(x)} \coth (x) \]
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Rubi [A] time = 0.0218042, antiderivative size = 13, 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, 2638} \[ \sqrt{-\sinh ^2(x)} \coth (x) \]
Antiderivative was successfully verified.
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Rule 3176
Rule 3207
Rule 2638
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*}
Mathematica [A] time = 0.0046484, size = 13, normalized size = 1. \[ \sqrt{-\sinh ^2(x)} \coth (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.073, size = 15, normalized size = 1.2 \begin{align*} -{\cosh \left ( x \right ) \sinh \left ( x \right ){\frac{1}{\sqrt{- \left ( \sinh \left ( x \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.64138, size = 15, normalized size = 1.15 \begin{align*} -\frac{1}{2} i \, e^{\left (-x\right )} - \frac{1}{2} i \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.20031, size = 4, normalized size = 0.31 \begin{align*} 0 \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 \sqrt{1 - \cosh ^{2}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.21349, size = 42, normalized size = 3.23 \begin{align*} -\frac{1}{2} i \, e^{\left (-x\right )} \mathrm{sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right ) - \frac{1}{2} i \, e^{x} \mathrm{sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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