Optimal. Leaf size=11 \[ \sqrt{\sinh ^2(x)} \coth (x) \]
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Rubi [A] time = 0.0175075, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, 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 = 0.0057719, size = 11, normalized size = 1. \[ \sqrt{\sinh ^2(x)} \coth (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.085, size = 14, normalized size = 1.3 \begin{align*}{\frac{\cosh \left ( x \right ) }{\sinh \left ( x \right ) }\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 [A] time = 1.70572, size = 15, normalized size = 1.36 \begin{align*} -\frac{1}{2} \, e^{\left (-x\right )} - \frac{1}{2} \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99559, size = 12, normalized size = 1.09 \begin{align*} \cosh \left (x\right ) \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{\cosh ^{2}{\left (x \right )} - 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.24778, size = 42, normalized size = 3.82 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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