Optimal. Leaf size=16 \[ \coth (x) \sqrt{a \tanh ^2(x)} \log (\cosh (x)) \]
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Rubi [A] time = 0.0153141, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3658, 3475} \[ \coth (x) \sqrt{a \tanh ^2(x)} \log (\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 3658
Rule 3475
Rubi steps
\begin{align*} \int \sqrt{a \tanh ^2(x)} \, dx &=\left (\coth (x) \sqrt{a \tanh ^2(x)}\right ) \int \tanh (x) \, dx\\ &=\coth (x) \log (\cosh (x)) \sqrt{a \tanh ^2(x)}\\ \end{align*}
Mathematica [A] time = 0.0063562, size = 16, normalized size = 1. \[ \coth (x) \sqrt{a \tanh ^2(x)} \log (\cosh (x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 26, normalized size = 1.6 \begin{align*} -{\frac{\ln \left ( \tanh \left ( x \right ) -1 \right ) +\ln \left ( 1+\tanh \left ( x \right ) \right ) }{2\,\tanh \left ( x \right ) }\sqrt{a \left ( \tanh \left ( x \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.57406, size = 26, normalized size = 1.62 \begin{align*} -\sqrt{a} x - \sqrt{a} \log \left (e^{\left (-2 \, x\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.24951, size = 196, normalized size = 12.25 \begin{align*} -\frac{{\left (x e^{\left (2 \, x\right )} -{\left (e^{\left (2 \, x\right )} + 1\right )} \log \left (\frac{2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + x\right )} \sqrt{\frac{a e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}}}{e^{\left (2 \, x\right )} - 1} \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{a \tanh ^{2}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19594, size = 42, normalized size = 2.62 \begin{align*} -{\left (x \mathrm{sgn}\left (e^{\left (4 \, x\right )} - 1\right ) - \log \left (e^{\left (2 \, x\right )} + 1\right ) \mathrm{sgn}\left (e^{\left (4 \, x\right )} - 1\right )\right )} \sqrt{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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