Optimal. Leaf size=16 \[ \coth (x) \sqrt {a \tanh ^2(x)} \log (\cosh (x)) \]
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Rubi [A] time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3658, 3475} \[ \coth (x) \sqrt {a \tanh ^2(x)} \log (\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3658
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*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 1.00 \[ \coth (x) \sqrt {a \tanh ^2(x)} \log (\cosh (x)) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 72, normalized size = 4.50 \[ -\frac {{\left (x e^{\left (2 \, x\right )} - {\left (e^{\left (2 \, x\right )} + 1\right )} \log \left (\frac {2 \, \cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) + x\right )} \sqrt {\frac {a e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}}}{e^{\left (2 \, x\right )} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.90, size = 31, normalized size = 1.94 \[ -{\left (x \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) - \log \left (e^{\left (2 \, x\right )} + 1\right ) \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right )\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 26, normalized size = 1.62 \[ -\frac {\sqrt {a \left (\tanh ^{2}\relax (x )\right )}\, \left (\ln \left (\tanh \relax (x )-1\right )+\ln \left (1+\tanh \relax (x )\right )\right )}{2 \tanh \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 19, normalized size = 1.19 \[ -\sqrt {a} x - \sqrt {a} \log \left (e^{\left (-2 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.06 \[ \int \sqrt {a\,{\mathrm {tanh}\relax (x)}^2} \,d x \]
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 \[ \int \sqrt {a \tanh ^{2}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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