Optimal. Leaf size=11 \[ a \tan ^{-1}(\sinh (x))+b \log (\cosh (x)) \]
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Rubi [A] time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3770, 3475} \[ a \tan ^{-1}(\sinh (x))+b \log (\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3770
Rubi steps
\begin {align*} \int (a \text {sech}(x)+b \tanh (x)) \, dx &=a \int \text {sech}(x) \, dx+b \int \tanh (x) \, dx\\ &=a \tan ^{-1}(\sinh (x))+b \log (\cosh (x))\\ \end {align*}
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Mathematica [A] time = 0.00, size = 16, normalized size = 1.45 \[ 2 a \tan ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right )+b \log (\cosh (x)) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 30, normalized size = 2.73 \[ -b x + 2 \, a \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right ) + b \log \left (\frac {2 \, \cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 21, normalized size = 1.91 \[ -b {\left (x - \log \left (e^{\left (2 \, x\right )} + 1\right )\right )} + 2 \, a \arctan \left (e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 12, normalized size = 1.09 \[ a \arctan \left (\sinh \relax (x )\right )+b \ln \left (\cosh \relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 11, normalized size = 1.00 \[ a \arctan \left (\sinh \relax (x)\right ) + b \log \left (\cosh \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 40, normalized size = 3.64 \[ b\,\ln \left (4\,a^2\,{\mathrm {e}}^{2\,x}+4\,a^2\right )-b\,x+2\,\mathrm {atan}\left (\frac {a\,{\mathrm {e}}^x}{\sqrt {a^2}}\right )\,\sqrt {a^2} \]
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 \[ \int \left (a \operatorname {sech}{\relax (x )} + b \tanh {\relax (x )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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