Optimal. Leaf size=14 \[ \frac {\text {sech}(x)}{a}+\frac {\log (\cosh (x))}{a} \]
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Rubi [A] time = 0.05, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3879, 43} \[ \frac {\text {sech}(x)}{a}+\frac {\log (\cosh (x))}{a} \]
Antiderivative was successfully verified.
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Rule 43
Rule 3879
Rubi steps
\begin {align*} \int \frac {\tanh ^3(x)}{a+a \text {sech}(x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {a-a x}{x^2} \, dx,x,\cosh (x)\right )}{a^2}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {a}{x^2}-\frac {a}{x}\right ) \, dx,x,\cosh (x)\right )}{a^2}\\ &=\frac {\log (\cosh (x))}{a}+\frac {\text {sech}(x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 10, normalized size = 0.71 \[ \frac {\text {sech}(x)+\log (\cosh (x))}{a} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 85, normalized size = 6.07 \[ -\frac {x \cosh \relax (x)^{2} + x \sinh \relax (x)^{2} - {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1\right )} \log \left (\frac {2 \, \cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) + 2 \, {\left (x \cosh \relax (x) - 1\right )} \sinh \relax (x) + x - 2 \, \cosh \relax (x)}{a \cosh \relax (x)^{2} + 2 \, a \cosh \relax (x) \sinh \relax (x) + a \sinh \relax (x)^{2} + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 35, normalized size = 2.50 \[ \frac {\log \left (e^{\left (-x\right )} + e^{x}\right )}{a} - \frac {e^{\left (-x\right )} + e^{x} - 2}{a {\left (e^{\left (-x\right )} + e^{x}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.13, size = 54, normalized size = 3.86 \[ -\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}+\frac {2}{a \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )}+\frac {\ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.71, size = 33, normalized size = 2.36 \[ \frac {x}{a} + \frac {2 \, e^{\left (-x\right )}}{a e^{\left (-2 \, x\right )} + a} + \frac {\log \left (e^{\left (-2 \, x\right )} + 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.36, size = 33, normalized size = 2.36 \[ \frac {\ln \left ({\mathrm {e}}^{2\,x}+1\right )}{a}-\frac {x}{a}+\frac {2\,{\mathrm {e}}^x}{a\,\left ({\mathrm {e}}^{2\,x}+1\right )} \]
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 \[ \frac {\int \frac {\tanh ^{3}{\relax (x )}}{\operatorname {sech}{\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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