Optimal. Leaf size=11 \[ \frac {\log (a+b \tanh (x))}{b} \]
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Rubi [A] time = 0.04, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3506, 31} \[ \frac {\log (a+b \tanh (x))}{b} \]
Antiderivative was successfully verified.
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Rule 31
Rule 3506
Rubi steps
\begin {align*} \int \frac {\text {sech}^2(x)}{a+b \tanh (x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \tanh (x)\right )}{b}\\ &=\frac {\log (a+b \tanh (x))}{b}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 20, normalized size = 1.82 \[ \frac {\log (a \cosh (x)+b \sinh (x))-\log (\cosh (x))}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.57, size = 42, normalized size = 3.82 \[ \frac {\log \left (\frac {2 \, {\left (a \cosh \relax (x) + b \sinh \relax (x)\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right ) - \log \left (\frac {2 \, \cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 45, normalized size = 4.09 \[ \frac {{\left (a + b\right )} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a b + b^{2}} - \frac {\log \left (e^{\left (2 \, x\right )} + 1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 12, normalized size = 1.09 \[ \frac {\ln \left (a +b \tanh \relax (x )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 11, normalized size = 1.00 \[ \frac {\log \left (b \tanh \relax (x) + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 50, normalized size = 4.55 \[ -\frac {2\,\mathrm {atan}\left (\frac {a\,\sqrt {-b^2}+a\,{\mathrm {e}}^{2\,x}\,\sqrt {-b^2}+b\,{\mathrm {e}}^{2\,x}\,\sqrt {-b^2}}{b^2}\right )}{\sqrt {-b^2}} \]
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 \[ \int \frac {\operatorname {sech}^{2}{\relax (x )}}{a + b \tanh {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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