Optimal. Leaf size=11 \[ \frac {\log (a+b \sinh (x))}{b} \]
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Rubi [A] time = 0.04, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3159, 2668, 31} \[ \frac {\log (a+b \sinh (x))}{b} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2668
Rule 3159
Rubi steps
\begin {align*} \int \frac {1}{a \text {sech}(x)+b \tanh (x)} \, dx &=\int \frac {\cosh (x)}{a+b \sinh (x)} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \sinh (x)\right )}{b}\\ &=\frac {\log (a+b \sinh (x))}{b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 11, normalized size = 1.00 \[ \frac {\log (a+b \sinh (x))}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 27, normalized size = 2.45 \[ -\frac {x - \log \left (\frac {2 \, {\left (b \sinh \relax (x) + a\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 22, normalized size = 2.00 \[ \frac {\log \left ({\left | -b {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, a \right |}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.21, size = 50, normalized size = 4.55 \[ -\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{b}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{b}+\frac {\ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 \tanh \left (\frac {x}{2}\right ) b -a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 28, normalized size = 2.55 \[ \frac {x}{b} + \frac {\log \left (-2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} - b\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 25, normalized size = 2.27 \[ -\frac {x-\ln \left (2\,a\,{\mathrm {e}}^x-b+b\,{\mathrm {e}}^{2\,x}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.44, size = 32, normalized size = 2.91 \[ \begin {cases} \frac {x}{b} + \frac {\log {\left (\frac {a \operatorname {sech}{\relax (x )}}{b} + \tanh {\relax (x )} \right )}}{b} - \frac {\log {\left (\tanh {\relax (x )} + 1 \right )}}{b} & \text {for}\: b \neq 0 \\\frac {\tanh {\relax (x )}}{a \operatorname {sech}{\relax (x )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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