Optimal. Leaf size=17 \[ -\frac {\log \left (1-2 \tan ^{-1}(x)\right )}{2 a b} \]
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Rubi [A] time = 0.04, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {4882} \[ -\frac {\log \left (1-2 \tan ^{-1}(x)\right )}{2 a b} \]
Antiderivative was successfully verified.
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Rule 4882
Rubi steps
\begin {align*} \int \frac {1}{\left (a+a x^2\right ) \left (b-2 b \tan ^{-1}(x)\right )} \, dx &=-\frac {\log \left (1-2 \tan ^{-1}(x)\right )}{2 a b}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 17, normalized size = 1.00 \[ -\frac {\log \left (2 \tan ^{-1}(x)-1\right )}{2 a b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 15, normalized size = 0.88 \[ -\frac {\log \left (2 \, \arctan \relax (x) - 1\right )}{2 \, a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 16, normalized size = 0.94 \[ -\frac {\log \left ({\left | 2 \, \arctan \relax (x) - 1 \right |}\right )}{2 \, a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 19, normalized size = 1.12 \[ -\frac {\ln \left (2 b \arctan \relax (x )-b \right )}{2 a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 16, normalized size = 0.94 \[ -\frac {\log \left ({\left | 2 \, \arctan \relax (x) - 1 \right |}\right )}{2 \, a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 15, normalized size = 0.88 \[ -\frac {\ln \left (2\,\mathrm {atan}\relax (x)-1\right )}{2\,a\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.51, size = 14, normalized size = 0.82 \[ - \frac {\log {\left (\operatorname {atan}{\relax (x )} - \frac {1}{2} \right )}}{2 a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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