Optimal. Leaf size=17 \[ \frac{\log \left (1-2 \cot ^{-1}(x)\right )}{2 a b} \]
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Rubi [A] time = 0.04, antiderivative size = 17, normalized size of antiderivative = 1., 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 = {4883} \[ \frac{\log \left (1-2 \cot ^{-1}(x)\right )}{2 a b} \]
Antiderivative was successfully verified.
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Rule 4883
Rubi steps
\begin{align*} \int \frac{1}{\left (a+a x^2\right ) \left (b-2 b \cot ^{-1}(x)\right )} \, dx &=\frac{\log \left (1-2 \cot ^{-1}(x)\right )}{2 a b}\\ \end{align*}
Mathematica [A] time = 0.0447506, size = 17, normalized size = 1. \[ \frac{\log \left (2 \cot ^{-1}(x)-1\right )}{2 a b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.128, size = 19, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( 2\,b{\rm arccot} \left (x\right )-b \right ) }{2\,ab}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01412, size = 23, normalized size = 1.35 \begin{align*} \frac{\log \left ({\left | 2 \, \arctan \left (1, x\right ) - 1 \right |}\right )}{2 \, a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.34995, size = 43, normalized size = 2.53 \begin{align*} \frac{\log \left (2 \, \operatorname{arccot}\left (x\right ) - 1\right )}{2 \, a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.74049, size = 12, normalized size = 0.71 \begin{align*} \frac{\log{\left (\operatorname{acot}{\left (x \right )} - \frac{1}{2} \right )}}{2 a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14587, size = 24, normalized size = 1.41 \begin{align*} \frac{\log \left ({\left | 2 \, \arctan \left (\frac{1}{x}\right ) - 1 \right |}\right )}{2 \, a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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