Optimal. Leaf size=27 \[ -\frac{1}{4} \log \left (x^2+1\right )+\frac{1}{2} \log (1-x)+\frac{1}{2} \tan ^{-1}(x) \]
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Rubi [A] time = 0.0180382, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {801, 635, 203, 260} \[ -\frac{1}{4} \log \left (x^2+1\right )+\frac{1}{2} \log (1-x)+\frac{1}{2} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 801
Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int \frac{x}{(-1+x) \left (1+x^2\right )} \, dx &=\int \left (\frac{1}{2 (-1+x)}+\frac{1-x}{2 \left (1+x^2\right )}\right ) \, dx\\ &=\frac{1}{2} \log (1-x)+\frac{1}{2} \int \frac{1-x}{1+x^2} \, dx\\ &=\frac{1}{2} \log (1-x)+\frac{1}{2} \int \frac{1}{1+x^2} \, dx-\frac{1}{2} \int \frac{x}{1+x^2} \, dx\\ &=\frac{1}{2} \tan ^{-1}(x)+\frac{1}{2} \log (1-x)-\frac{1}{4} \log \left (1+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0043409, size = 27, normalized size = 1. \[ -\frac{1}{4} \log \left (x^2+1\right )+\frac{1}{2} \log (1-x)+\frac{1}{2} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 20, normalized size = 0.7 \begin{align*} -{\frac{\ln \left ({x}^{2}+1 \right ) }{4}}+{\frac{\arctan \left ( x \right ) }{2}}+{\frac{\ln \left ( -1+x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.42416, size = 26, normalized size = 0.96 \begin{align*} \frac{1}{2} \, \arctan \left (x\right ) - \frac{1}{4} \, \log \left (x^{2} + 1\right ) + \frac{1}{2} \, \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01521, size = 69, normalized size = 2.56 \begin{align*} \frac{1}{2} \, \arctan \left (x\right ) - \frac{1}{4} \, \log \left (x^{2} + 1\right ) + \frac{1}{2} \, \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.113708, size = 19, normalized size = 0.7 \begin{align*} \frac{\log{\left (x - 1 \right )}}{2} - \frac{\log{\left (x^{2} + 1 \right )}}{4} + \frac{\operatorname{atan}{\left (x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08794, size = 27, normalized size = 1. \begin{align*} \frac{1}{2} \, \arctan \left (x\right ) - \frac{1}{4} \, \log \left (x^{2} + 1\right ) + \frac{1}{2} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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