Optimal. Leaf size=23 \[ \frac{1}{2} \log \left (x^2+1\right )+2 \log (1-x)-3 \tan ^{-1}(x) \]
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Rubi [A] time = 0.033852, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {1629, 635, 203, 260} \[ \frac{1}{2} \log \left (x^2+1\right )+2 \log (1-x)-3 \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 1629
Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int \frac{5-4 x+3 x^2}{(-1+x) \left (1+x^2\right )} \, dx &=\int \left (\frac{2}{-1+x}+\frac{-3+x}{1+x^2}\right ) \, dx\\ &=2 \log (1-x)+\int \frac{-3+x}{1+x^2} \, dx\\ &=2 \log (1-x)-3 \int \frac{1}{1+x^2} \, dx+\int \frac{x}{1+x^2} \, dx\\ &=-3 \tan ^{-1}(x)+2 \log (1-x)+\frac{1}{2} \log \left (1+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0080614, size = 28, normalized size = 1.22 \[ \frac{1}{2} \log \left ((x-1)^2+2 (x-1)+2\right )+2 \log (x-1)-3 \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 20, normalized size = 0.9 \begin{align*}{\frac{\ln \left ({x}^{2}+1 \right ) }{2}}-3\,\arctan \left ( x \right ) +2\,\ln \left ( x-1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.64397, size = 26, normalized size = 1.13 \begin{align*} -3 \, \arctan \left (x\right ) + \frac{1}{2} \, \log \left (x^{2} + 1\right ) + 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 = 1.77813, size = 65, normalized size = 2.83 \begin{align*} -3 \, \arctan \left (x\right ) + \frac{1}{2} \, \log \left (x^{2} + 1\right ) + 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 = 0.120955, size = 19, normalized size = 0.83 \begin{align*} 2 \log{\left (x - 1 \right )} + \frac{\log{\left (x^{2} + 1 \right )}}{2} - 3 \operatorname{atan}{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24783, size = 27, normalized size = 1.17 \begin{align*} -3 \, \arctan \left (x\right ) + \frac{1}{2} \, \log \left (x^{2} + 1\right ) + 2 \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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