Optimal. Leaf size=23 \[ -3 \tan ^{-1}(x)+2 \log (1-x)+\frac {1}{2} \log \left (1+x^2\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, 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 = {1643, 649, 209,
266} \begin {gather*} -3 \text {ArcTan}(x)+\frac {1}{2} \log \left (x^2+1\right )+2 \log (1-x) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 266
Rule 649
Rule 1643
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*}
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Mathematica [A]
time = 0.01, size = 28, normalized size = 1.22 \begin {gather*} -3 \tan ^{-1}(x)+\frac {1}{2} \log \left (2+2 (-1+x)+(-1+x)^2\right )+2 \log (-1+x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 20, normalized size = 0.87
method | result | size |
default | \(2 \ln \left (-1+x \right )+\frac {\ln \left (x^{2}+1\right )}{2}-3 \arctan \left (x \right )\) | \(20\) |
risch | \(2 \ln \left (-1+x \right )+\frac {\ln \left (9 x^{2}+9\right )}{2}-3 \arctan \left (x \right )\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.00, size = 19, normalized size = 0.83 \begin {gather*} -3 \, \arctan \left (x\right ) + \frac {1}{2} \, \log \left (x^{2} + 1\right ) + 2 \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.39, size = 19, normalized size = 0.83 \begin {gather*} -3 \, \arctan \left (x\right ) + \frac {1}{2} \, \log \left (x^{2} + 1\right ) + 2 \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 19, normalized size = 0.83 \begin {gather*} 2 \log {\left (x - 1 \right )} + \frac {\log {\left (x^{2} + 1 \right )}}{2} - 3 \operatorname {atan}{\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 20, normalized size = 0.87 \begin {gather*} -3 \, \arctan \left (x\right ) + \frac {1}{2} \, \log \left (x^{2} + 1\right ) + 2 \, \log \left ({\left | x - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 25, normalized size = 1.09 \begin {gather*} 2\,\ln \left (x-1\right )+\ln \left (x-\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {3}{2}{}\mathrm {i}\right )+\ln \left (x+1{}\mathrm {i}\right )\,\left (\frac {1}{2}-\frac {3}{2}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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