Optimal. Leaf size=24 \[ -\frac {1}{2} \log \left (x^2+1\right )+\frac {1}{x-1}+\log (1-x)+\tan ^{-1}(x) \]
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Rubi [A] time = 0.03, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {1629, 635, 203, 260} \begin {gather*} -\frac {1}{2} \log \left (x^2+1\right )+\frac {1}{x-1}+\log (1-x)+\tan ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 635
Rule 1629
Rubi steps
\begin {align*} \int \frac {-1-2 x+x^2}{(-1+x)^2 \left (1+x^2\right )} \, dx &=\int \left (-\frac {1}{(-1+x)^2}+\frac {1}{-1+x}+\frac {1-x}{1+x^2}\right ) \, dx\\ &=\frac {1}{-1+x}+\log (1-x)+\int \frac {1-x}{1+x^2} \, dx\\ &=\frac {1}{-1+x}+\log (1-x)+\int \frac {1}{1+x^2} \, dx-\int \frac {x}{1+x^2} \, dx\\ &=\frac {1}{-1+x}+\tan ^{-1}(x)+\log (1-x)-\frac {1}{2} \log \left (1+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 0.92 \begin {gather*} -\frac {1}{2} \log \left (x^2+1\right )+\frac {1}{x-1}+\log (x-1)+\tan ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-1-2 x+x^2}{(-1+x)^2 \left (1+x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.35, size = 36, normalized size = 1.50 \begin {gather*} \frac {2 \, {\left (x - 1\right )} \arctan \relax (x) - {\left (x - 1\right )} \log \left (x^{2} + 1\right ) + 2 \, {\left (x - 1\right )} \log \left (x - 1\right ) + 2}{2 \, {\left (x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.30, size = 47, normalized size = 1.96 \begin {gather*} \frac {1}{4} \, \pi - \pi \left \lfloor \frac {\pi + 4 \, \arctan \relax (x)}{4 \, \pi } + \frac {1}{2} \right \rfloor + \frac {1}{x - 1} + \arctan \relax (x) - \frac {1}{2} \, \log \left (\frac {2}{x - 1} + \frac {2}{{\left (x - 1\right )}^{2}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 21, normalized size = 0.88 \begin {gather*} \arctan \relax (x )+\ln \left (x -1\right )-\frac {\ln \left (x^{2}+1\right )}{2}+\frac {1}{x -1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.60, size = 20, normalized size = 0.83 \begin {gather*} \frac {1}{x - 1} + \arctan \relax (x) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) + \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.11, size = 28, normalized size = 1.17 \begin {gather*} \ln \left (x-1\right )+\frac {1}{x-1}+\ln \left (x-\mathrm {i}\right )\,\left (-\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )+\ln \left (x+1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 20, normalized size = 0.83 \begin {gather*} \log {\left (x - 1 \right )} - \frac {\log {\left (x^{2} + 1 \right )}}{2} + \operatorname {atan}{\relax (x )} + \frac {1}{x - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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