Optimal. Leaf size=49 \[ -\frac {1}{4 \left (x^2+1\right )}+\frac {1}{4} \log \left (x^2+1\right )+\frac {1}{4 (1-x)}-\frac {1}{2} \log (1-x)+\frac {1}{4} \tan ^{-1}(x) \]
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Rubi [A] time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {741, 801, 635, 203, 260} \[ -\frac {1}{4 \left (x^2+1\right )}+\frac {1}{4} \log \left (x^2+1\right )+\frac {1}{4 (1-x)}-\frac {1}{2} \log (1-x)+\frac {1}{4} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 635
Rule 741
Rule 801
Rubi steps
\begin {align*} \int \frac {1}{(-1+x)^2 \left (1+x^2\right )^2} \, dx &=-\frac {1}{4 \left (1+x^2\right )}-\frac {1}{4} \int \frac {-4+2 x}{(-1+x)^2 \left (1+x^2\right )} \, dx\\ &=-\frac {1}{4 \left (1+x^2\right )}-\frac {1}{4} \int \left (-\frac {1}{(-1+x)^2}+\frac {2}{-1+x}+\frac {-1-2 x}{1+x^2}\right ) \, dx\\ &=\frac {1}{4 (1-x)}-\frac {1}{4 \left (1+x^2\right )}-\frac {1}{2} \log (1-x)-\frac {1}{4} \int \frac {-1-2 x}{1+x^2} \, dx\\ &=\frac {1}{4 (1-x)}-\frac {1}{4 \left (1+x^2\right )}-\frac {1}{2} \log (1-x)+\frac {1}{4} \int \frac {1}{1+x^2} \, dx+\frac {1}{2} \int \frac {x}{1+x^2} \, dx\\ &=\frac {1}{4 (1-x)}-\frac {1}{4 \left (1+x^2\right )}+\frac {1}{4} \tan ^{-1}(x)-\frac {1}{2} \log (1-x)+\frac {1}{4} \log \left (1+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 35, normalized size = 0.71 \[ \frac {1}{4} \left (-\frac {1}{x^2+1}+\log \left (x^2+1\right )+\frac {1}{1-x}-2 \log (x-1)+\tan ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 71, normalized size = 1.45 \[ -\frac {x^{2} - {\left (x^{3} - x^{2} + x - 1\right )} \arctan \relax (x) - {\left (x^{3} - x^{2} + x - 1\right )} \log \left (x^{2} + 1\right ) + 2 \, {\left (x^{3} - x^{2} + x - 1\right )} \log \left (x - 1\right ) + x}{4 \, {\left (x^{3} - x^{2} + x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.04, size = 80, normalized size = 1.63 \[ \frac {1}{16} \, \pi - \frac {1}{4} \, \pi \left \lfloor \frac {\pi + 4 \, \arctan \relax (x)}{4 \, \pi } + \frac {1}{2} \right \rfloor + \frac {\frac {2}{x - 1} + 1}{8 \, {\left (\frac {2}{x - 1} + \frac {2}{{\left (x - 1\right )}^{2}} + 1\right )}} - \frac {1}{4 \, {\left (x - 1\right )}} + \frac {1}{4} \, \arctan \relax (x) + \frac {1}{4} \, \log \left (\frac {2}{x - 1} + \frac {2}{{\left (x - 1\right )}^{2}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 36, normalized size = 0.73 \[ \frac {\arctan \relax (x )}{4}-\frac {\ln \left (x -1\right )}{2}+\frac {\ln \left (x^{2}+1\right )}{4}-\frac {1}{4 \left (x -1\right )}-\frac {1}{4 \left (x^{2}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 39, normalized size = 0.80 \[ -\frac {x^{2} + x}{4 \, {\left (x^{3} - x^{2} + x - 1\right )}} + \frac {1}{4} \, \arctan \relax (x) + \frac {1}{4} \, \log \left (x^{2} + 1\right ) - \frac {1}{2} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 49, normalized size = 1.00 \[ -\frac {\ln \left (x-1\right )}{2}-\frac {\frac {x^2}{4}+\frac {x}{4}}{x^3-x^2+x-1}+\ln \left (x-\mathrm {i}\right )\,\left (\frac {1}{4}-\frac {1}{8}{}\mathrm {i}\right )+\ln \left (x+1{}\mathrm {i}\right )\,\left (\frac {1}{4}+\frac {1}{8}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 41, normalized size = 0.84 \[ \frac {- x^{2} - x}{4 x^{3} - 4 x^{2} + 4 x - 4} - \frac {\log {\left (x - 1 \right )}}{2} + \frac {\log {\left (x^{2} + 1 \right )}}{4} + \frac {\operatorname {atan}{\relax (x )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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