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 = 0.0288985, antiderivative size = 49, normalized size of antiderivative = 1., 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 741
Rule 801
Rule 635
Rule 203
Rule 260
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*}
Mathematica [A] time = 0.0195791, 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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Maple [A] time = 0.011, size = 36, normalized size = 0.7 \begin{align*} -{\frac{1}{4\,{x}^{2}+4}}+{\frac{\ln \left ({x}^{2}+1 \right ) }{4}}+{\frac{\arctan \left ( x \right ) }{4}}-{\frac{1}{-4+4\,x}}-{\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 = 1.43306, size = 53, normalized size = 1.08 \begin{align*} -\frac{x^{2} + x}{4 \,{\left (x^{3} - x^{2} + x - 1\right )}} + \frac{1}{4} \, \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 = 1.93164, size = 186, normalized size = 3.8 \begin{align*} -\frac{x^{2} -{\left (x^{3} - x^{2} + x - 1\right )} \arctan \left (x\right ) -{\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.161398, size = 39, normalized size = 0.8 \begin{align*} - \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}{\left (x \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.11861, size = 108, normalized size = 2.2 \begin{align*} \frac{1}{16} \, \pi - \frac{1}{4} \, \pi \left \lfloor \frac{\pi + 4 \, \arctan \left (x\right )}{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 \left (x\right ) + \frac{1}{4} \, \log \left (\frac{2}{x - 1} + \frac{2}{{\left (x - 1\right )}^{2}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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