Optimal. Leaf size=68 \[ -\frac {1}{2} \log \left (x^2-2 x+2\right )-\frac {1}{x}-\frac {\log \left (\frac {1-(1-x)^2}{(x-1)^2+1}\right )}{x}+\frac {1}{2} \log (2-x)+\frac {\log (x)}{2}+\tan ^{-1}(1-x) \]
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Rubi [A] time = 0.25, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2525, 12, 6728, 634, 617, 204, 628} \[ -\frac {1}{2} \log \left (x^2-2 x+2\right )-\frac {1}{x}-\frac {\log \left (\frac {1-(1-x)^2}{(x-1)^2+1}\right )}{x}+\frac {1}{2} \log (2-x)+\frac {\log (x)}{2}+\tan ^{-1}(1-x) \]
Antiderivative was successfully verified.
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Rule 12
Rule 204
Rule 617
Rule 628
Rule 634
Rule 2525
Rule 6728
Rubi steps
\begin {align*} \int \frac {\log \left (\frac {1-(-1+x)^2}{1+(-1+x)^2}\right )}{x^2} \, dx &=-\frac {\log \left (\frac {1-(1-x)^2}{1+(-1+x)^2}\right )}{x}+\int \frac {4 (1-x)}{(2-x) x^2 \left (2-2 x+x^2\right )} \, dx\\ &=-\frac {\log \left (\frac {1-(1-x)^2}{1+(-1+x)^2}\right )}{x}+4 \int \frac {1-x}{(2-x) x^2 \left (2-2 x+x^2\right )} \, dx\\ &=-\frac {\log \left (\frac {1-(1-x)^2}{1+(-1+x)^2}\right )}{x}+4 \int \left (\frac {1}{8 (-2+x)}+\frac {1}{4 x^2}+\frac {1}{8 x}-\frac {x}{4 \left (2-2 x+x^2\right )}\right ) \, dx\\ &=-\frac {1}{x}-\frac {\log \left (\frac {1-(1-x)^2}{1+(-1+x)^2}\right )}{x}+\frac {1}{2} \log (2-x)+\frac {\log (x)}{2}-\int \frac {x}{2-2 x+x^2} \, dx\\ &=-\frac {1}{x}-\frac {\log \left (\frac {1-(1-x)^2}{1+(-1+x)^2}\right )}{x}+\frac {1}{2} \log (2-x)+\frac {\log (x)}{2}-\frac {1}{2} \int \frac {-2+2 x}{2-2 x+x^2} \, dx-\int \frac {1}{2-2 x+x^2} \, dx\\ &=-\frac {1}{x}-\frac {\log \left (\frac {1-(1-x)^2}{1+(-1+x)^2}\right )}{x}+\frac {1}{2} \log (2-x)+\frac {\log (x)}{2}-\frac {1}{2} \log \left (2-2 x+x^2\right )-\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-x\right )\\ &=-\frac {1}{x}+\tan ^{-1}(1-x)-\frac {\log \left (\frac {1-(1-x)^2}{1+(-1+x)^2}\right )}{x}+\frac {1}{2} \log (2-x)+\frac {\log (x)}{2}-\frac {1}{2} \log \left (2-2 x+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 63, normalized size = 0.93 \[ -\frac {\log \left (-\frac {(x-2) x}{x^2-2 x+2}\right )}{x}-\frac {1}{2} \log \left (x^2-2 x+2\right )-\frac {1}{x}+\frac {1}{2} \log (2-x)+\frac {\log (x)}{2}+\tan ^{-1}(1-x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 58, normalized size = 0.85 \[ -\frac {2 \, x \arctan \left (x - 1\right ) + x \log \left (x^{2} - 2 \, x + 2\right ) - x \log \left (x^{2} - 2 \, x\right ) + 2 \, \log \left (-\frac {x^{2} - 2 \, x}{x^{2} - 2 \, x + 2}\right ) + 2}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 59, normalized size = 0.87 \[ -\frac {\log \left (-\frac {{\left (x - 1\right )}^{2} - 1}{{\left (x - 1\right )}^{2} + 1}\right )}{x} - \frac {1}{x} - \arctan \left (x - 1\right ) - \frac {1}{2} \, \log \left (x^{2} - 2 \, x + 2\right ) + \frac {1}{2} \, \log \left ({\left | x - 2 \right |}\right ) + \frac {1}{2} \, \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 57, normalized size = 0.84 \[ -\arctan \left (x -1\right )+\frac {\ln \relax (x )}{2}+\frac {\ln \left (x -2\right )}{2}-\frac {\ln \left (x^{2}-2 x +2\right )}{2}-\frac {\ln \left (\frac {\left (-x +2\right ) x}{x^{2}-2 x +2}\right )}{x}-\frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.29, size = 57, normalized size = 0.84 \[ -\frac {\log \left (-\frac {{\left (x - 1\right )}^{2} - 1}{{\left (x - 1\right )}^{2} + 1}\right )}{x} - \frac {1}{x} - \arctan \left (x - 1\right ) - \frac {1}{2} \, \log \left (x^{2} - 2 \, x + 2\right ) + \frac {1}{2} \, \log \left (x - 2\right ) + \frac {1}{2} \, \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.46, size = 59, normalized size = 0.87 \[ \frac {\ln \left (x\,\left (x-2\right )\right )}{2}-\mathrm {atan}\left (x-1\right )-\frac {\ln \left (x^2-2\,x+2\right )}{2}-\frac {\ln \left (2\,x-x^2\right )}{x}+\frac {\ln \left (x^2-2\,x+2\right )}{x}-\frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.25, size = 46, normalized size = 0.68 \[ \frac {\log {\left (x^{2} - 2 x \right )}}{2} - \frac {\log {\left (x^{2} - 2 x + 2 \right )}}{2} - \operatorname {atan}{\left (x - 1 \right )} - \frac {\log {\left (\frac {1 - \left (x - 1\right )^{2}}{\left (x - 1\right )^{2} + 1} \right )}}{x} - \frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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