Optimal. Leaf size=49 \[ \frac {1}{4 (1-x)}-\frac {1}{4 \left (1+x^2\right )}+\frac {1}{4} \tan ^{-1}(x)-\frac {1}{2} \log (1-x)+\frac {1}{4} \log \left (1+x^2\right ) \]
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Rubi [A]
time = 0.02, 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 = {755, 815, 649,
209, 266} \begin {gather*} -\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) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 266
Rule 649
Rule 755
Rule 815
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 \begin {gather*} \frac {1}{4} \left (\frac {1}{1-x}-\frac {1}{1+x^2}+\tan ^{-1}(x)-2 \log (-1+x)+\log \left (1+x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 2.00, size = 49, normalized size = 1.00 \begin {gather*} \frac {-\frac {x \left (1+x\right )}{4}+\frac {\left (-1+x-x^2+x^3\right ) \left (\text {ArcTan}\left [x\right ]+\text {Log}\left [1+x^2\right ]-2 \text {Log}\left [-1+x\right ]\right )}{4}}{-1+x-x^2+x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 36, normalized size = 0.73
method | result | size |
default | \(-\frac {1}{4 \left (-1+x \right )}-\frac {\ln \left (-1+x \right )}{2}-\frac {1}{4 \left (x^{2}+1\right )}+\frac {\ln \left (x^{2}+1\right )}{4}+\frac {\arctan \left (x \right )}{4}\) | \(36\) |
risch | \(\frac {-\frac {1}{4} x^{2}-\frac {1}{4} x}{\left (x^{2}+1\right ) \left (-1+x \right )}-\frac {\ln \left (-1+x \right )}{2}+\frac {\ln \left (x^{2}+1\right )}{4}+\frac {\arctan \left (x \right )}{4}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 39, normalized size = 0.80 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 71, normalized size = 1.45 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 41, normalized size = 0.84 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 45, normalized size = 0.92 \begin {gather*} -\frac {\ln \left |x-1\right |}{2}+\frac {\ln \left (x^{2}+1\right )}{4}+\frac {\arctan x}{4}+\frac {-x^{2}-x}{4 \left (x^{3}-x^{2}+x-1\right )} \end {gather*}
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 \begin {gather*} -\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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