Optimal. Leaf size=33 \[ \frac {1}{2 \left (x^2+1\right )}-\frac {1}{2} \log \left (x^2+1\right )-\frac {1}{x}+\log (x)-\tan ^{-1}(x) \]
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Rubi [A] time = 0.04, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {1805, 801, 635, 203, 260} \[ \frac {1}{2 \left (x^2+1\right )}-\frac {1}{2} \log \left (x^2+1\right )-\frac {1}{x}+\log (x)-\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 635
Rule 801
Rule 1805
Rubi steps
\begin {align*} \int \frac {1+x+x^2}{x^2 \left (1+x^2\right )^2} \, dx &=\frac {1}{2 \left (1+x^2\right )}-\frac {1}{2} \int \frac {-2-2 x}{x^2 \left (1+x^2\right )} \, dx\\ &=\frac {1}{2 \left (1+x^2\right )}-\frac {1}{2} \int \left (-\frac {2}{x^2}-\frac {2}{x}+\frac {2 (1+x)}{1+x^2}\right ) \, dx\\ &=-\frac {1}{x}+\frac {1}{2 \left (1+x^2\right )}+\log (x)-\int \frac {1+x}{1+x^2} \, dx\\ &=-\frac {1}{x}+\frac {1}{2 \left (1+x^2\right )}+\log (x)-\int \frac {1}{1+x^2} \, dx-\int \frac {x}{1+x^2} \, dx\\ &=-\frac {1}{x}+\frac {1}{2 \left (1+x^2\right )}-\tan ^{-1}(x)+\log (x)-\frac {1}{2} \log \left (1+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 1.00 \[ \frac {1}{2 \left (x^2+1\right )}-\frac {1}{2} \log \left (x^2+1\right )-\frac {1}{x}+\log (x)-\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 49, normalized size = 1.48 \[ -\frac {2 \, x^{2} + 2 \, {\left (x^{3} + x\right )} \arctan \relax (x) + {\left (x^{3} + x\right )} \log \left (x^{2} + 1\right ) - 2 \, {\left (x^{3} + x\right )} \log \relax (x) - x + 2}{2 \, {\left (x^{3} + x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 35, normalized size = 1.06 \[ -\frac {2 \, x^{2} - x + 2}{2 \, {\left (x^{3} + x\right )}} - \arctan \relax (x) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) + \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 30, normalized size = 0.91 \[ -\arctan \relax (x )+\ln \relax (x )-\frac {\ln \left (x^{2}+1\right )}{2}-\frac {1}{x}+\frac {1}{2 x^{2}+2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 34, normalized size = 1.03 \[ -\frac {2 \, x^{2} - x + 2}{2 \, {\left (x^{3} + x\right )}} - \arctan \relax (x) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.81, size = 38, normalized size = 1.15 \[ \ln \relax (x)-\frac {x^2-\frac {x}{2}+1}{x^3+x}+\ln \left (x-\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )+\ln \left (x+1{}\mathrm {i}\right )\,\left (-\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 31, normalized size = 0.94 \[ \log {\relax (x )} - \frac {\log {\left (x^{2} + 1 \right )}}{2} - \operatorname {atan}{\relax (x )} + \frac {- 2 x^{2} + x - 2}{2 x^{3} + 2 x} \]
Verification of antiderivative is not currently implemented for this CAS.
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