Optimal. Leaf size=45 \[ -\frac {x}{2 \left (x^2+1\right )}-\frac {1}{2 x^2}+\frac {1}{2} \log \left (x^2+1\right )-\frac {1}{x}-\log (x)-\frac {3}{2} \tan ^{-1}(x) \]
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Rubi [A] time = 0.06, antiderivative size = 45, 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, 1802, 635, 203, 260} \[ -\frac {x}{2 \left (x^2+1\right )}-\frac {1}{2 x^2}+\frac {1}{2} \log \left (x^2+1\right )-\frac {1}{x}-\log (x)-\frac {3}{2} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 635
Rule 1802
Rule 1805
Rubi steps
\begin {align*} \int \frac {1+x+x^2}{x^3 \left (1+x^2\right )^2} \, dx &=-\frac {x}{2 \left (1+x^2\right )}-\frac {1}{2} \int \frac {-2-2 x+x^3}{x^3 \left (1+x^2\right )} \, dx\\ &=-\frac {x}{2 \left (1+x^2\right )}-\frac {1}{2} \int \left (-\frac {2}{x^3}-\frac {2}{x^2}+\frac {2}{x}+\frac {3-2 x}{1+x^2}\right ) \, dx\\ &=-\frac {1}{2 x^2}-\frac {1}{x}-\frac {x}{2 \left (1+x^2\right )}-\log (x)-\frac {1}{2} \int \frac {3-2 x}{1+x^2} \, dx\\ &=-\frac {1}{2 x^2}-\frac {1}{x}-\frac {x}{2 \left (1+x^2\right )}-\log (x)-\frac {3}{2} \int \frac {1}{1+x^2} \, dx+\int \frac {x}{1+x^2} \, dx\\ &=-\frac {1}{2 x^2}-\frac {1}{x}-\frac {x}{2 \left (1+x^2\right )}-\frac {3}{2} \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 = 39, normalized size = 0.87 \[ \frac {1}{2} \left (-\frac {x}{x^2+1}-\frac {1}{x^2}+\log \left (x^2+1\right )-\frac {2}{x}-2 \log (x)-3 \tan ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.07, size = 61, normalized size = 1.36 \[ -\frac {3 \, x^{3} + x^{2} + 3 \, {\left (x^{4} + x^{2}\right )} \arctan \relax (x) - {\left (x^{4} + x^{2}\right )} \log \left (x^{2} + 1\right ) + 2 \, {\left (x^{4} + x^{2}\right )} \log \relax (x) + 2 \, x + 1}{2 \, {\left (x^{4} + x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 43, normalized size = 0.96 \[ -\frac {3 \, x^{3} + x^{2} + 2 \, x + 1}{2 \, {\left (x^{2} + 1\right )} x^{2}} - \frac {3}{2} \, \arctan \relax (x) + \frac {1}{2} \, \log \left (x^{2} + 1\right ) - \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 38, normalized size = 0.84 \[ -\frac {x}{2 \left (x^{2}+1\right )}-\frac {3 \arctan \relax (x )}{2}-\ln \relax (x )+\frac {\ln \left (x^{2}+1\right )}{2}-\frac {1}{x}-\frac {1}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 41, normalized size = 0.91 \[ -\frac {3 \, x^{3} + x^{2} + 2 \, x + 1}{2 \, {\left (x^{4} + x^{2}\right )}} - \frac {3}{2} \, \arctan \relax (x) + \frac {1}{2} \, \log \left (x^{2} + 1\right ) - \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 47, normalized size = 1.04 \[ -\ln \relax (x)-\frac {\frac {3\,x^3}{2}+\frac {x^2}{2}+x+\frac {1}{2}}{x^4+x^2}+\ln \left (x-\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {3}{4}{}\mathrm {i}\right )+\ln \left (x+1{}\mathrm {i}\right )\,\left (\frac {1}{2}-\frac {3}{4}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 42, normalized size = 0.93 \[ - \log {\relax (x )} + \frac {\log {\left (x^{2} + 1 \right )}}{2} - \frac {3 \operatorname {atan}{\relax (x )}}{2} + \frac {- 3 x^{3} - x^{2} - 2 x - 1}{2 x^{4} + 2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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