Optimal. Leaf size=29 \[ -\frac {x}{2 \left (x^2+1\right )}+\frac {1}{2} \log \left (x^2+1\right )-\frac {1}{2} \tan ^{-1}(x) \]
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Rubi [A] time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1814, 635, 203, 260} \[ -\frac {x}{2 \left (x^2+1\right )}+\frac {1}{2} \log \left (x^2+1\right )-\frac {1}{2} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 635
Rule 1814
Rubi steps
\begin {align*} \int \frac {-1+x+x^3}{\left (1+x^2\right )^2} \, dx &=-\frac {x}{2 \left (1+x^2\right )}-\frac {1}{2} \int \frac {1-2 x}{1+x^2} \, dx\\ &=-\frac {x}{2 \left (1+x^2\right )}-\frac {1}{2} \int \frac {1}{1+x^2} \, dx+\int \frac {x}{1+x^2} \, dx\\ &=-\frac {x}{2 \left (1+x^2\right )}-\frac {1}{2} \tan ^{-1}(x)+\frac {1}{2} \log \left (1+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 0.86 \[ \frac {1}{2} \left (-\frac {x}{x^2+1}+\log \left (x^2+1\right )-\tan ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 32, normalized size = 1.10 \[ -\frac {{\left (x^{2} + 1\right )} \arctan \relax (x) - {\left (x^{2} + 1\right )} \log \left (x^{2} + 1\right ) + x}{2 \, {\left (x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 23, normalized size = 0.79 \[ -\frac {x}{2 \, {\left (x^{2} + 1\right )}} - \frac {1}{2} \, \arctan \relax (x) + \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 24, normalized size = 0.83 \[ -\frac {x}{2 \left (x^{2}+1\right )}-\frac {\arctan \relax (x )}{2}+\frac {\ln \left (x^{2}+1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.33, size = 23, normalized size = 0.79 \[ -\frac {x}{2 \, {\left (x^{2} + 1\right )}} - \frac {1}{2} \, \arctan \relax (x) + \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.13, size = 25, normalized size = 0.86 \[ \frac {\ln \left (x^2+1\right )}{2}-\frac {\mathrm {atan}\relax (x)}{2}-\frac {x}{2\,\left (x^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 20, normalized size = 0.69 \[ - \frac {x}{2 x^{2} + 2} + \frac {\log {\left (x^{2} + 1 \right )}}{2} - \frac {\operatorname {atan}{\relax (x )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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