Optimal. Leaf size=16 \[ \frac {1}{2 (x+1)}+\frac {1}{2} \tan ^{-1}(x) \]
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Rubi [A] time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {801, 203} \[ \frac {1}{2 (x+1)}+\frac {1}{2} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 203
Rule 801
Rubi steps
\begin {align*} \int \frac {x}{(1+x)^2 \left (1+x^2\right )} \, dx &=\int \left (-\frac {1}{2 (1+x)^2}+\frac {1}{2 \left (1+x^2\right )}\right ) \, dx\\ &=\frac {1}{2 (1+x)}+\frac {1}{2} \int \frac {1}{1+x^2} \, dx\\ &=\frac {1}{2 (1+x)}+\frac {1}{2} \tan ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 12, normalized size = 0.75 \[ \frac {1}{2} \left (\frac {1}{x+1}+\tan ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 15, normalized size = 0.94 \[ \frac {{\left (x + 1\right )} \arctan \relax (x) + 1}{2 \, {\left (x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 32, normalized size = 2.00 \[ -\frac {1}{8} \, \pi - \frac {1}{2} \, \pi \left \lfloor -\frac {\pi - 4 \, \arctan \relax (x)}{4 \, \pi } + \frac {1}{2} \right \rfloor + \frac {1}{2 \, {\left (x + 1\right )}} + \frac {1}{2} \, \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 13, normalized size = 0.81 \[ \frac {\arctan \relax (x )}{2}+\frac {1}{2 x +2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.79, size = 12, normalized size = 0.75 \[ \frac {1}{2 \, {\left (x + 1\right )}} + \frac {1}{2} \, \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.22, size = 12, normalized size = 0.75 \[ \frac {\mathrm {atan}\relax (x)}{2}+\frac {1}{2\,\left (x+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 10, normalized size = 0.62 \[ \frac {\operatorname {atan}{\relax (x )}}{2} + \frac {1}{2 x + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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