Optimal. Leaf size=19 \[ \frac {x^3}{3}+\log \left (x^2+1\right )-x+\tan ^{-1}(x) \]
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Rubi [A] time = 0.03, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1593, 1802, 635, 203, 260} \begin {gather*} \frac {x^3}{3}+\log \left (x^2+1\right )-x+\tan ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 635
Rule 1593
Rule 1802
Rubi steps
\begin {align*} \int \frac {2 x+x^4}{1+x^2} \, dx &=\int \frac {x \left (2+x^3\right )}{1+x^2} \, dx\\ &=\int \left (-1+x^2+\frac {1+2 x}{1+x^2}\right ) \, dx\\ &=-x+\frac {x^3}{3}+\int \frac {1+2 x}{1+x^2} \, dx\\ &=-x+\frac {x^3}{3}+2 \int \frac {x}{1+x^2} \, dx+\int \frac {1}{1+x^2} \, dx\\ &=-x+\frac {x^3}{3}+\tan ^{-1}(x)+\log \left (1+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 19, normalized size = 1.00 \begin {gather*} \frac {x^3}{3}+\log \left (x^2+1\right )-x+\tan ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 x+x^4}{1+x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.81, size = 17, normalized size = 0.89 \begin {gather*} \frac {1}{3} \, x^{3} - x + \arctan \relax (x) + \log \left (x^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 17, normalized size = 0.89 \begin {gather*} \frac {1}{3} \, x^{3} - x + \arctan \relax (x) + \log \left (x^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 18, normalized size = 0.95 \begin {gather*} \frac {x^{3}}{3}-x +\arctan \relax (x )+\ln \left (x^{2}+1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.03, size = 17, normalized size = 0.89 \begin {gather*} \frac {1}{3} \, x^{3} - x + \arctan \relax (x) + \log \left (x^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.10, size = 17, normalized size = 0.89 \begin {gather*} \ln \left (x^2+1\right )-x+\mathrm {atan}\relax (x)+\frac {x^3}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 15, normalized size = 0.79 \begin {gather*} \frac {x^{3}}{3} - x + \log {\left (x^{2} + 1 \right )} + \operatorname {atan}{\relax (x )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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