Optimal. Leaf size=30 \[ \frac{x^3}{3}+\frac{x^2}{2}-\frac{1}{2} \log \left (x^2+1\right )-x+\tan ^{-1}(x) \]
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Rubi [A] time = 0.0252424, antiderivative size = 30, normalized size of antiderivative = 1., 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, 801, 635, 203, 260} \[ \frac{x^3}{3}+\frac{x^2}{2}-\frac{1}{2} \log \left (x^2+1\right )-x+\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 1593
Rule 801
Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int \frac{x^3+x^4}{1+x^2} \, dx &=\int \frac{x^3 (1+x)}{1+x^2} \, dx\\ &=\int \left (-1+x+x^2+\frac{1-x}{1+x^2}\right ) \, dx\\ &=-x+\frac{x^2}{2}+\frac{x^3}{3}+\int \frac{1-x}{1+x^2} \, dx\\ &=-x+\frac{x^2}{2}+\frac{x^3}{3}+\int \frac{1}{1+x^2} \, dx-\int \frac{x}{1+x^2} \, dx\\ &=-x+\frac{x^2}{2}+\frac{x^3}{3}+\tan ^{-1}(x)-\frac{1}{2} \log \left (1+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0053304, size = 30, normalized size = 1. \[ \frac{x^3}{3}+\frac{x^2}{2}-\frac{1}{2} \log \left (x^2+1\right )-x+\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 25, normalized size = 0.8 \begin{align*} -x+{\frac{{x}^{2}}{2}}+{\frac{{x}^{3}}{3}}+\arctan \left ( x \right ) -{\frac{\ln \left ({x}^{2}+1 \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52162, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{3} \, x^{3} + \frac{1}{2} \, x^{2} - x + \arctan \left (x\right ) - \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45196, size = 73, normalized size = 2.43 \begin{align*} \frac{1}{3} \, x^{3} + \frac{1}{2} \, x^{2} - x + \arctan \left (x\right ) - \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.089859, size = 22, normalized size = 0.73 \begin{align*} \frac{x^{3}}{3} + \frac{x^{2}}{2} - x - \frac{\log{\left (x^{2} + 1 \right )}}{2} + \operatorname{atan}{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13593, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{3} \, x^{3} + \frac{1}{2} \, x^{2} - x + \arctan \left (x\right ) - \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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