Optimal. Leaf size=30 \[ -x+\frac {x^2}{2}+\frac {x^3}{3}+\tan ^{-1}(x)-\frac {1}{2} \log \left (1+x^2\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 30, 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 = {1607, 815, 649,
209, 266} \begin {gather*} \text {ArcTan}(x)+\frac {x^3}{3}+\frac {x^2}{2}-\frac {1}{2} \log \left (x^2+1\right )-x \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 266
Rule 649
Rule 815
Rule 1607
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*}
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Mathematica [A]
time = 0.00, size = 30, normalized size = 1.00 \begin {gather*} -x+\frac {x^2}{2}+\frac {x^3}{3}+\tan ^{-1}(x)-\frac {1}{2} \log \left (1+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 25, normalized size = 0.83
method | result | size |
default | \(-x +\frac {x^{2}}{2}+\frac {x^{3}}{3}+\arctan \left (x \right )-\frac {\ln \left (x^{2}+1\right )}{2}\) | \(25\) |
risch | \(-x +\frac {x^{2}}{2}+\frac {x^{3}}{3}+\arctan \left (x \right )-\frac {\ln \left (x^{2}+1\right )}{2}\) | \(25\) |
meijerg | \(-\frac {x \left (-5 x^{2}+15\right )}{15}+\arctan \left (x \right )+\frac {x^{2}}{2}-\frac {\ln \left (x^{2}+1\right )}{2}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 24, normalized size = 0.80 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.25, size = 24, normalized size = 0.80 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 22, normalized size = 0.73 \begin {gather*} \frac {x^{3}}{3} + \frac {x^{2}}{2} - x - \frac {\log {\left (x^{2} + 1 \right )}}{2} + \operatorname {atan}{\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.96, size = 24, normalized size = 0.80 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 24, normalized size = 0.80 \begin {gather*} \mathrm {atan}\left (x\right )-\frac {\ln \left (x^2+1\right )}{2}-x+\frac {x^2}{2}+\frac {x^3}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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