Optimal. Leaf size=21 \[ -3 x+\frac {x^2}{2}+\frac {1}{2} \log \left (1+x^2\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1824, 266}
\begin {gather*} \frac {x^2}{2}+\frac {1}{2} \log \left (x^2+1\right )-3 x \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 1824
Rubi steps
\begin {align*} \int \frac {-3+2 x-3 x^2+x^3}{1+x^2} \, dx &=\int \left (-3+x+\frac {x}{1+x^2}\right ) \, dx\\ &=-3 x+\frac {x^2}{2}+\int \frac {x}{1+x^2} \, dx\\ &=-3 x+\frac {x^2}{2}+\frac {1}{2} \log \left (1+x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 21, normalized size = 1.00 \begin {gather*} -3 x+\frac {x^2}{2}+\frac {1}{2} \log \left (1+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 18, normalized size = 0.86
method | result | size |
default | \(-3 x +\frac {x^{2}}{2}+\frac {\ln \left (x^{2}+1\right )}{2}\) | \(18\) |
norman | \(-3 x +\frac {x^{2}}{2}+\frac {\ln \left (x^{2}+1\right )}{2}\) | \(18\) |
meijerg | \(-3 x +\frac {x^{2}}{2}+\frac {\ln \left (x^{2}+1\right )}{2}\) | \(18\) |
risch | \(-3 x +\frac {x^{2}}{2}+\frac {\ln \left (x^{2}+1\right )}{2}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 17, normalized size = 0.81 \begin {gather*} \frac {1}{2} \, x^{2} - 3 \, x + \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 = 0.40, size = 17, normalized size = 0.81 \begin {gather*} \frac {1}{2} \, x^{2} - 3 \, x + \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 = 0.02, size = 15, normalized size = 0.71 \begin {gather*} \frac {x^{2}}{2} - 3 x + \frac {\log {\left (x^{2} + 1 \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.51, size = 17, normalized size = 0.81 \begin {gather*} \frac {1}{2} \, x^{2} - 3 \, x + \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 = 17, normalized size = 0.81 \begin {gather*} \frac {\ln \left (x^2+1\right )}{2}-3\,x+\frac {x^2}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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