Optimal. Leaf size=22 \[ \frac {x^2}{2}-\frac {2}{7} \tanh ^{-1}\left (\frac {1}{7} (1+2 x)\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.18, number of steps
used = 5, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1671, 630, 31}
\begin {gather*} \frac {x^2}{2}+\frac {1}{7} \log (3-x)-\frac {1}{7} \log (x+4) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 630
Rule 1671
Rubi steps
\begin {align*} \int \frac {1-12 x+x^2+x^3}{-12+x+x^2} \, dx &=\int \left (x+\frac {1}{-12+x+x^2}\right ) \, dx\\ &=\frac {x^2}{2}+\int \frac {1}{-12+x+x^2} \, dx\\ &=\frac {x^2}{2}+\frac {1}{7} \int \frac {1}{-3+x} \, dx-\frac {1}{7} \int \frac {1}{4+x} \, dx\\ &=\frac {x^2}{2}+\frac {1}{7} \log (3-x)-\frac {1}{7} \log (4+x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 26, normalized size = 1.18 \begin {gather*} \frac {x^2}{2}+\frac {1}{7} \log (3-x)-\frac {1}{7} \log (4+x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 19, normalized size = 0.86
method | result | size |
default | \(\frac {x^{2}}{2}-\frac {\ln \left (x +4\right )}{7}+\frac {\ln \left (x -3\right )}{7}\) | \(19\) |
norman | \(\frac {x^{2}}{2}-\frac {\ln \left (x +4\right )}{7}+\frac {\ln \left (x -3\right )}{7}\) | \(19\) |
risch | \(\frac {x^{2}}{2}-\frac {\ln \left (x +4\right )}{7}+\frac {\ln \left (x -3\right )}{7}\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 18, normalized size = 0.82 \begin {gather*} \frac {1}{2} \, x^{2} - \frac {1}{7} \, \log \left (x + 4\right ) + \frac {1}{7} \, \log \left (x - 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 18, normalized size = 0.82 \begin {gather*} \frac {1}{2} \, x^{2} - \frac {1}{7} \, \log \left (x + 4\right ) + \frac {1}{7} \, \log \left (x - 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 17, normalized size = 0.77 \begin {gather*} \frac {x^{2}}{2} + \frac {\log {\left (x - 3 \right )}}{7} - \frac {\log {\left (x + 4 \right )}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.65, size = 20, normalized size = 0.91 \begin {gather*} \frac {1}{2} \, x^{2} - \frac {1}{7} \, \log \left ({\left | x + 4 \right |}\right ) + \frac {1}{7} \, \log \left ({\left | x - 3 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 14, normalized size = 0.64 \begin {gather*} \frac {x^2}{2}-\frac {2\,\mathrm {atanh}\left (\frac {2\,x}{7}+\frac {1}{7}\right )}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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