Optimal. Leaf size=15 \[ -4 x+\frac {3 x^2}{2}+4 \tan ^{-1}(x) \]
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Rubi [A]
time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1608, 1816,
209} \begin {gather*} 4 \text {ArcTan}(x)+\frac {3 x^2}{2}-4 x \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 1608
Rule 1816
Rubi steps
\begin {align*} \int \frac {3 x-4 x^2+3 x^3}{1+x^2} \, dx &=\int \frac {x \left (3-4 x+3 x^2\right )}{1+x^2} \, dx\\ &=\int \left (-4+3 x+\frac {4}{1+x^2}\right ) \, dx\\ &=-4 x+\frac {3 x^2}{2}+4 \int \frac {1}{1+x^2} \, dx\\ &=-4 x+\frac {3 x^2}{2}+4 \tan ^{-1}(x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} -4 x+\frac {3 x^2}{2}+4 \tan ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 14, normalized size = 0.93
method | result | size |
default | \(-4 x +\frac {3 x^{2}}{2}+4 \arctan \left (x \right )\) | \(14\) |
meijerg | \(-4 x +\frac {3 x^{2}}{2}+4 \arctan \left (x \right )\) | \(14\) |
risch | \(-4 x +\frac {3 x^{2}}{2}+4 \arctan \left (x \right )\) | \(14\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 13, normalized size = 0.87 \begin {gather*} \frac {3}{2} \, x^{2} - 4 \, x + 4 \, \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 13, normalized size = 0.87 \begin {gather*} \frac {3}{2} \, x^{2} - 4 \, x + 4 \, \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 14, normalized size = 0.93 \begin {gather*} \frac {3 x^{2}}{2} - 4 x + 4 \operatorname {atan}{\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.93, size = 13, normalized size = 0.87 \begin {gather*} \frac {3}{2} \, x^{2} - 4 \, x + 4 \, \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.13, size = 13, normalized size = 0.87 \begin {gather*} 4\,\mathrm {atan}\left (x\right )-4\,x+\frac {3\,x^2}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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