Optimal. Leaf size=15 \[ \frac {3 x^2}{2}-4 x+4 \tan ^{-1}(x) \]
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Rubi [A] time = 0.03, 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 = {1594, 1802, 203} \[ \frac {3 x^2}{2}-4 x+4 \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 203
Rule 1594
Rule 1802
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 \[ \frac {3 x^2}{2}-4 x+4 \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 13, normalized size = 0.87 \[ \frac {3}{2} \, x^{2} - 4 \, x + 4 \, \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 13, normalized size = 0.87 \[ \frac {3}{2} \, x^{2} - 4 \, x + 4 \, \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 14, normalized size = 0.93 \[ \frac {3 x^{2}}{2}-4 x +4 \arctan \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.30, size = 13, normalized size = 0.87 \[ \frac {3}{2} \, x^{2} - 4 \, x + 4 \, \arctan \relax (x) \]
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 \[ 4\,\mathrm {atan}\relax (x)-4\,x+\frac {3\,x^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 14, normalized size = 0.93 \[ \frac {3 x^{2}}{2} - 4 x + 4 \operatorname {atan}{\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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