3.267 \(\int \frac{3 x-4 x^2+3 x^3}{1+x^2} \, dx\)

Optimal. Leaf size=15 \[ \frac{3 x^2}{2}-4 x+4 \tan ^{-1}(x) \]

[Out]

-4*x + (3*x^2)/2 + 4*ArcTan[x]

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Rubi [A]  time = 0.0401547, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{3 x^2}{2}-4 x+4 \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(3*x - 4*x^2 + 3*x^3)/(1 + x^2),x]

[Out]

-4*x + (3*x^2)/2 + 4*ArcTan[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - 4 x + 4 \operatorname{atan}{\left (x \right )} + 3 \int x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3*x**3-4*x**2+3*x)/(x**2+1),x)

[Out]

-4*x + 4*atan(x) + 3*Integral(x, x)

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Mathematica [A]  time = 0.00745848, size = 15, normalized size = 1. \[ \frac{3 x^2}{2}-4 x+4 \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Integrate[(3*x - 4*x^2 + 3*x^3)/(1 + x^2),x]

[Out]

-4*x + (3*x^2)/2 + 4*ArcTan[x]

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Maple [A]  time = 0.004, size = 14, normalized size = 0.9 \[ -4\,x+{\frac{3\,{x}^{2}}{2}}+4\,\arctan \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3*x^3-4*x^2+3*x)/(x^2+1),x)

[Out]

-4*x+3/2*x^2+4*arctan(x)

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Maxima [A]  time = 0.904657, size = 18, normalized size = 1.2 \[ \frac{3}{2} \, x^{2} - 4 \, x + 4 \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^3 - 4*x^2 + 3*x)/(x^2 + 1),x, algorithm="maxima")

[Out]

3/2*x^2 - 4*x + 4*arctan(x)

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Fricas [A]  time = 0.253062, size = 18, normalized size = 1.2 \[ \frac{3}{2} \, x^{2} - 4 \, x + 4 \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^3 - 4*x^2 + 3*x)/(x^2 + 1),x, algorithm="fricas")

[Out]

3/2*x^2 - 4*x + 4*arctan(x)

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Sympy [A]  time = 0.169562, size = 14, normalized size = 0.93 \[ \frac{3 x^{2}}{2} - 4 x + 4 \operatorname{atan}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x**3-4*x**2+3*x)/(x**2+1),x)

[Out]

3*x**2/2 - 4*x + 4*atan(x)

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GIAC/XCAS [A]  time = 0.258449, size = 18, normalized size = 1.2 \[ \frac{3}{2} \, x^{2} - 4 \, x + 4 \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^3 - 4*x^2 + 3*x)/(x^2 + 1),x, algorithm="giac")

[Out]

3/2*x^2 - 4*x + 4*arctan(x)