3.309 \(\int \frac{1-3 x+2 x^2-x^3}{\left (1+x^2\right )^2} \, dx\)

Optimal. Leaf size=33 \[ \frac{2-x}{2 \left (x^2+1\right )}-\frac{1}{2} \log \left (x^2+1\right )+\frac{3}{2} \tan ^{-1}(x) \]

[Out]

(2 - x)/(2*(1 + x^2)) + (3*ArcTan[x])/2 - Log[1 + x^2]/2

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Rubi [A]  time = 0.0369622, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{2-x}{2 \left (x^2+1\right )}-\frac{1}{2} \log \left (x^2+1\right )+\frac{3}{2} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

(2 - x)/(2*(1 + x^2)) + (3*ArcTan[x])/2 - Log[1 + x^2]/2

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Rubi in Sympy [A]  time = 11.7374, size = 27, normalized size = 0.82 \[ - \frac{x \left (2 x + 1\right )}{2 \left (x^{2} + 1\right )} - \frac{\log{\left (x^{2} + 1 \right )}}{2} + \frac{3 \operatorname{atan}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*(2*x + 1)/(2*(x**2 + 1)) - log(x**2 + 1)/2 + 3*atan(x)/2

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Mathematica [A]  time = 0.0178586, size = 30, normalized size = 0.91 \[ \frac{1}{2} \left (\frac{2-x}{x^2+1}-\log \left (x^2+1\right )+3 \tan ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

((2 - x)/(1 + x^2) + 3*ArcTan[x] - Log[1 + x^2])/2

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Maple [A]  time = 0.007, size = 28, normalized size = 0.9 \[ -{\frac{1}{{x}^{2}+1} \left ({\frac{x}{2}}-1 \right ) }-{\frac{\ln \left ({x}^{2}+1 \right ) }{2}}+{\frac{3\,\arctan \left ( x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(1/2*x-1)/(x^2+1)-1/2*ln(x^2+1)+3/2*arctan(x)

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Maxima [A]  time = 0.883924, size = 34, normalized size = 1.03 \[ -\frac{x - 2}{2 \,{\left (x^{2} + 1\right )}} + \frac{3}{2} \, \arctan \left (x\right ) - \frac{1}{2} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(x - 2)/(x^2 + 1) + 3/2*arctan(x) - 1/2*log(x^2 + 1)

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Fricas [A]  time = 0.255712, size = 49, normalized size = 1.48 \[ \frac{3 \,{\left (x^{2} + 1\right )} \arctan \left (x\right ) -{\left (x^{2} + 1\right )} \log \left (x^{2} + 1\right ) - x + 2}{2 \,{\left (x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(3*(x^2 + 1)*arctan(x) - (x^2 + 1)*log(x^2 + 1) - x + 2)/(x^2 + 1)

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Sympy [A]  time = 0.280811, size = 24, normalized size = 0.73 \[ - \frac{x - 2}{2 x^{2} + 2} - \frac{\log{\left (x^{2} + 1 \right )}}{2} + \frac{3 \operatorname{atan}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(x - 2)/(2*x**2 + 2) - log(x**2 + 1)/2 + 3*atan(x)/2

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GIAC/XCAS [A]  time = 0.26147, size = 34, normalized size = 1.03 \[ -\frac{x - 2}{2 \,{\left (x^{2} + 1\right )}} + \frac{3}{2} \, \arctan \left (x\right ) - \frac{1}{2} \,{\rm ln}\left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(x - 2)/(x^2 + 1) + 3/2*arctan(x) - 1/2*ln(x^2 + 1)