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

Optimal. Leaf size=23 \[ \frac{2}{x^2+1}-\frac{1}{4 \left (x^2+1\right )^2}+\tan ^{-1}(x) \]

[Out]

-1/(4*(1 + x^2)^2) + 2/(1 + x^2) + ArcTan[x]

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Rubi [A]  time = 0.0407249, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{2}{x^2+1}-\frac{1}{4 \left (x^2+1\right )^2}+\tan ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

-1/(4*(1 + x^2)^2) + 2/(1 + x^2) + ArcTan[x]

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Rubi in Sympy [A]  time = 15.5979, size = 22, normalized size = 0.96 \[ \frac{x^{2}}{4 \left (x^{2} + 1\right )^{2}} + \operatorname{atan}{\left (x \right )} + \frac{7}{4 \left (x^{2} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2/(4*(x**2 + 1)**2) + atan(x) + 7/(4*(x**2 + 1))

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Mathematica [A]  time = 0.0189779, size = 23, normalized size = 1. \[ \frac{2}{x^2+1}-\frac{1}{4 \left (x^2+1\right )^2}+\tan ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

-1/(4*(1 + x^2)^2) + 2/(1 + x^2) + ArcTan[x]

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Maple [A]  time = 0.007, size = 19, normalized size = 0.8 \[{\frac{1}{ \left ({x}^{2}+1 \right ) ^{2}} \left ( 2\,{x}^{2}+{\frac{7}{4}} \right ) }+\arctan \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*x^2+7/4)/(x^2+1)^2+arctan(x)

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Maxima [A]  time = 0.893775, size = 32, normalized size = 1.39 \[ \frac{8 \, x^{2} + 7}{4 \,{\left (x^{4} + 2 \, x^{2} + 1\right )}} + \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(8*x^2 + 7)/(x^4 + 2*x^2 + 1) + arctan(x)

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Fricas [A]  time = 0.253124, size = 47, normalized size = 2.04 \[ \frac{8 \, x^{2} + 4 \,{\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \left (x\right ) + 7}{4 \,{\left (x^{4} + 2 \, x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(8*x^2 + 4*(x^4 + 2*x^2 + 1)*arctan(x) + 7)/(x^4 + 2*x^2 + 1)

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Sympy [A]  time = 0.32248, size = 20, normalized size = 0.87 \[ \frac{8 x^{2} + 7}{4 x^{4} + 8 x^{2} + 4} + \operatorname{atan}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(8*x**2 + 7)/(4*x**4 + 8*x**2 + 4) + atan(x)

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GIAC/XCAS [A]  time = 0.261317, size = 26, normalized size = 1.13 \[ \frac{8 \, x^{2} + 7}{4 \,{\left (x^{2} + 1\right )}^{2}} + \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(8*x^2 + 7)/(x^2 + 1)^2 + arctan(x)