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

Optimal. Leaf size=19 \[ -\frac{3 x}{2 \left (x^2+1\right )}-\frac{1}{2} \tan ^{-1}(x) \]

[Out]

(-3*x)/(2*(1 + x^2)) - ArcTan[x]/2

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Rubi [A]  time = 0.0153595, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{3 x}{2 \left (x^2+1\right )}-\frac{1}{2} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(-2 + x^2)/(1 + x^2)^2,x]

[Out]

(-3*x)/(2*(1 + x^2)) - ArcTan[x]/2

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Rubi in Sympy [A]  time = 3.87925, size = 15, normalized size = 0.79 \[ - \frac{3 x}{2 \left (x^{2} + 1\right )} - \frac{\operatorname{atan}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*x/(2*(x**2 + 1)) - atan(x)/2

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Mathematica [A]  time = 0.0119485, size = 19, normalized size = 1. \[ -\frac{3 x}{2 \left (x^2+1\right )}-\frac{1}{2} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

(-3*x)/(2*(1 + x^2)) - ArcTan[x]/2

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Maple [A]  time = 0.007, size = 16, normalized size = 0.8 \[ -{\frac{3\,x}{2\,{x}^{2}+2}}-{\frac{\arctan \left ( x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2-2)/(x^2+1)^2,x)

[Out]

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

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Maxima [A]  time = 1.49675, size = 20, normalized size = 1.05 \[ -\frac{3 \, x}{2 \,{\left (x^{2} + 1\right )}} - \frac{1}{2} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/2*x/(x^2 + 1) - 1/2*arctan(x)

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Fricas [A]  time = 0.223722, size = 28, normalized size = 1.47 \[ -\frac{{\left (x^{2} + 1\right )} \arctan \left (x\right ) + 3 \, x}{2 \,{\left (x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.219321, size = 15, normalized size = 0.79 \[ - \frac{3 x}{2 x^{2} + 2} - \frac{\operatorname{atan}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*x/(2*x**2 + 2) - atan(x)/2

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GIAC/XCAS [A]  time = 0.223116, size = 20, normalized size = 1.05 \[ -\frac{3 \, x}{2 \,{\left (x^{2} + 1\right )}} - \frac{1}{2} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/2*x/(x^2 + 1) - 1/2*arctan(x)