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

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

[Out]

-x/(2*(1 + x^2)) + ArcTan[x]/2

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Rubi [A]  time = 0.01292, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{1}{2} \tan ^{-1}(x)-\frac{x}{2 \left (x^2+1\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(1 + x^2)^2,x]

[Out]

-x/(2*(1 + x^2)) + ArcTan[x]/2

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Rubi in Sympy [A]  time = 1.35815, size = 12, normalized size = 0.63 \[ - \frac{x}{2 \left (x^{2} + 1\right )} + \frac{\operatorname{atan}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x/(2*(x**2 + 1)) + atan(x)/2

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Mathematica [A]  time = 0.0116371, size = 19, normalized size = 1. \[ \frac{1}{2} \tan ^{-1}(x)-\frac{x}{2 \left (x^2+1\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(1 + x^2)^2,x]

[Out]

-x/(2*(1 + x^2)) + ArcTan[x]/2

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Maple [A]  time = 0.019, size = 16, normalized size = 0.8 \[ -{\frac{x}{2\,{x}^{2}+2}}+{\frac{\arctan \left ( x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(x^2+1)^2,x)

[Out]

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

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Maxima [A]  time = 1.51569, size = 20, normalized size = 1.05 \[ -\frac{x}{2 \,{\left (x^{2} + 1\right )}} + \frac{1}{2} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*x/(x^2 + 1) + 1/2*arctan(x)

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Fricas [A]  time = 0.190376, size = 28, normalized size = 1.47 \[ \frac{{\left (x^{2} + 1\right )} \arctan \left (x\right ) - x}{2 \,{\left (x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.108322, size = 12, normalized size = 0.63 \[ - \frac{x}{2 x^{2} + 2} + \frac{\operatorname{atan}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(x**2+1)**2,x)

[Out]

-x/(2*x**2 + 2) + atan(x)/2

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GIAC/XCAS [A]  time = 0.20756, size = 20, normalized size = 1.05 \[ -\frac{x}{2 \,{\left (x^{2} + 1\right )}} + \frac{1}{2} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*x/(x^2 + 1) + 1/2*arctan(x)