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

Optimal. Leaf size=14 \[ \frac{x}{x^2+1}+2 \tan ^{-1}(x) \]

[Out]

x/(1 + x^2) + 2*ArcTan[x]

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Rubi [A]  time = 0.0153864, antiderivative size = 14, 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 \[ \frac{x}{x^2+1}+2 \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(3 + x^2)/(1 + x^2)^2,x]

[Out]

x/(1 + x^2) + 2*ArcTan[x]

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Rubi in Sympy [A]  time = 3.87684, size = 10, normalized size = 0.71 \[ \frac{x}{x^{2} + 1} + 2 \operatorname{atan}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(x**2 + 1) + 2*atan(x)

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Mathematica [A]  time = 0.00953869, size = 14, normalized size = 1. \[ \frac{x}{x^2+1}+2 \tan ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

x/(1 + x^2) + 2*ArcTan[x]

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Maple [A]  time = 0.01, size = 15, normalized size = 1.1 \[{\frac{x}{{x}^{2}+1}}+2\,\arctan \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+3)/(x^2+1)^2,x)

[Out]

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

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Maxima [A]  time = 1.4859, size = 19, normalized size = 1.36 \[ \frac{x}{x^{2} + 1} + 2 \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(x^2 + 1) + 2*arctan(x)

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Fricas [A]  time = 0.226713, size = 26, normalized size = 1.86 \[ \frac{2 \,{\left (x^{2} + 1\right )} \arctan \left (x\right ) + x}{x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*(x^2 + 1)*arctan(x) + x)/(x^2 + 1)

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Sympy [A]  time = 0.21219, size = 10, normalized size = 0.71 \[ \frac{x}{x^{2} + 1} + 2 \operatorname{atan}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(x**2 + 1) + 2*atan(x)

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GIAC/XCAS [A]  time = 0.221667, size = 19, normalized size = 1.36 \[ \frac{x}{x^{2} + 1} + 2 \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(x^2 + 1) + 2*arctan(x)