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

Optimal. Leaf size=19 \[ \frac{x}{2 \left (x^2+1\right )}+\frac{5}{2} \tan ^{-1}(x) \]

[Out]

x/(2*(1 + x^2)) + (5*ArcTan[x])/2

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Rubi [A]  time = 0.0167511, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{x}{2 \left (x^2+1\right )}+\frac{5}{2} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(3 + 2*x^2)/(1 + x^2)^2,x]

[Out]

x/(2*(1 + x^2)) + (5*ArcTan[x])/2

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Rubi in Sympy [A]  time = 4.10227, size = 14, normalized size = 0.74 \[ \frac{x}{2 \left (x^{2} + 1\right )} + \frac{5 \operatorname{atan}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(2*(x**2 + 1)) + 5*atan(x)/2

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Mathematica [A]  time = 0.0116605, size = 19, normalized size = 1. \[ \frac{x}{2 \left (x^2+1\right )}+\frac{5}{2} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

x/(2*(1 + x^2)) + (5*ArcTan[x])/2

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Maple [A]  time = 0.008, size = 16, normalized size = 0.8 \[{\frac{x}{2\,{x}^{2}+2}}+{\frac{5\,\arctan \left ( x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*x^2+3)/(x^2+1)^2,x)

[Out]

1/2*x/(x^2+1)+5/2*arctan(x)

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Maxima [A]  time = 1.48253, size = 20, normalized size = 1.05 \[ \frac{x}{2 \,{\left (x^{2} + 1\right )}} + \frac{5}{2} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x/(x^2 + 1) + 5/2*arctan(x)

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Fricas [A]  time = 0.236045, size = 27, normalized size = 1.42 \[ \frac{5 \,{\left (x^{2} + 1\right )} \arctan \left (x\right ) + x}{2 \,{\left (x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(5*(x^2 + 1)*arctan(x) + x)/(x^2 + 1)

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Sympy [A]  time = 0.221471, size = 14, normalized size = 0.74 \[ \frac{x}{2 x^{2} + 2} + \frac{5 \operatorname{atan}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(2*x**2 + 2) + 5*atan(x)/2

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GIAC/XCAS [A]  time = 0.221626, size = 20, normalized size = 1.05 \[ \frac{x}{2 \,{\left (x^{2} + 1\right )}} + \frac{5}{2} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x/(x^2 + 1) + 5/2*arctan(x)