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

Optimal. Leaf size=20 \[ 3 \tan ^{-1}(x)-\sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \]

[Out]

3*ArcTan[x] - Sqrt[2]*ArcTan[x/Sqrt[2]]

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Rubi [A]  time = 0.0411773, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ 3 \tan ^{-1}(x)-\sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

3*ArcTan[x] - Sqrt[2]*ArcTan[x/Sqrt[2]]

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Rubi in Sympy [A]  time = 8.12611, size = 19, normalized size = 0.95 \[ 3 \operatorname{atan}{\left (x \right )} - \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*atan(x) - sqrt(2)*atan(sqrt(2)*x/2)

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Mathematica [A]  time = 0.0152242, size = 20, normalized size = 1. \[ 3 \tan ^{-1}(x)-\sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

3*ArcTan[x] - Sqrt[2]*ArcTan[x/Sqrt[2]]

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Maple [A]  time = 0.009, size = 18, normalized size = 0.9 \[ 3\,\arctan \left ( x \right ) -\arctan \left ({\frac{\sqrt{2}x}{2}} \right ) \sqrt{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+4)/(x^2+1)/(x^2+2),x)

[Out]

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

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Maxima [A]  time = 0.889019, size = 23, normalized size = 1.15 \[ -\sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + 3 \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.260523, size = 23, normalized size = 1.15 \[ -\sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + 3 \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.404922, size = 19, normalized size = 0.95 \[ 3 \operatorname{atan}{\left (x \right )} - \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*atan(x) - sqrt(2)*atan(sqrt(2)*x/2)

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GIAC/XCAS [A]  time = 0.262265, size = 23, normalized size = 1.15 \[ -\sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + 3 \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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