3.471 \(\int \frac{-10+x^2}{4+9 x^2+2 x^4} \, dx\)

Optimal. Leaf size=22 \[ \tan ^{-1}\left (\frac{x}{2}\right )-\frac{3 \tan ^{-1}\left (\sqrt{2} x\right )}{\sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.031989, antiderivative size = 22, 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 \[ \tan ^{-1}\left (\frac{x}{2}\right )-\frac{3 \tan ^{-1}\left (\sqrt{2} x\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[(-10 + x^2)/(4 + 9*x^2 + 2*x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 8.63619, size = 20, normalized size = 0.91 \[ \operatorname{atan}{\left (\frac{x}{2} \right )} - \frac{3 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**2-10)/(2*x**4+9*x**2+4),x)

[Out]

atan(x/2) - 3*sqrt(2)*atan(sqrt(2)*x)/2

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Mathematica [A]  time = 0.0196729, size = 22, normalized size = 1. \[ \tan ^{-1}\left (\frac{x}{2}\right )-\frac{3 \tan ^{-1}\left (\sqrt{2} x\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(-10 + x^2)/(4 + 9*x^2 + 2*x^4),x]

[Out]

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

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Maple [A]  time = 0.01, size = 17, normalized size = 0.8 \[ \arctan \left ({\frac{x}{2}} \right ) -{\frac{3\,\arctan \left ( \sqrt{2}x \right ) \sqrt{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2-10)/(2*x^4+9*x^2+4),x)

[Out]

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

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Maxima [A]  time = 0.857709, size = 22, normalized size = 1. \[ -\frac{3}{2} \, \sqrt{2} \arctan \left (\sqrt{2} x\right ) + \arctan \left (\frac{1}{2} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 - 10)/(2*x^4 + 9*x^2 + 4),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.265483, size = 30, normalized size = 1.36 \[ \frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} \arctan \left (\frac{1}{2} \, x\right ) - 3 \, \arctan \left (\sqrt{2} x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 - 10)/(2*x^4 + 9*x^2 + 4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.411319, size = 20, normalized size = 0.91 \[ \operatorname{atan}{\left (\frac{x}{2} \right )} - \frac{3 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**2-10)/(2*x**4+9*x**2+4),x)

[Out]

atan(x/2) - 3*sqrt(2)*atan(sqrt(2)*x)/2

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GIAC/XCAS [A]  time = 0.259908, size = 22, normalized size = 1. \[ -\frac{3}{2} \, \sqrt{2} \arctan \left (\sqrt{2} x\right ) + \arctan \left (\frac{1}{2} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 - 10)/(2*x^4 + 9*x^2 + 4),x, algorithm="giac")

[Out]

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