3.100 \(\int -\frac{1+3 x^2}{1+2 x^2+9 x^4} \, dx\)

Optimal. Leaf size=43 \[ \frac{\tan ^{-1}\left (\frac{1-3 x}{\sqrt{2}}\right )}{2 \sqrt{2}}-\frac{\tan ^{-1}\left (\frac{3 x+1}{\sqrt{2}}\right )}{2 \sqrt{2}} \]

[Out]

ArcTan[(1 - 3*x)/Sqrt[2]]/(2*Sqrt[2]) - ArcTan[(1 + 3*x)/Sqrt[2]]/(2*Sqrt[2])

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Rubi [A]  time = 0.0685218, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{\tan ^{-1}\left (\frac{1-3 x}{\sqrt{2}}\right )}{2 \sqrt{2}}-\frac{\tan ^{-1}\left (\frac{3 x+1}{\sqrt{2}}\right )}{2 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[-((1 + 3*x^2)/(1 + 2*x^2 + 9*x^4)),x]

[Out]

ArcTan[(1 - 3*x)/Sqrt[2]]/(2*Sqrt[2]) - ArcTan[(1 + 3*x)/Sqrt[2]]/(2*Sqrt[2])

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Rubi in Sympy [A]  time = 8.38436, size = 42, normalized size = 0.98 \[ \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \left (- \frac{3 x}{2} + \frac{1}{2}\right ) \right )}}{4} - \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \left (\frac{3 x}{2} + \frac{1}{2}\right ) \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-3*x**2-1)/(9*x**4+2*x**2+1),x)

[Out]

sqrt(2)*atan(sqrt(2)*(-3*x/2 + 1/2))/4 - sqrt(2)*atan(sqrt(2)*(3*x/2 + 1/2))/4

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Mathematica [C]  time = 0.166557, size = 99, normalized size = 2.3 \[ -\frac{\left (\sqrt{2}-i\right ) \tan ^{-1}\left (\frac{3 x}{\sqrt{1-2 i \sqrt{2}}}\right )}{2 \sqrt{2 \left (1-2 i \sqrt{2}\right )}}-\frac{\left (\sqrt{2}+i\right ) \tan ^{-1}\left (\frac{3 x}{\sqrt{1+2 i \sqrt{2}}}\right )}{2 \sqrt{2 \left (1+2 i \sqrt{2}\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[-((1 + 3*x^2)/(1 + 2*x^2 + 9*x^4)),x]

[Out]

-((-I + Sqrt[2])*ArcTan[(3*x)/Sqrt[1 - (2*I)*Sqrt[2]]])/(2*Sqrt[2*(1 - (2*I)*Sqr
t[2])]) - ((I + Sqrt[2])*ArcTan[(3*x)/Sqrt[1 + (2*I)*Sqrt[2]]])/(2*Sqrt[2*(1 + (
2*I)*Sqrt[2])])

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Maple [A]  time = 0.01, size = 34, normalized size = 0.8 \[ -{\frac{\sqrt{2}}{4}\arctan \left ({\frac{ \left ( 6\,x+2 \right ) \sqrt{2}}{4}} \right ) }-{\frac{\sqrt{2}}{4}\arctan \left ({\frac{ \left ( 6\,x-2 \right ) \sqrt{2}}{4}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-3*x^2-1)/(9*x^4+2*x^2+1),x)

[Out]

-1/4*2^(1/2)*arctan(1/4*(6*x+2)*2^(1/2))-1/4*2^(1/2)*arctan(1/4*(6*x-2)*2^(1/2))

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Maxima [A]  time = 0.840625, size = 45, normalized size = 1.05 \[ -\frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x + 1\right )}\right ) - \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x - 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 1)/(9*x^4 + 2*x^2 + 1),x, algorithm="maxima")

[Out]

-1/4*sqrt(2)*arctan(1/2*sqrt(2)*(3*x + 1)) - 1/4*sqrt(2)*arctan(1/2*sqrt(2)*(3*x
- 1))

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Fricas [A]  time = 0.280519, size = 38, normalized size = 0.88 \[ -\frac{1}{4} \, \sqrt{2}{\left (\arctan \left (\frac{1}{4} \, \sqrt{2}{\left (9 \, x^{3} + 5 \, x\right )}\right ) + \arctan \left (\frac{3}{4} \, \sqrt{2} x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 1)/(9*x^4 + 2*x^2 + 1),x, algorithm="fricas")

[Out]

-1/4*sqrt(2)*(arctan(1/4*sqrt(2)*(9*x^3 + 5*x)) + arctan(3/4*sqrt(2)*x))

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Sympy [A]  time = 0.267402, size = 46, normalized size = 1.07 \[ - \frac{\sqrt{2} \left (2 \operatorname{atan}{\left (\frac{3 \sqrt{2} x}{4} \right )} + 2 \operatorname{atan}{\left (\frac{9 \sqrt{2} x^{3}}{4} + \frac{5 \sqrt{2} x}{4} \right )}\right )}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-3*x**2-1)/(9*x**4+2*x**2+1),x)

[Out]

-sqrt(2)*(2*atan(3*sqrt(2)*x/4) + 2*atan(9*sqrt(2)*x**3/4 + 5*sqrt(2)*x/4))/8

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GIAC/XCAS [A]  time = 0.269518, size = 45, normalized size = 1.05 \[ -\frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x + 1\right )}\right ) - \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x - 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 1)/(9*x^4 + 2*x^2 + 1),x, algorithm="giac")

[Out]

-1/4*sqrt(2)*arctan(1/2*sqrt(2)*(3*x + 1)) - 1/4*sqrt(2)*arctan(1/2*sqrt(2)*(3*x
- 1))