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

Optimal. Leaf size=51 \[ \frac{\log \left (3 x^2+2 \sqrt{3} x+2\right )}{4 \sqrt{3}}-\frac{\log \left (3 x^2-2 \sqrt{3} x+2\right )}{4 \sqrt{3}} \]

[Out]

-Log[2 - 2*Sqrt[3]*x + 3*x^2]/(4*Sqrt[3]) + Log[2 + 2*Sqrt[3]*x + 3*x^2]/(4*Sqrt
[3])

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Rubi [A]  time = 0.0483462, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{\log \left (3 x^2+2 \sqrt{3} x+2\right )}{4 \sqrt{3}}-\frac{\log \left (3 x^2-2 \sqrt{3} x+2\right )}{4 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(2 - 3*x^2)/(4 + 9*x^4),x]

[Out]

-Log[2 - 2*Sqrt[3]*x + 3*x^2]/(4*Sqrt[3]) + Log[2 + 2*Sqrt[3]*x + 3*x^2]/(4*Sqrt
[3])

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Rubi in Sympy [A]  time = 12.3444, size = 46, normalized size = 0.9 \[ - \frac{\sqrt{3} \log{\left (3 x^{2} - 2 \sqrt{3} x + 2 \right )}}{12} + \frac{\sqrt{3} \log{\left (3 x^{2} + 2 \sqrt{3} x + 2 \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(3)*log(3*x**2 - 2*sqrt(3)*x + 2)/12 + sqrt(3)*log(3*x**2 + 2*sqrt(3)*x + 2
)/12

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Mathematica [A]  time = 0.020653, size = 44, normalized size = 0.86 \[ \frac{\log \left (3 x^2+2 \sqrt{3} x+2\right )-\log \left (-3 x^2+2 \sqrt{3} x-2\right )}{4 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(-Log[-2 + 2*Sqrt[3]*x - 3*x^2] + Log[2 + 2*Sqrt[3]*x + 3*x^2])/(4*Sqrt[3])

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Maple [B]  time = 0.003, size = 82, normalized size = 1.6 \[{\frac{\sqrt{6}\sqrt{2}}{48}\ln \left ({1 \left ({x}^{2}+{\frac{\sqrt{6}x\sqrt{2}}{3}}+{\frac{2}{3}} \right ) \left ({x}^{2}-{\frac{\sqrt{6}x\sqrt{2}}{3}}+{\frac{2}{3}} \right ) ^{-1}} \right ) }-{\frac{\sqrt{6}\sqrt{2}}{48}\ln \left ({1 \left ({x}^{2}-{\frac{\sqrt{6}x\sqrt{2}}{3}}+{\frac{2}{3}} \right ) \left ({x}^{2}+{\frac{\sqrt{6}x\sqrt{2}}{3}}+{\frac{2}{3}} \right ) ^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-3*x^2+2)/(9*x^4+4),x)

[Out]

1/48*6^(1/2)*2^(1/2)*ln((x^2+1/3*6^(1/2)*x*2^(1/2)+2/3)/(x^2-1/3*6^(1/2)*x*2^(1/
2)+2/3))-1/48*6^(1/2)*2^(1/2)*ln((x^2-1/3*6^(1/2)*x*2^(1/2)+2/3)/(x^2+1/3*6^(1/2
)*x*2^(1/2)+2/3))

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Maxima [A]  time = 0.836451, size = 53, normalized size = 1.04 \[ \frac{1}{12} \, \sqrt{3} \log \left (3 \, x^{2} + 2 \, \sqrt{3} x + 2\right ) - \frac{1}{12} \, \sqrt{3} \log \left (3 \, x^{2} - 2 \, \sqrt{3} x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*sqrt(3)*log(3*x^2 + 2*sqrt(3)*x + 2) - 1/12*sqrt(3)*log(3*x^2 - 2*sqrt(3)*x
+ 2)

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Fricas [A]  time = 0.28456, size = 55, normalized size = 1.08 \[ \frac{1}{12} \, \sqrt{3} \log \left (\frac{36 \, x^{3} + \sqrt{3}{\left (9 \, x^{4} + 24 \, x^{2} + 4\right )} + 24 \, x}{9 \, x^{4} + 4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*sqrt(3)*log((36*x^3 + sqrt(3)*(9*x^4 + 24*x^2 + 4) + 24*x)/(9*x^4 + 4))

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Sympy [A]  time = 0.194143, size = 49, normalized size = 0.96 \[ - \frac{\sqrt{3} \log{\left (x^{2} - \frac{2 \sqrt{3} x}{3} + \frac{2}{3} \right )}}{12} + \frac{\sqrt{3} \log{\left (x^{2} + \frac{2 \sqrt{3} x}{3} + \frac{2}{3} \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(3)*log(x**2 - 2*sqrt(3)*x/3 + 2/3)/12 + sqrt(3)*log(x**2 + 2*sqrt(3)*x/3 +
2/3)/12

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GIAC/XCAS [A]  time = 0.276163, size = 54, normalized size = 1.06 \[ \frac{1}{12} \, \sqrt{3}{\rm ln}\left (x^{2} + \sqrt{2} \left (\frac{4}{9}\right )^{\frac{1}{4}} x + \frac{2}{3}\right ) - \frac{1}{12} \, \sqrt{3}{\rm ln}\left (x^{2} - \sqrt{2} \left (\frac{4}{9}\right )^{\frac{1}{4}} x + \frac{2}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*sqrt(3)*ln(x^2 + sqrt(2)*(4/9)^(1/4)*x + 2/3) - 1/12*sqrt(3)*ln(x^2 - sqrt(
2)*(4/9)^(1/4)*x + 2/3)