3.691 \(\int \frac{1}{x^3 \left (2+3 x^4\right )} \, dx\)

Optimal. Leaf size=31 \[ -\frac{1}{4 x^2}-\frac{1}{4} \sqrt{\frac{3}{2}} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right ) \]

[Out]

-1/(4*x^2) - (Sqrt[3/2]*ArcTan[Sqrt[3/2]*x^2])/4

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Rubi [A]  time = 0.0347719, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{1}{4 x^2}-\frac{1}{4} \sqrt{\frac{3}{2}} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(2 + 3*x^4)),x]

[Out]

-1/(4*x^2) - (Sqrt[3/2]*ArcTan[Sqrt[3/2]*x^2])/4

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Rubi in Sympy [A]  time = 5.17499, size = 26, normalized size = 0.84 \[ - \frac{\sqrt{6} \operatorname{atan}{\left (\frac{\sqrt{6} x^{2}}{2} \right )}}{8} - \frac{1}{4 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(6)*atan(sqrt(6)*x**2/2)/8 - 1/(4*x**2)

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Mathematica [A]  time = 0.0357927, size = 48, normalized size = 1.55 \[ \frac{\sqrt{6} x^2 \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+\sqrt{6} x^2 \tan ^{-1}\left (\sqrt [4]{6} x+1\right )-2}{8 x^2} \]

Antiderivative was successfully verified.

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

[Out]

(-2 + Sqrt[6]*x^2*ArcTan[1 - 6^(1/4)*x] + Sqrt[6]*x^2*ArcTan[1 + 6^(1/4)*x])/(8*
x^2)

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Maple [A]  time = 0.006, size = 21, normalized size = 0.7 \[ -{\frac{1}{4\,{x}^{2}}}-{\frac{\sqrt{6}}{8}\arctan \left ({\frac{{x}^{2}\sqrt{6}}{2}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(3*x^4+2),x)

[Out]

-1/4/x^2-1/8*arctan(1/2*x^2*6^(1/2))*6^(1/2)

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Maxima [A]  time = 1.59174, size = 27, normalized size = 0.87 \[ -\frac{1}{8} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x^{2}\right ) - \frac{1}{4 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*sqrt(6)*arctan(1/2*sqrt(6)*x^2) - 1/4/x^2

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Fricas [A]  time = 0.221196, size = 42, normalized size = 1.35 \[ -\frac{\sqrt{2}{\left (\sqrt{3} x^{2} \arctan \left (\frac{1}{2} \, \sqrt{3} \sqrt{2} x^{2}\right ) + \sqrt{2}\right )}}{8 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*sqrt(2)*(sqrt(3)*x^2*arctan(1/2*sqrt(3)*sqrt(2)*x^2) + sqrt(2))/x^2

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Sympy [A]  time = 0.279784, size = 26, normalized size = 0.84 \[ - \frac{\sqrt{6} \operatorname{atan}{\left (\frac{\sqrt{6} x^{2}}{2} \right )}}{8} - \frac{1}{4 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(3*x**4+2),x)

[Out]

-sqrt(6)*atan(sqrt(6)*x**2/2)/8 - 1/(4*x**2)

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GIAC/XCAS [A]  time = 0.219831, size = 27, normalized size = 0.87 \[ -\frac{1}{8} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x^{2}\right ) - \frac{1}{4 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*sqrt(6)*arctan(1/2*sqrt(6)*x^2) - 1/4/x^2