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

Optimal. Leaf size=47 \[ -\frac{3}{16 x^2}-\frac{3}{16} \sqrt{\frac{3}{2}} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )+\frac{1}{8 x^2 \left (3 x^4+2\right )} \]

[Out]

-3/(16*x^2) + 1/(8*x^2*(2 + 3*x^4)) - (3*Sqrt[3/2]*ArcTan[Sqrt[3/2]*x^2])/16

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Rubi [A]  time = 0.0534458, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{3}{16 x^2}-\frac{3}{16} \sqrt{\frac{3}{2}} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )+\frac{1}{8 x^2 \left (3 x^4+2\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(2 + 3*x^4)^2),x]

[Out]

-3/(16*x^2) + 1/(8*x^2*(2 + 3*x^4)) - (3*Sqrt[3/2]*ArcTan[Sqrt[3/2]*x^2])/16

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Rubi in Sympy [A]  time = 7.19162, size = 39, normalized size = 0.83 \[ - \frac{3 \sqrt{6} \operatorname{atan}{\left (\frac{\sqrt{6} x^{2}}{2} \right )}}{32} - \frac{3}{16 x^{2}} + \frac{1}{8 x^{2} \left (3 x^{4} + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*sqrt(6)*atan(sqrt(6)*x**2/2)/32 - 3/(16*x**2) + 1/(8*x**2*(3*x**4 + 2))

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Mathematica [A]  time = 0.0739957, size = 59, normalized size = 1.26 \[ \frac{1}{32} \left (-\frac{4}{x^2}-\frac{6 x^2}{3 x^4+2}+3 \sqrt{6} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+3 \sqrt{6} \tan ^{-1}\left (\sqrt [4]{6} x+1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-4/x^2 - (6*x^2)/(2 + 3*x^4) + 3*Sqrt[6]*ArcTan[1 - 6^(1/4)*x] + 3*Sqrt[6]*ArcT
an[1 + 6^(1/4)*x])/32

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Maple [A]  time = 0.016, size = 33, normalized size = 0.7 \[ -{\frac{1}{8\,{x}^{2}}}-{\frac{{x}^{2}}{16} \left ({x}^{4}+{\frac{2}{3}} \right ) ^{-1}}-{\frac{3\,\sqrt{6}}{32}\arctan \left ({\frac{{x}^{2}\sqrt{6}}{2}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(3*x^4+2)^2,x)

[Out]

-1/8/x^2-1/16*x^2/(x^4+2/3)-3/32*arctan(1/2*x^2*6^(1/2))*6^(1/2)

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Maxima [A]  time = 1.64916, size = 50, normalized size = 1.06 \[ -\frac{3}{32} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x^{2}\right ) - \frac{9 \, x^{4} + 4}{16 \,{\left (3 \, x^{6} + 2 \, x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/32*sqrt(6)*arctan(1/2*sqrt(6)*x^2) - 1/16*(9*x^4 + 4)/(3*x^6 + 2*x^2)

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Fricas [A]  time = 0.226613, size = 78, normalized size = 1.66 \[ -\frac{\sqrt{2}{\left (3 \, \sqrt{3}{\left (3 \, x^{6} + 2 \, x^{2}\right )} \arctan \left (\frac{1}{2} \, \sqrt{3} \sqrt{2} x^{2}\right ) + \sqrt{2}{\left (9 \, x^{4} + 4\right )}\right )}}{32 \,{\left (3 \, x^{6} + 2 \, x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/32*sqrt(2)*(3*sqrt(3)*(3*x^6 + 2*x^2)*arctan(1/2*sqrt(3)*sqrt(2)*x^2) + sqrt(
2)*(9*x^4 + 4))/(3*x^6 + 2*x^2)

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Sympy [A]  time = 0.440606, size = 37, normalized size = 0.79 \[ - \frac{9 x^{4} + 4}{48 x^{6} + 32 x^{2}} - \frac{3 \sqrt{6} \operatorname{atan}{\left (\frac{\sqrt{6} x^{2}}{2} \right )}}{32} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(9*x**4 + 4)/(48*x**6 + 32*x**2) - 3*sqrt(6)*atan(sqrt(6)*x**2/2)/32

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GIAC/XCAS [A]  time = 0.220241, size = 50, normalized size = 1.06 \[ -\frac{3}{32} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x^{2}\right ) - \frac{9 \, x^{4} + 4}{16 \,{\left (3 \, x^{6} + 2 \, x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/32*sqrt(6)*arctan(1/2*sqrt(6)*x^2) - 1/16*(9*x^4 + 4)/(3*x^6 + 2*x^2)