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

Optimal. Leaf size=131 \[ \frac{\log \left (\sqrt{3} x^2-2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{32\ 6^{3/4}}-\frac{\log \left (\sqrt{3} x^2+2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{32\ 6^{3/4}}+\frac{x^3}{8 \left (3 x^4+2\right )}-\frac{\tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{16\ 6^{3/4}}+\frac{\tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{16\ 6^{3/4}} \]

[Out]

x^3/(8*(2 + 3*x^4)) - ArcTan[1 - 6^(1/4)*x]/(16*6^(3/4)) + ArcTan[1 + 6^(1/4)*x]
/(16*6^(3/4)) + Log[Sqrt[2] - 2^(3/4)*3^(1/4)*x + Sqrt[3]*x^2]/(32*6^(3/4)) - Lo
g[Sqrt[2] + 2^(3/4)*3^(1/4)*x + Sqrt[3]*x^2]/(32*6^(3/4))

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Rubi [A]  time = 0.150667, antiderivative size = 113, normalized size of antiderivative = 0.86, number of steps used = 10, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538 \[ \frac{\log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{32\ 6^{3/4}}-\frac{\log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{32\ 6^{3/4}}+\frac{x^3}{8 \left (3 x^4+2\right )}-\frac{\tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{16\ 6^{3/4}}+\frac{\tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{16\ 6^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(2 + 3*x^4)^2,x]

[Out]

x^3/(8*(2 + 3*x^4)) - ArcTan[1 - 6^(1/4)*x]/(16*6^(3/4)) + ArcTan[1 + 6^(1/4)*x]
/(16*6^(3/4)) + Log[Sqrt[6] - 6^(3/4)*x + 3*x^2]/(32*6^(3/4)) - Log[Sqrt[6] + 6^
(3/4)*x + 3*x^2]/(32*6^(3/4))

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Rubi in Sympy [A]  time = 20.6163, size = 94, normalized size = 0.72 \[ \frac{x^{3}}{8 \left (3 x^{4} + 2\right )} + \frac{\sqrt [4]{6} \log{\left (3 x^{2} - 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{192} - \frac{\sqrt [4]{6} \log{\left (3 x^{2} + 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{192} + \frac{\sqrt [4]{6} \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{96} + \frac{\sqrt [4]{6} \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{96} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3/(8*(3*x**4 + 2)) + 6**(1/4)*log(3*x**2 - 6**(3/4)*x + sqrt(6))/192 - 6**(1/
4)*log(3*x**2 + 6**(3/4)*x + sqrt(6))/192 + 6**(1/4)*atan(6**(1/4)*x - 1)/96 + 6
**(1/4)*atan(6**(1/4)*x + 1)/96

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Mathematica [A]  time = 0.148017, size = 107, normalized size = 0.82 \[ \frac{1}{192} \left (\sqrt [4]{6} \log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )-\sqrt [4]{6} \log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )+\frac{24 x^3}{3 x^4+2}-2 \sqrt [4]{6} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2 \sqrt [4]{6} \tan ^{-1}\left (\sqrt [4]{6} x+1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(2 + 3*x^4)^2,x]

[Out]

((24*x^3)/(2 + 3*x^4) - 2*6^(1/4)*ArcTan[1 - 6^(1/4)*x] + 2*6^(1/4)*ArcTan[1 + 6
^(1/4)*x] + 6^(1/4)*Log[2 - 2*6^(1/4)*x + Sqrt[6]*x^2] - 6^(1/4)*Log[2 + 2*6^(1/
4)*x + Sqrt[6]*x^2])/192

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Maple [A]  time = 0.006, size = 125, normalized size = 1. \[{\frac{{x}^{3}}{24\,{x}^{4}+16}}+{\frac{\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{576}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{1152}\ln \left ({1 \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) }+{\frac{\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{576}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(3*x^4+2)^2,x)

[Out]

1/8*x^3/(3*x^4+2)+1/576*6^(3/4)*3^(1/2)*2^(1/2)*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/
4)*x-1)+1/1152*6^(3/4)*3^(1/2)*2^(1/2)*ln((x^2-1/3*3^(1/2)*6^(1/4)*x*2^(1/2)+1/3
*6^(1/2))/(x^2+1/3*3^(1/2)*6^(1/4)*x*2^(1/2)+1/3*6^(1/2)))+1/576*6^(3/4)*3^(1/2)
*2^(1/2)*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/4)*x+1)

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Maxima [A]  time = 1.61455, size = 182, normalized size = 1.39 \[ \frac{x^{3}}{8 \,{\left (3 \, x^{4} + 2\right )}} + \frac{1}{96} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x + 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{96} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x - 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) - \frac{1}{192} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{1}{192} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*x^3/(3*x^4 + 2) + 1/96*3^(1/4)*2^(1/4)*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)
*x + 3^(1/4)*2^(3/4))) + 1/96*3^(1/4)*2^(1/4)*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt
(3)*x - 3^(1/4)*2^(3/4))) - 1/192*3^(1/4)*2^(1/4)*log(sqrt(3)*x^2 + 3^(1/4)*2^(3
/4)*x + sqrt(2)) + 1/192*3^(1/4)*2^(1/4)*log(sqrt(3)*x^2 - 3^(1/4)*2^(3/4)*x + s
qrt(2))

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Fricas [A]  time = 0.241979, size = 284, normalized size = 2.17 \[ \frac{54^{\frac{3}{4}}{\left (8 \cdot 54^{\frac{1}{4}} x^{3} - 4 \, \sqrt{2}{\left (3 \, x^{4} + 2\right )} \arctan \left (\frac{54}{54^{\frac{3}{4}} \sqrt{2} \sqrt{\frac{1}{6}} \sqrt{\sqrt{6}{\left (9 \, \sqrt{6} x^{2} + 54^{\frac{3}{4}} \sqrt{2} x + 18\right )}} + 3 \cdot 54^{\frac{3}{4}} \sqrt{2} x + 54}\right ) - 4 \, \sqrt{2}{\left (3 \, x^{4} + 2\right )} \arctan \left (\frac{54}{54^{\frac{3}{4}} \sqrt{2} \sqrt{\frac{1}{6}} \sqrt{\sqrt{6}{\left (9 \, \sqrt{6} x^{2} - 54^{\frac{3}{4}} \sqrt{2} x + 18\right )}} + 3 \cdot 54^{\frac{3}{4}} \sqrt{2} x - 54}\right ) - \sqrt{2}{\left (3 \, x^{4} + 2\right )} \log \left (9 \, \sqrt{6} x^{2} + 54^{\frac{3}{4}} \sqrt{2} x + 18\right ) + \sqrt{2}{\left (3 \, x^{4} + 2\right )} \log \left (9 \, \sqrt{6} x^{2} - 54^{\frac{3}{4}} \sqrt{2} x + 18\right )\right )}}{3456 \,{\left (3 \, x^{4} + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3456*54^(3/4)*(8*54^(1/4)*x^3 - 4*sqrt(2)*(3*x^4 + 2)*arctan(54/(54^(3/4)*sqrt
(2)*sqrt(1/6)*sqrt(sqrt(6)*(9*sqrt(6)*x^2 + 54^(3/4)*sqrt(2)*x + 18)) + 3*54^(3/
4)*sqrt(2)*x + 54)) - 4*sqrt(2)*(3*x^4 + 2)*arctan(54/(54^(3/4)*sqrt(2)*sqrt(1/6
)*sqrt(sqrt(6)*(9*sqrt(6)*x^2 - 54^(3/4)*sqrt(2)*x + 18)) + 3*54^(3/4)*sqrt(2)*x
- 54)) - sqrt(2)*(3*x^4 + 2)*log(9*sqrt(6)*x^2 + 54^(3/4)*sqrt(2)*x + 18) + sqr
t(2)*(3*x^4 + 2)*log(9*sqrt(6)*x^2 - 54^(3/4)*sqrt(2)*x + 18))/(3*x^4 + 2)

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Sympy [A]  time = 1.69839, size = 97, normalized size = 0.74 \[ \frac{x^{3}}{24 x^{4} + 16} + \frac{\sqrt [4]{6} \log{\left (x^{2} - \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{192} - \frac{\sqrt [4]{6} \log{\left (x^{2} + \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{192} + \frac{\sqrt [4]{6} \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{96} + \frac{\sqrt [4]{6} \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{96} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(3*x**4+2)**2,x)

[Out]

x**3/(24*x**4 + 16) + 6**(1/4)*log(x**2 - 6**(3/4)*x/3 + sqrt(6)/3)/192 - 6**(1/
4)*log(x**2 + 6**(3/4)*x/3 + sqrt(6)/3)/192 + 6**(1/4)*atan(6**(1/4)*x - 1)/96 +
6**(1/4)*atan(6**(1/4)*x + 1)/96

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GIAC/XCAS [A]  time = 0.224579, size = 147, normalized size = 1.12 \[ \frac{x^{3}}{8 \,{\left (3 \, x^{4} + 2\right )}} + \frac{1}{96} \cdot 6^{\frac{1}{4}} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{96} \cdot 6^{\frac{1}{4}} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) - \frac{1}{192} \cdot 6^{\frac{1}{4}}{\rm ln}\left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) + \frac{1}{192} \cdot 6^{\frac{1}{4}}{\rm ln}\left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*x^3/(3*x^4 + 2) + 1/96*6^(1/4)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x + sqrt(2)
*(2/3)^(1/4))) + 1/96*6^(1/4)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x - sqrt(2)*(2/3
)^(1/4))) - 1/192*6^(1/4)*ln(x^2 + sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3)) + 1/192*6^
(1/4)*ln(x^2 - sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3))