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

Optimal. Leaf size=129 \[ -\frac{x}{12 \left (3 x^4+2\right )}-\frac{\log \left (\sqrt{3} x^2-2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{96 \sqrt [4]{6}}+\frac{\log \left (\sqrt{3} x^2+2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{96 \sqrt [4]{6}}-\frac{\tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{48 \sqrt [4]{6}}+\frac{\tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{48 \sqrt [4]{6}} \]

[Out]

-x/(12*(2 + 3*x^4)) - ArcTan[1 - 6^(1/4)*x]/(48*6^(1/4)) + ArcTan[1 + 6^(1/4)*x]
/(48*6^(1/4)) - Log[Sqrt[2] - 2^(3/4)*3^(1/4)*x + Sqrt[3]*x^2]/(96*6^(1/4)) + Lo
g[Sqrt[2] + 2^(3/4)*3^(1/4)*x + Sqrt[3]*x^2]/(96*6^(1/4))

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Rubi [A]  time = 0.146405, antiderivative size = 111, 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 \[ -\frac{x}{12 \left (3 x^4+2\right )}-\frac{\log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{96 \sqrt [4]{6}}+\frac{\log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{96 \sqrt [4]{6}}-\frac{\tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{48 \sqrt [4]{6}}+\frac{\tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{48 \sqrt [4]{6}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(2 + 3*x^4)^2,x]

[Out]

-x/(12*(2 + 3*x^4)) - ArcTan[1 - 6^(1/4)*x]/(48*6^(1/4)) + ArcTan[1 + 6^(1/4)*x]
/(48*6^(1/4)) - Log[Sqrt[6] - 6^(3/4)*x + 3*x^2]/(96*6^(1/4)) + Log[Sqrt[6] + 6^
(3/4)*x + 3*x^2]/(96*6^(1/4))

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Rubi in Sympy [A]  time = 19.6939, size = 92, normalized size = 0.71 \[ - \frac{x}{12 \left (3 x^{4} + 2\right )} - \frac{6^{\frac{3}{4}} \log{\left (3 x^{2} - 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{576} + \frac{6^{\frac{3}{4}} \log{\left (3 x^{2} + 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{576} + \frac{6^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{288} + \frac{6^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{288} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x/(12*(3*x**4 + 2)) - 6**(3/4)*log(3*x**2 - 6**(3/4)*x + sqrt(6))/576 + 6**(3/4
)*log(3*x**2 + 6**(3/4)*x + sqrt(6))/576 + 6**(3/4)*atan(6**(1/4)*x - 1)/288 + 6
**(3/4)*atan(6**(1/4)*x + 1)/288

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Mathematica [A]  time = 0.166623, size = 105, normalized size = 0.81 \[ \frac{1}{576} \left (-\frac{48 x}{3 x^4+2}-6^{3/4} \log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )+6^{3/4} \log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )-2\ 6^{3/4} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2\ 6^{3/4} \tan ^{-1}\left (\sqrt [4]{6} x+1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(2 + 3*x^4)^2,x]

[Out]

((-48*x)/(2 + 3*x^4) - 2*6^(3/4)*ArcTan[1 - 6^(1/4)*x] + 2*6^(3/4)*ArcTan[1 + 6^
(1/4)*x] - 6^(3/4)*Log[2 - 2*6^(1/4)*x + Sqrt[6]*x^2] + 6^(3/4)*Log[2 + 2*6^(1/4
)*x + Sqrt[6]*x^2])/576

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Maple [A]  time = 0.011, size = 121, normalized size = 0.9 \[ -{\frac{x}{36} \left ({x}^{4}+{\frac{2}{3}} \right ) ^{-1}}+{\frac{\sqrt{3}\sqrt [4]{6}\sqrt{2}}{288}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{\sqrt{3}\sqrt [4]{6}\sqrt{2}}{576}\ln \left ({1 \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) }+{\frac{\sqrt{3}\sqrt [4]{6}\sqrt{2}}{288}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(3*x^4+2)^2,x)

[Out]

-1/36*x/(x^4+2/3)+1/288*3^(1/2)*6^(1/4)*2^(1/2)*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/
4)*x-1)+1/576*3^(1/2)*6^(1/4)*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)))+1/288*3^(1/2)*6^(1/4)*
2^(1/2)*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/4)*x+1)

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Maxima [A]  time = 1.58328, size = 180, normalized size = 1.4 \[ \frac{1}{288} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x + 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{288} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x - 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{576} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) - \frac{1}{576} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) - \frac{x}{12 \,{\left (3 \, x^{4} + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/288*3^(3/4)*2^(3/4)*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x + 3^(1/4)*2^(3/4))
) + 1/288*3^(3/4)*2^(3/4)*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x - 3^(1/4)*2^(3
/4))) + 1/576*3^(3/4)*2^(3/4)*log(sqrt(3)*x^2 + 3^(1/4)*2^(3/4)*x + sqrt(2)) - 1
/576*3^(3/4)*2^(3/4)*log(sqrt(3)*x^2 - 3^(1/4)*2^(3/4)*x + sqrt(2)) - 1/12*x/(3*
x^4 + 2)

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Fricas [A]  time = 0.240542, size = 277, normalized size = 2.15 \[ -\frac{24^{\frac{3}{4}}{\left (4 \, \sqrt{2}{\left (3 \, x^{4} + 2\right )} \arctan \left (\frac{2}{24^{\frac{1}{4}} \sqrt{2} \sqrt{\frac{1}{6}} \sqrt{\sqrt{6}{\left (\sqrt{6} x^{2} + 24^{\frac{1}{4}} \sqrt{2} x + 2\right )}} + 24^{\frac{1}{4}} \sqrt{2} x + 2}\right ) + 4 \, \sqrt{2}{\left (3 \, x^{4} + 2\right )} \arctan \left (\frac{2}{24^{\frac{1}{4}} \sqrt{2} \sqrt{\frac{1}{6}} \sqrt{\sqrt{6}{\left (\sqrt{6} x^{2} - 24^{\frac{1}{4}} \sqrt{2} x + 2\right )}} + 24^{\frac{1}{4}} \sqrt{2} x - 2}\right ) - \sqrt{2}{\left (3 \, x^{4} + 2\right )} \log \left (2 \, \sqrt{6} x^{2} + 2 \cdot 24^{\frac{1}{4}} \sqrt{2} x + 4\right ) + \sqrt{2}{\left (3 \, x^{4} + 2\right )} \log \left (2 \, \sqrt{6} x^{2} - 2 \cdot 24^{\frac{1}{4}} \sqrt{2} x + 4\right ) + 8 \cdot 24^{\frac{1}{4}} x\right )}}{2304 \,{\left (3 \, x^{4} + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2304*24^(3/4)*(4*sqrt(2)*(3*x^4 + 2)*arctan(2/(24^(1/4)*sqrt(2)*sqrt(1/6)*sqr
t(sqrt(6)*(sqrt(6)*x^2 + 24^(1/4)*sqrt(2)*x + 2)) + 24^(1/4)*sqrt(2)*x + 2)) + 4
*sqrt(2)*(3*x^4 + 2)*arctan(2/(24^(1/4)*sqrt(2)*sqrt(1/6)*sqrt(sqrt(6)*(sqrt(6)*
x^2 - 24^(1/4)*sqrt(2)*x + 2)) + 24^(1/4)*sqrt(2)*x - 2)) - sqrt(2)*(3*x^4 + 2)*
log(2*sqrt(6)*x^2 + 2*24^(1/4)*sqrt(2)*x + 4) + sqrt(2)*(3*x^4 + 2)*log(2*sqrt(6
)*x^2 - 2*24^(1/4)*sqrt(2)*x + 4) + 8*24^(1/4)*x)/(3*x^4 + 2)

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Sympy [A]  time = 1.70627, size = 95, normalized size = 0.74 \[ - \frac{x}{36 x^{4} + 24} - \frac{6^{\frac{3}{4}} \log{\left (x^{2} - \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{288} + \frac{6^{\frac{3}{4}} \log{\left (x^{2} + \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{288} + \frac{6^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{144} + \frac{6^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{144} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(3*x**4+2)**2,x)

[Out]

-x/(36*x**4 + 24) - 6**(3/4)*log(x**2 - 6**(3/4)*x/3 + sqrt(6)/3)/288 + 6**(3/4)
*log(x**2 + 6**(3/4)*x/3 + sqrt(6)/3)/288 + 6**(3/4)*atan(6**(1/4)*x - 1)/144 +
6**(3/4)*atan(6**(1/4)*x + 1)/144

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GIAC/XCAS [A]  time = 0.228689, size = 144, normalized size = 1.12 \[ \frac{1}{288} \cdot 6^{\frac{3}{4}} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{288} \cdot 6^{\frac{3}{4}} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{576} \cdot 6^{\frac{3}{4}}{\rm ln}\left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) - \frac{1}{576} \cdot 6^{\frac{3}{4}}{\rm ln}\left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) - \frac{x}{12 \,{\left (3 \, x^{4} + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/288*6^(3/4)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x + sqrt(2)*(2/3)^(1/4))) + 1/28
8*6^(3/4)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x - sqrt(2)*(2/3)^(1/4))) + 1/576*6^
(3/4)*ln(x^2 + sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3)) - 1/576*6^(3/4)*ln(x^2 - sqrt(
2)*(2/3)^(1/4)*x + sqrt(2/3)) - 1/12*x/(3*x^4 + 2)