3.693 \(\int \frac{x^4}{2+3 x^4} \, dx\)

Optimal. Leaf size=120 \[ \frac{\log \left (\sqrt{3} x^2-2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{12 \sqrt [4]{6}}-\frac{\log \left (\sqrt{3} x^2+2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{12 \sqrt [4]{6}}+\frac{x}{3}+\frac{\tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{6 \sqrt [4]{6}}-\frac{\tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{6 \sqrt [4]{6}} \]

[Out]

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

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Rubi [A]  time = 0.149742, antiderivative size = 102, normalized size of antiderivative = 0.85, number of steps used = 10, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538 \[ \frac{\log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{12 \sqrt [4]{6}}-\frac{\log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{12 \sqrt [4]{6}}+\frac{x}{3}+\frac{\tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{6 \sqrt [4]{6}}-\frac{\tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{6 \sqrt [4]{6}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(2 + 3*x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 19.2207, size = 87, normalized size = 0.72 \[ \frac{x}{3} + \frac{6^{\frac{3}{4}} \log{\left (3 x^{2} - 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{72} - \frac{6^{\frac{3}{4}} \log{\left (3 x^{2} + 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{72} - \frac{6^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{36} - \frac{6^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{36} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/3 + 6**(3/4)*log(3*x**2 - 6**(3/4)*x + sqrt(6))/72 - 6**(3/4)*log(3*x**2 + 6**
(3/4)*x + sqrt(6))/72 - 6**(3/4)*atan(6**(1/4)*x - 1)/36 - 6**(3/4)*atan(6**(1/4
)*x + 1)/36

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Mathematica [A]  time = 0.0430889, size = 96, normalized size = 0.8 \[ \frac{1}{72} \left (6^{3/4} \log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )-6^{3/4} \log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )+24 x+2\ 6^{3/4} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )-2\ 6^{3/4} \tan ^{-1}\left (\sqrt [4]{6} x+1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(2 + 3*x^4),x]

[Out]

(24*x + 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])/72

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Maple [A]  time = 0.006, size = 114, normalized size = 1. \[{\frac{x}{3}}-{\frac{\sqrt{3}\sqrt [4]{6}\sqrt{2}}{36}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }-{\frac{\sqrt{3}\sqrt [4]{6}\sqrt{2}}{36}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }-{\frac{\sqrt{3}\sqrt [4]{6}\sqrt{2}}{72}\ln \left ({1 \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(3*x^4+2),x)

[Out]

1/3*x-1/36*3^(1/2)*6^(1/4)*2^(1/2)*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/4)*x+1)-1/36*
3^(1/2)*6^(1/4)*2^(1/2)*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/4)*x-1)-1/72*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)))

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Maxima [A]  time = 1.59697, size = 167, normalized size = 1.39 \[ -\frac{1}{36} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x + 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) - \frac{1}{36} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x - 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) - \frac{1}{72} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{1}{72} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{1}{3} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/36*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))
) - 1/36*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))) - 1/72*3^(3/4)*2^(3/4)*log(sqrt(3)*x^2 + 3^(1/4)*2^(3/4)*x + sqrt(2)) + 1/7
2*3^(3/4)*2^(3/4)*log(sqrt(3)*x^2 - 3^(1/4)*2^(3/4)*x + sqrt(2)) + 1/3*x

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Fricas [A]  time = 0.249149, size = 227, normalized size = 1.89 \[ \frac{1}{288} \cdot 24^{\frac{3}{4}}{\left (4 \, \sqrt{2} \arctan \left (\frac{2}{24^{\frac{1}{4}} \sqrt{2} \sqrt{\frac{1}{6}} \sqrt{\sqrt{6}{\left (\sqrt{6} x^{2} + 24^{\frac{1}{4}} \sqrt{2} x + 2\right )}} + 24^{\frac{1}{4}} \sqrt{2} x + 2}\right ) + 4 \, \sqrt{2} \arctan \left (\frac{2}{24^{\frac{1}{4}} \sqrt{2} \sqrt{\frac{1}{6}} \sqrt{\sqrt{6}{\left (\sqrt{6} x^{2} - 24^{\frac{1}{4}} \sqrt{2} x + 2\right )}} + 24^{\frac{1}{4}} \sqrt{2} x - 2}\right ) - \sqrt{2} \log \left (2 \, \sqrt{6} x^{2} + 2 \cdot 24^{\frac{1}{4}} \sqrt{2} x + 4\right ) + \sqrt{2} \log \left (2 \, \sqrt{6} x^{2} - 2 \cdot 24^{\frac{1}{4}} \sqrt{2} x + 4\right ) + 4 \cdot 24^{\frac{1}{4}} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/288*24^(3/4)*(4*sqrt(2)*arctan(2/(24^(1/4)*sqrt(2)*sqrt(1/6)*sqrt(sqrt(6)*(sqr
t(6)*x^2 + 24^(1/4)*sqrt(2)*x + 2)) + 24^(1/4)*sqrt(2)*x + 2)) + 4*sqrt(2)*arcta
n(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)*log(2*sqrt(6)*x^2 + 2*24^(1/4)*sqrt(2)
*x + 4) + sqrt(2)*log(2*sqrt(6)*x^2 - 2*24^(1/4)*sqrt(2)*x + 4) + 4*24^(1/4)*x)

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Sympy [A]  time = 1.57958, size = 90, normalized size = 0.75 \[ \frac{x}{3} + \frac{6^{\frac{3}{4}} \log{\left (x^{2} - \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{36} - \frac{6^{\frac{3}{4}} \log{\left (x^{2} + \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{36} - \frac{6^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{18} - \frac{6^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(3*x**4+2),x)

[Out]

x/3 + 6**(3/4)*log(x**2 - 6**(3/4)*x/3 + sqrt(6)/3)/36 - 6**(3/4)*log(x**2 + 6**
(3/4)*x/3 + sqrt(6)/3)/36 - 6**(3/4)*atan(6**(1/4)*x - 1)/18 - 6**(3/4)*atan(6**
(1/4)*x + 1)/18

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GIAC/XCAS [A]  time = 0.226606, size = 132, normalized size = 1.1 \[ -\frac{1}{36} \cdot 6^{\frac{3}{4}} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) - \frac{1}{36} \cdot 6^{\frac{3}{4}} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) - \frac{1}{72} \cdot 6^{\frac{3}{4}}{\rm ln}\left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) + \frac{1}{72} \cdot 6^{\frac{3}{4}}{\rm ln}\left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) + \frac{1}{3} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/36*6^(3/4)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x + sqrt(2)*(2/3)^(1/4))) - 1/36
*6^(3/4)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x - sqrt(2)*(2/3)^(1/4))) - 1/72*6^(3
/4)*ln(x^2 + sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3)) + 1/72*6^(3/4)*ln(x^2 - sqrt(2)*
(2/3)^(1/4)*x + sqrt(2/3)) + 1/3*x