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

Optimal. Leaf size=115 \[ -\frac{\log \left (\sqrt{3} x^2-2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{8 \sqrt [4]{6}}+\frac{\log \left (\sqrt{3} x^2+2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{8 \sqrt [4]{6}}-\frac{\tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac{\tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4 \sqrt [4]{6}} \]

[Out]

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

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Rubi [A]  time = 0.127433, antiderivative size = 97, normalized size of antiderivative = 0.84, number of steps used = 9, number of rules used = 6, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667 \[ -\frac{\log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{8 \sqrt [4]{6}}+\frac{\log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{8 \sqrt [4]{6}}-\frac{\tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac{\tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4 \sqrt [4]{6}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + 3*x^4)^(-1),x]

[Out]

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

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Rubi in Sympy [A]  time = 17.3884, size = 83, normalized size = 0.72 \[ - \frac{6^{\frac{3}{4}} \log{\left (3 x^{2} - 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{48} + \frac{6^{\frac{3}{4}} \log{\left (3 x^{2} + 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{48} + \frac{6^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{24} + \frac{6^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{24} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(3*x**4+2),x)

[Out]

-6**(3/4)*log(3*x**2 - 6**(3/4)*x + sqrt(6))/48 + 6**(3/4)*log(3*x**2 + 6**(3/4)
*x + sqrt(6))/48 + 6**(3/4)*atan(6**(1/4)*x - 1)/24 + 6**(3/4)*atan(6**(1/4)*x +
1)/24

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Mathematica [A]  time = 0.0265599, size = 77, normalized size = 0.67 \[ \frac{-\log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )+\log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )-2 \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2 \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{8 \sqrt [4]{6}} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + 3*x^4)^(-1),x]

[Out]

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

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Maple [A]  time = 0.003, size = 111, normalized size = 1. \[{\frac{\sqrt{3}\sqrt [4]{6}\sqrt{2}}{24}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }+{\frac{\sqrt{3}\sqrt [4]{6}\sqrt{2}}{24}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{\sqrt{3}\sqrt [4]{6}\sqrt{2}}{48}\ln \left ({1 \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(3*x^4+2),x)

[Out]

1/24*3^(1/2)*6^(1/4)*2^(1/2)*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/4)*x+1)+1/24*3^(1/2
)*6^(1/4)*2^(1/2)*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/4)*x-1)+1/48*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 = 1.60736, size = 163, normalized size = 1.42 \[ \frac{1}{24} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x + 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{24} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x - 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{48} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) - \frac{1}{48} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*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)))
+ 1/24*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
))) + 1/48*3^(3/4)*2^(3/4)*log(sqrt(3)*x^2 + 3^(1/4)*2^(3/4)*x + sqrt(2)) - 1/48
*3^(3/4)*2^(3/4)*log(sqrt(3)*x^2 - 3^(1/4)*2^(3/4)*x + sqrt(2))

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Fricas [A]  time = 0.241417, size = 219, normalized size = 1.9 \[ -\frac{1}{192} \cdot 24^{\frac{3}{4}}{\left (4 \, \sqrt{2} \arctan \left (\frac{2}{24^{\frac{1}{4}} \sqrt{2} \sqrt{\frac{1}{6}} \sqrt{\sqrt{6}{\left (\sqrt{6} x^{2} + 24^{\frac{1}{4}} \sqrt{2} x + 2\right )}} + 24^{\frac{1}{4}} \sqrt{2} x + 2}\right ) + 4 \, \sqrt{2} \arctan \left (\frac{2}{24^{\frac{1}{4}} \sqrt{2} \sqrt{\frac{1}{6}} \sqrt{\sqrt{6}{\left (\sqrt{6} x^{2} - 24^{\frac{1}{4}} \sqrt{2} x + 2\right )}} + 24^{\frac{1}{4}} \sqrt{2} x - 2}\right ) - \sqrt{2} \log \left (2 \, \sqrt{6} x^{2} + 2 \cdot 24^{\frac{1}{4}} \sqrt{2} x + 4\right ) + \sqrt{2} \log \left (2 \, \sqrt{6} x^{2} - 2 \cdot 24^{\frac{1}{4}} \sqrt{2} x + 4\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/192*24^(3/4)*(4*sqrt(2)*arctan(2/(24^(1/4)*sqrt(2)*sqrt(1/6)*sqrt(sqrt(6)*(sq
rt(6)*x^2 + 24^(1/4)*sqrt(2)*x + 2)) + 24^(1/4)*sqrt(2)*x + 2)) + 4*sqrt(2)*arct
an(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))

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Sympy [A]  time = 1.57107, size = 87, normalized size = 0.76 \[ - \frac{6^{\frac{3}{4}} \log{\left (x^{2} - \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{24} + \frac{6^{\frac{3}{4}} \log{\left (x^{2} + \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{24} + \frac{6^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{12} + \frac{6^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(3*x**4+2),x)

[Out]

-6**(3/4)*log(x**2 - 6**(3/4)*x/3 + sqrt(6)/3)/24 + 6**(3/4)*log(x**2 + 6**(3/4)
*x/3 + sqrt(6)/3)/24 + 6**(3/4)*atan(6**(1/4)*x - 1)/12 + 6**(3/4)*atan(6**(1/4)
*x + 1)/12

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GIAC/XCAS [A]  time = 0.23348, size = 128, normalized size = 1.11 \[ \frac{1}{24} \cdot 6^{\frac{3}{4}} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{24} \cdot 6^{\frac{3}{4}} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{48} \cdot 6^{\frac{3}{4}}{\rm ln}\left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) - \frac{1}{48} \cdot 6^{\frac{3}{4}}{\rm ln}\left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*6^(3/4)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x + sqrt(2)*(2/3)^(1/4))) + 1/24*
6^(3/4)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x - sqrt(2)*(2/3)^(1/4))) + 1/48*6^(3/
4)*ln(x^2 + sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3)) - 1/48*6^(3/4)*ln(x^2 - sqrt(2)*(
2/3)^(1/4)*x + sqrt(2/3))