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

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

[Out]

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

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Rubi [A]  time = 0.134494, antiderivative size = 97, normalized size of antiderivative = 0.84, number of steps used = 9, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462 \[ \frac{\log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{4\ 6^{3/4}}-\frac{\log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{4\ 6^{3/4}}-\frac{\tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac{\tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{2\ 6^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(2 + 3*x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 18.4263, size = 83, normalized size = 0.72 \[ \frac{\sqrt [4]{6} \log{\left (3 x^{2} - 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{24} - \frac{\sqrt [4]{6} \log{\left (3 x^{2} + 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{24} + \frac{\sqrt [4]{6} \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{12} + \frac{\sqrt [4]{6} \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0306633, size = 77, normalized size = 0.67 \[ \frac{\log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )-\log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )-2 \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2 \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4\ 6^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(2 + 3*x^4),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])/(4*6^(3/4))

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Maple [A]  time = 0.004, size = 111, normalized size = 1. \[{\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}}{72}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }+{\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}}{72}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}}{144}\ln \left ({1 \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(3*x^4+2),x)

[Out]

1/72*6^(3/4)*3^(1/2)*2^(1/2)*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/4)*x+1)+1/72*6^(3/4
)*3^(1/2)*2^(1/2)*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/4)*x-1)+1/144*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)))

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Maxima [A]  time = 1.60043, size = 163, normalized size = 1.42 \[ \frac{1}{12} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x + 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{12} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x - 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) - \frac{1}{24} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{1}{24} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.239131, size = 223, normalized size = 1.94 \[ -\frac{1}{432} \cdot 54^{\frac{3}{4}}{\left (4 \, \sqrt{2} \arctan \left (\frac{54}{54^{\frac{3}{4}} \sqrt{2} \sqrt{\frac{1}{6}} \sqrt{\sqrt{6}{\left (9 \, \sqrt{6} x^{2} + 54^{\frac{3}{4}} \sqrt{2} x + 18\right )}} + 3 \cdot 54^{\frac{3}{4}} \sqrt{2} x + 54}\right ) + 4 \, \sqrt{2} \arctan \left (\frac{54}{54^{\frac{3}{4}} \sqrt{2} \sqrt{\frac{1}{6}} \sqrt{\sqrt{6}{\left (9 \, \sqrt{6} x^{2} - 54^{\frac{3}{4}} \sqrt{2} x + 18\right )}} + 3 \cdot 54^{\frac{3}{4}} \sqrt{2} x - 54}\right ) + \sqrt{2} \log \left (9 \, \sqrt{6} x^{2} + 54^{\frac{3}{4}} \sqrt{2} x + 18\right ) - \sqrt{2} \log \left (9 \, \sqrt{6} x^{2} - 54^{\frac{3}{4}} \sqrt{2} x + 18\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/432*54^(3/4)*(4*sqrt(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)*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)*log(9*sqrt(6)*x^2 + 54^
(3/4)*sqrt(2)*x + 18) - sqrt(2)*log(9*sqrt(6)*x^2 - 54^(3/4)*sqrt(2)*x + 18))

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Sympy [A]  time = 1.54761, size = 87, normalized size = 0.76 \[ \frac{\sqrt [4]{6} \log{\left (x^{2} - \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{24} - \frac{\sqrt [4]{6} \log{\left (x^{2} + \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{24} + \frac{\sqrt [4]{6} \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{12} + \frac{\sqrt [4]{6} \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(3*x**4+2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.229129, size = 128, normalized size = 1.11 \[ \frac{1}{12} \cdot 6^{\frac{1}{4}} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{12} \cdot 6^{\frac{1}{4}} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) - \frac{1}{24} \cdot 6^{\frac{1}{4}}{\rm ln}\left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) + \frac{1}{24} \cdot 6^{\frac{1}{4}}{\rm ln}\left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*6^(1/4)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x + sqrt(2)*(2/3)^(1/4))) + 1/12*
6^(1/4)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x - sqrt(2)*(2/3)^(1/4))) - 1/24*6^(1/
4)*ln(x^2 + sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3)) + 1/24*6^(1/4)*ln(x^2 - sqrt(2)*(
2/3)^(1/4)*x + sqrt(2/3))