3.1034 \(\int \frac{x}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx\)

Optimal. Leaf size=91 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2}-\sqrt{2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{3\ 2^{3/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2-3 x^2}+\sqrt{2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{3\ 2^{3/4}} \]

[Out]

ArcTan[(Sqrt[2] - Sqrt[2 - 3*x^2])/(2^(3/4)*(2 - 3*x^2)^(1/4))]/(3*2^(3/4)) + Ar
cTanh[(Sqrt[2] + Sqrt[2 - 3*x^2])/(2^(3/4)*(2 - 3*x^2)^(1/4))]/(3*2^(3/4))

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Rubi [A]  time = 0.0589236, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2}-\sqrt{2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{3\ 2^{3/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2-3 x^2}+\sqrt{2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{3\ 2^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[x/((2 - 3*x^2)^(1/4)*(4 - 3*x^2)),x]

[Out]

ArcTan[(Sqrt[2] - Sqrt[2 - 3*x^2])/(2^(3/4)*(2 - 3*x^2)^(1/4))]/(3*2^(3/4)) + Ar
cTanh[(Sqrt[2] + Sqrt[2 - 3*x^2])/(2^(3/4)*(2 - 3*x^2)^(1/4))]/(3*2^(3/4))

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Rubi in Sympy [A]  time = 27.6946, size = 128, normalized size = 1.41 \[ - \frac{\sqrt [4]{2} \log{\left (- 2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2} + \sqrt{- 3 x^{2} + 2} + \sqrt{2} \right )}}{12} + \frac{\sqrt [4]{2} \log{\left (2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2} + \sqrt{- 3 x^{2} + 2} + \sqrt{2} \right )}}{12} - \frac{\sqrt [4]{2} \operatorname{atan}{\left (\sqrt [4]{2} \sqrt [4]{- 3 x^{2} + 2} - 1 \right )}}{6} - \frac{\sqrt [4]{2} \operatorname{atan}{\left (\sqrt [4]{2} \sqrt [4]{- 3 x^{2} + 2} + 1 \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(-3*x**2+2)**(1/4)/(-3*x**2+4),x)

[Out]

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

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Mathematica [A]  time = 0.0718851, size = 100, normalized size = 1.1 \[ \frac{-\log \left (\sqrt{4-6 x^2}-2 \sqrt [4]{4-6 x^2}+2\right )+\log \left (\sqrt{4-6 x^2}+2 \sqrt [4]{4-6 x^2}+2\right )+2 \tan ^{-1}\left (1-\sqrt [4]{4-6 x^2}\right )-2 \tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )}{6\ 2^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/((2 - 3*x^2)^(1/4)*(4 - 3*x^2)),x]

[Out]

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

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \[ \int{\frac{x}{-3\,{x}^{2}+4}{\frac{1}{\sqrt [4]{-3\,{x}^{2}+2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(-3*x^2+2)^(1/4)/(-3*x^2+4),x)

[Out]

int(x/(-3*x^2+2)^(1/4)/(-3*x^2+4),x)

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Maxima [A]  time = 1.50988, size = 159, normalized size = 1.75 \[ -\frac{1}{6} \cdot 2^{\frac{1}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} + 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{1}{6} \cdot 2^{\frac{1}{4}} \arctan \left (-\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} - 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) + \frac{1}{12} \cdot 2^{\frac{1}{4}} \log \left (2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) - \frac{1}{12} \cdot 2^{\frac{1}{4}} \log \left (-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.271445, size = 250, normalized size = 2.75 \[ \frac{1}{24} \cdot 2^{\frac{3}{4}}{\left (4 \, \sqrt{2} \arctan \left (\frac{1}{2^{\frac{1}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{\sqrt{2} \sqrt{-3 \, x^{2} + 2} + 2 \cdot 2^{\frac{1}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 2} + 1}\right ) + 4 \, \sqrt{2} \arctan \left (\frac{1}{2^{\frac{1}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{\sqrt{2} \sqrt{-3 \, x^{2} + 2} - 2 \cdot 2^{\frac{1}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 2} - 1}\right ) + \sqrt{2} \log \left (4 \, \sqrt{2} \sqrt{-3 \, x^{2} + 2} + 8 \cdot 2^{\frac{1}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 8\right ) - \sqrt{2} \log \left (4 \, \sqrt{2} \sqrt{-3 \, x^{2} + 2} - 8 \cdot 2^{\frac{1}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 8\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x}{3 x^{2} \sqrt [4]{- 3 x^{2} + 2} - 4 \sqrt [4]{- 3 x^{2} + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(-3*x**2+2)**(1/4)/(-3*x**2+4),x)

[Out]

-Integral(x/(3*x**2*(-3*x**2 + 2)**(1/4) - 4*(-3*x**2 + 2)**(1/4)), x)

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GIAC/XCAS [A]  time = 0.237091, size = 159, normalized size = 1.75 \[ -\frac{1}{6} \cdot 2^{\frac{1}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} + 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{1}{6} \cdot 2^{\frac{1}{4}} \arctan \left (-\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} - 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) + \frac{1}{12} \cdot 2^{\frac{1}{4}}{\rm ln}\left (2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) - \frac{1}{12} \cdot 2^{\frac{1}{4}}{\rm ln}\left (-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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