3.1064 \(\int \frac{x}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx\)

Optimal. Leaf size=143 \[ \frac{\log \left (\sqrt{2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{12 \sqrt [4]{2}}-\frac{\log \left (\sqrt{2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{12 \sqrt [4]{2}}-\frac{\tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )}{6 \sqrt [4]{2}}+\frac{\tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )}{6 \sqrt [4]{2}} \]

[Out]

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

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Rubi [A]  time = 0.265978, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ \frac{\log \left (\sqrt{2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{12 \sqrt [4]{2}}-\frac{\log \left (\sqrt{2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{12 \sqrt [4]{2}}-\frac{\tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )}{6 \sqrt [4]{2}}+\frac{\tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )}{6 \sqrt [4]{2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 27.3218, size = 128, normalized size = 0.9 \[ \frac{2^{\frac{3}{4}} \log{\left (- 2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2} + \sqrt{- 3 x^{2} + 2} + \sqrt{2} \right )}}{24} - \frac{2^{\frac{3}{4}} \log{\left (2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2} + \sqrt{- 3 x^{2} + 2} + \sqrt{2} \right )}}{24} - \frac{2^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{2} \sqrt [4]{- 3 x^{2} + 2} - 1 \right )}}{12} - \frac{2^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{2} \sqrt [4]{- 3 x^{2} + 2} + 1 \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0761332, size = 100, normalized size = 0.7 \[ \frac{\log \left (\sqrt{4-6 x^2}-2 \sqrt [4]{4-6 x^2}+2\right )-\log \left (\sqrt{4-6 x^2}+2 \sqrt [4]{4-6 x^2}+2\right )+2 \tan ^{-1}\left (1-\sqrt [4]{4-6 x^2}\right )-2 \tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )}{12 \sqrt [4]{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/((2 - 3*x^2)^(3/4)*(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]])/(12*2^(1/4))

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Maple [F]  time = 0.062, size = 0, normalized size = 0. \[ \int{\frac{x}{-3\,{x}^{2}+4} \left ( -3\,{x}^{2}+2 \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.50882, size = 159, normalized size = 1.11 \[ -\frac{1}{12} \cdot 2^{\frac{3}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} + 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{1}{12} \cdot 2^{\frac{3}{4}} \arctan \left (-\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} - 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{1}{24} \cdot 2^{\frac{3}{4}} \log \left (2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{1}{24} \cdot 2^{\frac{3}{4}} \log \left (-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.239979, size = 282, normalized size = 1.97 \[ \frac{1}{96} \cdot 8^{\frac{3}{4}}{\left (4 \, \sqrt{2} \arctan \left (\frac{2}{8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 2 \, \sqrt{8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} \sqrt{-3 \, x^{2} + 2} + 2} + 2}\right ) + 4 \, \sqrt{2} \arctan \left (\frac{2}{8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{-4 \cdot 8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 4 \, \sqrt{2} \sqrt{-3 \, x^{2} + 2} + 8} - 2}\right ) - \sqrt{2} \log \left (4 \cdot 8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 4 \, \sqrt{2} \sqrt{-3 \, x^{2} + 2} + 8\right ) + \sqrt{2} \log \left (-4 \cdot 8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 4 \, \sqrt{2} \sqrt{-3 \, x^{2} + 2} + 8\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/96*8^(3/4)*(4*sqrt(2)*arctan(2/(8^(1/4)*sqrt(2)*(-3*x^2 + 2)^(1/4) + 2*sqrt(8^
(1/4)*sqrt(2)*(-3*x^2 + 2)^(1/4) + sqrt(2)*sqrt(-3*x^2 + 2) + 2) + 2)) + 4*sqrt(
2)*arctan(2/(8^(1/4)*sqrt(2)*(-3*x^2 + 2)^(1/4) + sqrt(-4*8^(1/4)*sqrt(2)*(-3*x^
2 + 2)^(1/4) + 4*sqrt(2)*sqrt(-3*x^2 + 2) + 8) - 2)) - sqrt(2)*log(4*8^(1/4)*sqr
t(2)*(-3*x^2 + 2)^(1/4) + 4*sqrt(2)*sqrt(-3*x^2 + 2) + 8) + sqrt(2)*log(-4*8^(1/
4)*sqrt(2)*(-3*x^2 + 2)^(1/4) + 4*sqrt(2)*sqrt(-3*x^2 + 2) + 8))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.237306, size = 159, normalized size = 1.11 \[ -\frac{1}{12} \cdot 2^{\frac{3}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} + 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{1}{12} \cdot 2^{\frac{3}{4}} \arctan \left (-\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} - 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{1}{24} \cdot 2^{\frac{3}{4}}{\rm ln}\left (2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{1}{24} \cdot 2^{\frac{3}{4}}{\rm ln}\left (-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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