3.1035 \(\int \frac{1}{x \sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx\)

Optimal. Leaf size=145 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4 \sqrt [4]{2}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2}-\sqrt{2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{4\ 2^{3/4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4 \sqrt [4]{2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2-3 x^2}+\sqrt{2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{4\ 2^{3/4}} \]

[Out]

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

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Rubi [A]  time = 0.232747, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4 \sqrt [4]{2}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2}-\sqrt{2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{4\ 2^{3/4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4 \sqrt [4]{2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2-3 x^2}+\sqrt{2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{4\ 2^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 35.6925, size = 178, normalized size = 1.23 \[ - \frac{\sqrt [4]{2} \log{\left (- 2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2} + \sqrt{- 3 x^{2} + 2} + \sqrt{2} \right )}}{16} + \frac{\sqrt [4]{2} \log{\left (2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2} + \sqrt{- 3 x^{2} + 2} + \sqrt{2} \right )}}{16} + \frac{2^{\frac{3}{4}} \operatorname{atan}{\left (\frac{2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2}}{2} \right )}}{8} - \frac{\sqrt [4]{2} \operatorname{atan}{\left (\sqrt [4]{2} \sqrt [4]{- 3 x^{2} + 2} - 1 \right )}}{8} - \frac{\sqrt [4]{2} \operatorname{atan}{\left (\sqrt [4]{2} \sqrt [4]{- 3 x^{2} + 2} + 1 \right )}}{8} - \frac{2^{\frac{3}{4}} \operatorname{atanh}{\left (\frac{2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2}}{2} \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/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))/16 +
2**(1/4)*log(2**(3/4)*(-3*x**2 + 2)**(1/4) + sqrt(-3*x**2 + 2) + sqrt(2))/16 +
2**(3/4)*atan(2**(3/4)*(-3*x**2 + 2)**(1/4)/2)/8 - 2**(1/4)*atan(2**(1/4)*(-3*x*
*2 + 2)**(1/4) - 1)/8 - 2**(1/4)*atan(2**(1/4)*(-3*x**2 + 2)**(1/4) + 1)/8 - 2**
(3/4)*atanh(2**(3/4)*(-3*x**2 + 2)**(1/4)/2)/8

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Mathematica [C]  time = 0.239162, size = 140, normalized size = 0.97 \[ \frac{54 x^2 F_1\left (\frac{5}{4};\frac{1}{4},1;\frac{9}{4};\frac{2}{3 x^2},\frac{4}{3 x^2}\right )}{5 \sqrt [4]{2-3 x^2} \left (3 x^2-4\right ) \left (27 x^2 F_1\left (\frac{5}{4};\frac{1}{4},1;\frac{9}{4};\frac{2}{3 x^2},\frac{4}{3 x^2}\right )+2 \left (8 F_1\left (\frac{9}{4};\frac{1}{4},2;\frac{13}{4};\frac{2}{3 x^2},\frac{4}{3 x^2}\right )+F_1\left (\frac{9}{4};\frac{5}{4},1;\frac{13}{4};\frac{2}{3 x^2},\frac{4}{3 x^2}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(54*x^2*AppellF1[5/4, 1/4, 1, 9/4, 2/(3*x^2), 4/(3*x^2)])/(5*(2 - 3*x^2)^(1/4)*(
-4 + 3*x^2)*(27*x^2*AppellF1[5/4, 1/4, 1, 9/4, 2/(3*x^2), 4/(3*x^2)] + 2*(8*Appe
llF1[9/4, 1/4, 2, 13/4, 2/(3*x^2), 4/(3*x^2)] + AppellF1[9/4, 5/4, 1, 13/4, 2/(3
*x^2), 4/(3*x^2)])))

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Maple [F]  time = 0.077, size = 0, normalized size = 0. \[ \int{\frac{1}{x \left ( -3\,{x}^{2}+4 \right ) }{\frac{1}{\sqrt [4]{-3\,{x}^{2}+2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{1}{{\left (3 \, x^{2} - 4\right )}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.261023, size = 344, normalized size = 2.37 \[ -\frac{1}{16} \cdot 2^{\frac{1}{4}}{\left (4 \, \sqrt{2} \arctan \left (\frac{2}{2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2 \, \sqrt{2} \sqrt{-3 \, x^{2} + 2} + 4}}\right ) + \sqrt{2} \log \left (2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 2\right ) - \sqrt{2} \log \left (2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} - 2\right ) - 4 \, \arctan \left (\frac{1}{2^{\frac{1}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{\sqrt{2} \sqrt{-3 \, x^{2} + 2} + 2 \cdot 2^{\frac{1}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 2} + 1}\right ) - 4 \, \arctan \left (\frac{1}{2^{\frac{1}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{\sqrt{2} \sqrt{-3 \, x^{2} + 2} - 2 \cdot 2^{\frac{1}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 2} - 1}\right ) - \log \left (4 \, \sqrt{2} \sqrt{-3 \, x^{2} + 2} + 8 \cdot 2^{\frac{1}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 8\right ) + \log \left (4 \, \sqrt{2} \sqrt{-3 \, x^{2} + 2} - 8 \cdot 2^{\frac{1}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 8\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/16*2^(1/4)*(4*sqrt(2)*arctan(2/(2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2*sqrt(2)*s
qrt(-3*x^2 + 2) + 4))) + sqrt(2)*log(2^(3/4)*(-3*x^2 + 2)^(1/4) + 2) - sqrt(2)*l
og(2^(3/4)*(-3*x^2 + 2)^(1/4) - 2) - 4*arctan(1/(2^(1/4)*(-3*x^2 + 2)^(1/4) + sq
rt(sqrt(2)*sqrt(-3*x^2 + 2) + 2*2^(1/4)*(-3*x^2 + 2)^(1/4) + 2) + 1)) - 4*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)) - log(4*sqrt(2)*sqrt(-3*x^2 + 2) + 8*2^(1/4)*(-3*x^2 +
2)^(1/4) + 8) + 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. \[ - \int \frac{1}{3 x^{3} \sqrt [4]{- 3 x^{2} + 2} - 4 x \sqrt [4]{- 3 x^{2} + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.257897, size = 292, normalized size = 2.01 \[ -\frac{1}{16} \cdot 4^{\frac{3}{8}} \sqrt{2} \arctan \left (\frac{1}{8} \cdot 4^{\frac{7}{8}} \sqrt{2}{\left (4^{\frac{1}{8}} \sqrt{2} + 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{1}{16} \cdot 4^{\frac{3}{8}} \sqrt{2} \arctan \left (-\frac{1}{8} \cdot 4^{\frac{7}{8}} \sqrt{2}{\left (4^{\frac{1}{8}} \sqrt{2} - 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) + \frac{1}{32} \cdot 4^{\frac{3}{8}} \sqrt{2}{\rm ln}\left (4^{\frac{1}{8}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{-3 \, x^{2} + 2} + 4^{\frac{1}{4}}\right ) - \frac{1}{32} \cdot 4^{\frac{3}{8}} \sqrt{2}{\rm ln}\left (-4^{\frac{1}{8}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{-3 \, x^{2} + 2} + 4^{\frac{1}{4}}\right ) + \frac{1}{8} \cdot 4^{\frac{1}{8}} \sqrt{2} \arctan \left (\frac{1}{4} \cdot 4^{\frac{7}{8}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right ) + \frac{1}{16} \cdot 4^{\frac{1}{8}} \sqrt{2}{\rm ln}\left (-{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 4^{\frac{1}{8}}\right ) - \frac{1}{16} \cdot 4^{\frac{3}{8}}{\rm ln}\left ({\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 4^{\frac{1}{8}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/16*4^(3/8)*sqrt(2)*arctan(1/8*4^(7/8)*sqrt(2)*(4^(1/8)*sqrt(2) + 2*(-3*x^2 +
2)^(1/4))) - 1/16*4^(3/8)*sqrt(2)*arctan(-1/8*4^(7/8)*sqrt(2)*(4^(1/8)*sqrt(2) -
2*(-3*x^2 + 2)^(1/4))) + 1/32*4^(3/8)*sqrt(2)*ln(4^(1/8)*sqrt(2)*(-3*x^2 + 2)^(
1/4) + sqrt(-3*x^2 + 2) + 4^(1/4)) - 1/32*4^(3/8)*sqrt(2)*ln(-4^(1/8)*sqrt(2)*(-
3*x^2 + 2)^(1/4) + sqrt(-3*x^2 + 2) + 4^(1/4)) + 1/8*4^(1/8)*sqrt(2)*arctan(1/4*
4^(7/8)*(-3*x^2 + 2)^(1/4)) + 1/16*4^(1/8)*sqrt(2)*ln(-(-3*x^2 + 2)^(1/4) + 4^(1
/8)) - 1/16*4^(3/8)*ln((-3*x^2 + 2)^(1/4) + 4^(1/8))