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

Optimal. Leaf size=61 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{2 \sqrt{6}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{2 \sqrt{6}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0340955, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{2 \sqrt{6}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{2 \sqrt{6}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 50.2988, size = 168, normalized size = 2.75 \[ \frac{\sqrt{2} x \left (1 - i\right ) \Pi \left (i; \operatorname{asin}{\left (\frac{\sqrt{2} \left (1 + i\right ) \sqrt [4]{3 x^{2} - 1}}{2} \right )}\middle | -1\right )}{2 \sqrt{- i \sqrt{3 x^{2} - 1} + 1} \sqrt{i \sqrt{3 x^{2} - 1} + 1}} - \frac{\sqrt{3} \sqrt{\frac{x^{2}}{\left (\sqrt{3 x^{2} - 1} + 1\right )^{2}}} \left (\sqrt{3 x^{2} - 1} + 1\right ) F\left (2 \operatorname{atan}{\left (\sqrt [4]{3 x^{2} - 1} \right )}\middle | \frac{1}{2}\right )}{12 x} - \frac{\sqrt{6} \sqrt{x^{2}} \operatorname{atanh}{\left (\frac{\sqrt{6} \sqrt [4]{3 x^{2} - 1}}{3 \sqrt{x^{2}}} \right )}}{12 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*x*(1 - I)*elliptic_pi(I, asin(sqrt(2)*(1 + I)*(3*x**2 - 1)**(1/4)/2), -1
)/(2*sqrt(-I*sqrt(3*x**2 - 1) + 1)*sqrt(I*sqrt(3*x**2 - 1) + 1)) - sqrt(3)*sqrt(
x**2/(sqrt(3*x**2 - 1) + 1)**2)*(sqrt(3*x**2 - 1) + 1)*elliptic_f(2*atan((3*x**2
- 1)**(1/4)), 1/2)/(12*x) - sqrt(6)*sqrt(x**2)*atanh(sqrt(6)*(3*x**2 - 1)**(1/4
)/(3*sqrt(x**2)))/(12*x)

_______________________________________________________________________________________

Mathematica [C]  time = 0.210795, size = 127, normalized size = 2.08 \[ \frac{2 x F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};3 x^2,\frac{3 x^2}{2}\right )}{\left (3 x^2-2\right ) \sqrt [4]{3 x^2-1} \left (x^2 \left (2 F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right )+F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right )\right )+2 F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};3 x^2,\frac{3 x^2}{2}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*x*AppellF1[1/2, 1/4, 1, 3/2, 3*x^2, (3*x^2)/2])/((-2 + 3*x^2)*(-1 + 3*x^2)^(1
/4)*(2*AppellF1[1/2, 1/4, 1, 3/2, 3*x^2, (3*x^2)/2] + x^2*(2*AppellF1[3/2, 1/4,
2, 5/2, 3*x^2, (3*x^2)/2] + AppellF1[3/2, 5/4, 1, 5/2, 3*x^2, (3*x^2)/2])))

_______________________________________________________________________________________

Maple [F]  time = 0.095, size = 0, normalized size = 0. \[ \int{\frac{1}{3\,{x}^{2}-2}{\frac{1}{\sqrt [4]{3\,{x}^{2}-1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 2.70102, size = 139, normalized size = 2.28 \[ \frac{1}{24} \, \sqrt{6}{\left (2 \, \arctan \left (\frac{\sqrt{6}{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}}{3 \, x}\right ) + \log \left (\frac{36 \,{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} x^{3} - 12 \, \sqrt{6} \sqrt{3 \, x^{2} - 1} x^{2} + 24 \,{\left (3 \, x^{2} - 1\right )}^{\frac{3}{4}} x - \sqrt{6}{\left (9 \, x^{4} + 12 \, x^{2} - 4\right )}}{9 \, x^{4} - 12 \, x^{2} + 4}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*sqrt(6)*(2*arctan(1/3*sqrt(6)*(3*x^2 - 1)^(1/4)/x) + log((36*(3*x^2 - 1)^(1
/4)*x^3 - 12*sqrt(6)*sqrt(3*x^2 - 1)*x^2 + 24*(3*x^2 - 1)^(3/4)*x - sqrt(6)*(9*x
^4 + 12*x^2 - 4))/(9*x^4 - 12*x^2 + 4)))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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