3.1090 \(\int \frac{1}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=127 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{2 \sqrt{6}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{2 \sqrt{6}}-\frac{\sqrt{\frac{x^2}{\left (\sqrt{3 x^2-1}+1\right )^2}} \left (\sqrt{3 x^2-1}+1\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )|\frac{1}{2}\right )}{2 \sqrt{3} x} \]

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

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Rubi [A]  time = 0.143443, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{2 \sqrt{6}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{2 \sqrt{6}}-\frac{\sqrt{\frac{x^2}{\left (\sqrt{3 x^2-1}+1\right )^2}} \left (\sqrt{3 x^2-1}+1\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )|\frac{1}{2}\right )}{2 \sqrt{3} x} \]

Antiderivative was successfully verified.

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

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Rubi in Sympy [A]  time = 49.1533, size = 170, normalized size = 1.34 \[ - \frac{\sqrt{2} x \left (1 - i\right ) \Pi \left (i; \operatorname{asin}{\left (\frac{\sqrt{2} \left (1 + i\right ) \sqrt [4]{3 x^{2} - 1}}{2} \right )}\middle | -1\right )}{2 \sqrt{- i \sqrt{3 x^{2} - 1} + 1} \sqrt{i \sqrt{3 x^{2} - 1} + 1}} - \frac{\sqrt{3} \sqrt{\frac{x^{2}}{\left (\sqrt{3 x^{2} - 1} + 1\right )^{2}}} \left (\sqrt{3 x^{2} - 1} + 1\right ) F\left (2 \operatorname{atan}{\left (\sqrt [4]{3 x^{2} - 1} \right )}\middle | \frac{1}{2}\right )}{12 x} - \frac{\sqrt{6} \sqrt{x^{2}} \operatorname{atanh}{\left (\frac{\sqrt{6} \sqrt [4]{3 x^{2} - 1}}{3 \sqrt{x^{2}}} \right )}}{12 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Mathematica [C]  time = 0.159596, size = 129, normalized size = 1.02 \[ \frac{2 x F_1\left (\frac{1}{2};\frac{3}{4},1;\frac{3}{2};3 x^2,\frac{3 x^2}{2}\right )}{\left (3 x^2-2\right ) \left (3 x^2-1\right )^{3/4} \left (x^2 \left (2 F_1\left (\frac{3}{2};\frac{3}{4},2;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right )+3 F_1\left (\frac{3}{2};\frac{7}{4},1;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right )\right )+2 F_1\left (\frac{1}{2};\frac{3}{4},1;\frac{3}{2};3 x^2,\frac{3 x^2}{2}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*x*AppellF1[1/2, 3/4, 1, 3/2, 3*x^2, (3*x^2)/2])/((-2 + 3*x^2)*(-1 + 3*x^2)^(3
/4)*(2*AppellF1[1/2, 3/4, 1, 3/2, 3*x^2, (3*x^2)/2] + x^2*(2*AppellF1[3/2, 3/4,
2, 5/2, 3*x^2, (3*x^2)/2] + 3*AppellF1[3/2, 7/4, 1, 5/2, 3*x^2, (3*x^2)/2])))

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Maple [F]  time = 0.098, size = 0, normalized size = 0. \[ \int{\frac{1}{3\,{x}^{2}-2} \left ( 3\,{x}^{2}-1 \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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