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

Optimal. Leaf size=120 \[ \frac{\tan ^{-1}\left (\frac{2-\sqrt{2} \sqrt{2-3 x^2}}{\sqrt [4]{2} \sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{2\ 2^{3/4} \sqrt{3}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{2-3 x^2}+2}{\sqrt [4]{2} \sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{2\ 2^{3/4} \sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.054775, antiderivative size = 120, 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 \[ \frac{\tan ^{-1}\left (\frac{2^{3/4}-\sqrt [4]{2} \sqrt{2-3 x^2}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{2\ 2^{3/4} \sqrt{3}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt{2-3 x^2}+2^{3/4}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{2\ 2^{3/4} \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 73.619, size = 88, normalized size = 0.73 \[ - \frac{\sqrt [4]{2} \sqrt{3} i \sqrt{x^{2}} \Pi \left (- i; \operatorname{asin}{\left (\frac{2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2}}{2} \right )}\middle | -1\right )}{6 x} + \frac{\sqrt [4]{2} \sqrt{3} i \sqrt{x^{2}} \Pi \left (i; \operatorname{asin}{\left (\frac{2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2}}{2} \right )}\middle | -1\right )}{6 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.196143, size = 135, normalized size = 1.12 \[ -\frac{4 x F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )}{\sqrt [4]{2-3 x^2} \left (3 x^2-4\right ) \left (x^2 \left (2 F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )+F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )\right )+4 F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.076, size = 0, normalized size = 0. \[ \int{\frac{1}{-3\,{x}^{2}+4}{\frac{1}{\sqrt [4]{-3\,{x}^{2}+2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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