3.1070 \(\int \frac{1}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx\)

Optimal. Leaf size=148 \[ \frac{\tan ^{-1}\left (\frac{2^{3/4}-\sqrt [4]{2} \sqrt{2-3 x^2}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{4 \sqrt [4]{2} \sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt{2-3 x^2}+2^{3/4}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{4 \sqrt [4]{2} \sqrt{3}}+\frac{F\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{2 \sqrt [4]{2} \sqrt{3}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.13442, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{\tan ^{-1}\left (\frac{2^{3/4}-\sqrt [4]{2} \sqrt{2-3 x^2}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{4 \sqrt [4]{2} \sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt{2-3 x^2}+2^{3/4}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{4 \sqrt [4]{2} \sqrt{3}}+\frac{F\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{2 \sqrt [4]{2} \sqrt{3}} \]

Antiderivative was successfully verified.

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 81.7798, size = 87, normalized size = 0.59 \[ - \frac{2^{\frac{3}{4}} \sqrt{3} \sqrt{x^{2}} \Pi \left (- i; \operatorname{asin}{\left (\frac{2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2}}{2} \right )}\middle | -1\right )}{12 x} - \frac{2^{\frac{3}{4}} \sqrt{3} \sqrt{x^{2}} \Pi \left (i; \operatorname{asin}{\left (\frac{2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2}}{2} \right )}\middle | -1\right )}{12 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.185931, size = 137, normalized size = 0.93 \[ -\frac{4 x F_1\left (\frac{1}{2};\frac{3}{4},1;\frac{3}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )}{\left (2-3 x^2\right )^{3/4} \left (3 x^2-4\right ) \left (x^2 \left (2 F_1\left (\frac{3}{2};\frac{3}{4},2;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )+3 F_1\left (\frac{3}{2};\frac{7}{4},1;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )\right )+4 F_1\left (\frac{1}{2};\frac{3}{4},1;\frac{3}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.065, size = 0, normalized size = 0. \[ \int{\frac{1}{-3\,{x}^{2}+4} \left ( -3\,{x}^{2}+2 \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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)^(3/4)),x, algorithm="fricas")

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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