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

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])

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Rubi [A]  time = 0.0333361, 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])

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Rubi in Sympy [A]  time = 51.1358, size = 185, normalized size = 3.03 \[ \frac{\sqrt{2} x \left (1 - i\right ) \Pi \left (i; \operatorname{asin}{\left (\frac{\sqrt{2} \left (1 + i\right ) \sqrt [4]{- 3 x^{2} - 1}}{2} \right )}\middle | -1\right )}{2 \sqrt{- i \sqrt{- 3 x^{2} - 1} + 1} \sqrt{i \sqrt{- 3 x^{2} - 1} + 1}} + \frac{\sqrt{6} \sqrt{- x^{2}} \operatorname{atanh}{\left (\frac{\sqrt{6} \sqrt [4]{- 3 x^{2} - 1}}{3 \sqrt{- x^{2}}} \right )}}{12 x} + \frac{\sqrt{3} \sqrt{- \frac{x^{2}}{\left (\sqrt{- 3 x^{2} - 1} + 1\right )^{2}}} \left (\sqrt{- 3 x^{2} - 1} + 1\right ) F\left (2 \operatorname{atan}{\left (\sqrt [4]{- 3 x^{2} - 1} \right )}\middle | \frac{1}{2}\right )}{12 x} \]

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(6)*sq
rt(-x**2)*atanh(sqrt(6)*(-3*x**2 - 1)**(1/4)/(3*sqrt(-x**2)))/(12*x) + sqrt(3)*s
qrt(-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)

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Mathematica [C]  time = 0.221579, size = 127, normalized size = 2.08 \[ \frac{2 x F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-3 x^2,-\frac{3 x^2}{2}\right )}{\sqrt [4]{-3 x^2-1} \left (3 x^2+2\right ) \left (x^2 \left (2 F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-3 x^2,-\frac{3 x^2}{2}\right )+F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-3 x^2,-\frac{3 x^2}{2}\right )\right )-2 F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-3 x^2,-\frac{3 x^2}{2}\right )\right )} \]

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

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Maple [F]  time = 0.036, 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)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 2.76999, size = 332, normalized size = 5.44 \[ -\frac{1}{24} \, \sqrt{6}{\left (\log \left (\frac{\sqrt{6}{\left (3 \, \sqrt{-3 \, x^{2} - 1} x + \sqrt{6}{\left (-3 \, x^{2} - 1\right )}^{\frac{3}{4}} - 3 \, x - \sqrt{6}{\left (-3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right )}}{9 \,{\left (3 \, x^{2} + 2\right )}}\right ) - \log \left (-\frac{\sqrt{6}{\left (3 \, \sqrt{-3 \, x^{2} - 1} x - \sqrt{6}{\left (-3 \, x^{2} - 1\right )}^{\frac{3}{4}} - 3 \, x + \sqrt{6}{\left (-3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right )}}{9 \,{\left (3 \, x^{2} + 2\right )}}\right ) - i \, \log \left (\frac{\sqrt{6}{\left (6 i \, \sqrt{-3 \, x^{2} - 1} x + 2 \, \sqrt{6}{\left (-3 \, x^{2} - 1\right )}^{\frac{3}{4}} + 6 i \, x + 2 \, \sqrt{6}{\left (-3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right )}}{18 \,{\left (3 \, x^{2} + 2\right )}}\right ) + i \, \log \left (\frac{\sqrt{6}{\left (-6 i \, \sqrt{-3 \, x^{2} - 1} x + 2 \, \sqrt{6}{\left (-3 \, x^{2} - 1\right )}^{\frac{3}{4}} - 6 i \, x + 2 \, \sqrt{6}{\left (-3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right )}}{18 \,{\left (3 \, x^{2} + 2\right )}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*sqrt(6)*(log(1/9*sqrt(6)*(3*sqrt(-3*x^2 - 1)*x + sqrt(6)*(-3*x^2 - 1)^(3/4
) - 3*x - sqrt(6)*(-3*x^2 - 1)^(1/4))/(3*x^2 + 2)) - log(-1/9*sqrt(6)*(3*sqrt(-3
*x^2 - 1)*x - sqrt(6)*(-3*x^2 - 1)^(3/4) - 3*x + sqrt(6)*(-3*x^2 - 1)^(1/4))/(3*
x^2 + 2)) - I*log(1/18*sqrt(6)*(6*I*sqrt(-3*x^2 - 1)*x + 2*sqrt(6)*(-3*x^2 - 1)^
(3/4) + 6*I*x + 2*sqrt(6)*(-3*x^2 - 1)^(1/4))/(3*x^2 + 2)) + I*log(1/18*sqrt(6)*
(-6*I*sqrt(-3*x^2 - 1)*x + 2*sqrt(6)*(-3*x^2 - 1)^(3/4) - 6*I*x + 2*sqrt(6)*(-3*
x^2 - 1)^(1/4))/(3*x^2 + 2)))

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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