3.159 \(\int \frac{1}{\sqrt [3]{-2-3 x^2} \left (6 d+d x^2\right )} \, dx\)

Optimal. Leaf size=119 \[ -\frac{\tan ^{-1}\left (\frac{\left (\sqrt [3]{-3 x^2-2}+\sqrt [3]{2}\right )^2}{3 \sqrt [6]{2} \sqrt{3} x}\right )}{4\ 2^{5/6} \sqrt{3} d}+\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{-3 x^2-2}+\sqrt [3]{2}\right )}{x}\right )}{4\ 2^{5/6} d}-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{6}}\right )}{4\ 2^{5/6} \sqrt{3} d} \]

[Out]

-ArcTan[x/Sqrt[6]]/(4*2^(5/6)*Sqrt[3]*d) - ArcTan[(2^(1/3) + (-2 - 3*x^2)^(1/3))
^2/(3*2^(1/6)*Sqrt[3]*x)]/(4*2^(5/6)*Sqrt[3]*d) + ArcTanh[(2^(1/6)*(2^(1/3) + (-
2 - 3*x^2)^(1/3)))/x]/(4*2^(5/6)*d)

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Rubi [A]  time = 0.0567701, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ -\frac{\tan ^{-1}\left (\frac{\left (\sqrt [3]{-3 x^2-2}+\sqrt [3]{2}\right )^2}{3 \sqrt [6]{2} \sqrt{3} x}\right )}{4\ 2^{5/6} \sqrt{3} d}+\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{-3 x^2-2}+\sqrt [3]{2}\right )}{x}\right )}{4\ 2^{5/6} d}-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{6}}\right )}{4\ 2^{5/6} \sqrt{3} d} \]

Antiderivative was successfully verified.

[In]  Int[1/((-2 - 3*x^2)^(1/3)*(6*d + d*x^2)),x]

[Out]

-ArcTan[x/Sqrt[6]]/(4*2^(5/6)*Sqrt[3]*d) - ArcTan[(2^(1/3) + (-2 - 3*x^2)^(1/3))
^2/(3*2^(1/6)*Sqrt[3]*x)]/(4*2^(5/6)*Sqrt[3]*d) + ArcTanh[(2^(1/6)*(2^(1/3) + (-
2 - 3*x^2)^(1/3)))/x]/(4*2^(5/6)*d)

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Rubi in Sympy [A]  time = 17.1447, size = 49, normalized size = 0.41 \[ - \frac{x \left (- 3 x^{2} - 2\right )^{\frac{2}{3}} \operatorname{appellf_{1}}{\left (\frac{1}{2},\frac{1}{3},1,\frac{3}{2},- \frac{3 x^{2}}{2},- \frac{x^{2}}{6} \right )}}{12 d \left (\frac{3 x^{2}}{2} + 1\right )^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-3*x**2-2)**(1/3)/(d*x**2+6*d),x)

[Out]

-x*(-3*x**2 - 2)**(2/3)*appellf1(1/2, 1/3, 1, 3/2, -3*x**2/2, -x**2/6)/(12*d*(3*
x**2/2 + 1)**(2/3))

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Mathematica [C]  time = 0.203705, size = 136, normalized size = 1.14 \[ -\frac{9 x F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-\frac{3 x^2}{2},-\frac{x^2}{6}\right )}{d \sqrt [3]{-3 x^2-2} \left (x^2+6\right ) \left (x^2 \left (F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};-\frac{3 x^2}{2},-\frac{x^2}{6}\right )+3 F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};-\frac{3 x^2}{2},-\frac{x^2}{6}\right )\right )-9 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-\frac{3 x^2}{2},-\frac{x^2}{6}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((-2 - 3*x^2)^(1/3)*(6*d + d*x^2)),x]

[Out]

(-9*x*AppellF1[1/2, 1/3, 1, 3/2, (-3*x^2)/2, -x^2/6])/(d*(-2 - 3*x^2)^(1/3)*(6 +
x^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, (-3*x^2)/2, -x^2/6] + x^2*(AppellF1[3/2, 1/
3, 2, 5/2, (-3*x^2)/2, -x^2/6] + 3*AppellF1[3/2, 4/3, 1, 5/2, (-3*x^2)/2, -x^2/6
])))

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Maple [F]  time = 0.034, size = 0, normalized size = 0. \[ \int{\frac{1}{d{x}^{2}+6\,d}{\frac{1}{\sqrt [3]{-3\,{x}^{2}-2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-3*x^2-2)^(1/3)/(d*x^2+6*d),x)

[Out]

int(1/(-3*x^2-2)^(1/3)/(d*x^2+6*d),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (d x^{2} + 6 \, d\right )}{\left (-3 \, x^{2} - 2\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((d*x^2 + 6*d)*(-3*x^2 - 2)^(1/3)),x, algorithm="maxima")

[Out]

integrate(1/((d*x^2 + 6*d)*(-3*x^2 - 2)^(1/3)), 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/((d*x^2 + 6*d)*(-3*x^2 - 2)^(1/3)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int \frac{1}{x^{2} \sqrt [3]{- 3 x^{2} - 2} + 6 \sqrt [3]{- 3 x^{2} - 2}}\, dx}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-3*x**2-2)**(1/3)/(d*x**2+6*d),x)

[Out]

Integral(1/(x**2*(-3*x**2 - 2)**(1/3) + 6*(-3*x**2 - 2)**(1/3)), x)/d

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (d x^{2} + 6 \, d\right )}{\left (-3 \, x^{2} - 2\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((d*x^2 + 6*d)*(-3*x^2 - 2)^(1/3)),x, algorithm="giac")

[Out]

integrate(1/((d*x^2 + 6*d)*(-3*x^2 - 2)^(1/3)), x)