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

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

[Out]

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

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Rubi [A]  time = 0.0560293, 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 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{3 x^2-2}+\sqrt [3]{2}\right )}{x}\right )}{4\ 2^{5/6} d}-\frac{\tanh ^{-1}\left (\frac{\left (\sqrt [3]{3 x^2-2}+\sqrt [3]{2}\right )^2}{3 \sqrt [6]{2} \sqrt{3} x}\right )}{4\ 2^{5/6} \sqrt{3} d}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{6}}\right )}{4\ 2^{5/6} \sqrt{3} d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.1748, size = 42, normalized size = 0.35 \[ \frac{x \left (3 x^{2} - 2\right )^{\frac{2}{3}} \operatorname{appellf_{1}}{\left (\frac{1}{2},\frac{1}{3},1,\frac{3}{2},\frac{3 x^{2}}{2},\frac{x^{2}}{6} \right )}}{12 d \left (- \frac{3 x^{2}}{2} + 1\right )^{\frac{2}{3}}} \]

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 = 0.188333, size = 136, normalized size = 1.14 \[ \frac{9 x F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{3 x^2}{2},\frac{x^2}{6}\right )}{d \left (x^2-6\right ) \sqrt [3]{3 x^2-2} \left (x^2 \left (F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{3 x^2}{2},\frac{x^2}{6}\right )+3 F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{3 x^2}{2},\frac{x^2}{6}\right )\right )+9 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{3 x^2}{2},\frac{x^2}{6}\right )\right )} \]

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*(-6 + x^2)*(-2 + 3*x^2)^(1
/3)*(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 = 0.074, 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)