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

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/Sqrt[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 = 0.0170262, 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, Rules used = {395} \[ \frac{\tan ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{3 x^2-2}+\sqrt [3]{2}\right )}{x}\right )}{4\ 2^{5/6} d}-\frac{\tanh ^{-1}\left (\frac{\left (\sqrt [3]{3 x^2-2}+\sqrt [3]{2}\right )^2}{3 \sqrt [6]{2} \sqrt{3} x}\right )}{4\ 2^{5/6} \sqrt{3} d}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{6}}\right )}{4\ 2^{5/6} \sqrt{3} d} \]

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/Sqrt[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)

Rule 395

Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[-(b/a), 2]}, -Simp[(q*Arc
Tanh[(q*x)/3])/(12*Rt[a, 3]*d), x] + (Simp[(q*ArcTanh[(Rt[a, 3] - (a + b*x^2)^(1/3))^2/(3*Rt[a, 3]^2*q*x)])/(1
2*Rt[a, 3]*d), x] - Simp[(q*ArcTan[(Sqrt[3]*(Rt[a, 3] - (a + b*x^2)^(1/3)))/(Rt[a, 3]*q*x)])/(4*Sqrt[3]*Rt[a,
3]*d), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c - 9*a*d, 0] && NegQ[b/a]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt [3]{-2+3 x^2} \left (-6 d+d x^2\right )} \, dx &=\frac{\tan ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2}+\sqrt [3]{-2+3 x^2}\right )}{x}\right )}{4\ 2^{5/6} d}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{6}}\right )}{4\ 2^{5/6} \sqrt{3} d}-\frac{\tanh ^{-1}\left (\frac{\left (\sqrt [3]{2}+\sqrt [3]{-2+3 x^2}\right )^2}{3 \sqrt [6]{2} \sqrt{3} x}\right )}{4\ 2^{5/6} \sqrt{3} d}\\ \end{align*}

Mathematica [C]  time = 0.101261, 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 = 0.051, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{d{x}^{2}-6\,d}{\frac{1}{\sqrt [3]{3\,{x}^{2}-2}}}}\, dx \end{align*}

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. \begin{align*} \int \frac{1}{{\left (d x^{2} - 6 \, d\right )}{\left (3 \, x^{2} - 2\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

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, 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. \begin{align*} \text{Timed out} \end{align*}

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, algorithm="fricas")

[Out]

Timed out

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

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

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, algorithm="giac")

[Out]

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