3.55 \(\int \frac{1}{x \sqrt [3]{4-6 x+3 x^2}} \, dx\)

Optimal. Leaf size=97 \[ \frac{\log \left (-3 \sqrt [3]{2} \sqrt [3]{3 x^2-6 x+4}-3 x+6\right )}{2\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} (2-x)}{\sqrt{3} \sqrt [3]{3 x^2-6 x+4}}+\frac{1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}-\frac{\log (x)}{2\ 2^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.0133008, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {750} \[ \frac{\log \left (-3 \sqrt [3]{2} \sqrt [3]{3 x^2-6 x+4}-3 x+6\right )}{2\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} (2-x)}{\sqrt{3} \sqrt [3]{3 x^2-6 x+4}}+\frac{1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}-\frac{\log (x)}{2\ 2^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(4 - 6*x + 3*x^2)^(1/3)),x]

[Out]

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

Rule 750

Int[1/(((d_.) + (e_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(1/3)), x_Symbol] :> With[{q = Rt[3*c*e^2*(2*c*
d - b*e), 3]}, -Simp[(Sqrt[3]*c*e*ArcTan[1/Sqrt[3] + (2*(c*d - b*e - c*e*x))/(Sqrt[3]*q*(a + b*x + c*x^2)^(1/3
))])/q^2, x] + (-Simp[(3*c*e*Log[d + e*x])/(2*q^2), x] + Simp[(3*c*e*Log[c*d - b*e - c*e*x - q*(a + b*x + c*x^
2)^(1/3)])/(2*q^2), x])] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && EqQ[c^2*d^2 - b*c*d*e + b^2*e^
2 - 3*a*c*e^2, 0] && PosQ[c*e^2*(2*c*d - b*e)]

Rubi steps

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

Mathematica [C]  time = 0.0602103, size = 111, normalized size = 1.14 \[ -\frac{\sqrt [3]{\frac{3 x+i \sqrt{3}-3}{x}} \sqrt [3]{\frac{9 x-3 i \sqrt{3}-9}{x}} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{3-i \sqrt{3}}{3 x},\frac{3+i \sqrt{3}}{3 x}\right )}{2 \sqrt [3]{3 x^2-6 x+4}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(x*(4 - 6*x + 3*x^2)^(1/3)),x]

[Out]

-(((-3 + I*Sqrt[3] + 3*x)/x)^(1/3)*((-9 - (3*I)*Sqrt[3] + 9*x)/x)^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, (3 - I*Sq
rt[3])/(3*x), (3 + I*Sqrt[3])/(3*x)])/(2*(4 - 6*x + 3*x^2)^(1/3))

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Maple [F]  time = 0.166, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}{\frac{1}{\sqrt [3]{3\,{x}^{2}-6\,x+4}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(3*x^2-6*x+4)^(1/3),x)

[Out]

int(1/x/(3*x^2-6*x+4)^(1/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (3 \, x^{2} - 6 \, x + 4\right )}^{\frac{1}{3}} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(3*x^2-6*x+4)^(1/3),x, algorithm="maxima")

[Out]

integrate(1/((3*x^2 - 6*x + 4)^(1/3)*x), x)

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Fricas [B]  time = 20.7382, size = 495, normalized size = 5.1 \begin{align*} -\frac{1}{6} \cdot 4^{\frac{1}{6}} \sqrt{3} \arctan \left (\frac{4^{\frac{1}{6}} \sqrt{3}{\left (4^{\frac{1}{3}} x^{3} + 2 \cdot 4^{\frac{2}{3}}{\left (3 \, x^{2} - 6 \, x + 4\right )}^{\frac{2}{3}}{\left (x - 2\right )} + 4 \,{\left (3 \, x^{2} - 6 \, x + 4\right )}^{\frac{1}{3}}{\left (x^{2} - 4 \, x + 4\right )}\right )}}{6 \,{\left (x^{3} - 12 \, x^{2} + 24 \, x - 16\right )}}\right ) + \frac{1}{12} \cdot 4^{\frac{2}{3}} \log \left (\frac{4^{\frac{1}{3}}{\left (x - 2\right )} + 2 \,{\left (3 \, x^{2} - 6 \, x + 4\right )}^{\frac{1}{3}}}{x}\right ) - \frac{1}{24} \cdot 4^{\frac{2}{3}} \log \left (\frac{4^{\frac{2}{3}}{\left (3 \, x^{2} - 6 \, x + 4\right )}^{\frac{2}{3}} + 4^{\frac{1}{3}}{\left (x^{2} - 4 \, x + 4\right )} - 2 \,{\left (3 \, x^{2} - 6 \, x + 4\right )}^{\frac{1}{3}}{\left (x - 2\right )}}{x^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(3*x^2-6*x+4)^(1/3),x, algorithm="fricas")

[Out]

-1/6*4^(1/6)*sqrt(3)*arctan(1/6*4^(1/6)*sqrt(3)*(4^(1/3)*x^3 + 2*4^(2/3)*(3*x^2 - 6*x + 4)^(2/3)*(x - 2) + 4*(
3*x^2 - 6*x + 4)^(1/3)*(x^2 - 4*x + 4))/(x^3 - 12*x^2 + 24*x - 16)) + 1/12*4^(2/3)*log((4^(1/3)*(x - 2) + 2*(3
*x^2 - 6*x + 4)^(1/3))/x) - 1/24*4^(2/3)*log((4^(2/3)*(3*x^2 - 6*x + 4)^(2/3) + 4^(1/3)*(x^2 - 4*x + 4) - 2*(3
*x^2 - 6*x + 4)^(1/3)*(x - 2))/x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt [3]{3 x^{2} - 6 x + 4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(3*x**2-6*x+4)**(1/3),x)

[Out]

Integral(1/(x*(3*x**2 - 6*x + 4)**(1/3)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (3 \, x^{2} - 6 \, x + 4\right )}^{\frac{1}{3}} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(3*x^2-6*x+4)^(1/3),x, algorithm="giac")

[Out]

integrate(1/((3*x^2 - 6*x + 4)^(1/3)*x), x)