3.880 \(\int \frac{1}{\sqrt [3]{1-x} \sqrt [3]{2-x} x} \, dx\)

Optimal. Leaf size=99 \[ \frac{3 \log \left (\frac{(2-x)^{2/3}}{2^{2/3}}-\sqrt [3]{1-x}\right )}{4 \sqrt [3]{2}}-\frac{\log (x)}{2 \sqrt [3]{2}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{2} (2-x)^{2/3}}{\sqrt{3} \sqrt [3]{1-x}}+\frac{1}{\sqrt{3}}\right )}{2 \sqrt [3]{2}} \]

[Out]

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

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Rubi [A]  time = 0.0107965, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {123} \[ \frac{3 \log \left (\frac{(2-x)^{2/3}}{2^{2/3}}-\sqrt [3]{1-x}\right )}{4 \sqrt [3]{2}}-\frac{\log (x)}{2 \sqrt [3]{2}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{2} (2-x)^{2/3}}{\sqrt{3} \sqrt [3]{1-x}}+\frac{1}{\sqrt{3}}\right )}{2 \sqrt [3]{2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - x)^(1/3)*(2 - x)^(1/3)*x),x]

[Out]

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

Rule 123

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)*((e_.) + (f_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[
(b*(b*e - a*f))/(b*c - a*d)^2, 3]}, -Simp[Log[a + b*x]/(2*q*(b*c - a*d)), x] + (-Simp[(Sqrt[3]*ArcTan[1/Sqrt[3
] + (2*q*(c + d*x)^(2/3))/(Sqrt[3]*(e + f*x)^(1/3))])/(2*q*(b*c - a*d)), x] + Simp[(3*Log[q*(c + d*x)^(2/3) -
(e + f*x)^(1/3)])/(4*q*(b*c - a*d)), x])] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d*e - b*c*f - a*d*f, 0]

Rubi steps

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

Mathematica [C]  time = 0.0250392, size = 32, normalized size = 0.32 \[ -\frac{3}{2} (1-x)^{2/3} F_1\left (\frac{2}{3};\frac{1}{3},1;\frac{5}{3};x-1,1-x\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.04, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}{\frac{1}{\sqrt [3]{1-x}}}{\frac{1}{\sqrt [3]{2-x}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-x)^(1/3)/(2-x)^(1/3)/x,x)

[Out]

int(1/(1-x)^(1/3)/(2-x)^(1/3)/x,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(1/3)/(2-x)^(1/3)/x,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 13.261, size = 878, normalized size = 8.87 \begin{align*} \frac{1}{12} \, \sqrt{3} 2^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3} 2^{\frac{1}{6}}{\left (24 \cdot 2^{\frac{1}{6}}{\left (x^{4} - 36 \, x^{3} + 180 \, x^{2} - 288 \, x + 144\right )}{\left (-x + 2\right )}^{\frac{2}{3}}{\left (-x + 1\right )}^{\frac{2}{3}} - 12 \, \sqrt{2}{\left (x^{5} - 14 \, x^{4} + 36 \, x^{3} - 24 \, x^{2}\right )}{\left (-x + 2\right )}^{\frac{1}{3}}{\left (-x + 1\right )}^{\frac{1}{3}} + 2^{\frac{5}{6}}{\left (x^{6} - 72 \, x^{5} + 792 \, x^{4} - 3168 \, x^{3} + 5904 \, x^{2} - 5184 \, x + 1728\right )}\right )}}{6 \,{\left (x^{6} - 432 \, x^{4} + 2592 \, x^{3} - 5616 \, x^{2} + 5184 \, x - 1728\right )}}\right ) + \frac{1}{12} \cdot 2^{\frac{2}{3}} \log \left (\frac{2^{\frac{1}{3}} x^{2} + 6 \cdot 2^{\frac{2}{3}}{\left (-x + 2\right )}^{\frac{2}{3}}{\left (-x + 1\right )}^{\frac{2}{3}} + 6 \,{\left (x - 2\right )}{\left (-x + 2\right )}^{\frac{1}{3}}{\left (-x + 1\right )}^{\frac{1}{3}}}{x^{2}}\right ) - \frac{1}{24} \cdot 2^{\frac{2}{3}} \log \left (\frac{24 \,{\left (x^{2} - 6 \, x + 6\right )}{\left (-x + 2\right )}^{\frac{2}{3}}{\left (-x + 1\right )}^{\frac{2}{3}} - 6 \cdot 2^{\frac{1}{3}}{\left (x^{3} - 14 \, x^{2} + 36 \, x - 24\right )}{\left (-x + 2\right )}^{\frac{1}{3}}{\left (-x + 1\right )}^{\frac{1}{3}} + 2^{\frac{2}{3}}{\left (x^{4} - 36 \, x^{3} + 180 \, x^{2} - 288 \, x + 144\right )}}{x^{4}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(1/3)/(2-x)^(1/3)/x,x, algorithm="fricas")

[Out]

1/12*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(1/6)*(24*2^(1/6)*(x^4 - 36*x^3 + 180*x^2 - 288*x + 144)*(-x + 2)^(2
/3)*(-x + 1)^(2/3) - 12*sqrt(2)*(x^5 - 14*x^4 + 36*x^3 - 24*x^2)*(-x + 2)^(1/3)*(-x + 1)^(1/3) + 2^(5/6)*(x^6
- 72*x^5 + 792*x^4 - 3168*x^3 + 5904*x^2 - 5184*x + 1728))/(x^6 - 432*x^4 + 2592*x^3 - 5616*x^2 + 5184*x - 172
8)) + 1/12*2^(2/3)*log((2^(1/3)*x^2 + 6*2^(2/3)*(-x + 2)^(2/3)*(-x + 1)^(2/3) + 6*(x - 2)*(-x + 2)^(1/3)*(-x +
1)^(1/3))/x^2) - 1/24*2^(2/3)*log((24*(x^2 - 6*x + 6)*(-x + 2)^(2/3)*(-x + 1)^(2/3) - 6*2^(1/3)*(x^3 - 14*x^2
+ 36*x - 24)*(-x + 2)^(1/3)*(-x + 1)^(1/3) + 2^(2/3)*(x^4 - 36*x^3 + 180*x^2 - 288*x + 144))/x^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)**(1/3)/(2-x)**(1/3)/x,x)

[Out]

Integral(1/(x*(1 - x)**(1/3)*(2 - x)**(1/3)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(1/3)/(2-x)^(1/3)/x,x, algorithm="giac")

[Out]

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