\(\int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{2-x} x} \, dx\) [490]

Optimal result

Integrand size = 22, antiderivative size = 99 \[ \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{2-x} x} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{2} (2-x)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{2 \sqrt [3]{2}}+\frac {3 \log \left (-\sqrt [3]{1-x}+\frac {(2-x)^{2/3}}{2^{2/3}}\right )}{4 \sqrt [3]{2}}-\frac {\log (x)}{2 \sqrt [3]{2}} \] Output:

-1/4*3^(1/2)*arctan(1/3*3^(1/2)+1/3*2^(1/3)*(2-x)^(2/3)*3^(1/2)/(1-x)^(1/3 
))*2^(2/3)+3/8*ln(-(1-x)^(1/3)+1/2*(2-x)^(2/3)*2^(1/3))*2^(2/3)-1/4*ln(x)* 
2^(2/3)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.42 \[ \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{2-x} x} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{1-x}}{\sqrt [3]{1-x}+\sqrt [3]{2} (2-x)^{2/3}}\right )+2 \log \left (2 \sqrt [3]{1-x}-\sqrt [3]{2} (2-x)^{2/3}\right )-\log \left (4 (1-x)^{2/3}+2 \sqrt [3]{2-2 x} (2-x)^{2/3}+2^{2/3} (2-x)^{4/3}\right )}{4 \sqrt [3]{2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {133}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 133

\(\displaystyle -\frac {\sqrt {3} \arctan \left (\frac {\sqrt [3]{2} (2-x)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt [3]{2}}+\frac {3 \log \left (\frac {(2-x)^{2/3}}{2^{2/3}}-\sqrt [3]{1-x}\right )}{4 \sqrt [3]{2}}-\frac {\log (x)}{2 \sqrt [3]{2}}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 133

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)*((e_.) + (f_.)*(x_)) 
^(1/3)), x_] :> 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]
 

Maple [F]

\[\int \frac {1}{\left (1-x \right )^{\frac {1}{3}} \left (-x +2\right )^{\frac {1}{3}} x}d x\]

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 293 vs. \(2 (73) = 146\).

Time = 0.92 (sec) , antiderivative size = 293, normalized size of antiderivative = 2.96 \[ \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{2-x} x} \, dx=\frac {1}{6} \cdot 2^{\frac {1}{6}} \sqrt {\frac {3}{2}} \arctan \left (\frac {2^{\frac {1}{6}} \sqrt {\frac {3}{2}} {\left (12 \cdot 2^{\frac {2}{3}} {\left (x^{4} - 36 \, x^{3} + 180 \, x^{2} - 288 \, x + 144\right )} {\left (-x + 2\right )}^{\frac {2}{3}} {\left (-x + 1\right )}^{\frac {2}{3}} - 12 \, {\left (x^{5} - 14 \, x^{4} + 36 \, x^{3} - 24 \, x^{2}\right )} {\left (-x + 2\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}} + 2^{\frac {1}{3}} {\left (x^{6} - 72 \, x^{5} + 792 \, x^{4} - 3168 \, x^{3} + 5904 \, x^{2} - 5184 \, x + 1728\right )}\right )}}{3 \, {\left (x^{6} - 432 \, x^{4} + 2592 \, x^{3} - 5616 \, x^{2} + 5184 \, x - 1728\right )}}\right ) + \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (\frac {2^{\frac {1}{3}} x^{2} + 6 \cdot 2^{\frac {2}{3}} {\left (-x + 2\right )}^{\frac {2}{3}} {\left (-x + 1\right )}^{\frac {2}{3}} + 6 \, {\left (x - 2\right )} {\left (-x + 2\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}}}{x^{2}}\right ) - \frac {1}{24} \cdot 2^{\frac {2}{3}} \log \left (\frac {24 \, {\left (x^{2} - 6 \, x + 6\right )} {\left (-x + 2\right )}^{\frac {2}{3}} {\left (-x + 1\right )}^{\frac {2}{3}} - 6 \cdot 2^{\frac {1}{3}} {\left (x^{3} - 14 \, x^{2} + 36 \, x - 24\right )} {\left (-x + 2\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}} + 2^{\frac {2}{3}} {\left (x^{4} - 36 \, x^{3} + 180 \, x^{2} - 288 \, x + 144\right )}}{x^{4}}\right ) \] Input:

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

Output:

1/6*2^(1/6)*sqrt(3/2)*arctan(1/3*2^(1/6)*sqrt(3/2)*(12*2^(2/3)*(x^4 - 36*x 
^3 + 180*x^2 - 288*x + 144)*(-x + 2)^(2/3)*(-x + 1)^(2/3) - 12*(x^5 - 14*x 
^4 + 36*x^3 - 24*x^2)*(-x + 2)^(1/3)*(-x + 1)^(1/3) + 2^(1/3)*(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 - 1728)) + 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)
 

Sympy [F]

\[ \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{2-x} x} \, dx=\int \frac {1}{x \sqrt [3]{1 - x} \sqrt [3]{2 - x}}\, dx \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{2-x} x} \, dx=\int \frac {1}{x\,{\left (1-x\right )}^{1/3}\,{\left (2-x\right )}^{1/3}} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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