Integrand size = 18, antiderivative size = 198 \[ \int \frac {1}{x \sqrt [3]{4-6 x+3 x^2}} \, dx=-\frac {\arctan \left (\frac {\frac {2\ 2^{2/3}}{\sqrt {3}}-\frac {2^{2/3} x}{\sqrt {3}}+\frac {\sqrt [3]{4-6 x+3 x^2}}{\sqrt {3}}}{\sqrt [3]{4-6 x+3 x^2}}\right )}{2^{2/3} \sqrt {3}}+\frac {\log \left (-2 2^{2/3}+2^{2/3} x+2 \sqrt [3]{4-6 x+3 x^2}\right )}{3\ 2^{2/3}}-\frac {\log \left (-4 \sqrt [3]{2}+4 \sqrt [3]{2} x-\sqrt [3]{2} x^2+\left (-2 2^{2/3}+2^{2/3} x\right ) \sqrt [3]{4-6 x+3 x^2}-2 \left (4-6 x+3 x^2\right )^{2/3}\right )}{6\ 2^{2/3}} \] Output:
-1/6*arctan((2/3*2^(2/3)*3^(1/2)-1/3*2^(2/3)*x*3^(1/2)+1/3*(3*x^2-6*x+4)^( 1/3)*3^(1/2))/(3*x^2-6*x+4)^(1/3))*2^(1/3)*3^(1/2)+1/6*ln(-2*2^(2/3)+2^(2/ 3)*x+2*(3*x^2-6*x+4)^(1/3))*2^(1/3)-1/12*ln(-4*2^(1/3)+4*2^(1/3)*x-2^(1/3) *x^2+(-2*2^(2/3)+2^(2/3)*x)*(3*x^2-6*x+4)^(1/3)-2*(3*x^2-6*x+4)^(2/3))*2^( 1/3)
Time = 0.21 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x \sqrt [3]{4-6 x+3 x^2}} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {2\ 2^{2/3}-2^{2/3} x+\sqrt [3]{4-6 x+3 x^2}}{\sqrt {3} \sqrt [3]{4-6 x+3 x^2}}\right )-2 \log \left (-2 2^{2/3}+2^{2/3} x+2 \sqrt [3]{4-6 x+3 x^2}\right )+\log \left (-4 \sqrt [3]{2}+4 \sqrt [3]{2} x-\sqrt [3]{2} x^2+2^{2/3} (-2+x) \sqrt [3]{4-6 x+3 x^2}-2 \left (4-6 x+3 x^2\right )^{2/3}\right )}{6\ 2^{2/3}} \] Input:
Integrate[1/(x*(4 - 6*x + 3*x^2)^(1/3)),x]
Output:
-1/6*(2*Sqrt[3]*ArcTan[(2*2^(2/3) - 2^(2/3)*x + (4 - 6*x + 3*x^2)^(1/3))/( Sqrt[3]*(4 - 6*x + 3*x^2)^(1/3))] - 2*Log[-2*2^(2/3) + 2^(2/3)*x + 2*(4 - 6*x + 3*x^2)^(1/3)] + Log[-4*2^(1/3) + 4*2^(1/3)*x - 2^(1/3)*x^2 + 2^(2/3) *(-2 + x)*(4 - 6*x + 3*x^2)^(1/3) - 2*(4 - 6*x + 3*x^2)^(2/3)])/2^(2/3)
Time = 0.19 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.49, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1175}
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.
\(\displaystyle \int \frac {1}{x \sqrt [3]{3 x^2-6 x+4}} \, dx\) |
\(\Big \downarrow \) 1175 |
\(\displaystyle -\frac {\arctan \left (\frac {2^{2/3} (2-x)}{\sqrt {3} \sqrt [3]{3 x^2-6 x+4}}+\frac {1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\log \left (-3 \sqrt [3]{2} \sqrt [3]{3 x^2-6 x+4}-3 x+6\right )}{2\ 2^{2/3}}-\frac {\log (x)}{2\ 2^{2/3}}\) |
Input:
Int[1/(x*(4 - 6*x + 3*x^2)^(1/3)),x]
Output:
-(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))
Int[1/(((d_.) + (e_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(1/3)), x_Sy mbol] :> With[{q = Rt[3*c*e^2*(2*c*d - b*e), 3]}, Simp[(-Sqrt[3])*c*e*(ArcT an[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] && 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)]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 12.93 (sec) , antiderivative size = 2380, normalized size of antiderivative = 12.02
Input:
int(1/x/(3*x^2-6*x+4)^(1/3),x,method=_RETURNVERBOSE)
Output:
1/6*RootOf(_Z^3-2)*ln((-48*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_ Z^2)^2*RootOf(_Z^3-2)^2*x^2-30*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2) +4*_Z^2)*RootOf(_Z^3-2)^3*x^2-8*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2 )+4*_Z^2)*x^3-40*RootOf(_Z^3-2)^3*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3 -2)+4*_Z^2)+192*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^2-38 4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x-36*(3*x^2-6*x+4)^( 2/3)-5*RootOf(_Z^3-2)*x^3+120*RootOf(_Z^3-2)*x^2-240*RootOf(_Z^3-2)*x+160* RootOf(_Z^3-2)+256*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)+96* RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x+6 0*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x-6 4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2+1 6*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x ^3+10*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3 *x^3+18*(3*x^2-6*x+4)^(2/3)*x-60*RootOf(_Z^3-2)^2*(3*x^2-6*x+4)^(1/3)-96*( 3*x^2-6*x+4)^(2/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*Roo tOf(_Z^3-2)^2-15*(3*x^2-6*x+4)^(1/3)*RootOf(_Z^3-2)^2*x^2+60*RootOf(_Z^3-2 )^2*(3*x^2-6*x+4)^(1/3)*x+72*(3*x^2-6*x+4)^(1/3)*RootOf(_Z^3-2)*RootOf(Roo tOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)+48*(3*x^2-6*x+4)^(2/3)*RootOf(Ro otOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^2*x+18*(3*x^2-6* x+4)^(1/3)*RootOf(_Z^3-2)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4...
Time = 1.05 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x \sqrt [3]{4-6 x+3 x^2}} \, dx=-\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (4^{\frac {1}{3}} x^{3} + 2 \cdot 4^{\frac {2}{3}} {\left (3 \, x^{2} - 6 \, x + 4\right )}^{\frac {2}{3}} {\left (x - 2\right )} + 4 \, {\left (3 \, x^{2} - 6 \, x + 4\right )}^{\frac {1}{3}} {\left (x^{2} - 4 \, x + 4\right )}\right )}}{6 \, {\left (x^{3} - 12 \, x^{2} + 24 \, x - 16\right )}}\right ) + \frac {1}{12} \cdot 4^{\frac {2}{3}} \log \left (\frac {4^{\frac {1}{3}} {\left (x - 2\right )} + 2 \, {\left (3 \, x^{2} - 6 \, x + 4\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{24} \cdot 4^{\frac {2}{3}} \log \left (\frac {4^{\frac {2}{3}} {\left (3 \, x^{2} - 6 \, x + 4\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (x^{2} - 4 \, x + 4\right )} - 2 \, {\left (3 \, x^{2} - 6 \, x + 4\right )}^{\frac {1}{3}} {\left (x - 2\right )}}{x^{2}}\right ) \] Input:
integrate(1/x/(3*x^2-6*x+4)^(1/3),x, algorithm="fricas")
Output:
-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)
\[ \int \frac {1}{x \sqrt [3]{4-6 x+3 x^2}} \, dx=\int \frac {1}{x \sqrt [3]{3 x^{2} - 6 x + 4}}\, dx \] Input:
integrate(1/x/(3*x**2-6*x+4)**(1/3),x)
Output:
Integral(1/(x*(3*x**2 - 6*x + 4)**(1/3)), x)
\[ \int \frac {1}{x \sqrt [3]{4-6 x+3 x^2}} \, dx=\int { \frac {1}{{\left (3 \, x^{2} - 6 \, x + 4\right )}^{\frac {1}{3}} x} \,d x } \] Input:
integrate(1/x/(3*x^2-6*x+4)^(1/3),x, algorithm="maxima")
Output:
integrate(1/((3*x^2 - 6*x + 4)^(1/3)*x), x)
\[ \int \frac {1}{x \sqrt [3]{4-6 x+3 x^2}} \, dx=\int { \frac {1}{{\left (3 \, x^{2} - 6 \, x + 4\right )}^{\frac {1}{3}} x} \,d x } \] Input:
integrate(1/x/(3*x^2-6*x+4)^(1/3),x, algorithm="giac")
Output:
integrate(1/((3*x^2 - 6*x + 4)^(1/3)*x), x)
Timed out. \[ \int \frac {1}{x \sqrt [3]{4-6 x+3 x^2}} \, dx=\int \frac {1}{x\,{\left (3\,x^2-6\,x+4\right )}^{1/3}} \,d x \] Input:
int(1/(x*(3*x^2 - 6*x + 4)^(1/3)),x)
Output:
int(1/(x*(3*x^2 - 6*x + 4)^(1/3)), x)
\[ \int \frac {1}{x \sqrt [3]{4-6 x+3 x^2}} \, dx=\int \frac {1}{\left (3 x^{2}-6 x +4\right )^{\frac {1}{3}} x}d x \] Input:
int(1/x/(3*x^2-6*x+4)^(1/3),x)
Output:
int(1/((3*x**2 - 6*x + 4)**(1/3)*x),x)