\(\int \frac {1}{x \sqrt [3]{2-3 x+x^2}} \, dx\) [39]

Optimal result

Integrand size = 16, antiderivative size = 110 \[ \int \frac {1}{x \sqrt [3]{2-3 x+x^2}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{2} (2-x)}{\sqrt {3} \sqrt [3]{2-3 x+x^2}}\right )}{2 \sqrt [3]{2}}-\frac {\log (2-x)}{4 \sqrt [3]{2}}-\frac {\log (x)}{2 \sqrt [3]{2}}+\frac {3 \log \left (2-x-2^{2/3} \sqrt [3]{2-3 x+x^2}\right )}{4 \sqrt [3]{2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.47 \[ \int \frac {1}{x \sqrt [3]{2-3 x+x^2}} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{2-3 x+x^2}}{2 \sqrt [3]{2}-\sqrt [3]{2} x+\sqrt [3]{2-3 x+x^2}}\right )+2 \log \left (-2 \sqrt [3]{2}+\sqrt [3]{2} x+2 \sqrt [3]{2-3 x+x^2}\right )-\log \left (4\ 2^{2/3}-4\ 2^{2/3} x+2^{2/3} x^2-2 \sqrt [3]{2} (-2+x) \sqrt [3]{2-3 x+x^2}+4 \left (2-3 x+x^2\right )^{2/3}\right )}{4 \sqrt [3]{2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1177, 27, 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.

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

\(\Big \downarrow \) 1177

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 133

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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]
 

rule 1177

Int[1/(((d_.) + (e_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(1/3)), x_Sy 
mbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(b + q + 2*c*x)^(1/3)*((b - q 
+ 2*c*x)^(1/3)/(a + b*x + c*x^2)^(1/3))   Int[1/((d + e*x)*(b + q + 2*c*x)^ 
(1/3)*(b - q + 2*c*x)^(1/3)), x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c 
^2*d^2 - b*c*d*e - 2*b^2*e^2 + 9*a*c*e^2, 0]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 7.76 (sec) , antiderivative size = 1592, normalized size of antiderivative = 14.47

method	result	size
trager	\(\text {Expression too large to display}\)	\(1592\)

Input:

int(1/x/(x^2-3*x+2)^(1/3),x,method=_RETURNVERBOSE)
 

Output:

-1/4*ln((12*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^ 
3-4)^3*x^2+136*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)^2*RootO 
f(_Z^3-4)^2*x^2-54*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*Roo 
tOf(_Z^3-4)^3*x-612*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)^2* 
RootOf(_Z^3-4)^2*x+216*(x^2-3*x+2)^(2/3)*RootOf(_Z^3-4)^2*RootOf(RootOf(_Z 
^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)+54*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf 
(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^3+612*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf( 
_Z^3-4)+4*_Z^2)^2*RootOf(_Z^3-4)^2+129*(x^2-3*x+2)^(1/3)*RootOf(_Z^3-4)^2* 
x+474*(x^2-3*x+2)^(1/3)*RootOf(_Z^3-4)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf 
(_Z^3-4)+4*_Z^2)*x-258*(x^2-3*x+2)^(1/3)*RootOf(_Z^3-4)^2-948*(x^2-3*x+2)^ 
(1/3)*RootOf(_Z^3-4)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)+2 
1*RootOf(_Z^3-4)*x^2+238*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^ 
2)*x^2-180*RootOf(_Z^3-4)*x-2040*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3- 
4)+4*_Z^2)*x+948*(x^2-3*x+2)^(2/3)+180*RootOf(_Z^3-4)+2040*RootOf(RootOf(_ 
Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2))/x^2)*RootOf(_Z^3-4)-1/2*ln((12*RootO 
f(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^3*x^2+136*Ro 
otOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)^2*RootOf(_Z^3-4)^2*x^2-5 
4*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^3*x-6 
12*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)^2*RootOf(_Z^3-4)^2* 
x+216*(x^2-3*x+2)^(2/3)*RootOf(_Z^3-4)^2*RootOf(RootOf(_Z^3-4)^2+2*_Z*R...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (81) = 162\).

Time = 1.00 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.49 \[ \int \frac {1}{x \sqrt [3]{2-3 x+x^2}} \, dx=-\frac {1}{6} \cdot 2^{\frac {1}{6}} \sqrt {\frac {3}{2}} \arctan \left (\frac {2^{\frac {1}{6}} \sqrt {\frac {3}{2}} {\left (24 \cdot 2^{\frac {2}{3}} {\left (x^{4} - 6 \, x^{3} + 6 \, x^{2}\right )} {\left (x^{2} - 3 \, x + 2\right )}^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (x^{6} + 36 \, x^{5} - 612 \, x^{4} + 2880 \, x^{3} - 5760 \, x^{2} + 5184 \, x - 1728\right )} + 12 \, {\left (x^{5} - 38 \, x^{4} + 252 \, x^{3} - 648 \, x^{2} + 720 \, x - 288\right )} {\left (x^{2} - 3 \, x + 2\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (x^{6} - 108 \, x^{5} + 972 \, x^{4} - 3456 \, x^{3} + 6048 \, x^{2} - 5184 \, x + 1728\right )}}\right ) + \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (\frac {2^{\frac {2}{3}} x^{2} + 6 \cdot 2^{\frac {1}{3}} {\left (x^{2} - 3 \, x + 2\right )}^{\frac {1}{3}} {\left (x - 2\right )} + 12 \, {\left (x^{2} - 3 \, x + 2\right )}^{\frac {2}{3}}}{x^{2}}\right ) - \frac {1}{24} \cdot 2^{\frac {2}{3}} \log \left (\frac {12 \cdot 2^{\frac {2}{3}} {\left (x^{2} - 3 \, x + 2\right )}^{\frac {2}{3}} {\left (x^{2} - 6 \, x + 6\right )} + 2^{\frac {1}{3}} {\left (x^{4} - 36 \, x^{3} + 180 \, x^{2} - 288 \, x + 144\right )} - 6 \, {\left (x^{3} - 14 \, x^{2} + 36 \, x - 24\right )} {\left (x^{2} - 3 \, x + 2\right )}^{\frac {1}{3}}}{x^{4}}\right ) \] Input:

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

Output:

-1/6*2^(1/6)*sqrt(3/2)*arctan(1/3*2^(1/6)*sqrt(3/2)*(24*2^(2/3)*(x^4 - 6*x 
^3 + 6*x^2)*(x^2 - 3*x + 2)^(2/3) + 2^(1/3)*(x^6 + 36*x^5 - 612*x^4 + 2880 
*x^3 - 5760*x^2 + 5184*x - 1728) + 12*(x^5 - 38*x^4 + 252*x^3 - 648*x^2 + 
720*x - 288)*(x^2 - 3*x + 2)^(1/3))/(x^6 - 108*x^5 + 972*x^4 - 3456*x^3 + 
6048*x^2 - 5184*x + 1728)) + 1/12*2^(2/3)*log((2^(2/3)*x^2 + 6*2^(1/3)*(x^ 
2 - 3*x + 2)^(1/3)*(x - 2) + 12*(x^2 - 3*x + 2)^(2/3))/x^2) - 1/24*2^(2/3) 
*log((12*2^(2/3)*(x^2 - 3*x + 2)^(2/3)*(x^2 - 6*x + 6) + 2^(1/3)*(x^4 - 36 
*x^3 + 180*x^2 - 288*x + 144) - 6*(x^3 - 14*x^2 + 36*x - 24)*(x^2 - 3*x + 
2)^(1/3))/x^4)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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