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

Optimal result

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

[Out]

-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)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.60, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {769, 124} \[ \int \frac {1}{x \sqrt [3]{2-3 x+x^2}} \, dx=-\frac {\sqrt {3} \sqrt [3]{x-2} \sqrt [3]{x-1} \arctan \left (\frac {1}{\sqrt {3}}-\frac {\sqrt [3]{2} (x-2)^{2/3}}{\sqrt {3} \sqrt [3]{x-1}}\right )}{2 \sqrt [3]{2} \sqrt [3]{x^2-3 x+2}}+\frac {3 \sqrt [3]{x-2} \sqrt [3]{x-1} \log \left (-\frac {(x-2)^{2/3}}{\sqrt [3]{2}}-\sqrt [3]{2} \sqrt [3]{x-1}\right )}{4 \sqrt [3]{2} \sqrt [3]{x^2-3 x+2}}-\frac {\sqrt [3]{x-2} \sqrt [3]{x-1} \log (x)}{2 \sqrt [3]{2} \sqrt [3]{x^2-3 x+2}} \]

[In]

Int[1/(x*(2 - 3*x + x^2)^(1/3)),x]

[Out]

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

Rule 124

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]

Rule 769

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

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

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.47 \[ \int \frac {1}{x \sqrt [3]{2-3 x+x^2}} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{2-3 x+x^2}}{2 \sqrt [3]{2}-\sqrt [3]{2} x+\sqrt [3]{2-3 x+x^2}}\right )+2 \log \left (-2 \sqrt [3]{2}+\sqrt [3]{2} x+2 \sqrt [3]{2-3 x+x^2}\right )-\log \left (4\ 2^{2/3}-4\ 2^{2/3} x+2^{2/3} x^2-2 \sqrt [3]{2} (-2+x) \sqrt [3]{2-3 x+x^2}+4 \left (2-3 x+x^2\right )^{2/3}\right )}{4 \sqrt [3]{2}} \]

[In]

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

[Out]

(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))

Maple [C] (warning: unable to verify)

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

Time = 6.43 (sec) , antiderivative size = 1593, normalized size of antiderivative = 14.48

method	result	size
trager	\(\text {Expression too large to display}\)	\(1593\)

[In]

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

[Out]

1/4*RootOf(_Z^3-4)*ln(-(-12*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^3*x^2+112*RootO
f(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)^2*RootOf(_Z^3-4)^2*x^2+216*(x^2-3*x+2)^(2/3)*RootOf(RootOf(_Z^3
-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^2+54*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*Root
Of(_Z^3-4)^3*x-504*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)^2*RootOf(_Z^3-4)^2*x-237*(x^2-3*x+2)^(1
/3)*RootOf(_Z^3-4)^2*x-258*(x^2-3*x+2)^(1/3)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4
)*x-54*RootOf(_Z^3-4)^3*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)+504*RootOf(RootOf(_Z^3-4)^2+2*_Z*R
ootOf(_Z^3-4)+4*_Z^2)^2*RootOf(_Z^3-4)^2+474*(x^2-3*x+2)^(1/3)*RootOf(_Z^3-4)^2+516*(x^2-3*x+2)^(1/3)*RootOf(R
ootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)-3*RootOf(_Z^3-4)*x^2+28*RootOf(RootOf(_Z^3-4)^2+2*_
Z*RootOf(_Z^3-4)+4*_Z^2)*x^2-516*(x^2-3*x+2)^(2/3)-72*RootOf(_Z^3-4)*x+672*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf
(_Z^3-4)+4*_Z^2)*x+72*RootOf(_Z^3-4)-672*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2))/x^2)-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*Root
Of(_Z^3-4)+4*_Z^2)^2*RootOf(_Z^3-4)^2*x^2+216*(x^2-3*x+2)^(2/3)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*
_Z^2)*RootOf(_Z^3-4)^2-54*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^3*x-612*RootOf(Ro
otOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)^2*RootOf(_Z^3-4)^2*x+129*(x^2-3*x+2)^(1/3)*RootOf(_Z^3-4)^2*x+474*(
x^2-3*x+2)^(1/3)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)*x+54*RootOf(_Z^3-4)^3*Root
Of(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)+612*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)^2*Root
Of(_Z^3-4)^2-258*(x^2-3*x+2)^(1/3)*RootOf(_Z^3-4)^2-948*(x^2-3*x+2)^(1/3)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(
_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)+21*RootOf(_Z^3-4)*x^2+238*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x
^2+948*(x^2-3*x+2)^(2/3)-180*RootOf(_Z^3-4)*x-2040*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x+180*R
ootOf(_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*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*RootOf(_Z^3-4)^2*x^2+216*(x^2-3*x+2)^(2/3)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*R
ootOf(_Z^3-4)^2-54*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_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+129*(x^2-3*x+2)^(1/3)*RootOf(_Z^3-4)^2*x+474*(x^2-3*x
+2)^(1/3)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)*x+54*RootOf(_Z^3-4)^3*RootOf(Root
Of(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)+612*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)^2*RootOf(_Z^3
-4)^2-258*(x^2-3*x+2)^(1/3)*RootOf(_Z^3-4)^2-948*(x^2-3*x+2)^(1/3)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)
+4*_Z^2)*RootOf(_Z^3-4)+21*RootOf(_Z^3-4)*x^2+238*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x^2+948*
(x^2-3*x+2)^(2/3)-180*RootOf(_Z^3-4)*x-2040*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x+180*RootOf(_
Z^3-4)+2040*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2))/x^2)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3
-4)+4*_Z^2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 277 vs. \(2 (81) = 162\).

Time = 1.23 (sec) , antiderivative size = 277, normalized size of antiderivative = 2.52 \[ \int \frac {1}{x \sqrt [3]{2-3 x+x^2}} \, dx=-\frac {1}{12} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {1}{6}} {\left (2^{\frac {5}{6}} {\left (x^{6} + 36 \, x^{5} - 612 \, x^{4} + 2880 \, x^{3} - 5760 \, x^{2} + 5184 \, x - 1728\right )} + 12 \, \sqrt {2} {\left (x^{5} - 38 \, x^{4} + 252 \, x^{3} - 648 \, x^{2} + 720 \, x - 288\right )} {\left (x^{2} - 3 \, x + 2\right )}^{\frac {1}{3}} + 48 \cdot 2^{\frac {1}{6}} {\left (x^{4} - 6 \, x^{3} + 6 \, x^{2}\right )} {\left (x^{2} - 3 \, x + 2\right )}^{\frac {2}{3}}\right )}}{6 \, {\left (x^{6} - 108 \, x^{5} + 972 \, x^{4} - 3456 \, x^{3} + 6048 \, x^{2} - 5184 \, x + 1728\right )}}\right ) + \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (\frac {2^{\frac {2}{3}} x^{2} + 6 \cdot 2^{\frac {1}{3}} {\left (x^{2} - 3 \, x + 2\right )}^{\frac {1}{3}} {\left (x - 2\right )} + 12 \, {\left (x^{2} - 3 \, x + 2\right )}^{\frac {2}{3}}}{x^{2}}\right ) - \frac {1}{24} \cdot 2^{\frac {2}{3}} \log \left (\frac {12 \cdot 2^{\frac {2}{3}} {\left (x^{2} - 3 \, x + 2\right )}^{\frac {2}{3}} {\left (x^{2} - 6 \, x + 6\right )} + 2^{\frac {1}{3}} {\left (x^{4} - 36 \, x^{3} + 180 \, x^{2} - 288 \, x + 144\right )} - 6 \, {\left (x^{3} - 14 \, x^{2} + 36 \, x - 24\right )} {\left (x^{2} - 3 \, x + 2\right )}^{\frac {1}{3}}}{x^{4}}\right ) \]

[In]

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

[Out]

-1/12*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(1/6)*(2^(5/6)*(x^6 + 36*x^5 - 612*x^4 + 2880*x^3 - 5760*x^2 + 5184
*x - 1728) + 12*sqrt(2)*(x^5 - 38*x^4 + 252*x^3 - 648*x^2 + 720*x - 288)*(x^2 - 3*x + 2)^(1/3) + 48*2^(1/6)*(x
^4 - 6*x^3 + 6*x^2)*(x^2 - 3*x + 2)^(2/3))/(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) - 1/2
4*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 + 14
4) - 6*(x^3 - 14*x^2 + 36*x - 24)*(x^2 - 3*x + 2)^(1/3))/x^4)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(1/(x*(x^2 - 3*x + 2)^(1/3)),x)

[Out]

int(1/(x*(x^2 - 3*x + 2)^(1/3)), x)