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

Optimal. Leaf size=110 \[ -\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{2} (2-x)}{\sqrt {3} \sqrt [3]{2-3 x+x^2}}\right )}{2 \sqrt [3]{2}}-\frac {\log (2-x)}{4 \sqrt [3]{2}}-\frac {\log (x)}{2 \sqrt [3]{2}}+\frac {3 \log \left (2-x-2^{2/3} \sqrt [3]{2-3 x+x^2}\right )}{4 \sqrt [3]{2}} \]

[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)

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Rubi [A]
time = 0.02, antiderivative size = 176, normalized size of antiderivative = 1.60, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {769, 124} \begin {gather*} \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}}-\frac {\sqrt {3} \sqrt [3]{x-2} \sqrt [3]{x-1} \tan ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {1}{x \sqrt [3]{2-3 x+x^2}} \, dx &=\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} \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} \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*}

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Mathematica [A]
time = 0.18, size = 162, normalized size = 1.47 \begin {gather*} \frac {2 \sqrt {3} \tan ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[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))

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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded in __instancecheck__} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

cought exception: maximum recursion depth exceeded in __instancecheck__

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 5.38, size = 1069, normalized size = 9.72

method	result	size
trager	\(\text {Expression too large to display}\)	\(1069\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*RootOf(_Z^3-4)*ln(-(112*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)^2*RootOf(_Z^3-4)^2*x^2+68*Root
Of(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^3*x^2-504*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_
Z^3-4)+4*_Z^2)^2*RootOf(_Z^3-4)^2*x-216*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^2*(
x^2-3*x+2)^(2/3)-306*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_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+258*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^
2)*RootOf(_Z^3-4)*(x^2-3*x+2)^(1/3)*x+306*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^3
-108*RootOf(_Z^3-4)^2*(x^2-3*x+2)^(1/3)*x-516*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-
4)*(x^2-3*x+2)^(1/3)+196*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x^2+216*RootOf(_Z^3-4)^2*(x^2-3*x
+2)^(1/3)+119*RootOf(_Z^3-4)*x^2-1680*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x-948*(x^2-3*x+2)^(2
/3)-1020*RootOf(_Z^3-4)*x+1680*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)+1020*RootOf(_Z^3-4))/x^2)+1
/2*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*ln((68*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z
^2)^2*RootOf(_Z^3-4)^2*x^2+28*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^3*x^2-306*Roo
tOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)^2*RootOf(_Z^3-4)^2*x-108*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(
_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^2*(x^2-3*x+2)^(2/3)-126*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*Roo
tOf(_Z^3-4)^3*x+306*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)^2*RootOf(_Z^3-4)^2-237*RootOf(RootOf(_
Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)*(x^2-3*x+2)^(1/3)*x+126*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootO
f(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^3-54*RootOf(_Z^3-4)^2*(x^2-3*x+2)^(1/3)*x+474*RootOf(RootOf(_Z^3-4)^2+2*_Z*Ro
otOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)*(x^2-3*x+2)^(1/3)+17*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x
^2+108*RootOf(_Z^3-4)^2*(x^2-3*x+2)^(1/3)+7*RootOf(_Z^3-4)*x^2+408*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)
+4*_Z^2)*x+258*(x^2-3*x+2)^(2/3)+168*RootOf(_Z^3-4)*x-408*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)-
168*RootOf(_Z^3-4))/x^2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 277 vs. \(2 (81) = 162\).
time = 1.11, size = 277, normalized size = 2.52 \begin {gather*} -\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \sqrt [3]{\left (x - 2\right ) \left (x - 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Could not integrate

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x\,{\left (x^2-3\,x+2\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(x^2 - 3*x + 2)^(1/3)),x)

[Out]

int(1/(x*(x^2 - 3*x + 2)^(1/3)), x)

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