3.55 \(\int \frac {1}{x \sqrt [3]{4-6 x+3 x^2}} \, dx\)
Optimal. Leaf size=97 \[ \frac {\log \left (-3 \sqrt [3]{2} \sqrt [3]{3 x^2-6 x+4}-3 x+6\right )}{2\ 2^{2/3}}-\frac {\tan ^{-1}\left (\frac {2^{2/3} (2-x)}{\sqrt {3} \sqrt [3]{3 x^2-6 x+4}}+\frac {1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log (x)}{2\ 2^{2/3}} \]
[Out]
-1/4*ln(x)*2^(1/3)+1/4*ln(6-3*x-3*2^(1/3)*(3*x^2-6*x+4)^(1/3))*2^(1/3)+1/6*arctan(-1/3*3^(1/2)-1/3*2^(2/3)*(2-
x)/(3*x^2-6*x+4)^(1/3)*3^(1/2))*2^(1/3)*3^(1/2)
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Rubi [A] time = 0.01, antiderivative size = 97, normalized size of antiderivative = 1.00,
number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used =
{750} \[ \frac {\log \left (-3 \sqrt [3]{2} \sqrt [3]{3 x^2-6 x+4}-3 x+6\right )}{2\ 2^{2/3}}-\frac {\tan ^{-1}\left (\frac {2^{2/3} (2-x)}{\sqrt {3} \sqrt [3]{3 x^2-6 x+4}}+\frac {1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log (x)}{2\ 2^{2/3}} \]
Antiderivative was successfully verified.
[In]
Int[1/(x*(4 - 6*x + 3*x^2)^(1/3)),x]
[Out]
-(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))
Rule 750
Int[1/(((d_.) + (e_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(1/3)), x_Symbol] :> With[{q = Rt[3*c*e^2*(2*c*
d - b*e), 3]}, -Simp[(Sqrt[3]*c*e*ArcTan[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] && NeQ[2*c*d - b*e, 0] && 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)]
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt [3]{4-6 x+3 x^2}} \, dx &=-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (2-x)}{\sqrt {3} \sqrt [3]{4-6 x+3 x^2}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log (x)}{2\ 2^{2/3}}+\frac {\log \left (6-3 x-3 \sqrt [3]{2} \sqrt [3]{4-6 x+3 x^2}\right )}{2\ 2^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 111, normalized size = 1.14 \[ -\frac {\sqrt [3]{\frac {3 x+i \sqrt {3}-3}{x}} \sqrt [3]{\frac {9 x-3 i \sqrt {3}-9}{x}} F_1\left (\frac {2}{3};\frac {1}{3},\frac {1}{3};\frac {5}{3};\frac {3-i \sqrt {3}}{3 x},\frac {3+i \sqrt {3}}{3 x}\right )}{2 \sqrt [3]{3 x^2-6 x+4}} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[1/(x*(4 - 6*x + 3*x^2)^(1/3)),x]
[Out]
-1/2*(((-3 + I*Sqrt[3] + 3*x)/x)^(1/3)*((-9 - (3*I)*Sqrt[3] + 9*x)/x)^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, (3 -
I*Sqrt[3])/(3*x), (3 + I*Sqrt[3])/(3*x)])/(4 - 6*x + 3*x^2)^(1/3)
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fricas [B] time = 4.15, size = 171, normalized size = 1.76 \[ -\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (4^{\frac {1}{3}} x^{3} + 2 \cdot 4^{\frac {2}{3}} {\left (3 \, x^{2} - 6 \, x + 4\right )}^{\frac {2}{3}} {\left (x - 2\right )} + 4 \, {\left (3 \, x^{2} - 6 \, x + 4\right )}^{\frac {1}{3}} {\left (x^{2} - 4 \, x + 4\right )}\right )}}{6 \, {\left (x^{3} - 12 \, x^{2} + 24 \, x - 16\right )}}\right ) + \frac {1}{12} \cdot 4^{\frac {2}{3}} \log \left (\frac {4^{\frac {1}{3}} {\left (x - 2\right )} + 2 \, {\left (3 \, x^{2} - 6 \, x + 4\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{24} \cdot 4^{\frac {2}{3}} \log \left (\frac {4^{\frac {2}{3}} {\left (3 \, x^{2} - 6 \, x + 4\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (x^{2} - 4 \, x + 4\right )} - 2 \, {\left (3 \, x^{2} - 6 \, x + 4\right )}^{\frac {1}{3}} {\left (x - 2\right )}}{x^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/(3*x^2-6*x+4)^(1/3),x, algorithm="fricas")
[Out]
-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)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (3 \, x^{2} - 6 \, x + 4\right )}^{\frac {1}{3}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/(3*x^2-6*x+4)^(1/3),x, algorithm="giac")
[Out]
integrate(1/((3*x^2 - 6*x + 4)^(1/3)*x), x)
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maple [C] time = 14.13, size = 2378, normalized size = 24.52 \[ \text {Expression too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x/(3*x^2-6*x+4)^(1/3),x)
[Out]
1/6*RootOf(_Z^3-2)*ln((160*RootOf(_Z^3-2)-8*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^3+192*RootOf
(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^2-384*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)+256*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)-5*RootOf(_Z^3-2)*x^3+120*RootOf(_
Z^3-2)*x^2-240*RootOf(_Z^3-2)*x-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^2-64*RootOf(RootOf(_Z^3-2)^2+2*_Z*Ro
otOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2-40*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^
3+48*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^2*(3*x^2-6*x+4)^(2/3)*x+18*RootOf(Root
Of(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*(3*x^2-6*x+4)^(1/3)*x^2-72*RootOf(RootOf(_Z^3-2)^2+2*_
Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*(3*x^2-6*x+4)^(1/3)*x-96*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4
*_Z^2)*RootOf(_Z^3-2)^2*(3*x^2-6*x+4)^(2/3)-15*RootOf(_Z^3-2)^2*(3*x^2-6*x+4)^(1/3)*x^2+60*RootOf(_Z^3-2)^2*(3
*x^2-6*x+4)^(1/3)*x+72*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*(3*x^2-6*x+4)^(1/3)+
96*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x+60*RootOf(RootOf(_Z^3-2)^2+2*_Z*Ro
otOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x-60*RootOf(_Z^3-2)^2*(3*x^2-6*x+4)^(1/3)+18*(3*x^2-6*x+4)^(2/3)*x+16*Ro
otOf(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*RootO
f(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^3)/x^3)-1/3*ln((-200*RootOf(_Z^3-2)+6*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf
(_Z^3-2)+4*_Z^2)*x^3-60*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^2+120*RootOf(RootOf(_Z^3-2)^2+2*
_Z*RootOf(_Z^3-2)+4*_Z^2)*x+60*(3*x^2-6*x+4)^(2/3)-80*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)+15*R
ootOf(_Z^3-2)*x^3-150*RootOf(_Z^3-2)*x^2+300*RootOf(_Z^3-2)*x-12*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^2-16*R
ootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2-40*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_
Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3-48*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^2*(3*x^2
-6*x+4)^(2/3)*x-30*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*(3*x^2-6*x+4)^(1/3)*x^2+
120*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*(3*x^2-6*x+4)^(1/3)*x+96*RootOf(RootOf(
_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^2*(3*x^2-6*x+4)^(2/3)+9*RootOf(_Z^3-2)^2*(3*x^2-6*x+4)^(1
/3)*x^2-36*RootOf(_Z^3-2)^2*(3*x^2-6*x+4)^(1/3)*x-120*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*Root
Of(_Z^3-2)*(3*x^2-6*x+4)^(1/3)+24*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x+60*
RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x+36*RootOf(_Z^3-2)^2*(3*x^2-6*x+4)^(1/3)
-30*(3*x^2-6*x+4)^(2/3)*x+4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^3+10*Root
Of(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^3)/x^3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf
(_Z^3-2)+4*_Z^2)-1/6*ln((-200*RootOf(_Z^3-2)+6*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^3-60*Root
Of(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^2+120*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x+
60*(3*x^2-6*x+4)^(2/3)-80*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)+15*RootOf(_Z^3-2)*x^3-150*RootOf
(_Z^3-2)*x^2+300*RootOf(_Z^3-2)*x-12*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^2-16*RootOf(RootOf(_Z^3-2)^2+2*_Z*
RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2-40*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2
)^3-48*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^2*(3*x^2-6*x+4)^(2/3)*x-30*RootOf(Ro
otOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*(3*x^2-6*x+4)^(1/3)*x^2+120*RootOf(RootOf(_Z^3-2)^2+
2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*(3*x^2-6*x+4)^(1/3)*x+96*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2
)+4*_Z^2)*RootOf(_Z^3-2)^2*(3*x^2-6*x+4)^(2/3)+9*RootOf(_Z^3-2)^2*(3*x^2-6*x+4)^(1/3)*x^2-36*RootOf(_Z^3-2)^2*
(3*x^2-6*x+4)^(1/3)*x-120*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*(3*x^2-6*x+4)^(1/
3)+24*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x+60*RootOf(RootOf(_Z^3-2)^2+2*_Z
*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x+36*RootOf(_Z^3-2)^2*(3*x^2-6*x+4)^(1/3)-30*(3*x^2-6*x+4)^(2/3)*x+4*
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*Roo
tOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^3)/x^3)*RootOf(_Z^3-2)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (3 \, x^{2} - 6 \, x + 4\right )}^{\frac {1}{3}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/(3*x^2-6*x+4)^(1/3),x, algorithm="maxima")
[Out]
integrate(1/((3*x^2 - 6*x + 4)^(1/3)*x), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x\,{\left (3\,x^2-6\,x+4\right )}^{1/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(x*(3*x^2 - 6*x + 4)^(1/3)),x)
[Out]
int(1/(x*(3*x^2 - 6*x + 4)^(1/3)), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sqrt [3]{3 x^{2} - 6 x + 4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/(3*x**2-6*x+4)**(1/3),x)
[Out]
Integral(1/(x*(3*x**2 - 6*x + 4)**(1/3)), x)
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