∫ 1_____ x 3√_____

3.25.38 \(\int \frac {1}{x \sqrt [3]{4-6 x+3 x^2}} \, dx\)

Optimal. Leaf size=198 \[ \frac {\log \left (2 \sqrt [3]{3 x^2-6 x+4}+2^{2/3} x-2\ 2^{2/3}\right )}{3\ 2^{2/3}}-\frac {\log \left (-\sqrt [3]{2} x^2-2 \left (3 x^2-6 x+4\right )^{2/3}+\left (2^{2/3} x-2\ 2^{2/3}\right ) \sqrt [3]{3 x^2-6 x+4}+4 \sqrt [3]{2} x-4 \sqrt [3]{2}\right )}{6\ 2^{2/3}}-\frac {\tan ^{-1}\left (\frac {\frac {\sqrt [3]{3 x^2-6 x+4}}{\sqrt {3}}-\frac {2^{2/3} x}{\sqrt {3}}+\frac {2\ 2^{2/3}}{\sqrt {3}}}{\sqrt [3]{3 x^2-6 x+4}}\right )}{2^{2/3} \sqrt {3}} \]

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Rubi [A] time = 0.01, antiderivative size = 97, normalized size of antiderivative = 0.49, 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} \begin {gather*} \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}} \end {gather*}

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 = 0.06, size = 111, normalized size = 0.56 \begin {gather*} -\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}} \end {gather*}

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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IntegrateAlgebraic [A] time = 0.22, size = 198, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\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}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

IntegrateAlgebraic[1/(x*(4 - 6*x + 3*x^2)^(1/3)),x]

[Out]

-(ArcTan[((2*2^(2/3))/Sqrt[3] - (2^(2/3)*x)/Sqrt[3] + (4 - 6*x + 3*x^2)^(1/3)/Sqrt[3])/(4 - 6*x + 3*x^2)^(1/3)

]/(2^(2/3)*Sqrt[3])) + Log[-2*2^(2/3) + 2^(2/3)*x + 2*(4 - 6*x + 3*x^2)^(1/3)]/(3*2^(2/3)) - Log[-4*2^(1/3) +

4*2^(1/3)*x - 2^(1/3)*x^2 + (-2*2^(2/3) + 2^(2/3)*x)*(4 - 6*x + 3*x^2)^(1/3) - 2*(4 - 6*x + 3*x^2)^(2/3)]/(6*2

^(2/3))

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fricas [A] time = 2.13, size = 171, normalized size = 0.86 \begin {gather*} -\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 ) \end {gather*}

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 \begin {gather*} \int \frac {1}{{\left (3 \, x^{2} - 6 \, x + 4\right )}^{\frac {1}{3}} x}\,{d x} \end {gather*}

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 = 14.91, size = 2399, normalized size = 12.12


method	result	size

trager	\(\text {Expression too large to display}\)	\(2399\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(3*x^2-6*x+4)^(1/3),x,method=_RETURNVERBOSE)

[Out]

1/3*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*ln(-(12*RootOf(_Z^3-2)*x^3-120*RootOf(_Z^3-2)*x^2+240*

RootOf(_Z^3-2)*x+30*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^3-300*RootOf(RootOf(_Z^3-2)^2+2*_Z*R

ootOf(_Z^3-2)+4*_Z^2)*x^2+600*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)-400

*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)+20*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*

RootOf(_Z^3-2)^2*x^3-80*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2-32*RootOf(RootO

f(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3-160*RootOf(_Z^3-2)+48*(3*x^2-6*x+4)^(2/3)*RootOf(_Z^3

-2)^2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x-48*RootOf(_Z^3-2)*RootOf(RootOf(_Z^3-2)^2+2*_Z*Roo

tOf(_Z^3-2)+4*_Z^2)*(3*x^2-6*x+4)^(1/3)*x^2+192*(3*x^2-6*x+4)^(1/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2

)+4*_Z^2)*RootOf(_Z^3-2)*x+8*RootOf(_Z^3-2)^3*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^3-60*RootO

f(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^2-24*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_

Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^2-192*(3*x^2-6*x+4)^(1/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2

)*RootOf(_Z^3-2)-96*(3*x^2-6*x+4)^(2/3)*RootOf(_Z^3-2)^2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)-1

5*RootOf(_Z^3-2)^2*(3*x^2-6*x+4)^(1/3)*x^2+60*(3*x^2-6*x+4)^(1/3)*RootOf(_Z^3-2)^2*x-60*(3*x^2-6*x+4)^(1/3)*Ro

otOf(_Z^3-2)^2+30*(3*x^2-6*x+4)^(2/3)*x+120*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-

2)^2*x+48*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x)/x^3)-1/6*ln(-(-RootOf(_Z^3-2

)*x^3+24*RootOf(_Z^3-2)*x^2-48*RootOf(_Z^3-2)*x-10*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^3+240

*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^2-480*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)+320*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)+20*RootOf(RootOf(_Z^3-2)^2

+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^3-80*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*R

ootOf(_Z^3-2)^2-8*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3+32*RootOf(_Z^3-2)-48*(3

*x^2-6*x+4)^(2/3)*RootOf(_Z^3-2)^2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x+48*RootOf(_Z^3-2)*Roo

tOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*(3*x^2-6*x+4)^(1/3)*x^2-192*(3*x^2-6*x+4)^(1/3)*RootOf(RootOf

(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*x+2*RootOf(_Z^3-2)^3*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)^2*RootOf(_Z^3-2)^2*x^2-6*RootOf(Ro

otOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^2+192*(3*x^2-6*x+4)^(1/3)*RootOf(RootOf(_Z^3-2)^

2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)+96*(3*x^2-6*x+4)^(2/3)*RootOf(_Z^3-2)^2*RootOf(RootOf(_Z^3-2)^2+2

*_Z*RootOf(_Z^3-2)+4*_Z^2)+9*RootOf(_Z^3-2)^2*(3*x^2-6*x+4)^(1/3)*x^2-36*(3*x^2-6*x+4)^(1/3)*RootOf(_Z^3-2)^2*

x+36*(3*x^2-6*x+4)^(1/3)*RootOf(_Z^3-2)^2-18*(3*x^2-6*x+4)^(2/3)*x+120*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^

3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x+12*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x)/x

^3)*RootOf(_Z^3-2)-1/3*ln(-(-RootOf(_Z^3-2)*x^3+24*RootOf(_Z^3-2)*x^2-48*RootOf(_Z^3-2)*x-10*RootOf(RootOf(_Z^

3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^3+240*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^2-480*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)+320*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(

_Z^3-2)+4*_Z^2)+20*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^3-80*RootOf(RootOf

(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2-8*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2

)*RootOf(_Z^3-2)^3+32*RootOf(_Z^3-2)-48*(3*x^2-6*x+4)^(2/3)*RootOf(_Z^3-2)^2*RootOf(RootOf(_Z^3-2)^2+2*_Z*Root

Of(_Z^3-2)+4*_Z^2)*x+48*RootOf(_Z^3-2)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*(3*x^2-6*x+4)^(1/3)

*x^2-192*(3*x^2-6*x+4)^(1/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*x+2*RootOf(_Z^

3-2)^3*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)^2*RootOf(_Z^3-2)^2*x^2-6*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^2+192*

(3*x^2-6*x+4)^(1/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)+96*(3*x^2-6*x+4)^(2/3)*

RootOf(_Z^3-2)^2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)+9*RootOf(_Z^3-2)^2*(3*x^2-6*x+4)^(1/3)*x^

2-36*(3*x^2-6*x+4)^(1/3)*RootOf(_Z^3-2)^2*x+36*(3*x^2-6*x+4)^(1/3)*RootOf(_Z^3-2)^2-18*(3*x^2-6*x+4)^(2/3)*x+1

20*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x+12*RootOf(RootOf(_Z^3-2)^2+2*_Z*Ro

otOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x)/x^3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)

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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (3 \, x^{2} - 6 \, x + 4\right )}^{\frac {1}{3}} x}\,{d x} \end {gather*}

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 \begin {gather*} \int \frac {1}{x\,{\left (3\,x^2-6\,x+4\right )}^{1/3}} \,d x \end {gather*}

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 \begin {gather*} \int \frac {1}{x \sqrt [3]{3 x^{2} - 6 x + 4}}\, dx \end {gather*}

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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