3.1.55 \(\int \frac {1}{x \sqrt [3]{4-6 x+3 x^2}} \, dx\) [55]
Optimal. Leaf size=97 \[ -\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (2-x)}{\sqrt {3} \sqrt [3]{4-6 x+3 x^2}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log (x)}{2\ 2^{2/3}}+\frac {\log \left (6-3 x-3 \sqrt [3]{2} \sqrt [3]{4-6 x+3 x^2}\right )}{2\ 2^{2/3}} \]
[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 = {764}
\begin {gather*} -\frac {\text {ArcTan}\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 \left (-3 \sqrt [3]{2} \sqrt [3]{3 x^2-6 x+4}-3 x+6\right )}{2\ 2^{2/3}}-\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 764
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 [A]
time = 0.21, size = 168, normalized size = 1.73 \begin {gather*} -\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {2\ 2^{2/3}-2^{2/3} x+\sqrt [3]{4-6 x+3 x^2}}{\sqrt {3} \sqrt [3]{4-6 x+3 x^2}}\right )-2 \log \left (-2 2^{2/3}+2^{2/3} x+2 \sqrt [3]{4-6 x+3 x^2}\right )+\log \left (-4 \sqrt [3]{2}+4 \sqrt [3]{2} x-\sqrt [3]{2} x^2+2^{2/3} (-2+x) \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]
Integrate[1/(x*(4 - 6*x + 3*x^2)^(1/3)),x]
[Out]
-1/6*(2*Sqrt[3]*ArcTan[(2*2^(2/3) - 2^(2/3)*x + (4 - 6*x + 3*x^2)^(1/3))/(Sqrt[3]*(4 - 6*x + 3*x^2)^(1/3))] -
2*Log[-2*2^(2/3) + 2^(2/3)*x + 2*(4 - 6*x + 3*x^2)^(1/3)] + Log[-4*2^(1/3) + 4*2^(1/3)*x - 2^(1/3)*x^2 + 2^(2/
3)*(-2 + x)*(4 - 6*x + 3*x^2)^(1/3) - 2*(4 - 6*x + 3*x^2)^(2/3)])/2^(2/3)
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 8.73, size = 1592, normalized size = 16.41
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result |
size |
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\(\text {Expression too large to display}\) |
\(1592\) |
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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]
1/6*RootOf(_Z^3-2)*ln((96*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x-12*RootOf(R
ootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x-96*RootOf(_Z^3-2)^2*(3*x^2-6*x+4)^(1/3)-30*(3*x
^2-6*x+4)^(2/3)*x+24*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+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)-24*RootOf(_Z^3-2)^2*(3*x^2-6*x+4)^(1/3)*x^2
+96*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)-48*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^2-3*RootOf(_Z^3-2)*x^3+30*RootOf(_Z^3-2)*x^2-6
0*RootOf(_Z^3-2)*x-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(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+72*RootOf(Roo
tOf(_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-320*RootOf(RootOf(_Z^3-2)^2+2*_
Z*RootOf(_Z^3-2)+4*_Z^2)+8*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3+40*RootOf(_Z^3
-2)+16*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^3-2*RootOf(RootOf(_Z^3-2)^2+2*
_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^3+60*(3*x^2-6*x+4)^(2/3)-64*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_
Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2)/x^3)+1/3*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*ln((12*RootOf(
RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x-24*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-
2)+4*_Z^2)*RootOf(_Z^3-2)^3*x+48*RootOf(_Z^3-2)^2*(3*x^2-6*x+4)^(1/3)+9*(3*x^2-6*x+4)^(2/3)*x-RootOf(RootOf(_Z
^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^3+24*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^2-48*RootOf(R
ootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x-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)+12*RootOf(_Z^3-2)^2*(3*x^2-6*x+4)^(1/3)*x^2-48*RootOf(_Z^3-2)^2*(3*x^2-6*x+4)^(1/3
)*x+60*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)-6*RootOf(RootOf(
_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^2+12*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*
_Z^2)*RootOf(_Z^3-2)^3*x^2+2*RootOf(_Z^3-2)*x^3-48*RootOf(_Z^3-2)*x^2+96*RootOf(_Z^3-2)*x+24*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+15*RootOf(RootOf(_Z^3-2)^2+2*_Z*Root
Of(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*(3*x^2-6*x+4)^(1/3)*x^2-60*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+32*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)+16*RootOf(RootO
f(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3-64*RootOf(_Z^3-2)+2*RootOf(RootOf(_Z^3-2)^2+2*_Z*Root
Of(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^3-4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)
^3*x^3-18*(3*x^2-6*x+4)^(2/3)-8*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2)/x^3)
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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/(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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 171 vs.
\(2 (74) = 148\).
time = 3.67, size = 171, normalized size = 1.76 \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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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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Giac [F]
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/(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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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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