\(\int \frac {1}{x \sqrt [3]{3+3 x+x^2}} \, dx\) [2291]
Optimal result
Integrand size = 16, antiderivative size = 175 \[
\int \frac {1}{x \sqrt [3]{3+3 x+x^2}} \, dx=-\frac {\arctan \left (\frac {\frac {2}{\sqrt [6]{3}}+\frac {2 x}{3 \sqrt [6]{3}}+\frac {\sqrt [3]{3+3 x+x^2}}{\sqrt {3}}}{\sqrt [3]{3+3 x+x^2}}\right )}{3^{5/6}}+\frac {\log \left (3 \sqrt [3]{3}+\sqrt [3]{3} x-3 \sqrt [3]{3+3 x+x^2}\right )}{3 \sqrt [3]{3}}-\frac {\log \left (9\ 3^{2/3}+6\ 3^{2/3} x+3^{2/3} x^2+\left (9 \sqrt [3]{3}+3 \sqrt [3]{3} x\right ) \sqrt [3]{3+3 x+x^2}+9 \left (3+3 x+x^2\right )^{2/3}\right )}{6 \sqrt [3]{3}}
\]
[Out]
-1/3*arctan((2/3*3^(5/6)+2/9*x*3^(5/6)+1/3*(x^2+3*x+3)^(1/3)*3^(1/2))/(x^2+3*x+3)^(1/3))*3^(1/6)+1/9*ln(3*3^(1
/3)+3^(1/3)*x-3*(x^2+3*x+3)^(1/3))*3^(2/3)-1/18*ln(9*3^(2/3)+6*3^(2/3)*x+3^(2/3)*x^2+(9*3^(1/3)+3*3^(1/3)*x)*(
x^2+3*x+3)^(1/3)+9*(x^2+3*x+3)^(2/3))*3^(2/3)
Rubi [A] (verified)
Time = 0.01 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.47,
number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {766}
\[
\int \frac {1}{x \sqrt [3]{3+3 x+x^2}} \, dx=-\frac {\arctan \left (\frac {2 (x+3)}{3 \sqrt [6]{3} \sqrt [3]{x^2+3 x+3}}+\frac {1}{\sqrt {3}}\right )}{3^{5/6}}+\frac {\log \left (3^{2/3} \sqrt [3]{x^2+3 x+3}-x-3\right )}{2 \sqrt [3]{3}}-\frac {\log (x)}{2 \sqrt [3]{3}}
\]
[In]
Int[1/(x*(3 + 3*x + x^2)^(1/3)),x]
[Out]
-(ArcTan[1/Sqrt[3] + (2*(3 + x))/(3*3^(1/6)*(3 + 3*x + x^2)^(1/3))]/3^(5/6)) - Log[x]/(2*3^(1/3)) + Log[-3 - x
+ 3^(2/3)*(3 + 3*x + x^2)^(1/3)]/(2*3^(1/3))
Rule 766
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] && NegQ[c*e^2*(2*c*d - b*e)]
Rubi steps \begin{align*}
\text {integral}& = -\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {2 (3+x)}{3 \sqrt [6]{3} \sqrt [3]{3+3 x+x^2}}\right )}{3^{5/6}}-\frac {\log (x)}{2 \sqrt [3]{3}}+\frac {\log \left (-3-x+3^{2/3} \sqrt [3]{3+3 x+x^2}\right )}{2 \sqrt [3]{3}} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.20 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.93
\[
\int \frac {1}{x \sqrt [3]{3+3 x+x^2}} \, dx=\frac {-6 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2}{\sqrt [6]{3} \sqrt [3]{3+3 x+x^2}}+\frac {2 x}{3 \sqrt [6]{3} \sqrt [3]{3+3 x+x^2}}\right )+\sqrt {3} \left (2 \log \left (3 \sqrt [3]{3}+\sqrt [3]{3} x-3 \sqrt [3]{3+3 x+x^2}\right )-\log \left (9\ 3^{2/3}+6\ 3^{2/3} x+3^{2/3} x^2+3 \sqrt [3]{3} (3+x) \sqrt [3]{3+3 x+x^2}+9 \left (3+3 x+x^2\right )^{2/3}\right )\right )}{6\ 3^{5/6}}
\]
[In]
Integrate[1/(x*(3 + 3*x + x^2)^(1/3)),x]
[Out]
(-6*ArcTan[1/Sqrt[3] + 2/(3^(1/6)*(3 + 3*x + x^2)^(1/3)) + (2*x)/(3*3^(1/6)*(3 + 3*x + x^2)^(1/3))] + Sqrt[3]*
(2*Log[3*3^(1/3) + 3^(1/3)*x - 3*(3 + 3*x + x^2)^(1/3)] - Log[9*3^(2/3) + 6*3^(2/3)*x + 3^(2/3)*x^2 + 3*3^(1/3
)*(3 + x)*(3 + 3*x + x^2)^(1/3) + 9*(3 + 3*x + x^2)^(2/3)]))/(6*3^(5/6))
Maple [C] (warning: unable to verify)
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 11.44 (sec) , antiderivative size = 1553, normalized size of antiderivative =
8.87
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| method | result | size |
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| trager |
\(\text {Expression too large to display}\) |
\(1553\) |
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[In]
int(1/x/(x^2+3*x+3)^(1/3),x,method=_RETURNVERBOSE)
[Out]
1/3*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*ln((-24305106415545*RootOf(_Z^3-9)^3*RootOf(RootOf(_Z^
3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)+18684300290294655*(x^2+3*x+3)^(2/3)-13926825976107285*RootOf(_Z^3-9)-172836
312288320*RootOf(_Z^3-9)*x^3-4642275325369095*RootOf(_Z^3-9)*x^2-13926825976107285*RootOf(_Z^3-9)*x-5612801888
6371134*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)-32651552580786*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf
(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2*x^2-8101702138515*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*Root
Of(_Z^3-9)^3*x^2-97954657742358*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2*x-24305
106415545*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^3*x-696566455056768*RootOf(RootOf
(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*x^3-18709339628790378*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^
2)*x^2-56128018886371134*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*x+539492107992192*(x^2+3*x+3)^(2/
3)*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^2*x+1536541257596103*RootOf(_Z^3-9)*Root
Of(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*(x^2+3*x+3)^(1/3)*x^2+9219247545576618*(x^2+3*x+3)^(1/3)*RootO
f(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)*x+56711914969605*RootOf(RootOf(_Z^3-9)^2+3*_Z*Ro
otOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^3*x^3+228560868065502*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^
2*RootOf(_Z^3-9)^2*x^3-97954657742358*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2-1
618476323976576*(x^2+3*x+3)^(1/3)*RootOf(_Z^3-9)^2+6228100096764885*(x^2+3*x+3)^(2/3)*x+1618476323976576*(x^2+
3*x+3)^(2/3)*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^2-179830702664064*RootOf(_Z^3-
9)^2*(x^2+3*x+3)^(1/3)*x^2-1078984215984384*(x^2+3*x+3)^(1/3)*RootOf(_Z^3-9)^2*x+13828871318364927*(x^2+3*x+3)
^(1/3)*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9))/x^3)+1/9*RootOf(_Z^3-9)*ln(-(-32651
552580786*RootOf(_Z^3-9)^3*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)-13828871318364927*(x^2+3*x+3)^(
2/3)+18611384971048020*RootOf(_Z^3-9)+460749686417758*RootOf(_Z^3-9)*x^3+6203794990349340*RootOf(_Z^3-9)*x^2+1
8611384971048020*RootOf(_Z^3-9)*x+41561731970581950*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)-243051
06415545*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2*x^2-10883850860262*RootOf(Root
Of(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^3*x^2-72915319246635*RootOf(RootOf(_Z^3-9)^2+3*_Z*Root
Of(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2*x-32651552580786*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*Roo
tOf(_Z^3-9)^3*x+1028916171591405*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*x^3+13853910656860650*Roo
tOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*x^2+41561731970581950*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^
3-9)+9*_Z^2)*x+539492107992192*(x^2+3*x+3)^(2/3)*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z
^3-9)^2*x-2076033365588295*RootOf(_Z^3-9)*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*(x^2+3*x+3)^(1/3
)*x^2-12456200193529770*(x^2+3*x+3)^(1/3)*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)*x
+76186956021834*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^3*x^3+170135744908815*RootO
f(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2*x^3-72915319246635*RootOf(RootOf(_Z^3-9)^2+3
*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2-1618476323976576*(x^2+3*x+3)^(1/3)*RootOf(_Z^3-9)^2-460962377278
8309*(x^2+3*x+3)^(2/3)*x+1618476323976576*(x^2+3*x+3)^(2/3)*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2
)*RootOf(_Z^3-9)^2-179830702664064*RootOf(_Z^3-9)^2*(x^2+3*x+3)^(1/3)*x^2-1078984215984384*(x^2+3*x+3)^(1/3)*R
ootOf(_Z^3-9)^2*x-18684300290294655*(x^2+3*x+3)^(1/3)*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*Root
Of(_Z^3-9))/x^3)
Fricas [A] (verification not implemented)
none
Time = 1.12 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.89
\[
\int \frac {1}{x \sqrt [3]{3+3 x+x^2}} \, dx=\frac {1}{9} \cdot 3^{\frac {2}{3}} \log \left (\frac {3^{\frac {1}{3}} {\left (x + 3\right )} - 3 \, {\left (x^{2} + 3 \, x + 3\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{18} \cdot 3^{\frac {2}{3}} \log \left (\frac {3^{\frac {1}{3}} {\left (x^{2} + 6 \, x + 9\right )} + 3 \cdot 3^{\frac {2}{3}} {\left (x^{2} + 3 \, x + 3\right )}^{\frac {2}{3}} + 3 \, {\left (x^{2} + 3 \, x + 3\right )}^{\frac {1}{3}} {\left (x + 3\right )}}{x^{2}}\right ) - \frac {1}{3} \cdot 3^{\frac {1}{6}} \arctan \left (\frac {3^{\frac {1}{6}} {\left (3^{\frac {1}{3}} x^{3} + 6 \cdot 3^{\frac {2}{3}} {\left (x^{2} + 3 \, x + 3\right )}^{\frac {2}{3}} {\left (x + 3\right )} - 6 \, {\left (x^{2} + 6 \, x + 9\right )} {\left (x^{2} + 3 \, x + 3\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (x^{3} + 18 \, x^{2} + 54 \, x + 54\right )}}\right )
\]
[In]
integrate(1/x/(x^2+3*x+3)^(1/3),x, algorithm="fricas")
[Out]
1/9*3^(2/3)*log((3^(1/3)*(x + 3) - 3*(x^2 + 3*x + 3)^(1/3))/x) - 1/18*3^(2/3)*log((3^(1/3)*(x^2 + 6*x + 9) + 3
*3^(2/3)*(x^2 + 3*x + 3)^(2/3) + 3*(x^2 + 3*x + 3)^(1/3)*(x + 3))/x^2) - 1/3*3^(1/6)*arctan(1/3*3^(1/6)*(3^(1/
3)*x^3 + 6*3^(2/3)*(x^2 + 3*x + 3)^(2/3)*(x + 3) - 6*(x^2 + 6*x + 9)*(x^2 + 3*x + 3)^(1/3))/(x^3 + 18*x^2 + 54
*x + 54))
Sympy [F]
\[
\int \frac {1}{x \sqrt [3]{3+3 x+x^2}} \, dx=\int \frac {1}{x \sqrt [3]{x^{2} + 3 x + 3}}\, dx
\]
[In]
integrate(1/x/(x**2+3*x+3)**(1/3),x)
[Out]
Integral(1/(x*(x**2 + 3*x + 3)**(1/3)), x)
Maxima [F]
\[
\int \frac {1}{x \sqrt [3]{3+3 x+x^2}} \, dx=\int { \frac {1}{{\left (x^{2} + 3 \, x + 3\right )}^{\frac {1}{3}} x} \,d x }
\]
[In]
integrate(1/x/(x^2+3*x+3)^(1/3),x, algorithm="maxima")
[Out]
integrate(1/((x^2 + 3*x + 3)^(1/3)*x), x)
Giac [F]
\[
\int \frac {1}{x \sqrt [3]{3+3 x+x^2}} \, dx=\int { \frac {1}{{\left (x^{2} + 3 \, x + 3\right )}^{\frac {1}{3}} x} \,d x }
\]
[In]
integrate(1/x/(x^2+3*x+3)^(1/3),x, algorithm="giac")
[Out]
integrate(1/((x^2 + 3*x + 3)^(1/3)*x), x)
Mupad [F(-1)]
Timed out. \[
\int \frac {1}{x \sqrt [3]{3+3 x+x^2}} \, dx=\int \frac {1}{x\,{\left (x^2+3\,x+3\right )}^{1/3}} \,d x
\]
[In]
int(1/(x*(3*x + x^2 + 3)^(1/3)),x)
[Out]
int(1/(x*(3*x + x^2 + 3)^(1/3)), x)