3.23.91 \(\int \frac {1}{x \sqrt [3]{3+3 x+x^2}} \, dx\) [2291]

3.23.91.1 Optimal result

Integrand size = 16, antiderivative size = 175 \[ \int \frac {1}{x \sqrt [3]{3+3 x+x^2}} \, dx=-\frac {\arctan \left (\frac {\frac {2}{\sqrt [6]{3}}+\frac {2 x}{3 \sqrt [6]{3}}+\frac {\sqrt [3]{3+3 x+x^2}}{\sqrt {3}}}{\sqrt [3]{3+3 x+x^2}}\right )}{3^{5/6}}+\frac {\log \left (3 \sqrt [3]{3}+\sqrt [3]{3} x-3 \sqrt [3]{3+3 x+x^2}\right )}{3 \sqrt [3]{3}}-\frac {\log \left (9\ 3^{2/3}+6\ 3^{2/3} x+3^{2/3} x^2+\left (9 \sqrt [3]{3}+3 \sqrt [3]{3} x\right ) \sqrt [3]{3+3 x+x^2}+9 \left (3+3 x+x^2\right )^{2/3}\right )}{6 \sqrt [3]{3}} \]

output

-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)
 

3.23.91.2 Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x \sqrt [3]{3+3 x+x^2}} \, dx=\frac {-6 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2}{\sqrt [6]{3} \sqrt [3]{3+3 x+x^2}}+\frac {2 x}{3 \sqrt [6]{3} \sqrt [3]{3+3 x+x^2}}\right )+\sqrt {3} \left (2 \log \left (3 \sqrt [3]{3}+\sqrt [3]{3} x-3 \sqrt [3]{3+3 x+x^2}\right )-\log \left (9\ 3^{2/3}+6\ 3^{2/3} x+3^{2/3} x^2+3 \sqrt [3]{3} (3+x) \sqrt [3]{3+3 x+x^2}+9 \left (3+3 x+x^2\right )^{2/3}\right )\right )}{6\ 3^{5/6}} \]

input

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

output

(-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))
 

3.23.91.3 Rubi [A] (verified)

Time = 0.17 (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 = {1176}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 1176

\(\displaystyle -\frac {\arctan \left (\frac {2 (x+3)}{3 \sqrt [6]{3} \sqrt [3]{x^2+3 x+3}}+\frac {1}{\sqrt {3}}\right )}{3^{5/6}}+\frac {\log \left (3^{2/3} \sqrt [3]{x^2+3 x+3}-x-3\right )}{2 \sqrt [3]{3}}-\frac {\log (x)}{2 \sqrt [3]{3}}\)

input

Int[1/(x*(3 + 3*x + x^2)^(1/3)),x]
 

output

-(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))
 

3.23.91.3.1 Defintions of rubi rules used

rule 1176

Int[1/(((d_.) + (e_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(1/3)), x_Sy 
mbol] :> With[{q = Rt[-3*c*e^2*(2*c*d - b*e), 3]}, Simp[(-Sqrt[3])*c*e*(Arc 
Tan[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] && 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)]
 

3.23.91.4 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

method	result	size
trager	\(\text {Expression too large to display}\)	\(1553\)

input

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

output

1/3*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*ln((-2430510641554 
5*RootOf(_Z^3-9)^3*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)+186 
84300290294655*(x^2+3*x+3)^(2/3)-13926825976107285*RootOf(_Z^3-9)-17283631 
2288320*RootOf(_Z^3-9)*x^3-4642275325369095*RootOf(_Z^3-9)*x^2-13926825976 
107285*RootOf(_Z^3-9)*x-56128018886371134*RootOf(RootOf(_Z^3-9)^2+3*_Z*Roo 
tOf(_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)*RootOf(_Z^3-9)^3*x^2-97954657742358*RootOf(Roo 
tOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2*x-24305106415 
545*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^3-1 
8709339628790378*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*x^2-5 
6128018886371134*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*x+539 
492107992192*(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)*RootOf(RootOf( 
_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*(x^2+3*x+3)^(1/3)*x^2+92192475455766 
18*(x^2+3*x+3)^(1/3)*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*R 
ootOf(_Z^3-9)*x[phone]*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_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(Ro...
 

3.23.91.5 Fricas [A] (verification not implemented)

Time = 1.12 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x \sqrt [3]{3+3 x+x^2}} \, dx=\frac {1}{9} \cdot 3^{\frac {2}{3}} \log \left (\frac {3^{\frac {1}{3}} {\left (x + 3\right )} - 3 \, {\left (x^{2} + 3 \, x + 3\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{18} \cdot 3^{\frac {2}{3}} \log \left (\frac {3^{\frac {1}{3}} {\left (x^{2} + 6 \, x + 9\right )} + 3 \cdot 3^{\frac {2}{3}} {\left (x^{2} + 3 \, x + 3\right )}^{\frac {2}{3}} + 3 \, {\left (x^{2} + 3 \, x + 3\right )}^{\frac {1}{3}} {\left (x + 3\right )}}{x^{2}}\right ) - \frac {1}{3} \cdot 3^{\frac {1}{6}} \arctan \left (\frac {3^{\frac {1}{6}} {\left (3^{\frac {1}{3}} x^{3} + 6 \cdot 3^{\frac {2}{3}} {\left (x^{2} + 3 \, x + 3\right )}^{\frac {2}{3}} {\left (x + 3\right )} - 6 \, {\left (x^{2} + 6 \, x + 9\right )} {\left (x^{2} + 3 \, x + 3\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (x^{3} + 18 \, x^{2} + 54 \, x + 54\right )}}\right ) \]

input

integrate(1/x/(x^2+3*x+3)^(1/3),x, algorithm="fricas")
 

output

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))
 

3.23.91.6 Sympy [F]

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

input

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

output

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

3.23.91.7 Maxima [F]

\[ \int \frac {1}{x \sqrt [3]{3+3 x+x^2}} \, dx=\int { \frac {1}{{\left (x^{2} + 3 \, x + 3\right )}^{\frac {1}{3}} x} \,d x } \]

input

integrate(1/x/(x^2+3*x+3)^(1/3),x, algorithm="maxima")
 

output

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

3.23.91.8 Giac [F]

\[ \int \frac {1}{x \sqrt [3]{3+3 x+x^2}} \, dx=\int { \frac {1}{{\left (x^{2} + 3 \, x + 3\right )}^{\frac {1}{3}} x} \,d x } \]

input

integrate(1/x/(x^2+3*x+3)^(1/3),x, algorithm="giac")
 

output

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

3.23.91.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x \sqrt [3]{3+3 x+x^2}} \, dx=\int \frac {1}{x\,{\left (x^2+3\,x+3\right )}^{1/3}} \,d x \]

input

int(1/(x*(3*x + x^2 + 3)^(1/3)),x)
 

output

int(1/(x*(3*x + x^2 + 3)^(1/3)), x)