3.23.15 \(\int \frac {1}{(2+x) \sqrt [3]{1+x+x^2}} \, dx\)
Optimal. Leaf size=165 \[ \frac {\log \left (3 \sqrt [3]{x^2+x+1}+\sqrt [3]{3} x-\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}-\frac {\log \left (3^{2/3} x^2+9 \left (x^2+x+1\right )^{2/3}+\left (3 \sqrt [3]{3}-3 \sqrt [3]{3} x\right ) \sqrt [3]{x^2+x+1}-2\ 3^{2/3} x+3^{2/3}\right )}{6 \sqrt [3]{3}}-\frac {\tan ^{-1}\left (\frac {\frac {\sqrt [3]{x^2+x+1}}{\sqrt {3}}-\frac {2 x}{3 \sqrt [6]{3}}+\frac {2}{3 \sqrt [6]{3}}}{\sqrt [3]{x^2+x+1}}\right )}{3^{5/6}} \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 84, normalized size of antiderivative = 0.51,
number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used =
{750} \begin {gather*} \frac {\log \left (-3^{2/3} \sqrt [3]{x^2+x+1}-x+1\right )}{2 \sqrt [3]{3}}-\frac {\tan ^{-1}\left (\frac {2 (1-x)}{3 \sqrt [6]{3} \sqrt [3]{x^2+x+1}}+\frac {1}{\sqrt {3}}\right )}{3^{5/6}}-\frac {\log (x+2)}{2 \sqrt [3]{3}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/((2 + x)*(1 + x + x^2)^(1/3)),x]
[Out]
-(ArcTan[1/Sqrt[3] + (2*(1 - x))/(3*3^(1/6)*(1 + x + x^2)^(1/3))]/3^(5/6)) - Log[2 + x]/(2*3^(1/3)) + Log[1 -
x - 3^(2/3)*(1 + x + x^2)^(1/3)]/(2*3^(1/3))
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}{(2+x) \sqrt [3]{1+x+x^2}} \, dx &=-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 (1-x)}{3 \sqrt [6]{3} \sqrt [3]{1+x+x^2}}\right )}{3^{5/6}}-\frac {\log (2+x)}{2 \sqrt [3]{3}}+\frac {\log \left (1-x-3^{2/3} \sqrt [3]{1+x+x^2}\right )}{2 \sqrt [3]{3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.06, size = 118, normalized size = 0.72 \begin {gather*} -\frac {3 \sqrt [3]{\frac {2 x-i \sqrt {3}+1}{x+2}} \sqrt [3]{\frac {2 x+i \sqrt {3}+1}{x+2}} F_1\left (\frac {2}{3};\frac {1}{3},\frac {1}{3};\frac {5}{3};\frac {3-i \sqrt {3}}{2 x+4},\frac {3+i \sqrt {3}}{2 x+4}\right )}{2\ 2^{2/3} \sqrt [3]{x^2+x+1}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[1/((2 + x)*(1 + x + x^2)^(1/3)),x]
[Out]
(-3*((1 - I*Sqrt[3] + 2*x)/(2 + x))^(1/3)*((1 + I*Sqrt[3] + 2*x)/(2 + x))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, (
3 - I*Sqrt[3])/(4 + 2*x), (3 + I*Sqrt[3])/(4 + 2*x)])/(2*2^(2/3)*(1 + x + x^2)^(1/3))
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.21, size = 165, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\frac {2}{3 \sqrt [6]{3}}-\frac {2 x}{3 \sqrt [6]{3}}+\frac {\sqrt [3]{1+x+x^2}}{\sqrt {3}}}{\sqrt [3]{1+x+x^2}}\right )}{3^{5/6}}+\frac {\log \left (-\sqrt [3]{3}+\sqrt [3]{3} x+3 \sqrt [3]{1+x+x^2}\right )}{3 \sqrt [3]{3}}-\frac {\log \left (3^{2/3}-2\ 3^{2/3} x+3^{2/3} x^2+\left (3 \sqrt [3]{3}-3 \sqrt [3]{3} x\right ) \sqrt [3]{1+x+x^2}+9 \left (1+x+x^2\right )^{2/3}\right )}{6 \sqrt [3]{3}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[1/((2 + x)*(1 + x + x^2)^(1/3)),x]
[Out]
-(ArcTan[(2/(3*3^(1/6)) - (2*x)/(3*3^(1/6)) + (1 + x + x^2)^(1/3)/Sqrt[3])/(1 + x + x^2)^(1/3)]/3^(5/6)) + Log
[-3^(1/3) + 3^(1/3)*x + 3*(1 + x + x^2)^(1/3)]/(3*3^(1/3)) - Log[3^(2/3) - 2*3^(2/3)*x + 3^(2/3)*x^2 + (3*3^(1
/3) - 3*3^(1/3)*x)*(1 + x + x^2)^(1/3) + 9*(1 + x + x^2)^(2/3)]/(6*3^(1/3))
________________________________________________________________________________________
fricas [A] time = 2.43, size = 165, normalized size = 1.00 \begin {gather*} -\frac {1}{18} \cdot 3^{\frac {2}{3}} \log \left (\frac {3 \cdot 3^{\frac {2}{3}} {\left (x^{2} + x + 1\right )}^{\frac {2}{3}} + 3^{\frac {1}{3}} {\left (x^{2} - 2 \, x + 1\right )} - 3 \, {\left (x^{2} + x + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )}}{x^{2} + 4 \, x + 4}\right ) + \frac {1}{9} \cdot 3^{\frac {2}{3}} \log \left (\frac {3^{\frac {1}{3}} {\left (x - 1\right )} + 3 \, {\left (x^{2} + x + 1\right )}^{\frac {1}{3}}}{x + 2}\right ) - \frac {1}{3} \cdot 3^{\frac {1}{6}} \arctan \left (\frac {3^{\frac {1}{6}} {\left (6 \cdot 3^{\frac {2}{3}} {\left (x^{2} + x + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )} + 3^{\frac {1}{3}} {\left (x^{3} + 6 \, x^{2} + 12 \, x + 8\right )} + 6 \, {\left (x^{2} + x + 1\right )}^{\frac {1}{3}} {\left (x^{2} - 2 \, x + 1\right )}\right )}}{3 \, {\left (x^{3} - 12 \, x^{2} - 6 \, x - 10\right )}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(2+x)/(x^2+x+1)^(1/3),x, algorithm="fricas")
[Out]
-1/18*3^(2/3)*log((3*3^(2/3)*(x^2 + x + 1)^(2/3) + 3^(1/3)*(x^2 - 2*x + 1) - 3*(x^2 + x + 1)^(1/3)*(x - 1))/(x
^2 + 4*x + 4)) + 1/9*3^(2/3)*log((3^(1/3)*(x - 1) + 3*(x^2 + x + 1)^(1/3))/(x + 2)) - 1/3*3^(1/6)*arctan(1/3*3
^(1/6)*(6*3^(2/3)*(x^2 + x + 1)^(2/3)*(x - 1) + 3^(1/3)*(x^3 + 6*x^2 + 12*x + 8) + 6*(x^2 + x + 1)^(1/3)*(x^2
- 2*x + 1))/(x^3 - 12*x^2 - 6*x - 10))
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{2} + x + 1\right )}^{\frac {1}{3}} {\left (x + 2\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(2+x)/(x^2+x+1)^(1/3),x, algorithm="giac")
[Out]
integrate(1/((x^2 + x + 1)^(1/3)*(x + 2)), x)
________________________________________________________________________________________
maple [C] time = 14.96, size = 2074, normalized size = 12.57
| | |
| method |
result |
size |
| | |
| trager |
\(\text {Expression too large to display}\) |
\(2074\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(2+x)/(x^2+x+1)^(1/3),x,method=_RETURNVERBOSE)
[Out]
1/3*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*ln((14529*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+
9*_Z^2)*RootOf(_Z^3-9)^3*x^3+4257*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2*x^3-2
9058*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^3*x^2-8514*RootOf(RootOf(_Z^3-9)^2+3*_
Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2*x^2+131841*(x^2+x+1)^(2/3)*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^
3-9)+9*_Z^2)*RootOf(_Z^3-9)^2*x+58116*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^3*x+1
7028*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2*x-131841*(x^2+x+1)^(2/3)*RootOf(Ro
otOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^2-88233*(x^2+x+1)^(1/3)*RootOf(_Z^3-9)^2*x^2-131841*
(x^2+x+1)^(1/3)*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)*x^2+176466*(x^2+x+1)^(1/3)*
RootOf(_Z^3-9)^2*x+263682*(x^2+x+1)^(1/3)*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)*x
+48430*RootOf(_Z^3-9)*x^3+14190*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*x^3-88233*(x^2+x+1)^(1/3)*
RootOf(_Z^3-9)^2-131841*(x^2+x+1)^(1/3)*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)-537
573*RootOf(_Z^3-9)*x^2-157509*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*x^2+794097*(x^2+x+1)^(2/3)*x
-246993*RootOf(_Z^3-9)*x-72369*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*x-794097*(x^2+x+1)^(2/3)-44
0713*RootOf(_Z^3-9)-129129*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2))/(2+x)^3)-1/9*ln(-(13110*RootOf
(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^3*x^3-4257*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z
^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2*x^3-26220*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^
3*x^2+8514*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2*x^2+131841*(x^2+x+1)^(2/3)*R
ootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^2*x+52440*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootO
f(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^3*x-17028*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9
)^2*x-131841*(x^2+x+1)^(2/3)*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^2+44286*(x^2+x
+1)^(1/3)*RootOf(_Z^3-9)^2*x^2-131841*(x^2+x+1)^(1/3)*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*Root
Of(_Z^3-9)*x^2-88572*(x^2+x+1)^(1/3)*RootOf(_Z^3-9)^2*x+263682*(x^2+x+1)^(1/3)*RootOf(RootOf(_Z^3-9)^2+3*_Z*Ro
otOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)*x-4370*RootOf(_Z^3-9)*x^3+1419*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)
+9*_Z^2)*x^3+44286*(x^2+x+1)^(1/3)*RootOf(_Z^3-9)^2-131841*(x^2+x+1)^(1/3)*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf
(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)+406410*RootOf(_Z^3-9)*x^2-131967*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9
*_Z^2)*x^2-398574*(x^2+x+1)^(2/3)*x+380190*RootOf(_Z^3-9)*x-123453*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)
+9*_Z^2)*x+398574*(x^2+x+1)^(2/3)+397670*RootOf(_Z^3-9)-129129*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_
Z^2))/(2+x)^3)*RootOf(_Z^3-9)-1/3*ln(-(13110*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9
)^3*x^3-4257*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2*x^3-26220*RootOf(RootOf(_Z
^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^3*x^2+8514*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z
^2)^2*RootOf(_Z^3-9)^2*x^2+131841*(x^2+x+1)^(2/3)*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_
Z^3-9)^2*x+52440*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^3*x-17028*RootOf(RootOf(_Z
^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2*x-131841*(x^2+x+1)^(2/3)*RootOf(RootOf(_Z^3-9)^2+3*_Z*R
ootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^2+44286*(x^2+x+1)^(1/3)*RootOf(_Z^3-9)^2*x^2-131841*(x^2+x+1)^(1/3)*RootO
f(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)*x^2-88572*(x^2+x+1)^(1/3)*RootOf(_Z^3-9)^2*x+263
682*(x^2+x+1)^(1/3)*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)*x-4370*RootOf(_Z^3-9)*x
^3+1419*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*x^3+44286*(x^2+x+1)^(1/3)*RootOf(_Z^3-9)^2-131841*
(x^2+x+1)^(1/3)*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)+406410*RootOf(_Z^3-9)*x^2-1
31967*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*x^2-398574*(x^2+x+1)^(2/3)*x+380190*RootOf(_Z^3-9)*x
-123453*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*x+398574*(x^2+x+1)^(2/3)+397670*RootOf(_Z^3-9)-129
129*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2))/(2+x)^3)*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+
9*_Z^2)
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{2} + x + 1\right )}^{\frac {1}{3}} {\left (x + 2\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(2+x)/(x^2+x+1)^(1/3),x, algorithm="maxima")
[Out]
integrate(1/((x^2 + x + 1)^(1/3)*(x + 2)), x)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\left (x+2\right )\,{\left (x^2+x+1\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((x + 2)*(x + x^2 + 1)^(1/3)),x)
[Out]
int(1/((x + 2)*(x + x^2 + 1)^(1/3)), x)
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (x + 2\right ) \sqrt [3]{x^{2} + x + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(2+x)/(x**2+x+1)**(1/3),x)
[Out]
Integral(1/((x + 2)*(x**2 + x + 1)**(1/3)), x)
________________________________________________________________________________________