Integrand size = 20, antiderivative size = 45 \[ \int \frac {1}{(1-x)^{2/3} \left (1+x+x^2\right )^{2/3}} \, dx=\frac {x \left (1-x^3\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},x^3\right )}{(1-x)^{2/3} \left (1+x+x^2\right )^{2/3}} \] Output:
x*(-x^3+1)^(2/3)*hypergeom([1/3, 2/3],[4/3],x^3)/(1-x)^(2/3)/(x^2+x+1)^(2/ 3)
Result contains complex when optimal does not.
Time = 10.09 (sec) , antiderivative size = 145, normalized size of antiderivative = 3.22 \[ \int \frac {1}{(1-x)^{2/3} \left (1+x+x^2\right )^{2/3}} \, dx=-\frac {3 \sqrt [3]{1-x} \left (i+\sqrt {3}+2 i x\right ) \left (\frac {3 i-\sqrt {3}+\left (3 i+\sqrt {3}\right ) x}{3 i+\sqrt {3}-\left (-3 i+\sqrt {3}\right ) x}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {4 i \sqrt {3} (-1+x)}{\left (-3 i+\sqrt {3}\right ) \left (i+\sqrt {3}+2 i x\right )}\right )}{\left (3 i+\sqrt {3}\right ) \left (1+x+x^2\right )^{2/3}} \] Input:
Integrate[1/((1 - x)^(2/3)*(1 + x + x^2)^(2/3)),x]
Output:
(-3*(1 - x)^(1/3)*(I + Sqrt[3] + (2*I)*x)*((3*I - Sqrt[3] + (3*I + Sqrt[3] )*x)/(3*I + Sqrt[3] - (-3*I + Sqrt[3])*x))^(2/3)*Hypergeometric2F1[1/3, 2/ 3, 4/3, ((4*I)*Sqrt[3]*(-1 + x))/((-3*I + Sqrt[3])*(I + Sqrt[3] + (2*I)*x) )])/((3*I + Sqrt[3])*(1 + x + x^2)^(2/3))
Time = 0.17 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1151, 778}
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.
\(\displaystyle \int \frac {1}{(1-x)^{2/3} \left (x^2+x+1\right )^{2/3}} \, dx\) |
\(\Big \downarrow \) 1151 |
\(\displaystyle \frac {\left (1-x^3\right )^{2/3} \int \frac {1}{\left (1-x^3\right )^{2/3}}dx}{(1-x)^{2/3} \left (x^2+x+1\right )^{2/3}}\) |
\(\Big \downarrow \) 778 |
\(\displaystyle \frac {x \left (1-x^3\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},x^3\right )}{(1-x)^{2/3} \left (x^2+x+1\right )^{2/3}}\) |
Input:
Int[1/((1 - x)^(2/3)*(1 + x + x^2)^(2/3)),x]
Output:
(x*(1 - x^3)^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, x^3])/((1 - x)^(2/3)*( 1 + x + x^2)^(2/3))
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F 1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, n, p}, x] && !IGtQ[p , 0] && !IntegerQ[1/n] && !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p] || GtQ[a, 0])
Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Sy mbol] :> Simp[(d + e*x)^FracPart[p]*((a + b*x + c*x^2)^FracPart[p]/(a*d + c *e*x^3)^FracPart[p]) Int[(d + e*x)^(m - p)*(a*d + c*e*x^3)^p, x], x] /; F reeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*d + a*e, 0] && EqQ[c*d + b*e, 0] && IGtQ[m - p + 1, 0] && !IntegerQ[p]
\[\int \frac {1}{\left (1-x \right )^{\frac {2}{3}} \left (x^{2}+x +1\right )^{\frac {2}{3}}}d x\]
Input:
int(1/(1-x)^(2/3)/(x^2+x+1)^(2/3),x)
Output:
int(1/(1-x)^(2/3)/(x^2+x+1)^(2/3),x)
\[ \int \frac {1}{(1-x)^{2/3} \left (1+x+x^2\right )^{2/3}} \, dx=\int { \frac {1}{{\left (x^{2} + x + 1\right )}^{\frac {2}{3}} {\left (-x + 1\right )}^{\frac {2}{3}}} \,d x } \] Input:
integrate(1/(1-x)^(2/3)/(x^2+x+1)^(2/3),x, algorithm="fricas")
Output:
integral(-(x^2 + x + 1)^(1/3)*(-x + 1)^(1/3)/(x^3 - 1), x)
\[ \int \frac {1}{(1-x)^{2/3} \left (1+x+x^2\right )^{2/3}} \, dx=\int \frac {1}{\left (1 - x\right )^{\frac {2}{3}} \left (x^{2} + x + 1\right )^{\frac {2}{3}}}\, dx \] Input:
integrate(1/(1-x)**(2/3)/(x**2+x+1)**(2/3),x)
Output:
Integral(1/((1 - x)**(2/3)*(x**2 + x + 1)**(2/3)), x)
\[ \int \frac {1}{(1-x)^{2/3} \left (1+x+x^2\right )^{2/3}} \, dx=\int { \frac {1}{{\left (x^{2} + x + 1\right )}^{\frac {2}{3}} {\left (-x + 1\right )}^{\frac {2}{3}}} \,d x } \] Input:
integrate(1/(1-x)^(2/3)/(x^2+x+1)^(2/3),x, algorithm="maxima")
Output:
integrate(1/((x^2 + x + 1)^(2/3)*(-x + 1)^(2/3)), x)
\[ \int \frac {1}{(1-x)^{2/3} \left (1+x+x^2\right )^{2/3}} \, dx=\int { \frac {1}{{\left (x^{2} + x + 1\right )}^{\frac {2}{3}} {\left (-x + 1\right )}^{\frac {2}{3}}} \,d x } \] Input:
integrate(1/(1-x)^(2/3)/(x^2+x+1)^(2/3),x, algorithm="giac")
Output:
integrate(1/((x^2 + x + 1)^(2/3)*(-x + 1)^(2/3)), x)
Timed out. \[ \int \frac {1}{(1-x)^{2/3} \left (1+x+x^2\right )^{2/3}} \, dx=\int \frac {1}{{\left (1-x\right )}^{2/3}\,{\left (x^2+x+1\right )}^{2/3}} \,d x \] Input:
int(1/((1 - x)^(2/3)*(x + x^2 + 1)^(2/3)),x)
Output:
int(1/((1 - x)^(2/3)*(x + x^2 + 1)^(2/3)), x)
\[ \int \frac {1}{(1-x)^{2/3} \left (1+x+x^2\right )^{2/3}} \, dx=\int \frac {1}{\left (1-x \right )^{\frac {2}{3}} \left (x^{2}+x +1\right )^{\frac {2}{3}}}d x \] Input:
int(1/(1-x)^(2/3)/(x^2+x+1)^(2/3),x)
Output:
int(1/(( - x + 1)**(2/3)*(x**2 + x + 1)**(2/3)),x)