∫ 1−x_______ (1+x+x2) 3 √ ___

Optimal. Leaf size=43 \[ \frac {1}{\sqrt [3]{1-x^3}}+\frac {x}{\sqrt [3]{1-x^3}}-x^2 \, _2F_1\left (\frac {2}{3},\frac {4}{3};\frac {5}{3};x^3\right ) \]

1/(-x^3+1)^(1/3)+x/(-x^3+1)^(1/3)-x^2*hypergeom([2/3, 4/3],[5/3],x^3)

________________________________________________________________________________________

Rubi [A]
time = 0.05, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2183, 197, 371, 267} \begin {gather*} x^2 \left (-\, _2F_1\left (\frac {2}{3},\frac {4}{3};\frac {5}{3};x^3\right )\right )+\frac {x}{\sqrt [3]{1-x^3}}+\frac {1}{\sqrt [3]{1-x^3}} \end {gather*}

(1 - x^3)^(-1/3) + x/(1 - x^3)^(1/3) - x^2*Hypergeometric2F1[2/3, 4/3, 5/3, x^3]

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ

[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg

eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] &&

Int[(Px_.)*((c_) + (d_.)*(x_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^3)^(p_.), x_Symbol] :> Dist[1/c^q, Int[E

xpandIntegrand[(c^3 - d^3*x^3)^q*(a + b*x^3)^p, Px/(c - d*x)^q, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] &&

PolyQ[Px, x] && EqQ[d^2 - c*e, 0] && ILtQ[q, 0] && RationalQ[p] && EqQ[Denominator[p], 3]

\begin {align*} \int \frac {1-x}{\left (1+x+x^2\right ) \sqrt [3]{1-x^3}} \, dx &=\int \left (\frac {-1-i \sqrt {3}}{\left (1-i \sqrt {3}+2 x\right ) \sqrt [3]{1-x^3}}+\frac {-1+i \sqrt {3}}{\left (1+i \sqrt {3}+2 x\right ) \sqrt [3]{1-x^3}}\right ) \, dx\\ &=\left (-1-i \sqrt {3}\right ) \int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \sqrt [3]{1-x^3}} \, dx+\left (-1+i \sqrt {3}\right ) \int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt [3]{1-x^3}} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 10.06, size = 43, normalized size = 1.00 \begin {gather*} \frac {(1+2 x) \left (1-x^3\right )^{2/3}}{1+x+x^2}+x^2 \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};x^3\right ) \end {gather*}

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

((1 + 2*x)*(1 - x^3)^(2/3))/(1 + x + x^2) + x^2*Hypergeometric2F1[1/3, 2/3, 5/3, x^3]

________________________________________________________________________________________

Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded} \end {gather*}

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

cought exception: maximum recursion depth exceeded

________________________________________________________________________________________


method	result	size

risch	\(-\frac {\left (-1+x \right ) \left (1+2 x \right )}{\left (-x^{3}+1\right )^{\frac {1}{3}}}+x^{2} \hypergeom \left (\left [\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{3}\right )\)	\(34\)

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

-(-1+x)*(1+2*x)/(-x^3+1)^(1/3)+x^2*hypergeom([1/3,2/3],[5/3],x^3)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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

________________________________________________________________________________________

Fricas [F]
time = 0.32, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

integral((-x^3 + 1)^(2/3)/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1), x)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x}{x^{2} \sqrt [3]{1 - x^{3}} + x \sqrt [3]{1 - x^{3}} + \sqrt [3]{1 - x^{3}}}\, dx - \int \left (- \frac {1}{x^{2} \sqrt [3]{1 - x^{3}} + x \sqrt [3]{1 - x^{3}} + \sqrt [3]{1 - x^{3}}}\right )\, dx \end {gather*}

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

*3)**(1/3) + x*(1 - x**3)**(1/3) + (1 - x**3)**(1/3)), x)

________________________________________________________________________________________

Giac [F] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Could not integrate} \end {gather*}

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} -\int \frac {x-1}{{\left (1-x^3\right )}^{1/3}\,\left (x^2+x+1\right )} \,d x \end {gather*}

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

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

________________________________________________________________________________________

3.2.4 \(\int \frac {1-x}{(1+x+x^2) \sqrt [3]{1-x^3}} \, dx\) [104]