### 3.2583 $$\int \frac{1}{(1+x)^{2/3} (1-x+x^2)^{2/3}} \, dx$$

Optimal. Leaf size=45 $\frac{x \left (x^3+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-x^3\right )}{(x+1)^{2/3} \left (x^2-x+1\right )^{2/3}}$

[Out]

(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))

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Rubi [A]  time = 0.0152849, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.1, Rules used = {713, 245} $\frac{x \left (x^3+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-x^3\right )}{(x+1)^{2/3} \left (x^2-x+1\right )^{2/3}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(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))

Rule 713

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[((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]
/; FreeQ[{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] &&  !Intege
rQ[p]

Rule 245

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-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])

Rubi steps

\begin{align*} \int \frac{1}{(1+x)^{2/3} \left (1-x+x^2\right )^{2/3}} \, dx &=\frac{\left (1+x^3\right )^{2/3} \int \frac{1}{\left (1+x^3\right )^{2/3}} \, dx}{(1+x)^{2/3} \left (1-x+x^2\right )^{2/3}}\\ &=\frac{x \left (1+x^3\right )^{2/3} \, _2F_1\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}}\\ \end{align*}

Mathematica [C]  time = 0.0740744, size = 143, normalized size = 3.18 $\frac{3 \left (2 i x+\sqrt{3}-i\right ) \sqrt [3]{x+1} \left (-\frac{\left (\sqrt{3}-3 i\right ) x+\sqrt{3}+3 i}{\left (\sqrt{3}+3 i\right ) x+\sqrt{3}-3 i}\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};\frac{4 i \sqrt{3} (x+1)}{\left (3 i+\sqrt{3}\right ) \left (2 i x+\sqrt{3}-i\right )}\right )}{\left (\sqrt{3}-3 i\right ) \left (x^2-x+1\right )^{2/3}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(3*(-I + Sqrt[3] + (2*I)*x)*(1 + x)^(1/3)*(-((3*I + Sqrt[3] + (-3*I + Sqrt[3])*x)/(-3*I + Sqrt[3] + (3*I + Sqr
t[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))

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Maple [F]  time = 1.603, size = 0, normalized size = 0. \begin{align*} \int{ \left ( 1+x \right ) ^{-{\frac{2}{3}}} \left ({x}^{2}-x+1 \right ) ^{-{\frac{2}{3}}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{2} - x + 1\right )}^{\frac{2}{3}}{\left (x + 1\right )}^{\frac{2}{3}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (x^{2} - x + 1\right )}^{\frac{1}{3}}{\left (x + 1\right )}^{\frac{1}{3}}}{x^{3} + 1}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{2} - x + 1\right )}^{\frac{2}{3}}{\left (x + 1\right )}^{\frac{2}{3}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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