### 3.2582 $$\int \frac{1}{\sqrt [3]{1+x} \sqrt [3]{1-x+x^2}} \, dx$$

Optimal. Leaf size=102 $\frac{\sqrt [3]{x^3+1} \tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{x+1} \sqrt [3]{x^2-x+1}}-\frac{\sqrt [3]{x^3+1} \log \left (\sqrt [3]{x^3+1}-x\right )}{2 \sqrt [3]{x+1} \sqrt [3]{x^2-x+1}}$

[Out]

((1 + x^3)^(1/3)*ArcTan[(1 + (2*x)/(1 + x^3)^(1/3))/Sqrt[3]])/(Sqrt[3]*(1 + x)^(1/3)*(1 - x + x^2)^(1/3)) - ((
1 + x^3)^(1/3)*Log[-x + (1 + x^3)^(1/3)])/(2*(1 + x)^(1/3)*(1 - x + x^2)^(1/3))

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Rubi [A]  time = 0.0171366, antiderivative size = 102, 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, 239} $\frac{\sqrt [3]{x^3+1} \tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{x+1} \sqrt [3]{x^2-x+1}}-\frac{\sqrt [3]{x^3+1} \log \left (\sqrt [3]{x^3+1}-x\right )}{2 \sqrt [3]{x+1} \sqrt [3]{x^2-x+1}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((1 + x^3)^(1/3)*ArcTan[(1 + (2*x)/(1 + x^3)^(1/3))/Sqrt[3]])/(Sqrt[3]*(1 + x)^(1/3)*(1 - x + x^2)^(1/3)) - ((
1 + x^3)^(1/3)*Log[-x + (1 + x^3)^(1/3)])/(2*(1 + x)^(1/3)*(1 - x + x^2)^(1/3))

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 239

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + (2*Rt[b, 3]*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [C]  time = 0.0704854, size = 132, normalized size = 1.29 $\frac{3 \sqrt [3]{\frac{-2 i x+\sqrt{3}+i}{\sqrt{3}+3 i}} \sqrt [3]{\frac{2 i x+\sqrt{3}-i}{\sqrt{3}-3 i}} (x+1)^{2/3} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{2 i (x+1)}{3 i+\sqrt{3}},-\frac{2 i (x+1)}{-3 i+\sqrt{3}}\right )}{2 \sqrt [3]{x^2-x+1}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(3*((I + Sqrt[3] - (2*I)*x)/(3*I + Sqrt[3]))^(1/3)*((-I + Sqrt[3] + (2*I)*x)/(-3*I + Sqrt[3]))^(1/3)*(1 + x)^(
2/3)*AppellF1[2/3, 1/3, 1/3, 5/3, ((2*I)*(1 + x))/(3*I + Sqrt[3]), ((-2*I)*(1 + x))/(-3*I + Sqrt[3])])/(2*(1 -
x + x^2)^(1/3))

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Maple [F]  time = 1.582, size = 0, normalized size = 0. \begin{align*} \int{{\frac{1}{\sqrt [3]{1+x}}}{\frac{1}{\sqrt [3]{{x}^{2}-x+1}}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(1+x)^(1/3)/(x^2-x+1)^(1/3),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{1}{3}}{\left (x + 1\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 3.30881, size = 333, normalized size = 3.26 \begin{align*} \frac{1}{3} \, \sqrt{3} \arctan \left (-\frac{4 \, \sqrt{3}{\left (x^{2} - x + 1\right )}^{\frac{1}{3}}{\left (x + 1\right )}^{\frac{1}{3}} x^{2} - 2 \, \sqrt{3}{\left (x^{2} - x + 1\right )}^{\frac{2}{3}}{\left (x + 1\right )}^{\frac{2}{3}} x + \sqrt{3}{\left (x^{3} + 1\right )}}{9 \, x^{3} + 1}\right ) - \frac{1}{6} \, \log \left (3 \,{\left (x^{2} - x + 1\right )}^{\frac{1}{3}}{\left (x + 1\right )}^{\frac{1}{3}} x^{2} - 3 \,{\left (x^{2} - x + 1\right )}^{\frac{2}{3}}{\left (x + 1\right )}^{\frac{2}{3}} x + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sqrt(3)*arctan(-(4*sqrt(3)*(x^2 - x + 1)^(1/3)*(x + 1)^(1/3)*x^2 - 2*sqrt(3)*(x^2 - x + 1)^(2/3)*(x + 1)^(
2/3)*x + sqrt(3)*(x^3 + 1))/(9*x^3 + 1)) - 1/6*log(3*(x^2 - x + 1)^(1/3)*(x + 1)^(1/3)*x^2 - 3*(x^2 - x + 1)^(
2/3)*(x + 1)^(2/3)*x + 1)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/((x + 1)**(1/3)*(x**2 - x + 1)**(1/3)), 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{1}{3}}{\left (x + 1\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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