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

Optimal. Leaf size=135 \[ \frac{\log \left (\frac{2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{2 \sqrt [3]{2}}-\frac{\log \left (\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{\sqrt [3]{2}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt [3]{2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 0.336726, antiderivative size = 409, normalized size of antiderivative = 3.03, number of steps used = 4, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {6728, 2148} \[ -\frac{3 \left (-\sqrt{3}+i\right ) \log \left (2\ 2^{2/3} \sqrt [3]{1-x^3}+2 x-i \sqrt{3}+1\right )}{4 \sqrt [3]{2} \left (\sqrt{3}+i\right )}-\frac{3 \left (\sqrt{3}+i\right ) \log \left (2\ 2^{2/3} \sqrt [3]{1-x^3}+2 x+i \sqrt{3}+1\right )}{4 \sqrt [3]{2} \left (-\sqrt{3}+i\right )}-\frac{\left (3-i \sqrt{3}\right ) \tan ^{-1}\left (\frac{2-\frac{\sqrt [3]{2} \left (2 x-i \sqrt{3}+1\right )}{\sqrt [3]{1-x^3}}}{2 \sqrt{3}}\right )}{2 \sqrt [3]{2} \left (\sqrt{3}+i\right )}+\frac{\left (3+i \sqrt{3}\right ) \tan ^{-1}\left (\frac{2-\frac{\sqrt [3]{2} \left (2 x+i \sqrt{3}+1\right )}{\sqrt [3]{1-x^3}}}{2 \sqrt{3}}\right )}{2 \sqrt [3]{2} \left (-\sqrt{3}+i\right )}+\frac{\left (-\sqrt{3}+i\right ) \log \left (-\left (-2 x-i \sqrt{3}+1\right )^2 \left (2 x-i \sqrt{3}+1\right )\right )}{4 \sqrt [3]{2} \left (\sqrt{3}+i\right )}+\frac{\left (\sqrt{3}+i\right ) \log \left (-\left (-2 x+i \sqrt{3}+1\right )^2 \left (2 x+i \sqrt{3}+1\right )\right )}{4 \sqrt [3]{2} \left (-\sqrt{3}+i\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((3 - I*Sqrt[3])*ArcTan[(2 - (2^(1/3)*(1 - I*Sqrt[3] + 2*x))/(1 - x^3)^(1/3))/(2*Sqrt[3])])/(2*2^(1/3)*(I + S
qrt[3])) + ((3 + I*Sqrt[3])*ArcTan[(2 - (2^(1/3)*(1 + I*Sqrt[3] + 2*x))/(1 - x^3)^(1/3))/(2*Sqrt[3])])/(2*2^(1
/3)*(I - Sqrt[3])) + ((I - Sqrt[3])*Log[-((1 - I*Sqrt[3] - 2*x)^2*(1 - I*Sqrt[3] + 2*x))])/(4*2^(1/3)*(I + Sqr
t[3])) + ((I + Sqrt[3])*Log[-((1 + I*Sqrt[3] - 2*x)^2*(1 + I*Sqrt[3] + 2*x))])/(4*2^(1/3)*(I - Sqrt[3])) - (3*
(I - Sqrt[3])*Log[1 - I*Sqrt[3] + 2*x + 2*2^(2/3)*(1 - x^3)^(1/3)])/(4*2^(1/3)*(I + Sqrt[3])) - (3*(I + Sqrt[3
])*Log[1 + I*Sqrt[3] + 2*x + 2*2^(2/3)*(1 - x^3)^(1/3)])/(4*2^(1/3)*(I - Sqrt[3]))

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rule 2148

Int[1/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)), x_Symbol] :> Simp[(Sqrt[3]*ArcTan[(1 - (2^(1/3)*Rt[b,
3]*(c - d*x))/(d*(a + b*x^3)^(1/3)))/Sqrt[3]])/(2^(4/3)*Rt[b, 3]*c), x] + (Simp[Log[(c + d*x)^2*(c - d*x)]/(2
^(7/3)*Rt[b, 3]*c), x] - Simp[(3*Log[Rt[b, 3]*(c - d*x) + 2^(2/3)*d*(a + b*x^3)^(1/3)])/(2^(7/3)*Rt[b, 3]*c),
x]) /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^3 + a*d^3, 0]

Rubi steps

\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\\ &=-\frac{\left (3-i \sqrt{3}\right ) \tan ^{-1}\left (\frac{2-\frac{\sqrt [3]{2} \left (1-i \sqrt{3}+2 x\right )}{\sqrt [3]{1-x^3}}}{2 \sqrt{3}}\right )}{2 \sqrt [3]{2} \left (i+\sqrt{3}\right )}+\frac{\left (3+i \sqrt{3}\right ) \tan ^{-1}\left (\frac{2-\frac{\sqrt [3]{2} \left (1+i \sqrt{3}+2 x\right )}{\sqrt [3]{1-x^3}}}{2 \sqrt{3}}\right )}{2 \sqrt [3]{2} \left (i-\sqrt{3}\right )}+\frac{\left (i-\sqrt{3}\right ) \log \left (-\left (1-i \sqrt{3}-2 x\right )^2 \left (1-i \sqrt{3}+2 x\right )\right )}{4 \sqrt [3]{2} \left (i+\sqrt{3}\right )}+\frac{\left (i+\sqrt{3}\right ) \log \left (-\left (1+i \sqrt{3}-2 x\right )^2 \left (1+i \sqrt{3}+2 x\right )\right )}{4 \sqrt [3]{2} \left (i-\sqrt{3}\right )}-\frac{3 \left (i-\sqrt{3}\right ) \log \left (1-i \sqrt{3}+2 x+2\ 2^{2/3} \sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2} \left (i+\sqrt{3}\right )}-\frac{3 \left (i+\sqrt{3}\right ) \log \left (1+i \sqrt{3}+2 x+2\ 2^{2/3} \sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2} \left (i-\sqrt{3}\right )}\\ \end{align*}

Mathematica [F]  time = 0.19016, size = 0, normalized size = 0. \[ \int \frac{1+x}{\left (1-x+x^2\right ) \sqrt [3]{1-x^3}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.079, size = 0, normalized size = 0. \begin{align*} \int{\frac{1+x}{{x}^{2}-x+1}{\frac{1}{\sqrt [3]{-{x}^{3}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 111.657, size = 882, normalized size = 6.53 \begin{align*} \frac{1}{6} \, \sqrt{3} 2^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3} 2^{\frac{1}{6}}{\left (4 \cdot 2^{\frac{1}{6}} \left (-1\right )^{\frac{2}{3}}{\left (x^{4} - 4 \, x^{3} + 5 \, x^{2} - 4 \, x + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} - 4 \, \sqrt{2} \left (-1\right )^{\frac{1}{3}}{\left (x^{5} - x^{4} - 3 \, x^{3} + 3 \, x^{2} + x - 1\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 2^{\frac{5}{6}}{\left (x^{6} - 7 \, x^{5} + 10 \, x^{4} - 7 \, x^{3} + 10 \, x^{2} - 7 \, x + 1\right )}\right )}}{6 \,{\left (3 \, x^{6} - 9 \, x^{5} + 6 \, x^{4} - x^{3} + 6 \, x^{2} - 9 \, x + 3\right )}}\right ) - \frac{1}{12} \cdot 2^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} \log \left (-\frac{2^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}}{\left (x^{2} - 3 \, x + 1\right )} + 2^{\frac{1}{3}} \left (-1\right )^{\frac{2}{3}}{\left (x^{4} - 3 \, x^{2} + 1\right )} + 4 \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}}{\left (x^{2} - x\right )}}{x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1}\right ) + \frac{1}{6} \cdot 2^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} \log \left (-\frac{2 \cdot 2^{\frac{1}{3}} \left (-1\right )^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}{\left (x - 1\right )} + 2^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}}{\left (x^{2} - x + 1\right )} - 2 \,{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{x^{2} - x + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(3)*2^(2/3)*(-1)^(1/3)*arctan(1/6*sqrt(3)*2^(1/6)*(4*2^(1/6)*(-1)^(2/3)*(x^4 - 4*x^3 + 5*x^2 - 4*x + 1
)*(-x^3 + 1)^(2/3) - 4*sqrt(2)*(-1)^(1/3)*(x^5 - x^4 - 3*x^3 + 3*x^2 + x - 1)*(-x^3 + 1)^(1/3) + 2^(5/6)*(x^6
- 7*x^5 + 10*x^4 - 7*x^3 + 10*x^2 - 7*x + 1))/(3*x^6 - 9*x^5 + 6*x^4 - x^3 + 6*x^2 - 9*x + 3)) - 1/12*2^(2/3)*
(-1)^(1/3)*log(-(2^(2/3)*(-1)^(1/3)*(-x^3 + 1)^(2/3)*(x^2 - 3*x + 1) + 2^(1/3)*(-1)^(2/3)*(x^4 - 3*x^2 + 1) +
4*(-x^3 + 1)^(1/3)*(x^2 - x))/(x^4 - 2*x^3 + 3*x^2 - 2*x + 1)) + 1/6*2^(2/3)*(-1)^(1/3)*log(-(2*2^(1/3)*(-1)^(
2/3)*(-x^3 + 1)^(1/3)*(x - 1) + 2^(2/3)*(-1)^(1/3)*(x^2 - x + 1) - 2*(-x^3 + 1)^(2/3))/(x^2 - x + 1))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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