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

Optimal. Leaf size=119 \[ -\frac{\log \left (\frac{2^{2/3} (x+1)^2}{\left (x^3+1\right )^{2/3}}-\frac{\sqrt [3]{2} (x+1)}{\sqrt [3]{x^3+1}}+1\right )}{2 \sqrt [3]{2}}+\frac{\log \left (\frac{\sqrt [3]{2} (x+1)}{\sqrt [3]{x^3+1}}+1\right )}{\sqrt [3]{2}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} (x+1)}{\sqrt [3]{x^3+1}}}{\sqrt{3}}\right )}{\sqrt [3]{2}} \]

[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)

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Rubi [C]  time = 0.299049, antiderivative size = 399, normalized size of antiderivative = 3.35, number of steps used = 4, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {6728, 2148} \[ \frac{3 \left (-\sqrt{3}+i\right ) \log \left (2\ 2^{2/3} \sqrt [3]{x^3+1}-2 x-i \sqrt{3}+1\right )}{4 \sqrt [3]{2} \left (\sqrt{3}+i\right )}+\frac{3 \left (\sqrt{3}+i\right ) \log \left (2\ 2^{2/3} \sqrt [3]{x^3+1}-2 x+i \sqrt{3}+1\right )}{4 \sqrt [3]{2} \left (-\sqrt{3}+i\right )}+\frac{\left (3-i \sqrt{3}\right ) \tan ^{-1}\left (\frac{2-\frac{\sqrt [3]{2} \left (-2 x-i \sqrt{3}+1\right )}{\sqrt [3]{x^3+1}}}{2 \sqrt{3}}\right )}{2 \sqrt [3]{2} \left (\sqrt{3}+i\right )}-\frac{\left (3+i \sqrt{3}\right ) \tan ^{-1}\left (\frac{2-\frac{\sqrt [3]{2} \left (-2 x+i \sqrt{3}+1\right )}{\sqrt [3]{x^3+1}}}{2 \sqrt{3}}\right )}{2 \sqrt [3]{2} \left (-\sqrt{3}+i\right )}-\frac{\left (-\sqrt{3}+i\right ) \log \left (\left (-2 x-i \sqrt{3}+1\right ) \left (2 x-i \sqrt{3}+1\right )^2\right )}{4 \sqrt [3]{2} \left (\sqrt{3}+i\right )}-\frac{\left (\sqrt{3}+i\right ) \log \left (\left (-2 x+i \sqrt{3}+1\right ) \left (2 x+i \sqrt{3}+1\right )^2\right )}{4 \sqrt [3]{2} \left (-\sqrt{3}+i\right )} \]

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 + Sq
rt[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)*(1 - I*Sqrt[3] + 2*x)^2])/(4*2^(1/3)*(I + Sqrt[3]
)) - ((I + Sqrt[3])*Log[(1 + I*Sqrt[3] - 2*x)*(1 + I*Sqrt[3] + 2*x)^2])/(4*2^(1/3)*(I - Sqrt[3])) + (3*(I - Sq
rt[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 ) \left (1-i \sqrt{3}+2 x\right )^2\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 ) \left (1+i \sqrt{3}+2 x\right )^2\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 = 0.162654, 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]

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Maple [F]  time = 0.069, 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)

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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)

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Fricas [B]  time = 93.4274, size = 737, normalized size = 6.19 \begin{align*} \frac{1}{6} \, \sqrt{3} 2^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3} 2^{\frac{1}{6}}{\left (2^{\frac{5}{6}}{\left (x^{6} + 7 \, x^{5} + 10 \, x^{4} + 7 \, x^{3} + 10 \, x^{2} + 7 \, x + 1\right )} - 4 \, \sqrt{2}{\left (x^{5} + x^{4} - 3 \, x^{3} - 3 \, x^{2} + x + 1\right )}{\left (x^{3} + 1\right )}^{\frac{1}{3}} + 4 \cdot 2^{\frac{1}{6}}{\left (x^{4} + 4 \, x^{3} + 5 \, x^{2} + 4 \, x + 1\right )}{\left (x^{3} + 1\right )}^{\frac{2}{3}}\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}} \log \left (\frac{2^{\frac{2}{3}}{\left (x^{3} + 1\right )}^{\frac{2}{3}}{\left (x^{2} + 3 \, x + 1\right )} - 2^{\frac{1}{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}} \log \left (\frac{2^{\frac{2}{3}}{\left (x^{2} + x + 1\right )} + 2 \cdot 2^{\frac{1}{3}}{\left (x^{3} + 1\right )}^{\frac{1}{3}}{\left (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)*arctan(1/6*sqrt(3)*2^(1/6)*(2^(5/6)*(x^6 + 7*x^5 + 10*x^4 + 7*x^3 + 10*x^2 + 7*x + 1) - 4*
sqrt(2)*(x^5 + x^4 - 3*x^3 - 3*x^2 + x + 1)*(x^3 + 1)^(1/3) + 4*2^(1/6)*(x^4 + 4*x^3 + 5*x^2 + 4*x + 1)*(x^3 +
1)^(2/3))/(3*x^6 + 9*x^5 + 6*x^4 + x^3 + 6*x^2 + 9*x + 3)) - 1/12*2^(2/3)*log((2^(2/3)*(x^3 + 1)^(2/3)*(x^2 +
3*x + 1) - 2^(1/3)*(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)*log((2^(2/3)*(x^2 + x + 1) + 2*2^(1/3)*(x^3 + 1)^(1/3)*(x + 1) + 2*(x^3 + 1)^(2/3))/(x^2 + x + 1))

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

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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)