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]

1/4*ln(1+2^(2/3)*(1-x)^2/(-x^3+1)^(2/3)-2^(1/3)*(1-x)/(-x^3+1)^(1/3))*2^(2/3)-1/2*ln(1+2^(1/3)*(1-x)/(-x^3+1)^
(1/3))*2^(2/3)+1/2*arctan(1/3*(1-2*2^(1/3)*(1-x)/(-x^3+1)^(1/3))*3^(1/2))*2^(2/3)*3^(1/2)

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Rubi [C]  time = 0.34, antiderivative size = 409, normalized size of antiderivative = 3.03, number of steps used = 4, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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 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]

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]

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*}

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Mathematica [F]  time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {1+x}{\left (1-x+x^2\right ) \sqrt [3]{1-x^3}} \, dx \]

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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fricas [B]  time = 12.83, size = 318, normalized size = 2.36 \[ \frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {1}{6}} {\left (4 \cdot 2^{\frac {1}{6}} \left (-1\right )^{\frac {2}{3}} {\left (x^{4} - 4 \, x^{3} + 5 \, x^{2} - 4 \, x + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - 4 \, \sqrt {2} \left (-1\right )^{\frac {1}{3}} {\left (x^{5} - x^{4} - 3 \, x^{3} + 3 \, x^{2} + x - 1\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 2^{\frac {5}{6}} {\left (x^{6} - 7 \, x^{5} + 10 \, x^{4} - 7 \, x^{3} + 10 \, x^{2} - 7 \, x + 1\right )}\right )}}{6 \, {\left (3 \, x^{6} - 9 \, x^{5} + 6 \, x^{4} - x^{3} + 6 \, x^{2} - 9 \, x + 3\right )}}\right ) - \frac {1}{12} \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{2} - 3 \, x + 1\right )} + 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{4} - 3 \, x^{2} + 1\right )} + 4 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{2} - x\right )}}{x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1}\right ) + \frac {1}{6} \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {2 \cdot 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} + 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{2} - x + 1\right )} - 2 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2} - x + 1}\right ) \]

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))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x + 1}{{\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{2} - x + 1\right )}}\,{d x} \]

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)

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maple [C]  time = 7.80, size = 720, normalized size = 5.33 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*ln((RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*Ro
otOf(_Z^3+4)*(-x^3+1)^(2/3)+RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*RootOf(_Z^3+4)^2*x-2*RootOf(Ro
otOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*(-x^3+1)^(1/3)*x+2*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z
^2)*(-x^3+1)^(1/3)-x^2+x-1)/(x^2-x+1))-1/2*ln(-((-x^3+1)^(2/3)*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_
Z^2)*RootOf(_Z^3+4)^2+RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*RootOf(_Z^3+4)^3*x-(-x^3+1)^(1/3)*Ro
otOf(_Z^3+4)^2*x-2*(-x^3+1)^(1/3)*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*RootOf(_Z^3+4)*x+(-x^3+1
)^(1/3)*RootOf(_Z^3+4)^2+2*(-x^3+1)^(1/3)*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*RootOf(_Z^3+4)+R
ootOf(_Z^3+4)*x^2-2*(-x^3+1)^(2/3)-3*RootOf(_Z^3+4)*x+RootOf(_Z^3+4))/(x^2-x+1))*RootOf(_Z^3+4)-ln(-((-x^3+1)^
(2/3)*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*RootOf(_Z^3+4)^2+RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf
(_Z^3+4)+4*_Z^2)*RootOf(_Z^3+4)^3*x-(-x^3+1)^(1/3)*RootOf(_Z^3+4)^2*x-2*(-x^3+1)^(1/3)*RootOf(RootOf(_Z^3+4)^2
+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*RootOf(_Z^3+4)*x+(-x^3+1)^(1/3)*RootOf(_Z^3+4)^2+2*(-x^3+1)^(1/3)*RootOf(RootOf(_
Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*RootOf(_Z^3+4)+RootOf(_Z^3+4)*x^2-2*(-x^3+1)^(2/3)-3*RootOf(_Z^3+4)*x+Roo
tOf(_Z^3+4))/(x^2-x+1))*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x + 1}{{\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{2} - x + 1\right )}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x+1}{{\left (1-x^3\right )}^{1/3}\,\left (x^2-x+1\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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)

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