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]

-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))*3^(1/2)*2^(2/3)

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Rubi [C]  time = 0.30, 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 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 ) \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*}

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Mathematica [F]  time = 0.13, 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 = 13.54, size = 268, normalized size = 2.25 \[ \frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {1}{6}} {\left (2^{\frac {5}{6}} {\left (x^{6} + 7 \, x^{5} + 10 \, x^{4} + 7 \, x^{3} + 10 \, x^{2} + 7 \, x + 1\right )} - 4 \, \sqrt {2} {\left (x^{5} + x^{4} - 3 \, x^{3} - 3 \, x^{2} + x + 1\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 4 \cdot 2^{\frac {1}{6}} {\left (x^{4} + 4 \, x^{3} + 5 \, x^{2} + 4 \, x + 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}\right )}}{6 \, {\left (3 \, x^{6} + 9 \, x^{5} + 6 \, x^{4} + x^{3} + 6 \, x^{2} + 9 \, x + 3\right )}}\right ) - \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (\frac {2^{\frac {2}{3}} {\left (x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{2} + 3 \, x + 1\right )} - 2^{\frac {1}{3}} {\left (x^{4} - 3 \, x^{2} + 1\right )} - 4 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{2} + x\right )}}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1}\right ) + \frac {1}{6} \cdot 2^{\frac {2}{3}} \log \left (\frac {2^{\frac {2}{3}} {\left (x^{2} + x + 1\right )} + 2 \cdot 2^{\frac {1}{3}} {\left (x^{3} + 1\right )}^{\frac {1}{3}} {\left (x + 1\right )} + 2 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2} + x + 1}\right ) \]

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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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 = 7.94, size = 652, normalized size = 5.48 \[ \text {result too large to display} \]

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]

1/2*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-Roo
tOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^3*x-2*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3
-4)+4*_Z^2)^2*RootOf(_Z^3-4)^2*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*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+
4*_Z^2)*x^2+RootOf(_Z^3-4)*x+2*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x+RootOf(_Z^3-4)+2*RootOf(R
ootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2))/(x^2+x+1))+RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^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-2*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)^2*RootOf(_
Z^3-4)^2*x+(x^3+1)^(1/3)*RootOf(_Z^3-4)^2*x+(x^3+1)^(1/3)*RootOf(_Z^3-4)^2-RootOf(_Z^3-4)*x^2-2*RootOf(RootOf(
_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x^2+2*(x^3+1)^(2/3)-3*RootOf(_Z^3-4)*x-6*RootOf(RootOf(_Z^3-4)^2+2*_Z*Ro
otOf(_Z^3-4)+4*_Z^2)*x-RootOf(_Z^3-4)-2*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2))/(x^2+x+1))

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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 \[ -\int \frac {x-1}{{\left (x^3+1\right )}^{1/3}\,\left (x^2+x+1\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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/(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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