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

Optimal. Leaf size=141 \[ \frac {(x-1)^{2/3} \sqrt [3]{x+1} \left (\sqrt [3]{x-1} (x+1)^{2/3}+\frac {2}{3} \log \left (\sqrt [3]{x-1}-\sqrt [3]{x+1}\right )-\frac {1}{3} \log \left ((x-1)^{2/3}+\sqrt [3]{x+1} \sqrt [3]{x-1}+(x+1)^{2/3}\right )+\frac {2 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x+1}}{2 \sqrt [3]{x-1}+\sqrt [3]{x+1}}\right )}{\sqrt {3}}\right )}{\sqrt [3]{(x-1)^2 (x+1)}} \]

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Rubi [A]  time = 0.20, antiderivative size = 214, normalized size of antiderivative = 1.52, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2081, 2077, 21, 50, 60} \begin {gather*} -\frac {(1-x) (x+1)}{\sqrt [3]{x^3-x^2-x+1}}+\frac {(3-3 x)^{2/3} \sqrt [3]{x+1} \log \left (-\frac {8}{3} (x-1)\right )}{3\ 3^{2/3} \sqrt [3]{x^3-x^2-x+1}}+\frac {(3-3 x)^{2/3} \sqrt [3]{x+1} \log \left (\frac {\sqrt [3]{3} \sqrt [3]{x+1}}{\sqrt [3]{3-3 x}}+1\right )}{3^{2/3} \sqrt [3]{x^3-x^2-x+1}}+\frac {2 (3-3 x)^{2/3} \sqrt [3]{x+1} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x+1}}{\sqrt [6]{3} \sqrt [3]{3-3 x}}\right )}{3 \sqrt [6]{3} \sqrt [3]{x^3-x^2-x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((1 - x)*(1 + x))/(1 - x - x^2 + x^3)^(1/3)) + (2*(3 - 3*x)^(2/3)*(1 + x)^(1/3)*ArcTan[1/Sqrt[3] - (2*(1 + x
)^(1/3))/(3^(1/6)*(3 - 3*x)^(1/3))])/(3*3^(1/6)*(1 - x - x^2 + x^3)^(1/3)) + ((3 - 3*x)^(2/3)*(1 + x)^(1/3)*Lo
g[(-8*(-1 + x))/3])/(3*3^(2/3)*(1 - x - x^2 + x^3)^(1/3)) + ((3 - 3*x)^(2/3)*(1 + x)^(1/3)*Log[1 + (3^(1/3)*(1
+ x)^(1/3))/(3 - 3*x)^(1/3)])/(3^(2/3)*(1 - x - x^2 + x^3)^(1/3))

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 60

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[-(d/b), 3]}, Simp[(Sq
rt[3]*q*ArcTan[1/Sqrt[3] - (2*q*(a + b*x)^(1/3))/(Sqrt[3]*(c + d*x)^(1/3))])/d, x] + (Simp[(3*q*Log[(q*(a + b*
x)^(1/3))/(c + d*x)^(1/3) + 1])/(2*d), x] + Simp[(q*Log[c + d*x])/(2*d), x])] /; FreeQ[{a, b, c, d}, x] && NeQ
[b*c - a*d, 0] && NegQ[d/b]

Rule 2077

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> Dist[(a + b*x + d*x^3)^p/
((3*a - b*x)^p*(3*a + 2*b*x)^(2*p)), Int[(e + f*x)^m*(3*a - b*x)^p*(3*a + 2*b*x)^(2*p), x], x] /; FreeQ[{a, b,
d, e, f, m, p}, x] && EqQ[4*b^3 + 27*a^2*d, 0] &&  !IntegerQ[p]

Rule 2081

Int[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3, x, 0], b = Coeff[P3, x, 1], c = C
oeff[P3, x, 2], d = Coeff[P3, x, 3]}, Subst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2
)/(27*d^2) - ((c^2 - 3*b*d)*x)/(3*d) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c, 0]] /; FreeQ[{e, f, m, p}, x
] && PolyQ[P3, x, 3]

Rubi steps

\begin {align*} \int \frac {-1+x}{\sqrt [3]{1-x-x^2+x^3}} \, dx &=\operatorname {Subst}\left (\int \frac {-\frac {2}{3}+x}{\sqrt [3]{\frac {16}{27}-\frac {4 x}{3}+x^3}} \, dx,x,-\frac {1}{3}+x\right )\\ &=\frac {\left (4\ 2^{2/3} (1-x)^{2/3} \sqrt [3]{1+x}\right ) \operatorname {Subst}\left (\int \frac {-\frac {2}{3}+x}{\left (\frac {16}{9}-\frac {8 x}{3}\right )^{2/3} \sqrt [3]{\frac {16}{9}+\frac {4 x}{3}}} \, dx,x,-\frac {1}{3}+x\right )}{3 \sqrt [3]{1-x-x^2+x^3}}\\ &=-\frac {\left ((1-x)^{2/3} \sqrt [3]{1+x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{\frac {16}{9}-\frac {8 x}{3}}}{\sqrt [3]{\frac {16}{9}+\frac {4 x}{3}}} \, dx,x,-\frac {1}{3}+x\right )}{\sqrt [3]{2} \sqrt [3]{1-x-x^2+x^3}}\\ &=-\frac {(1-x) (1+x)}{\sqrt [3]{1-x-x^2+x^3}}-\frac {\left (8\ 2^{2/3} (1-x)^{2/3} \sqrt [3]{1+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {16}{9}-\frac {8 x}{3}\right )^{2/3} \sqrt [3]{\frac {16}{9}+\frac {4 x}{3}}} \, dx,x,-\frac {1}{3}+x\right )}{9 \sqrt [3]{1-x-x^2+x^3}}\\ &=-\frac {(1-x) (1+x)}{\sqrt [3]{1-x-x^2+x^3}}+\frac {2 (1-x)^{2/3} \sqrt [3]{1+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1+x}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{\sqrt {3} \sqrt [3]{1-x-x^2+x^3}}+\frac {(1-x)^{2/3} \sqrt [3]{1+x} \log (1-x)}{3 \sqrt [3]{1-x-x^2+x^3}}+\frac {(1-x)^{2/3} \sqrt [3]{1+x} \log \left (\frac {3 \left (\sqrt [3]{1-x}+\sqrt [3]{1+x}\right )}{\sqrt [3]{1-x}}\right )}{\sqrt [3]{1-x-x^2+x^3}}\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 48, normalized size = 0.34 \begin {gather*} \frac {3 \left ((x-1)^2 (x+1)\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {4}{3};\frac {7}{3};\frac {1-x}{2}\right )}{4 \sqrt [3]{2} (x+1)^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*((-1 + x)^2*(1 + x))^(2/3)*Hypergeometric2F1[1/3, 4/3, 7/3, (1 - x)/2])/(4*2^(1/3)*(1 + x)^(2/3))

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IntegrateAlgebraic [A]  time = 5.04, size = 141, normalized size = 1.00 \begin {gather*} \frac {(-1+x)^{2/3} \sqrt [3]{1+x} \left (\sqrt [3]{-1+x} (1+x)^{2/3}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{1+x}}{2 \sqrt [3]{-1+x}+\sqrt [3]{1+x}}\right )}{\sqrt {3}}+\frac {2}{3} \log \left (\sqrt [3]{-1+x}-\sqrt [3]{1+x}\right )-\frac {1}{3} \log \left ((-1+x)^{2/3}+\sqrt [3]{-1+x} \sqrt [3]{1+x}+(1+x)^{2/3}\right )\right )}{\sqrt [3]{(-1+x)^2 (1+x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.53, size = 161, normalized size = 1.14 \begin {gather*} \frac {2 \, \sqrt {3} {\left (x - 1\right )} \arctan \left (\frac {\sqrt {3} {\left (x - 1\right )} + 2 \, \sqrt {3} {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {1}{3}}}{3 \, {\left (x - 1\right )}}\right ) - {\left (x - 1\right )} \log \left (\frac {x^{2} + {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} - 2 \, x + {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {2}{3}} + 1}{x^{2} - 2 \, x + 1}\right ) + 2 \, {\left (x - 1\right )} \log \left (-\frac {x - {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {1}{3}} - 1}{x - 1}\right ) + 3 \, {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {2}{3}}}{3 \, {\left (x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(2*sqrt(3)*(x - 1)*arctan(1/3*(sqrt(3)*(x - 1) + 2*sqrt(3)*(x^3 - x^2 - x + 1)^(1/3))/(x - 1)) - (x - 1)*l
og((x^2 + (x^3 - x^2 - x + 1)^(1/3)*(x - 1) - 2*x + (x^3 - x^2 - x + 1)^(2/3) + 1)/(x^2 - 2*x + 1)) + 2*(x - 1
)*log(-(x - (x^3 - x^2 - x + 1)^(1/3) - 1)/(x - 1)) + 3*(x^3 - x^2 - x + 1)^(2/3))/(x - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.82, size = 361, normalized size = 2.56

method	result	size
risch	\(\frac {\left (-1+x \right ) \left (1+x \right )}{\left (\left (-1+x \right )^{2} \left (1+x \right )\right )^{\frac {1}{3}}}+\frac {2 \ln \left (-\frac {-4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x +4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x -4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}-3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x +2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -x^{2}+3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1}{-1+x}\right )}{3}+\frac {2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x -5 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+3 \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}-3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x +6 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -2 x^{2}+3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+4 x -2}{-1+x}\right )}{3}\)	\(361\)
trager	\(\frac {\left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}}{-1+x}+2 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-\frac {18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}+9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}+9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x -18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x +15 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}-12 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x +2 x^{2}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-2}{-1+x}\right )-\frac {2 \ln \left (\frac {-18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}+9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}+9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x +18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x +3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+3 \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}-9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}+3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x +x^{2}-3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-2 x +1}{-1+x}\right )}{3}-2 \ln \left (\frac {-18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}+9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}+9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x +18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x +3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+3 \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}-9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}+3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x +x^{2}-3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-2 x +1}{-1+x}\right ) \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )\)	\(658\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-1+x)*(1+x)/((-1+x)^2*(1+x))^(1/3)+2/3*ln(-(-4*RootOf(_Z^2+_Z+1)^2*x^2+3*RootOf(_Z^2+_Z+1)*(x^3-x^2-x+1)^(2/3
)-3*RootOf(_Z^2+_Z+1)*(x^3-x^2-x+1)^(1/3)*x+4*RootOf(_Z^2+_Z+1)^2*x-4*RootOf(_Z^2+_Z+1)*x^2+3*RootOf(_Z^2+_Z+1
)*(x^3-x^2-x+1)^(1/3)-3*(x^3-x^2-x+1)^(1/3)*x+2*RootOf(_Z^2+_Z+1)*x-x^2+3*(x^3-x^2-x+1)^(1/3)+2*RootOf(_Z^2+_Z
+1)+1)/(-1+x))+2/3*RootOf(_Z^2+_Z+1)*ln((-2*RootOf(_Z^2+_Z+1)^2*x^2+3*RootOf(_Z^2+_Z+1)*(x^3-x^2-x+1)^(2/3)+2*
RootOf(_Z^2+_Z+1)^2*x-5*RootOf(_Z^2+_Z+1)*x^2+3*(x^3-x^2-x+1)^(2/3)-3*(x^3-x^2-x+1)^(1/3)*x+6*RootOf(_Z^2+_Z+1
)*x-2*x^2+3*(x^3-x^2-x+1)^(1/3)-RootOf(_Z^2+_Z+1)+4*x-2)/(-1+x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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