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

Optimal. Leaf size=141 \[ \frac {\sqrt [3]{x-1} (x+1)^{2/3} \left ((x-1)^{2/3} \sqrt [3]{x+1}+\frac {1}{3} \log \left (\sqrt [3]{x-1}-\sqrt [3]{x+1}\right )-\frac {1}{6} \log \left ((x-1)^{2/3}+\sqrt [3]{x+1} \sqrt [3]{x-1}+(x+1)^{2/3}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x+1}}{2 \sqrt [3]{x-1}+\sqrt [3]{x+1}}\right )}{\sqrt {3}}\right )}{\sqrt [3]{(x-1) (x+1)^2}} \]

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Rubi [A]  time = 0.20, antiderivative size = 191, normalized size of antiderivative = 1.35, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2081, 2077, 80, 60} \begin {gather*} -\frac {(1-x) (x+1)}{\sqrt [3]{x^3+x^2-x-1}}+\frac {(-x-1)^{2/3} \sqrt [3]{x-1} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{-x-1}}+1\right )}{2 \sqrt [3]{x^3+x^2-x-1}}+\frac {(-x-1)^{2/3} \sqrt [3]{x-1} \log \left (-\frac {8}{3} (x+1)\right )}{6 \sqrt [3]{x^3+x^2-x-1}}+\frac {(-x-1)^{2/3} \sqrt [3]{x-1} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x-1}}{\sqrt {3} \sqrt [3]{-x-1}}\right )}{\sqrt {3} \sqrt [3]{x^3+x^2-x-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(-1 - x + x^2 + x^3)^(1/3),x]

[Out]

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

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 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

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 {x}{\sqrt [3]{-1-x+x^2+x^3}} \, dx &=\operatorname {Subst}\left (\int \frac {-\frac {1}{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 {1}{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 {(1-x) (1+x)}{\sqrt [3]{-1-x+x^2+x^3}}-\frac {\left (4\ 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 {(-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 \left (1+\frac {\sqrt [3]{-1+x}}{\sqrt [3]{-1-x}}\right )}{2 \sqrt [3]{-1-x+x^2+x^3}}+\frac {(-1-x)^{2/3} \sqrt [3]{-1+x} \log (1+x)}{6 \sqrt [3]{-1-x+x^2+x^3}}\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 58, normalized size = 0.41 \begin {gather*} \frac {(x-1) \left (-\sqrt [3]{2} (x+1)^{2/3} \, _2F_1\left (\frac {2}{3},\frac {2}{3};\frac {5}{3};\frac {1-x}{2}\right )+4 x+4\right )}{4 \sqrt [3]{(x-1) (x+1)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1 + x)*(4 + 4*x - 2^(1/3)*(1 + x)^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, (1 - x)/2]))/(4*((-1 + x)*(1 + x)^
2)^(1/3))

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IntegrateAlgebraic [A]  time = 5.07, size = 152, normalized size = 1.08 \begin {gather*} \frac {\sqrt [3]{-1+x} (1+x)^{2/3} \left (\frac {2 \sqrt [3]{1+x}}{\sqrt [3]{-1+x} \left (-1+\frac {1+x}{-1+x}\right )}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x}}{\sqrt {3} \sqrt [3]{-1+x}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (-1+\frac {\sqrt [3]{1+x}}{\sqrt [3]{-1+x}}\right )-\frac {1}{6} \log \left (1+\frac {\sqrt [3]{1+x}}{\sqrt [3]{-1+x}}+\frac {(1+x)^{2/3}}{(-1+x)^{2/3}}\right )\right )}{\sqrt [3]{(-1+x) (1+x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.47, size = 151, normalized size = 1.07 \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 ) + 6 \, {\left (x^{3} + x^{2} - x - 1\right )}^{\frac {2}{3}}}{6 \, {\left (x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(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)) + 6*(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}{{\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(x/(x^3+x^2-x-1)^(1/3),x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.81, size = 403, normalized size = 2.86

method	result	size
trager	\(\frac {\left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}}{1+x}+\frac {\ln \left (\frac {-36 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}+9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}-9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x -36 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x -12 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-3 x \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -x^{2}-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}+6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1}{1+x}\right )}{3}+\RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-\frac {-18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}+9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}-18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x -15 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+3 \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}-3 x \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -2 x^{2}-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-4 x -2}{1+x}\right )\)	\(403\)
risch	\(\frac {\left (-1+x \right ) \left (1+x \right )}{\left (\left (-1+x \right ) \left (1+x \right )^{2}\right )^{\frac {1}{3}}}+\frac {\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}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x +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 +5 \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}}+4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +2 x^{2}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2}{1+x}\right )}{3}-\frac {\ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x -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 -\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 {2}{3}}-3 x \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-2 x -1}{1+x}\right ) \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{3}-\frac {\ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x -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 -\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 {2}{3}}-3 x \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-2 x -1}{1+x}\right )}{3}\)	\(528\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(1+x)*(x^3+x^2-x-1)^(2/3)+1/3*ln((-36*RootOf(9*_Z^2+3*_Z+1)^2*x^2+9*RootOf(9*_Z^2+3*_Z+1)*(x^3+x^2-x-1)^(2/3
)-9*RootOf(9*_Z^2+3*_Z+1)*(x^3+x^2-x-1)^(1/3)*x-36*RootOf(9*_Z^2+3*_Z+1)^2*x-12*RootOf(9*_Z^2+3*_Z+1)*x^2-9*Ro
otOf(9*_Z^2+3*_Z+1)*(x^3+x^2-x-1)^(1/3)-3*x*(x^3+x^2-x-1)^(1/3)-6*RootOf(9*_Z^2+3*_Z+1)*x-x^2-3*(x^3+x^2-x-1)^
(1/3)+6*RootOf(9*_Z^2+3*_Z+1)+1)/(1+x))+RootOf(9*_Z^2+3*_Z+1)*ln(-(-18*RootOf(9*_Z^2+3*_Z+1)^2*x^2+9*RootOf(9*
_Z^2+3*_Z+1)*(x^3+x^2-x-1)^(2/3)-18*RootOf(9*_Z^2+3*_Z+1)^2*x-15*RootOf(9*_Z^2+3*_Z+1)*x^2+3*(x^3+x^2-x-1)^(2/
3)-3*x*(x^3+x^2-x-1)^(1/3)-18*RootOf(9*_Z^2+3*_Z+1)*x-2*x^2-3*(x^3+x^2-x-1)^(1/3)-3*RootOf(9*_Z^2+3*_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}{{\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(x/(x^3+x^2-x-1)^(1/3),x, algorithm="maxima")

[Out]

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

[Out]

int(x/(x^2 - x + x^3 - 1)^(1/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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