\(\int \frac {1+x}{(1+4 x+x^2) \sqrt [3]{1-x^3}} \, dx\) [2261]

Optimal result

Integrand size = 25, antiderivative size = 171 \[ \int \frac {1+x}{\left (1+4 x+x^2\right ) \sqrt [3]{1-x^3}} \, dx=\frac {\arctan \left (\frac {\sqrt {3} \sqrt [3]{1-x^3}}{2^{2/3}-2^{2/3} x+\sqrt [3]{1-x^3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\log \left (-2^{2/3}+2^{2/3} x+2 \sqrt [3]{1-x^3}\right )}{3\ 2^{2/3}}-\frac {\log \left (-\sqrt [3]{2}+2 \sqrt [3]{2} x-\sqrt [3]{2} x^2+\left (-2^{2/3}+2^{2/3} x\right ) \sqrt [3]{1-x^3}-2 \left (1-x^3\right )^{2/3}\right )}{6\ 2^{2/3}} \]

[Out]

1/6*arctan(3^(1/2)*(-x^3+1)^(1/3)/(2^(2/3)-2^(2/3)*x+(-x^3+1)^(1/3)))*2^(1/3)*3^(1/2)+1/6*ln(-2^(2/3)+2^(2/3)*
x+2*(-x^3+1)^(1/3))*2^(1/3)-1/12*ln(-2^(1/3)+2*2^(1/3)*x-2^(1/3)*x^2+(-2^(2/3)+2^(2/3)*x)*(-x^3+1)^(1/3)-2*(-x
^3+1)^(2/3))*2^(1/3)

Rubi [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 0.84 (sec) , antiderivative size = 559, normalized size of antiderivative = 3.27, number of steps used = 20, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {6860, 2181, 384, 524, 455, 57, 631, 210, 31} \[ \int \frac {1+x}{\left (1+4 x+x^2\right ) \sqrt [3]{1-x^3}} \, dx=-\frac {1}{12} \left (9+5 \sqrt {3}\right ) x^2 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^3,-\frac {x^3}{\left (2-\sqrt {3}\right )^3}\right )-\frac {1}{12} \left (9-5 \sqrt {3}\right ) x^2 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^3,-\frac {x^3}{\left (2+\sqrt {3}\right )^3}\right )-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{3 \left (9-5 \sqrt {3}\right )} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3}}-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{3 \left (9+5 \sqrt {3}\right )} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{1-x^3}}{\sqrt [3]{3 \left (9-5 \sqrt {3}\right )}}+1}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{1-x^3}}{\sqrt [3]{3 \left (9+5 \sqrt {3}\right )}}+1}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3}}-\frac {\log \left (x^3-15 \sqrt {3}+26\right )}{18\ 2^{2/3}}-\frac {\log \left (x^3+15 \sqrt {3}+26\right )}{18\ 2^{2/3}}-\frac {\log \left (8 x^3+8 \left (2-\sqrt {3}\right )^3\right )}{18\ 2^{2/3}}-\frac {\log \left (8 x^3+8 \left (2+\sqrt {3}\right )^3\right )}{18\ 2^{2/3}}+\frac {\log \left (\sqrt [3]{3 \left (9-5 \sqrt {3}\right )}-\sqrt [3]{1-x^3}\right )}{6\ 2^{2/3}}+\frac {\log \left (\sqrt [3]{3 \left (9+5 \sqrt {3}\right )}-\sqrt [3]{1-x^3}\right )}{6\ 2^{2/3}}+\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{3 \left (9-5 \sqrt {3}\right )} x\right )}{6\ 2^{2/3}}+\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{3 \left (9+5 \sqrt {3}\right )} x\right )}{6\ 2^{2/3}} \]

[In]

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

[Out]

-1/12*((9 + 5*Sqrt[3])*x^2*AppellF1[2/3, 1/3, 1, 5/3, x^3, -(x^3/(2 - Sqrt[3])^3)]) - ((9 - 5*Sqrt[3])*x^2*App
ellF1[2/3, 1/3, 1, 5/3, x^3, -(x^3/(2 + Sqrt[3])^3)])/12 - ArcTan[(1 - (2*(3*(9 - 5*Sqrt[3]))^(1/3)*x)/(1 - x^
3)^(1/3))/Sqrt[3]]/(3*2^(2/3)*Sqrt[3]) - ArcTan[(1 - (2*(3*(9 + 5*Sqrt[3]))^(1/3)*x)/(1 - x^3)^(1/3))/Sqrt[3]]
/(3*2^(2/3)*Sqrt[3]) + ArcTan[(1 + (2*(1 - x^3)^(1/3))/(3*(9 - 5*Sqrt[3]))^(1/3))/Sqrt[3]]/(3*2^(2/3)*Sqrt[3])
+ ArcTan[(1 + (2*(1 - x^3)^(1/3))/(3*(9 + 5*Sqrt[3]))^(1/3))/Sqrt[3]]/(3*2^(2/3)*Sqrt[3]) - Log[26 - 15*Sqrt[
3] + x^3]/(18*2^(2/3)) - Log[26 + 15*Sqrt[3] + x^3]/(18*2^(2/3)) - Log[8*(2 - Sqrt[3])^3 + 8*x^3]/(18*2^(2/3))
- Log[8*(2 + Sqrt[3])^3 + 8*x^3]/(18*2^(2/3)) + Log[(3*(9 - 5*Sqrt[3]))^(1/3) - (1 - x^3)^(1/3)]/(6*2^(2/3))
+ Log[(3*(9 + 5*Sqrt[3]))^(1/3) - (1 - x^3)^(1/3)]/(6*2^(2/3)) + Log[-((3*(9 - 5*Sqrt[3]))^(1/3)*x) - (1 - x^3
)^(1/3)]/(6*2^(2/3)) + Log[-((3*(9 + 5*Sqrt[3]))^(1/3)*x) - (1 - x^3)^(1/3)]/(6*2^(2/3))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 57

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, Simp[-L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && PosQ
[(b*c - a*d)/b]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 384

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[
ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q
), x] + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 524

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*
((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2181

Int[(Px_.)*((c_) + (d_.)*(x_))^(q_)*((a_) + (b_.)*(x_)^3)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c^3 + d^3*x
^3)^q*(a + b*x^3)^p, Px/(c^2 - c*d*x + d^2*x^2)^q, x], x] /; FreeQ[{a, b, c, d, p}, x] && PolyQ[Px, x] && ILtQ
[q, 0] && RationalQ[p] && EqQ[Denominator[p], 3]

Rule 6860

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*} \text {integral}& = \int \left (\frac {1-\frac {1}{\sqrt {3}}}{\left (4-2 \sqrt {3}+2 x\right ) \sqrt [3]{1-x^3}}+\frac {1+\frac {1}{\sqrt {3}}}{\left (4+2 \sqrt {3}+2 x\right ) \sqrt [3]{1-x^3}}\right ) \, dx \\ & = \frac {1}{3} \left (3-\sqrt {3}\right ) \int \frac {1}{\left (4-2 \sqrt {3}+2 x\right ) \sqrt [3]{1-x^3}} \, dx+\frac {1}{3} \left (3+\sqrt {3}\right ) \int \frac {1}{\left (4+2 \sqrt {3}+2 x\right ) \sqrt [3]{1-x^3}} \, dx \\ & = \frac {1}{3} \left (3-\sqrt {3}\right ) \int \left (-\frac {4 \left (-7+4 \sqrt {3}\right )}{\sqrt [3]{1-x^3} \left (\left (4-2 \sqrt {3}\right )^3+8 x^3\right )}+\frac {4 \left (-2+\sqrt {3}\right ) x}{\sqrt [3]{1-x^3} \left (\left (4-2 \sqrt {3}\right )^3+8 x^3\right )}+\frac {4 x^2}{\sqrt [3]{1-x^3} \left (\left (4-2 \sqrt {3}\right )^3+8 x^3\right )}\right ) \, dx+\frac {1}{3} \left (3+\sqrt {3}\right ) \int \left (\frac {4 \left (7+4 \sqrt {3}\right )}{\sqrt [3]{1-x^3} \left (\left (4+2 \sqrt {3}\right )^3+8 x^3\right )}-\frac {4 \left (2+\sqrt {3}\right ) x}{\sqrt [3]{1-x^3} \left (\left (4+2 \sqrt {3}\right )^3+8 x^3\right )}+\frac {4 x^2}{\sqrt [3]{1-x^3} \left (\left (4+2 \sqrt {3}\right )^3+8 x^3\right )}\right ) \, dx \\ & = \frac {1}{3} \left (4 \left (33-19 \sqrt {3}\right )\right ) \int \frac {1}{\sqrt [3]{1-x^3} \left (\left (4-2 \sqrt {3}\right )^3+8 x^3\right )} \, dx-\frac {1}{3} \left (4 \left (9-5 \sqrt {3}\right )\right ) \int \frac {x}{\sqrt [3]{1-x^3} \left (\left (4-2 \sqrt {3}\right )^3+8 x^3\right )} \, dx+\frac {1}{3} \left (4 \left (3-\sqrt {3}\right )\right ) \int \frac {x^2}{\sqrt [3]{1-x^3} \left (\left (4-2 \sqrt {3}\right )^3+8 x^3\right )} \, dx+\frac {1}{3} \left (4 \left (3+\sqrt {3}\right )\right ) \int \frac {x^2}{\sqrt [3]{1-x^3} \left (\left (4+2 \sqrt {3}\right )^3+8 x^3\right )} \, dx-\frac {1}{3} \left (4 \left (9+5 \sqrt {3}\right )\right ) \int \frac {x}{\sqrt [3]{1-x^3} \left (\left (4+2 \sqrt {3}\right )^3+8 x^3\right )} \, dx+\frac {1}{3} \left (4 \left (33+19 \sqrt {3}\right )\right ) \int \frac {1}{\sqrt [3]{1-x^3} \left (\left (4+2 \sqrt {3}\right )^3+8 x^3\right )} \, dx \\ & = -\frac {\left (9-5 \sqrt {3}\right ) x^2 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^3,-\frac {x^3}{\left (2-\sqrt {3}\right )^3}\right )}{12 \left (2-\sqrt {3}\right )^3}-\frac {\left (9+5 \sqrt {3}\right ) x^2 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^3,-\frac {x^3}{\left (2+\sqrt {3}\right )^3}\right )}{12 \left (2+\sqrt {3}\right )^3}-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{3 \left (9-5 \sqrt {3}\right )} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3}}-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{3 \left (9+5 \sqrt {3}\right )} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3}}-\frac {\log \left (8 \left (2-\sqrt {3}\right )^3+8 x^3\right )}{18\ 2^{2/3}}-\frac {\log \left (8 \left (2+\sqrt {3}\right )^3+8 x^3\right )}{18\ 2^{2/3}}+\frac {\log \left (-\sqrt [3]{3 \left (9-5 \sqrt {3}\right )} x-\sqrt [3]{1-x^3}\right )}{6\ 2^{2/3}}+\frac {\log \left (-\sqrt [3]{3 \left (9+5 \sqrt {3}\right )} x-\sqrt [3]{1-x^3}\right )}{6\ 2^{2/3}}+\frac {1}{9} \left (4 \left (3-\sqrt {3}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} \left (\left (4-2 \sqrt {3}\right )^3+8 x\right )} \, dx,x,x^3\right )+\frac {1}{9} \left (4 \left (3+\sqrt {3}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} \left (\left (4+2 \sqrt {3}\right )^3+8 x\right )} \, dx,x,x^3\right ) \\ & = -\frac {\left (9-5 \sqrt {3}\right ) x^2 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^3,-\frac {x^3}{\left (2-\sqrt {3}\right )^3}\right )}{12 \left (2-\sqrt {3}\right )^3}-\frac {\left (9+5 \sqrt {3}\right ) x^2 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^3,-\frac {x^3}{\left (2+\sqrt {3}\right )^3}\right )}{12 \left (2+\sqrt {3}\right )^3}-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{3 \left (9-5 \sqrt {3}\right )} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3}}-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{3 \left (9+5 \sqrt {3}\right )} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3}}-\frac {\log \left (26-15 \sqrt {3}+x^3\right )}{18\ 2^{2/3}}-\frac {\log \left (26+15 \sqrt {3}+x^3\right )}{18\ 2^{2/3}}-\frac {\log \left (8 \left (2-\sqrt {3}\right )^3+8 x^3\right )}{18\ 2^{2/3}}-\frac {\log \left (8 \left (2+\sqrt {3}\right )^3+8 x^3\right )}{18\ 2^{2/3}}+\frac {\log \left (-\sqrt [3]{3 \left (9-5 \sqrt {3}\right )} x-\sqrt [3]{1-x^3}\right )}{6\ 2^{2/3}}+\frac {\log \left (-\sqrt [3]{3 \left (9+5 \sqrt {3}\right )} x-\sqrt [3]{1-x^3}\right )}{6\ 2^{2/3}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt [3]{3 \left (9-5 \sqrt {3}\right )}-x} \, dx,x,\sqrt [3]{1-x^3}\right )}{6\ 2^{2/3}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt [3]{3 \left (9+5 \sqrt {3}\right )}-x} \, dx,x,\sqrt [3]{1-x^3}\right )}{6\ 2^{2/3}}+\frac {1}{12} \left (3-\sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\left (3 \left (9-5 \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{3 \left (9-5 \sqrt {3}\right )} x+x^2} \, dx,x,\sqrt [3]{1-x^3}\right )+\frac {1}{12} \left (3+\sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\left (3 \left (9+5 \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{3 \left (9+5 \sqrt {3}\right )} x+x^2} \, dx,x,\sqrt [3]{1-x^3}\right ) \\ & = -\frac {\left (9-5 \sqrt {3}\right ) x^2 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^3,-\frac {x^3}{\left (2-\sqrt {3}\right )^3}\right )}{12 \left (2-\sqrt {3}\right )^3}-\frac {\left (9+5 \sqrt {3}\right ) x^2 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^3,-\frac {x^3}{\left (2+\sqrt {3}\right )^3}\right )}{12 \left (2+\sqrt {3}\right )^3}-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{3 \left (9-5 \sqrt {3}\right )} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3}}-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{3 \left (9+5 \sqrt {3}\right )} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3}}-\frac {\log \left (26-15 \sqrt {3}+x^3\right )}{18\ 2^{2/3}}-\frac {\log \left (26+15 \sqrt {3}+x^3\right )}{18\ 2^{2/3}}-\frac {\log \left (8 \left (2-\sqrt {3}\right )^3+8 x^3\right )}{18\ 2^{2/3}}-\frac {\log \left (8 \left (2+\sqrt {3}\right )^3+8 x^3\right )}{18\ 2^{2/3}}+\frac {\log \left (\sqrt [3]{3 \left (9-5 \sqrt {3}\right )}-\sqrt [3]{1-x^3}\right )}{6\ 2^{2/3}}+\frac {\log \left (\sqrt [3]{3 \left (9+5 \sqrt {3}\right )}-\sqrt [3]{1-x^3}\right )}{6\ 2^{2/3}}+\frac {\log \left (-\sqrt [3]{3 \left (9-5 \sqrt {3}\right )} x-\sqrt [3]{1-x^3}\right )}{6\ 2^{2/3}}+\frac {\log \left (-\sqrt [3]{3 \left (9+5 \sqrt {3}\right )} x-\sqrt [3]{1-x^3}\right )}{6\ 2^{2/3}}-\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{1-x^3}}{\sqrt [3]{3 \left (9-5 \sqrt {3}\right )}}\right )}{3\ 2^{2/3}}-\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{1-x^3}}{\sqrt [3]{3 \left (9+5 \sqrt {3}\right )}}\right )}{3\ 2^{2/3}} \\ & = -\frac {\left (9-5 \sqrt {3}\right ) x^2 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^3,-\frac {x^3}{\left (2-\sqrt {3}\right )^3}\right )}{12 \left (2-\sqrt {3}\right )^3}-\frac {\left (9+5 \sqrt {3}\right ) x^2 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^3,-\frac {x^3}{\left (2+\sqrt {3}\right )^3}\right )}{12 \left (2+\sqrt {3}\right )^3}-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{3 \left (9-5 \sqrt {3}\right )} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3}}-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{3 \left (9+5 \sqrt {3}\right )} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3}}+\frac {\arctan \left (\frac {1+\frac {2 \sqrt [3]{1-x^3}}{\sqrt [3]{3 \left (9-5 \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3}}+\frac {\arctan \left (\frac {1+\frac {2 \sqrt [3]{1-x^3}}{\sqrt [3]{3 \left (9+5 \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3}}-\frac {\log \left (26-15 \sqrt {3}+x^3\right )}{18\ 2^{2/3}}-\frac {\log \left (26+15 \sqrt {3}+x^3\right )}{18\ 2^{2/3}}-\frac {\log \left (8 \left (2-\sqrt {3}\right )^3+8 x^3\right )}{18\ 2^{2/3}}-\frac {\log \left (8 \left (2+\sqrt {3}\right )^3+8 x^3\right )}{18\ 2^{2/3}}+\frac {\log \left (\sqrt [3]{3 \left (9-5 \sqrt {3}\right )}-\sqrt [3]{1-x^3}\right )}{6\ 2^{2/3}}+\frac {\log \left (\sqrt [3]{3 \left (9+5 \sqrt {3}\right )}-\sqrt [3]{1-x^3}\right )}{6\ 2^{2/3}}+\frac {\log \left (-\sqrt [3]{3 \left (9-5 \sqrt {3}\right )} x-\sqrt [3]{1-x^3}\right )}{6\ 2^{2/3}}+\frac {\log \left (-\sqrt [3]{3 \left (9+5 \sqrt {3}\right )} x-\sqrt [3]{1-x^3}\right )}{6\ 2^{2/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.27 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.91 \[ \int \frac {1+x}{\left (1+4 x+x^2\right ) \sqrt [3]{1-x^3}} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{1-x^3}}{2^{2/3}-2^{2/3} x+\sqrt [3]{1-x^3}}\right )+2 \log \left (-2^{2/3}+2^{2/3} x+2 \sqrt [3]{1-x^3}\right )-\log \left (-\sqrt [3]{2}+2 \sqrt [3]{2} x-\sqrt [3]{2} x^2+2^{2/3} (-1+x) \sqrt [3]{1-x^3}-2 \left (1-x^3\right )^{2/3}\right )}{6\ 2^{2/3}} \]

[In]

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

[Out]

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 7.40 (sec) , antiderivative size = 1176, normalized size of antiderivative = 6.88

method	result	size
trager	\(\text {Expression too large to display}\)	\(1176\)

[In]

int((1+x)/(x^2+4*x+1)/(-x^3+1)^(1/3),x,method=_RETURNVERBOSE)

[Out]

RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*ln((108*(-x^3+1)^(2/3)*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootO
f(_Z^3-2)+36*_Z^2)*RootOf(_Z^3-2)^2+36*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*RootOf(_Z^3-2)^3*x
-540*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)^2*RootOf(_Z^3-2)^2*x+15*(-x^3+1)^(1/3)*RootOf(_Z^3-2
)^2*x-108*(-x^3+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*RootOf(_Z^3-2)*x-15*(-x^3+1)^(1/
3)*RootOf(_Z^3-2)^2+108*(-x^3+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*RootOf(_Z^3-2)-14*
RootOf(_Z^3-2)*x^2+210*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*x^2-30*(-x^3+1)^(2/3)+4*RootOf(_Z^
3-2)*x-60*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*x-14*RootOf(_Z^3-2)+210*RootOf(RootOf(_Z^3-2)^2
+6*_Z*RootOf(_Z^3-2)+36*_Z^2))/(x^2+4*x+1))-1/6*ln(-(108*(-x^3+1)^(2/3)*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z
^3-2)+36*_Z^2)*RootOf(_Z^3-2)^2+126*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*RootOf(_Z^3-2)^3*x+54
0*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)^2*RootOf(_Z^3-2)^2*x-33*(-x^3+1)^(1/3)*RootOf(_Z^3-2)^2
*x-108*(-x^3+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*RootOf(_Z^3-2)*x+33*(-x^3+1)^(1/3)*
RootOf(_Z^3-2)^2+108*(-x^3+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*RootOf(_Z^3-2)+49*Roo
tOf(_Z^3-2)*x^2+210*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*x^2+66*(-x^3+1)^(2/3)+28*RootOf(_Z^3-
2)*x+120*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*x+49*RootOf(_Z^3-2)+210*RootOf(RootOf(_Z^3-2)^2+
6*_Z*RootOf(_Z^3-2)+36*_Z^2))/(x^2+4*x+1))*RootOf(_Z^3-2)-ln(-(108*(-x^3+1)^(2/3)*RootOf(RootOf(_Z^3-2)^2+6*_Z
*RootOf(_Z^3-2)+36*_Z^2)*RootOf(_Z^3-2)^2+126*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*RootOf(_Z^3
-2)^3*x+540*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)^2*RootOf(_Z^3-2)^2*x-33*(-x^3+1)^(1/3)*RootOf
(_Z^3-2)^2*x-108*(-x^3+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*RootOf(_Z^3-2)*x+33*(-x^3
+1)^(1/3)*RootOf(_Z^3-2)^2+108*(-x^3+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*RootOf(_Z^3
-2)+49*RootOf(_Z^3-2)*x^2+210*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*x^2+66*(-x^3+1)^(2/3)+28*Ro
otOf(_Z^3-2)*x+120*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*x+49*RootOf(_Z^3-2)+210*RootOf(RootOf(
_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2))/(x^2+4*x+1))*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (134) = 268\).

Time = 5.54 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.85 \[ \int \frac {1+x}{\left (1+4 x+x^2\right ) \sqrt [3]{1-x^3}} \, dx=\frac {1}{18} \cdot 4^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {4^{\frac {1}{6}} {\left (12 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (2 \, x^{4} + 7 \, x^{3} + 7 \, x + 2\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} \sqrt {3} {\left (91 \, x^{6} - 42 \, x^{5} + 105 \, x^{4} - 92 \, x^{3} + 105 \, x^{2} - 42 \, x + 91\right )} - 12 \, \sqrt {3} {\left (19 \, x^{5} - 29 \, x^{4} + 28 \, x^{3} - 28 \, x^{2} + 29 \, x - 19\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (53 \, x^{6} - 174 \, x^{5} + 111 \, x^{4} - 196 \, x^{3} + 111 \, x^{2} - 174 \, x + 53\right )}}\right ) - \frac {1}{72} \cdot 4^{\frac {2}{3}} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} {\left (2 \, x^{2} - x + 2\right )} + 4^{\frac {1}{3}} {\left (19 \, x^{4} - 10 \, x^{3} + 18 \, x^{2} - 10 \, x + 19\right )} - 6 \, {\left (5 \, x^{3} - 3 \, x^{2} + 3 \, x - 5\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{4} + 8 \, x^{3} + 18 \, x^{2} + 8 \, x + 1}\right ) + \frac {1}{36} \cdot 4^{\frac {2}{3}} \log \left (\frac {4^{\frac {2}{3}} {\left (x^{2} + 4 \, x + 1\right )} - 6 \cdot 4^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} - 12 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2} + 4 \, x + 1}\right ) \]

[In]

integrate((1+x)/(x^2+4*x+1)/(-x^3+1)^(1/3),x, algorithm="fricas")

[Out]

1/18*4^(1/6)*sqrt(3)*arctan(1/6*4^(1/6)*(12*4^(2/3)*sqrt(3)*(2*x^4 + 7*x^3 + 7*x + 2)*(-x^3 + 1)^(2/3) + 4^(1/
3)*sqrt(3)*(91*x^6 - 42*x^5 + 105*x^4 - 92*x^3 + 105*x^2 - 42*x + 91) - 12*sqrt(3)*(19*x^5 - 29*x^4 + 28*x^3 -
28*x^2 + 29*x - 19)*(-x^3 + 1)^(1/3))/(53*x^6 - 174*x^5 + 111*x^4 - 196*x^3 + 111*x^2 - 174*x + 53)) - 1/72*4
^(2/3)*log((6*4^(2/3)*(-x^3 + 1)^(2/3)*(2*x^2 - x + 2) + 4^(1/3)*(19*x^4 - 10*x^3 + 18*x^2 - 10*x + 19) - 6*(5
*x^3 - 3*x^2 + 3*x - 5)*(-x^3 + 1)^(1/3))/(x^4 + 8*x^3 + 18*x^2 + 8*x + 1)) + 1/36*4^(2/3)*log((4^(2/3)*(x^2 +
4*x + 1) - 6*4^(1/3)*(-x^3 + 1)^(1/3)*(x - 1) - 12*(-x^3 + 1)^(2/3))/(x^2 + 4*x + 1))

Sympy [F]

\[ \int \frac {1+x}{\left (1+4 x+x^2\right ) \sqrt [3]{1-x^3}} \, dx=\int \frac {x + 1}{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} + 4 x + 1\right )}\, dx \]

[In]

integrate((1+x)/(x**2+4*x+1)/(-x**3+1)**(1/3),x)

[Out]

Integral((x + 1)/((-(x - 1)*(x**2 + x + 1))**(1/3)*(x**2 + 4*x + 1)), x)

Maxima [F]

\[ \int \frac {1+x}{\left (1+4 x+x^2\right ) \sqrt [3]{1-x^3}} \, dx=\int { \frac {x + 1}{{\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{2} + 4 \, x + 1\right )}} \,d x } \]

[In]

integrate((1+x)/(x^2+4*x+1)/(-x^3+1)^(1/3),x, algorithm="maxima")

[Out]

integrate((x + 1)/((-x^3 + 1)^(1/3)*(x^2 + 4*x + 1)), x)

Giac [F]

\[ \int \frac {1+x}{\left (1+4 x+x^2\right ) \sqrt [3]{1-x^3}} \, dx=\int { \frac {x + 1}{{\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{2} + 4 \, x + 1\right )}} \,d x } \]

[In]

integrate((1+x)/(x^2+4*x+1)/(-x^3+1)^(1/3),x, algorithm="giac")

[Out]

integrate((x + 1)/((-x^3 + 1)^(1/3)*(x^2 + 4*x + 1)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1+x}{\left (1+4 x+x^2\right ) \sqrt [3]{1-x^3}} \, dx=\int \frac {x+1}{{\left (1-x^3\right )}^{1/3}\,\left (x^2+4\,x+1\right )} \,d x \]

[In]

int((x + 1)/((1 - x^3)^(1/3)*(4*x + x^2 + 1)),x)

[Out]

int((x + 1)/((1 - x^3)^(1/3)*(4*x + x^2 + 1)), x)