\(\int \frac {(-1+x) (1+3 x)}{(-1+3 x) (-x+x^3)^{2/3}} \, dx\) [11]

Optimal result

Integrand size = 27, antiderivative size = 125 \[ \int \frac {(-1+x) (1+3 x)}{(-1+3 x) \left (-x+x^3\right )^{2/3}} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} \left (124616800+1235276981 x+1609127381 x^2\right )[phone] \sqrt {3} (-1+x) \sqrt [3]{-x+x^3}[phone] \sqrt {3} \left (-x+x^3\right )^{2/3}}{-39304000+3108349623 x+2990437623 x^2}\right )-\frac {1}{2} \log \left (\frac {-1+3 x+3 (-1+x) \sqrt [3]{-x+x^3}-3 \left (-x+x^3\right )^{2/3}}{-1+3 x}\right ) \]

[Out]

-3^(1/2)*arctan((612314840*3^(1/2)*(x^3-x)^(1/3)*(-1+x)+3^(1/2)*(1609127381*x^2+1235276981*x+124616800)+260593
9922*3^(1/2)*(x^3-x)^(2/3))/(2990437623*x^2+3108349623*x-39304000))-1/2*ln((3*(-1+x)*(x^3-x)^(1/3)+3*x-3*(x^3-
x)^(2/3)-1)/(-1+3*x))

Rubi [C] (warning: unable to verify)

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

Time = 1.01 (sec) , antiderivative size = 537, normalized size of antiderivative = 4.30, number of steps used = 14, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {2081, 6865, 6857, 247, 231, 281, 337, 1452, 441, 440, 476, 503} \[ \int \frac {(-1+x) (1+3 x)}{(-1+3 x) \left (-x+x^3\right )^{2/3}} \, dx=\frac {4 x \left (1-x^2\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{6},\frac {2}{3},1,\frac {7}{6},x^2,9 x^2\right )}{\left (x^3-x\right )^{2/3}}-\frac {x \left (1-x^2\right ) \left (1-\frac {x^{2/3}}{\sqrt [3]{x^2-1}}\right ) \sqrt {\frac {\frac {x^{4/3}}{\left (x^2-1\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\left (1-\frac {\left (1+\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{x^2-1}}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {1-\frac {\left (1-\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{x^2-1}}}{1-\frac {\left (1+\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{x^2-1}}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2 \sqrt [4]{3} \left (x^3-x\right )^{2/3} \sqrt {-\frac {x^{2/3} \left (1-\frac {x^{2/3}}{\sqrt [3]{x^2-1}}\right )}{\sqrt [3]{x^2-1} \left (1-\frac {\left (1+\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{x^2-1}}\right )^2}}}-\frac {\sqrt {3} x^{2/3} \left (x^2-1\right )^{2/3} \arctan \left (\frac {1-\frac {4 x^{2/3}}{\sqrt [3]{x^2-1}}}{\sqrt {3}}\right )}{2 \left (x^3-x\right )^{2/3}}-\frac {\sqrt {3} x^{2/3} \left (x^2-1\right )^{2/3} \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{2 \left (x^3-x\right )^{2/3}}+\frac {x^{2/3} \left (x^2-1\right )^{2/3} \log \left (1-9 x^2\right )}{4 \left (x^3-x\right )^{2/3}}-\frac {3 x^{2/3} \left (x^2-1\right )^{2/3} \log \left (x^{2/3}-\sqrt [3]{x^2-1}\right )}{4 \left (x^3-x\right )^{2/3}}-\frac {3 x^{2/3} \left (x^2-1\right )^{2/3} \log \left (2 x^{2/3}+\sqrt [3]{x^2-1}\right )}{4 \left (x^3-x\right )^{2/3}} \]

[In]

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

[Out]

(4*x*(1 - x^2)^(2/3)*AppellF1[1/6, 2/3, 1, 7/6, x^2, 9*x^2])/(-x + x^3)^(2/3) - (Sqrt[3]*x^(2/3)*(-1 + x^2)^(2
/3)*ArcTan[(1 - (4*x^(2/3))/(-1 + x^2)^(1/3))/Sqrt[3]])/(2*(-x + x^3)^(2/3)) - (Sqrt[3]*x^(2/3)*(-1 + x^2)^(2/
3)*ArcTan[(1 + (2*x^(2/3))/(-1 + x^2)^(1/3))/Sqrt[3]])/(2*(-x + x^3)^(2/3)) - (x*(1 - x^2)*(1 - x^(2/3)/(-1 +
x^2)^(1/3))*Sqrt[(1 + x^(4/3)/(-1 + x^2)^(2/3) + x^(2/3)/(-1 + x^2)^(1/3))/(1 - ((1 + Sqrt[3])*x^(2/3))/(-1 +
x^2)^(1/3))^2]*EllipticF[ArcCos[(1 - ((1 - Sqrt[3])*x^(2/3))/(-1 + x^2)^(1/3))/(1 - ((1 + Sqrt[3])*x^(2/3))/(-
1 + x^2)^(1/3))], (2 + Sqrt[3])/4])/(2*3^(1/4)*(-x + x^3)^(2/3)*Sqrt[-((x^(2/3)*(1 - x^(2/3)/(-1 + x^2)^(1/3))
)/((-1 + x^2)^(1/3)*(1 - ((1 + Sqrt[3])*x^(2/3))/(-1 + x^2)^(1/3))^2))]) + (x^(2/3)*(-1 + x^2)^(2/3)*Log[1 - 9
*x^2])/(4*(-x + x^3)^(2/3)) - (3*x^(2/3)*(-1 + x^2)^(2/3)*Log[x^(2/3) - (-1 + x^2)^(1/3)])/(4*(-x + x^3)^(2/3)
) - (3*x^(2/3)*(-1 + x^2)^(2/3)*Log[2*x^(2/3) + (-1 + x^2)^(1/3)])/(4*(-x + x^3)^(2/3))

Rule 231

Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[x*(s +
r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[r*x^2*(
(s + r*x^2)/(s + (1 + Sqrt[3])*r*x^2)^2)]))*EllipticF[ArcCos[(s + (1 - Sqrt[3])*r*x^2)/(s + (1 + Sqrt[3])*r*x^
2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b}, x]

Rule 247

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a/(a + b*x^n))^(p + 1/n)*(a + b*x^n)^(p + 1/n), Subst[In
t[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p,
0] && NeQ[p, -2^(-1)] && LtQ[Denominator[p + 1/n], Denominator[p]]

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 337

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

Rule 440

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

Rule 441

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^F
racPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 476

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

Rule 503

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

Rule 1452

Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + c*x^(2
*n))^p, (d/(d^2 - e^2*x^(2*n)) - e*(x^n/(d^2 - e^2*x^(2*n))))^(-q), x], x] /; FreeQ[{a, c, d, e, n, p}, x] &&
EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[q, 0]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 6865

Int[(u_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k, Subst[Int[x^(k*(m + 1) - 1)*(u /. x -> x^k
), x], x, x^(1/k)], x]] /; FractionQ[m]

Rubi steps \begin{align*} \text {integral}= \frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \int \frac {(-1+x) (1+3 x)}{x^{2/3} (-1+3 x) \left (-1+x^2\right )^{2/3}} \, dx}{\left (-x+x^3\right )^{2/3}} \\ = \frac {\left (3 x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {\left (-1+x^3\right ) \left (1+3 x^3\right )}{\left (-1+3 x^3\right ) \left (-1+x^6\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\left (-x+x^3\right )^{2/3}} \\ = \frac {\left (3 x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \left (-\frac {1}{3 \left (-1+x^6\right )^{2/3}}+\frac {x^3}{\left (-1+x^6\right )^{2/3}}-\frac {4}{3 \left (-1+3 x^3\right ) \left (-1+x^6\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (-x+x^3\right )^{2/3}} \\ = -\frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+x^6\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\left (-x+x^3\right )^{2/3}}+\frac {\left (3 x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-1+x^6\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\left (-x+x^3\right )^{2/3}}-\frac {\left (4 x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+3 x^3\right ) \left (-1+x^6\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\left (-x+x^3\right )^{2/3}} \\ = -\frac {\left (x^{2/3} \sqrt [6]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^6}} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [6]{-1+x^2}}\right )}{\sqrt {-\frac {1}{-1+x^2}} \left (-x+x^3\right )^{2/3}}+\frac {\left (3 x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx,x,x^{2/3}\right )}{2 \left (-x+x^3\right )^{2/3}}-\frac {\left (4 x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \left (\frac {1}{\left (-1+x^6\right )^{2/3} \left (-1+9 x^6\right )}+\frac {3 x^3}{\left (-1+x^6\right )^{2/3} \left (-1+9 x^6\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (-x+x^3\right )^{2/3}} \\ = -\frac {\sqrt {3} x^{2/3} \left (-1+x^2\right )^{2/3} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{2 \left (-x+x^3\right )^{2/3}}-\frac {x \left (1-x^2\right ) \left (1-\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right ) \sqrt {\frac {1+\frac {x^{4/3}}{\left (-1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}}{\left (1-\frac {\left (1+\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{-1+x^2}}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {1-\frac {\left (1-\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{-1+x^2}}}{1-\frac {\left (1+\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{-1+x^2}}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2 \sqrt [4]{3} \left (-x+x^3\right )^{2/3} \sqrt {-\frac {x^{2/3} \left (1-\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{\sqrt [3]{-1+x^2} \left (1-\frac {\left (1+\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{-1+x^2}}\right )^2}}}-\frac {3 x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (x^{2/3}-\sqrt [3]{-1+x^2}\right )}{4 \left (-x+x^3\right )^{2/3}}-\frac {\left (4 x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+x^6\right )^{2/3} \left (-1+9 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (-x+x^3\right )^{2/3}}-\frac {\left (12 x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-1+x^6\right )^{2/3} \left (-1+9 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (-x+x^3\right )^{2/3}} \\ = -\frac {\sqrt {3} x^{2/3} \left (-1+x^2\right )^{2/3} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{2 \left (-x+x^3\right )^{2/3}}-\frac {x \left (1-x^2\right ) \left (1-\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right ) \sqrt {\frac {1+\frac {x^{4/3}}{\left (-1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}}{\left (1-\frac {\left (1+\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{-1+x^2}}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {1-\frac {\left (1-\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{-1+x^2}}}{1-\frac {\left (1+\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{-1+x^2}}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2 \sqrt [4]{3} \left (-x+x^3\right )^{2/3} \sqrt {-\frac {x^{2/3} \left (1-\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{\sqrt [3]{-1+x^2} \left (1-\frac {\left (1+\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{-1+x^2}}\right )^2}}}-\frac {3 x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (x^{2/3}-\sqrt [3]{-1+x^2}\right )}{4 \left (-x+x^3\right )^{2/3}}-\frac {\left (4 x^{2/3} \left (1-x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^6\right )^{2/3} \left (-1+9 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (-x+x^3\right )^{2/3}}-\frac {\left (6 x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x}{\left (-1+x^3\right )^{2/3} \left (-1+9 x^3\right )} \, dx,x,x^{2/3}\right )}{\left (-x+x^3\right )^{2/3}} \\ = \frac {4 x \left (1-x^2\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{6},\frac {2}{3},1,\frac {7}{6},x^2,9 x^2\right )}{\left (-x+x^3\right )^{2/3}}-\frac {\sqrt {3} x^{2/3} \left (-1+x^2\right )^{2/3} \arctan \left (\frac {1-\frac {4 x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{2 \left (-x+x^3\right )^{2/3}}-\frac {\sqrt {3} x^{2/3} \left (-1+x^2\right )^{2/3} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{2 \left (-x+x^3\right )^{2/3}}-\frac {x \left (1-x^2\right ) \left (1-\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right ) \sqrt {\frac {1+\frac {x^{4/3}}{\left (-1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}}{\left (1-\frac {\left (1+\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{-1+x^2}}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {1-\frac {\left (1-\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{-1+x^2}}}{1-\frac {\left (1+\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{-1+x^2}}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2 \sqrt [4]{3} \left (-x+x^3\right )^{2/3} \sqrt {-\frac {x^{2/3} \left (1-\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{\sqrt [3]{-1+x^2} \left (1-\frac {\left (1+\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{-1+x^2}}\right )^2}}}+\frac {x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (1-9 x^2\right )}{4 \left (-x+x^3\right )^{2/3}}-\frac {3 x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (x^{2/3}-\sqrt [3]{-1+x^2}\right )}{4 \left (-x+x^3\right )^{2/3}}-\frac {3 x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (2 x^{2/3}+\sqrt [3]{-1+x^2}\right )}{4 \left (-x+x^3\right )^{2/3}} \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [C] (verified)

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

Time = 0.60 (sec) , antiderivative size = 606, normalized size of antiderivative = 4.85

method	result	size
trager	\(\ln \left (-\frac {1415 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}+3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}} x -3679 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +1819 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-2127 \left (x^{3}-x \right )^{\frac {2}{3}}-3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}}-2127 \left (x^{3}-x \right )^{\frac {1}{3}} x +1698 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+2572 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -712 x^{2}+2127 \left (x^{3}-x \right )^{\frac {1}{3}}-251 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-445 x -89}{-1+3 x}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {-1415 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}+3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}} x +3679 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +4649 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-1107 \left (x^{3}-x \right )^{\frac {2}{3}}-3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}}-1107 \left (x^{3}-x \right )^{\frac {1}{3}} x -1698 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-4786 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -2522 x^{2}+1107 \left (x^{3}-x \right )^{\frac {1}{3}}+3145 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1552 x -1358}{-1+3 x}\right )+\ln \left (\frac {-1415 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}+3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}} x +3679 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +4649 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-1107 \left (x^{3}-x \right )^{\frac {2}{3}}-3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}}-1107 \left (x^{3}-x \right )^{\frac {1}{3}} x -1698 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-4786 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -2522 x^{2}+1107 \left (x^{3}-x \right )^{\frac {1}{3}}+3145 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1552 x -1358}{-1+3 x}\right )\)	\(606\)

[In]

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

[Out]

ln(-(1415*RootOf(_Z^2-_Z+1)^2*x^2+3234*RootOf(_Z^2-_Z+1)*(x^3-x)^(2/3)+3234*RootOf(_Z^2-_Z+1)*(x^3-x)^(1/3)*x-
3679*RootOf(_Z^2-_Z+1)^2*x+1819*RootOf(_Z^2-_Z+1)*x^2-2127*(x^3-x)^(2/3)-3234*RootOf(_Z^2-_Z+1)*(x^3-x)^(1/3)-
2127*(x^3-x)^(1/3)*x+1698*RootOf(_Z^2-_Z+1)^2+2572*RootOf(_Z^2-_Z+1)*x-712*x^2+2127*(x^3-x)^(1/3)-251*RootOf(_
Z^2-_Z+1)-445*x-89)/(-1+3*x))*RootOf(_Z^2-_Z+1)-RootOf(_Z^2-_Z+1)*ln((-1415*RootOf(_Z^2-_Z+1)^2*x^2+3234*RootO
f(_Z^2-_Z+1)*(x^3-x)^(2/3)+3234*RootOf(_Z^2-_Z+1)*(x^3-x)^(1/3)*x+3679*RootOf(_Z^2-_Z+1)^2*x+4649*RootOf(_Z^2-
_Z+1)*x^2-1107*(x^3-x)^(2/3)-3234*RootOf(_Z^2-_Z+1)*(x^3-x)^(1/3)-1107*(x^3-x)^(1/3)*x-1698*RootOf(_Z^2-_Z+1)^
2-4786*RootOf(_Z^2-_Z+1)*x-2522*x^2+1107*(x^3-x)^(1/3)+3145*RootOf(_Z^2-_Z+1)+1552*x-1358)/(-1+3*x))+ln((-1415
*RootOf(_Z^2-_Z+1)^2*x^2+3234*RootOf(_Z^2-_Z+1)*(x^3-x)^(2/3)+3234*RootOf(_Z^2-_Z+1)*(x^3-x)^(1/3)*x+3679*Root
Of(_Z^2-_Z+1)^2*x+4649*RootOf(_Z^2-_Z+1)*x^2-1107*(x^3-x)^(2/3)-3234*RootOf(_Z^2-_Z+1)*(x^3-x)^(1/3)-1107*(x^3
-x)^(1/3)*x-1698*RootOf(_Z^2-_Z+1)^2-4786*RootOf(_Z^2-_Z+1)*x-2522*x^2+1107*(x^3-x)^(1/3)+3145*RootOf(_Z^2-_Z+
1)+1552*x-1358)/(-1+3*x))

Fricas [A] (verification not implemented)

none

Time = 0.50 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.86 \[ \int \frac {(-1+x) (1+3 x)}{(-1+3 x) \left (-x+x^3\right )^{2/3}} \, dx=-\sqrt {3} \arctan \left (\frac {612314840 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (x - 1\right )} + \sqrt {3} {\left (1609127381 \, x^{2} + 1235276981 \, x + 124616800\right )} + 2605939922 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {2}{3}}}{2990437623 \, x^{2} + 3108349623 \, x - 39304000}\right ) - \frac {1}{2} \, \log \left (\frac {3 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (x - 1\right )} + 3 \, x - 3 \, {\left (x^{3} - x\right )}^{\frac {2}{3}} - 1}{3 \, x - 1}\right ) \]

[In]

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

[Out]

-sqrt(3)*arctan((612314840*sqrt(3)*(x^3 - x)^(1/3)*(x - 1) + sqrt(3)*(1609127381*x^2 + 1235276981*x + 12461680
0) + 2605939922*sqrt(3)*(x^3 - x)^(2/3))/(2990437623*x^2 + 3108349623*x - 39304000)) - 1/2*log((3*(x^3 - x)^(1
/3)*(x - 1) + 3*x - 3*(x^3 - x)^(2/3) - 1)/(3*x - 1))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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