3.81.33 \(\int \frac {-24+63 x-30 x^2-12 x^3+6 x^4+(-6+15 x-8 x^2-2 x^3+x^4) \log (\frac {-6+9 x+x^2-x^3}{-x+x^2})}{-6+15 x-8 x^2-2 x^3+x^4} \, dx\)

Optimal. Leaf size=23 \[ -16+x \left (5+\log \left (\frac {3 \left (3+\frac {1}{-1+x}\right )}{x}-x\right )\right ) \]

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Rubi [C]  time = 32.10, antiderivative size = 4489, normalized size of antiderivative = 195.17, number of steps used = 69, number of rules used = 12, integrand size = 84, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6742, 2100, 2081, 2079, 800, 634, 618, 206, 628, 2058, 2074, 2523}

result too large to display

Warning: Unable to verify antiderivative.

[In]

Int[(-24 + 63*x - 30*x^2 - 12*x^3 + 6*x^4 + (-6 + 15*x - 8*x^2 - 2*x^3 + x^4)*Log[(-6 + 9*x + x^2 - x^3)/(-x +
x^2)])/(-6 + 15*x - 8*x^2 - 2*x^3 + x^4),x]

[Out]

5*x - (3*2^(2/3)*(79*I + 9*Sqrt[1007])*(28*2^(2/3) - (79 - (9*I)*Sqrt[1007])^(2/3))*ArcTan[(2^(2/3)*(79 - (9*I
)*Sqrt[1007]) + 56*(2*(79 - (9*I)*Sqrt[1007]))^(1/3) + 4*(79 - (9*I)*Sqrt[1007])^(2/3)*(1 - 3*x))/(2*Sqrt[3*((
56*I)*(79 - (9*I)*Sqrt[1007])^(1/3)*(79*I + 9*Sqrt[1007]) - 2^(1/3)*(37663 + (711*I)*Sqrt[1007] - 784*2^(1/3)*
(79 - (9*I)*Sqrt[1007])^(2/3)))])])/((1568*2^(2/3) + 56*(79 - (9*I)*Sqrt[1007])^(2/3) + 2^(1/3)*(79 - (9*I)*Sq
rt[1007])^(4/3))*Sqrt[((56*I)*(79 - (9*I)*Sqrt[1007])^(1/3)*(79*I + 9*Sqrt[1007]) - 2^(1/3)*(37663 + (711*I)*S
qrt[1007] - 784*2^(1/3)*(79 - (9*I)*Sqrt[1007])^(2/3)))/3021]) - (63*((2*(79 - (9*I)*Sqrt[1007]))^(2/3)*(129 +
(25*I)*Sqrt[1007]) - 4*2^(1/3)*(7257 - (271*I)*Sqrt[1007]))*ArcTan[(2^(2/3)*(79 - (9*I)*Sqrt[1007]) + 56*(2*(
79 - (9*I)*Sqrt[1007]))^(1/3) + 4*(79 - (9*I)*Sqrt[1007])^(2/3)*(1 - 3*x))/(2*Sqrt[3*((56*I)*(79 - (9*I)*Sqrt[
1007])^(1/3)*(79*I + 9*Sqrt[1007]) - 2^(1/3)*(37663 + (711*I)*Sqrt[1007] - 784*2^(1/3)*(79 - (9*I)*Sqrt[1007])
^(2/3)))])])/((1568*2^(2/3) + 56*(79 - (9*I)*Sqrt[1007])^(2/3) + 2^(1/3)*(79 - (9*I)*Sqrt[1007])^(4/3))*Sqrt[(
(56*I)*(79 - (9*I)*Sqrt[1007])^(1/3)*(79*I + 9*Sqrt[1007]) - 2^(1/3)*(37663 + (711*I)*Sqrt[1007] - 784*2^(1/3)
*(79 - (9*I)*Sqrt[1007])^(2/3)))/3]) - (96*((2*(79 - (9*I)*Sqrt[1007]))^(2/3)*(27 - (13*I)*Sqrt[1007]) + 2^(1/
3)*(10575 - (649*I)*Sqrt[1007]))*ArcTan[(2^(2/3)*(79 - (9*I)*Sqrt[1007]) + 56*(2*(79 - (9*I)*Sqrt[1007]))^(1/3
) + 4*(79 - (9*I)*Sqrt[1007])^(2/3)*(1 - 3*x))/(2*Sqrt[3*((56*I)*(79 - (9*I)*Sqrt[1007])^(1/3)*(79*I + 9*Sqrt[
1007]) - 2^(1/3)*(37663 + (711*I)*Sqrt[1007] - 784*2^(1/3)*(79 - (9*I)*Sqrt[1007])^(2/3)))])])/((1568*2^(2/3)
+ 56*(79 - (9*I)*Sqrt[1007])^(2/3) + 2^(1/3)*(79 - (9*I)*Sqrt[1007])^(4/3))*Sqrt[((56*I)*(79 - (9*I)*Sqrt[1007
])^(1/3)*(79*I + 9*Sqrt[1007]) - 2^(1/3)*(37663 + (711*I)*Sqrt[1007] - 784*2^(1/3)*(79 - (9*I)*Sqrt[1007])^(2/
3)))/3]) - (6*(7*(2*(79 - (9*I)*Sqrt[1007]))^(2/3)*(2685 - (11*I)*Sqrt[1007]) - 8*2^(1/3)*(104073 + (2593*I)*S
qrt[1007]))*ArcTan[(2^(2/3)*(79 - (9*I)*Sqrt[1007]) + 56*(2*(79 - (9*I)*Sqrt[1007]))^(1/3) + 4*(79 - (9*I)*Sqr
t[1007])^(2/3)*(1 - 3*x))/(2*Sqrt[3*((56*I)*(79 - (9*I)*Sqrt[1007])^(1/3)*(79*I + 9*Sqrt[1007]) - 2^(1/3)*(376
63 + (711*I)*Sqrt[1007] - 784*2^(1/3)*(79 - (9*I)*Sqrt[1007])^(2/3)))])])/((1568*2^(2/3) + 56*(79 - (9*I)*Sqrt
[1007])^(2/3) + 2^(1/3)*(79 - (9*I)*Sqrt[1007])^(4/3))*Sqrt[((56*I)*(79 - (9*I)*Sqrt[1007])^(1/3)*(79*I + 9*Sq
rt[1007]) - 2^(1/3)*(37663 + (711*I)*Sqrt[1007] - 784*2^(1/3)*(79 - (9*I)*Sqrt[1007])^(2/3)))/3]) + (12*((2*(7
9 - (9*I)*Sqrt[1007]))^(2/3)*(3585 - (71*I)*Sqrt[1007]) - 2*2^(1/3)*(53691 + (3331*I)*Sqrt[1007]))*ArcTan[(2^(
2/3)*(79 - (9*I)*Sqrt[1007]) + 56*(2*(79 - (9*I)*Sqrt[1007]))^(1/3) + 4*(79 - (9*I)*Sqrt[1007])^(2/3)*(1 - 3*x
))/(2*Sqrt[3*((56*I)*(79 - (9*I)*Sqrt[1007])^(1/3)*(79*I + 9*Sqrt[1007]) - 2^(1/3)*(37663 + (711*I)*Sqrt[1007]
- 784*2^(1/3)*(79 - (9*I)*Sqrt[1007])^(2/3)))])])/((1568*2^(2/3) + 56*(79 - (9*I)*Sqrt[1007])^(2/3) + 2^(1/3)
*(79 - (9*I)*Sqrt[1007])^(4/3))*Sqrt[((56*I)*(79 - (9*I)*Sqrt[1007])^(1/3)*(79*I + 9*Sqrt[1007]) - 2^(1/3)*(37
663 + (711*I)*Sqrt[1007] - 784*2^(1/3)*(79 - (9*I)*Sqrt[1007])^(2/3)))/3]) - (60*2^(1/3)*Sqrt[3]*(49965 + (421
*I)*Sqrt[1007] - 8*2^(1/3)*(111 + I*Sqrt[1007])*(79 - (9*I)*Sqrt[1007])^(2/3))*ArcTan[(2^(2/3)*(79 - (9*I)*Sqr
t[1007]) + 56*(2*(79 - (9*I)*Sqrt[1007]))^(1/3) + 4*(79 - (9*I)*Sqrt[1007])^(2/3)*(1 - 3*x))/(2*Sqrt[3*((56*I)
*(79 - (9*I)*Sqrt[1007])^(1/3)*(79*I + 9*Sqrt[1007]) - 2^(1/3)*(37663 + (711*I)*Sqrt[1007] - 784*2^(1/3)*(79 -
(9*I)*Sqrt[1007])^(2/3)))])])/((1568*2^(2/3) + 56*(79 - (9*I)*Sqrt[1007])^(2/3) + 2^(1/3)*(79 - (9*I)*Sqrt[10
07])^(4/3))*Sqrt[(56*I)*(79 - (9*I)*Sqrt[1007])^(1/3)*(79*I + 9*Sqrt[1007]) - 2^(1/3)*(37663 + (711*I)*Sqrt[10
07] - 784*2^(1/3)*(79 - (9*I)*Sqrt[1007])^(2/3))]) - (14*(79 - (9*I)*Sqrt[1007])^(1/3)*(56*2^(1/3) + 25*(79 -
(9*I)*Sqrt[1007])^(1/3) + (2*(79 - (9*I)*Sqrt[1007]))^(2/3))*Log[56*2^(1/3) + (2*(79 - (9*I)*Sqrt[1007]))^(2/3
) - 2*(79 - (9*I)*Sqrt[1007])^(1/3)*(1 - 3*x)])/(1568*2^(2/3) + 56*(79 - (9*I)*Sqrt[1007])^(2/3) + 2^(1/3)*(79
- (9*I)*Sqrt[1007])^(4/3)) + (16*(79 - (9*I)*Sqrt[1007])^(1/3)*(56*2^(1/3) + 52*(79 - (9*I)*Sqrt[1007])^(1/3)
+ (2*(79 - (9*I)*Sqrt[1007]))^(2/3))*Log[56*2^(1/3) + (2*(79 - (9*I)*Sqrt[1007]))^(2/3) - 2*(79 - (9*I)*Sqrt[
1007])^(1/3)*(1 - 3*x)])/(3*(1568*2^(2/3) + 56*(79 - (9*I)*Sqrt[1007])^(2/3) + 2^(1/3)*(79 - (9*I)*Sqrt[1007])
^(4/3))) + (10*(79 - (9*I)*Sqrt[1007])^(1/3)*(616*2^(1/3) + 32*(79 - (9*I)*Sqrt[1007])^(1/3) + 11*(2*(79 - (9*
I)*Sqrt[1007]))^(2/3))*Log[56*2^(1/3) + (2*(79 - (9*I)*Sqrt[1007]))^(2/3) - 2*(79 - (9*I)*Sqrt[1007])^(1/3)*(1
- 3*x)])/(3*(1568*2^(2/3) + 56*(79 - (9*I)*Sqrt[1007])^(2/3) + 2^(1/3)*(79 - (9*I)*Sqrt[1007])^(4/3))) + (2*(
79 - (9*I)*Sqrt[1007])^(1/3)*(1568*2^(1/3) + 79*(79 - (9*I)*Sqrt[1007])^(1/3) + 28*(2*(79 - (9*I)*Sqrt[1007]))
^(2/3))*Log[56*2^(1/3) + (2*(79 - (9*I)*Sqrt[1007]))^(2/3) - 2*(79 - (9*I)*Sqrt[1007])^(1/3)*(1 - 3*x)])/(3*(1
568*2^(2/3) + 56*(79 - (9*I)*Sqrt[1007])^(2/3) + 2^(1/3)*(79 - (9*I)*Sqrt[1007])^(4/3))) + (4*(79 - (9*I)*Sqrt
[1007])^(1/3)*(77*(79 - (9*I)*Sqrt[1007])^(1/3) - 52*(56*2^(1/3) + (2*(79 - (9*I)*Sqrt[1007]))^(2/3)))*Log[56*
2^(1/3) + (2*(79 - (9*I)*Sqrt[1007]))^(2/3) - 2*(79 - (9*I)*Sqrt[1007])^(1/3)*(1 - 3*x)])/(3*(1568*2^(2/3) + 5
6*(79 - (9*I)*Sqrt[1007])^(2/3) + 2^(1/3)*(79 - (9*I)*Sqrt[1007])^(4/3))) - (4*(79 - (9*I)*Sqrt[1007])^(1/3)*(
142*(79 - (9*I)*Sqrt[1007])^(1/3) - 17*(56*2^(1/3) + (2*(79 - (9*I)*Sqrt[1007]))^(2/3)))*Log[56*2^(1/3) + (2*(
79 - (9*I)*Sqrt[1007]))^(2/3) - 2*(79 - (9*I)*Sqrt[1007])^(1/3)*(1 - 3*x)])/(3*(1568*2^(2/3) + 56*(79 - (9*I)*
Sqrt[1007])^(2/3) + 2^(1/3)*(79 - (9*I)*Sqrt[1007])^(4/3))) + (2*(2*(79 - (9*I)*Sqrt[1007]))^(2/3)*(71*2^(1/3)
- (476*2^(2/3))/(79 - (9*I)*Sqrt[1007])^(1/3) - 17*(79 - (9*I)*Sqrt[1007])^(1/3))*Log[-3*((6*I)*(79 - (9*I)*S
qrt[1007])^(2/3) - (2*(79 - (9*I)*Sqrt[1007]))^(1/3)*(15*I + Sqrt[1007]) - 2^(2/3)*(183*I + Sqrt[1007])) - (79
*I)*2^(2/3)*x - 9*2^(2/3)*Sqrt[1007]*x - (4*I)*(79 - (9*I)*Sqrt[1007])^(2/3)*x - (56*I)*(2*(79 - (9*I)*Sqrt[10
07]))^(1/3)*x + (6*I)*(79 - (9*I)*Sqrt[1007])^(2/3)*x^2])/(3*(1568*2^(2/3) + 56*(79 - (9*I)*Sqrt[1007])^(2/3)
+ 2^(1/3)*(79 - (9*I)*Sqrt[1007])^(4/3))) - (5*2^(2/3)*(79 - (9*I)*Sqrt[1007])^(1/3)*(308*2^(2/3) + 11*(79 - (
9*I)*Sqrt[1007])^(2/3) + 16*(2*(79 - (9*I)*Sqrt[1007]))^(1/3))*Log[-3*((6*I)*(79 - (9*I)*Sqrt[1007])^(2/3) - (
2*(79 - (9*I)*Sqrt[1007]))^(1/3)*(15*I + Sqrt[1007]) - 2^(2/3)*(183*I + Sqrt[1007])) - (79*I)*2^(2/3)*x - 9*2^
(2/3)*Sqrt[1007]*x - (4*I)*(79 - (9*I)*Sqrt[1007])^(2/3)*x - (56*I)*(2*(79 - (9*I)*Sqrt[1007]))^(1/3)*x + (6*I
)*(79 - (9*I)*Sqrt[1007])^(2/3)*x^2])/(3*(1568*2^(2/3) + 56*(79 - (9*I)*Sqrt[1007])^(2/3) + 2^(1/3)*(79 - (9*I
)*Sqrt[1007])^(4/3))) - (8*2^(2/3)*(79 - (9*I)*Sqrt[1007])^(1/3)*(28*2^(2/3) + (79 - (9*I)*Sqrt[1007])^(2/3) +
26*(2*(79 - (9*I)*Sqrt[1007]))^(1/3))*Log[-3*((6*I)*(79 - (9*I)*Sqrt[1007])^(2/3) - (2*(79 - (9*I)*Sqrt[1007]
))^(1/3)*(15*I + Sqrt[1007]) - 2^(2/3)*(183*I + Sqrt[1007])) - (79*I)*2^(2/3)*x - 9*2^(2/3)*Sqrt[1007]*x - (4*
I)*(79 - (9*I)*Sqrt[1007])^(2/3)*x - (56*I)*(2*(79 - (9*I)*Sqrt[1007]))^(1/3)*x + (6*I)*(79 - (9*I)*Sqrt[1007]
)^(2/3)*x^2])/(3*(1568*2^(2/3) + 56*(79 - (9*I)*Sqrt[1007])^(2/3) + 2^(1/3)*(79 - (9*I)*Sqrt[1007])^(4/3))) +
(7*(79 - (9*I)*Sqrt[1007])^(1/3)*(56*2^(1/3) + 25*(79 - (9*I)*Sqrt[1007])^(1/3) + (2*(79 - (9*I)*Sqrt[1007]))^
(2/3))*Log[-3*((6*I)*(79 - (9*I)*Sqrt[1007])^(2/3) - (2*(79 - (9*I)*Sqrt[1007]))^(1/3)*(15*I + Sqrt[1007]) - 2
^(2/3)*(183*I + Sqrt[1007])) - (79*I)*2^(2/3)*x - 9*2^(2/3)*Sqrt[1007]*x - (4*I)*(79 - (9*I)*Sqrt[1007])^(2/3)
*x - (56*I)*(2*(79 - (9*I)*Sqrt[1007]))^(1/3)*x + (6*I)*(79 - (9*I)*Sqrt[1007])^(2/3)*x^2])/(1568*2^(2/3) + 56
*(79 - (9*I)*Sqrt[1007])^(2/3) + 2^(1/3)*(79 - (9*I)*Sqrt[1007])^(4/3)) - ((79 - (9*I)*Sqrt[1007])^(1/3)*(1568
*2^(1/3) + 79*(79 - (9*I)*Sqrt[1007])^(1/3) + 28*(2*(79 - (9*I)*Sqrt[1007]))^(2/3))*Log[-3*((6*I)*(79 - (9*I)*
Sqrt[1007])^(2/3) - (2*(79 - (9*I)*Sqrt[1007]))^(1/3)*(15*I + Sqrt[1007]) - 2^(2/3)*(183*I + Sqrt[1007])) - (7
9*I)*2^(2/3)*x - 9*2^(2/3)*Sqrt[1007]*x - (4*I)*(79 - (9*I)*Sqrt[1007])^(2/3)*x - (56*I)*(2*(79 - (9*I)*Sqrt[1
007]))^(1/3)*x + (6*I)*(79 - (9*I)*Sqrt[1007])^(2/3)*x^2])/(3*(1568*2^(2/3) + 56*(79 - (9*I)*Sqrt[1007])^(2/3)
+ 2^(1/3)*(79 - (9*I)*Sqrt[1007])^(4/3))) + (2*(79 - (9*I)*Sqrt[1007])^(1/3)*(2912*2^(1/3) - 77*(79 - (9*I)*S
qrt[1007])^(1/3) + 52*(2*(79 - (9*I)*Sqrt[1007]))^(2/3))*Log[-3*((6*I)*(79 - (9*I)*Sqrt[1007])^(2/3) - (2*(79
- (9*I)*Sqrt[1007]))^(1/3)*(15*I + Sqrt[1007]) - 2^(2/3)*(183*I + Sqrt[1007])) - (79*I)*2^(2/3)*x - 9*2^(2/3)*
Sqrt[1007]*x - (4*I)*(79 - (9*I)*Sqrt[1007])^(2/3)*x - (56*I)*(2*(79 - (9*I)*Sqrt[1007]))^(1/3)*x + (6*I)*(79
- (9*I)*Sqrt[1007])^(2/3)*x^2])/(3*(1568*2^(2/3) + 56*(79 - (9*I)*Sqrt[1007])^(2/3) + 2^(1/3)*(79 - (9*I)*Sqrt
[1007])^(4/3))) + x*Log[(6 - 9*x - x^2 + x^3)/((1 - x)*x)]

Rule 206

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

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 2058

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P]}, Int[ExpandIntegrand[u^p, x], x] /;  !SumQ[NonfreeFactors[u,
x]]] /; PolyQ[P, x] && ILtQ[p, 0]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rule 2079

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> With[{r = Rt[-9*a*d^2 + S
qrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]}, Dist[1/d^(2*p), Int[(e + f*x)^m*Simp[(18^(1/3)*b*d)/(3*r) - r/18^(1/
3) + d*x, x]^p*Simp[(b*d)/3 + (12^(1/3)*b^2*d^2)/(3*r^2) + r^2/(3*12^(1/3)) - d*((2^(1/3)*b*d)/(3^(1/3)*r) - r
/18^(1/3))*x + d^2*x^2, x]^p, x], x]] /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[4*b^3 + 27*a^2*d, 0] && ILtQ[p, 0
]

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]

Rule 2100

Int[(Pm_)/(Qn_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Simp[(Coeff[Pm, x, m]*Log[Qn])/(n*Coe
ff[Qn, x, n]), x] + Dist[1/(n*Coeff[Qn, x, n]), Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[Pm, x, m]*D[Qn, x
], x]/Qn, x], x] /; EqQ[m, n - 1]] /; PolyQ[Pm, x] && PolyQ[Qn, x]

Rule 2523

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Log[c*RFx^p])^n, x] - Dist[b*n*p
, Int[SimplifyIntegrand[(x*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, p}, x] &
& RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {63 x}{(-1+x) \left (6-9 x-x^2+x^3\right )}+\frac {6 x^4}{(-1+x) \left (6-9 x-x^2+x^3\right )}-\frac {24}{-6+15 x-8 x^2-2 x^3+x^4}-\frac {30 x^2}{-6+15 x-8 x^2-2 x^3+x^4}-\frac {12 x^3}{-6+15 x-8 x^2-2 x^3+x^4}+\log \left (\frac {6-9 x-x^2+x^3}{(1-x) x}\right )\right ) \, dx\\ &=6 \int \frac {x^4}{(-1+x) \left (6-9 x-x^2+x^3\right )} \, dx-12 \int \frac {x^3}{-6+15 x-8 x^2-2 x^3+x^4} \, dx-24 \int \frac {1}{-6+15 x-8 x^2-2 x^3+x^4} \, dx-30 \int \frac {x^2}{-6+15 x-8 x^2-2 x^3+x^4} \, dx+63 \int \frac {x}{(-1+x) \left (6-9 x-x^2+x^3\right )} \, dx+\int \log \left (\frac {6-9 x-x^2+x^3}{(1-x) x}\right ) \, dx\\ &=x \log \left (\frac {6-9 x-x^2+x^3}{(1-x) x}\right )+6 \int \left (1-\frac {1}{3 (-1+x)}+\frac {-24+30 x+7 x^2}{3 \left (6-9 x-x^2+x^3\right )}\right ) \, dx-12 \int \left (-\frac {1}{3 (-1+x)}+\frac {-6+3 x+4 x^2}{3 \left (6-9 x-x^2+x^3\right )}\right ) \, dx-24 \int \left (-\frac {1}{3 (-1+x)}+\frac {-9+x^2}{3 \left (6-9 x-x^2+x^3\right )}\right ) \, dx-30 \int \left (-\frac {1}{3 (-1+x)}+\frac {-6+3 x+x^2}{3 \left (6-9 x-x^2+x^3\right )}\right ) \, dx+63 \int \left (-\frac {1}{3 (-1+x)}+\frac {-6+x^2}{3 \left (6-9 x-x^2+x^3\right )}\right ) \, dx-\int \frac {-6+12 x-10 x^2+2 x^3-x^4}{(1-x) \left (6-9 x-x^2+x^3\right )} \, dx\\ &=6 x-\log (1-x)+x \log \left (\frac {6-9 x-x^2+x^3}{(1-x) x}\right )+2 \int \frac {-24+30 x+7 x^2}{6-9 x-x^2+x^3} \, dx-4 \int \frac {-6+3 x+4 x^2}{6-9 x-x^2+x^3} \, dx-8 \int \frac {-9+x^2}{6-9 x-x^2+x^3} \, dx-10 \int \frac {-6+3 x+x^2}{6-9 x-x^2+x^3} \, dx+21 \int \frac {-6+x^2}{6-9 x-x^2+x^3} \, dx-\int \left (1+\frac {1}{1-x}+\frac {-18+18 x+x^2}{6-9 x-x^2+x^3}\right ) \, dx\\ &=5 x+\frac {1}{3} \log \left (6-9 x-x^2+x^3\right )+x \log \left (\frac {6-9 x-x^2+x^3}{(1-x) x}\right )+\frac {2}{3} \int \frac {-9+104 x}{6-9 x-x^2+x^3} \, dx-\frac {4}{3} \int \frac {18+17 x}{6-9 x-x^2+x^3} \, dx-\frac {8}{3} \int \frac {-18+2 x}{6-9 x-x^2+x^3} \, dx-\frac {10}{3} \int \frac {-9+11 x}{6-9 x-x^2+x^3} \, dx+7 \int \frac {-9+2 x}{6-9 x-x^2+x^3} \, dx-\int \frac {-18+18 x+x^2}{6-9 x-x^2+x^3} \, dx\\ &=5 x+x \log \left (\frac {6-9 x-x^2+x^3}{(1-x) x}\right )-\frac {1}{3} \int \frac {-45+56 x}{6-9 x-x^2+x^3} \, dx+\frac {2}{3} \operatorname {Subst}\left (\int \frac {\frac {77}{3}+104 x}{\frac {79}{27}-\frac {28 x}{3}+x^3} \, dx,x,-\frac {1}{3}+x\right )-\frac {4}{3} \operatorname {Subst}\left (\int \frac {\frac {71}{3}+17 x}{\frac {79}{27}-\frac {28 x}{3}+x^3} \, dx,x,-\frac {1}{3}+x\right )-\frac {8}{3} \operatorname {Subst}\left (\int \frac {-\frac {52}{3}+2 x}{\frac {79}{27}-\frac {28 x}{3}+x^3} \, dx,x,-\frac {1}{3}+x\right )-\frac {10}{3} \operatorname {Subst}\left (\int \frac {-\frac {16}{3}+11 x}{\frac {79}{27}-\frac {28 x}{3}+x^3} \, dx,x,-\frac {1}{3}+x\right )+7 \operatorname {Subst}\left (\int \frac {-\frac {25}{3}+2 x}{\frac {79}{27}-\frac {28 x}{3}+x^3} \, dx,x,-\frac {1}{3}+x\right )\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 30, normalized size = 1.30 \begin {gather*} 5 x+x \log \left (\frac {6-9 x-x^2+x^3}{x-x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-24 + 63*x - 30*x^2 - 12*x^3 + 6*x^4 + (-6 + 15*x - 8*x^2 - 2*x^3 + x^4)*Log[(-6 + 9*x + x^2 - x^3)
/(-x + x^2)])/(-6 + 15*x - 8*x^2 - 2*x^3 + x^4),x]

[Out]

5*x + x*Log[(6 - 9*x - x^2 + x^3)/(x - x^2)]

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fricas [A]  time = 0.62, size = 31, normalized size = 1.35 \begin {gather*} x \log \left (-\frac {x^{3} - x^{2} - 9 \, x + 6}{x^{2} - x}\right ) + 5 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^4-2*x^3-8*x^2+15*x-6)*log((-x^3+x^2+9*x-6)/(x^2-x))+6*x^4-12*x^3-30*x^2+63*x-24)/(x^4-2*x^3-8*x^
2+15*x-6),x, algorithm="fricas")

[Out]

x*log(-(x^3 - x^2 - 9*x + 6)/(x^2 - x)) + 5*x

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giac [A]  time = 0.26, size = 31, normalized size = 1.35 \begin {gather*} x \log \left (-\frac {x^{3} - x^{2} - 9 \, x + 6}{x^{2} - x}\right ) + 5 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^4-2*x^3-8*x^2+15*x-6)*log((-x^3+x^2+9*x-6)/(x^2-x))+6*x^4-12*x^3-30*x^2+63*x-24)/(x^4-2*x^3-8*x^
2+15*x-6),x, algorithm="giac")

[Out]

x*log(-(x^3 - x^2 - 9*x + 6)/(x^2 - x)) + 5*x

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maple [A]  time = 0.07, size = 30, normalized size = 1.30

method	result	size
default	\(5 x +\ln \left (\frac {-x^{3}+x^{2}+9 x -6}{x \left (x -1\right )}\right ) x\)	\(30\)
norman	\(x \ln \left (\frac {-x^{3}+x^{2}+9 x -6}{x^{2}-x}\right )+5 x\)	\(31\)
risch	\(x \ln \left (\frac {-x^{3}+x^{2}+9 x -6}{x^{2}-x}\right )+5 x\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^4-2*x^3-8*x^2+15*x-6)*ln((-x^3+x^2+9*x-6)/(x^2-x))+6*x^4-12*x^3-30*x^2+63*x-24)/(x^4-2*x^3-8*x^2+15*x-
6),x,method=_RETURNVERBOSE)

[Out]

5*x+ln((-x^3+x^2+9*x-6)/x/(x-1))*x

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maxima [A]  time = 0.37, size = 32, normalized size = 1.39 \begin {gather*} x \log \left (-x^{3} + x^{2} + 9 \, x - 6\right ) - x \log \left (x - 1\right ) - x \log \relax (x) + 5 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^4-2*x^3-8*x^2+15*x-6)*log((-x^3+x^2+9*x-6)/(x^2-x))+6*x^4-12*x^3-30*x^2+63*x-24)/(x^4-2*x^3-8*x^
2+15*x-6),x, algorithm="maxima")

[Out]

x*log(-x^3 + x^2 + 9*x - 6) - x*log(x - 1) - x*log(x) + 5*x

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mupad [B]  time = 5.49, size = 29, normalized size = 1.26 \begin {gather*} x\,\left (\ln \left (-\frac {-x^3+x^2+9\,x-6}{x-x^2}\right )+5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(-(9*x + x^2 - x^3 - 6)/(x - x^2))*(8*x^2 - 15*x + 2*x^3 - x^4 + 6) - 63*x + 30*x^2 + 12*x^3 - 6*x^4 +
24)/(8*x^2 - 15*x + 2*x^3 - x^4 + 6),x)

[Out]

x*(log(-(9*x + x^2 - x^3 - 6)/(x - x^2)) + 5)

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sympy [A]  time = 0.29, size = 22, normalized size = 0.96 \begin {gather*} x \log {\left (\frac {- x^{3} + x^{2} + 9 x - 6}{x^{2} - x} \right )} + 5 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**4-2*x**3-8*x**2+15*x-6)*ln((-x**3+x**2+9*x-6)/(x**2-x))+6*x**4-12*x**3-30*x**2+63*x-24)/(x**4-2
*x**3-8*x**2+15*x-6),x)

[Out]

x*log((-x**3 + x**2 + 9*x - 6)/(x**2 - x)) + 5*x

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