3.97.100 \(\int \frac {(e^3 (-4 x^3+4 x^4)+e^6 (2 x^3-6 x^4+4 x^5)) \log (\frac {-1-2 e^3 x+e^6 (x-x^2)}{e^6})+(-3 x^2+2 x^3+e^3 (-6 x^3+4 x^4)+e^6 (3 x^3-5 x^4+2 x^5)) \log ^2(\frac {-1-2 e^3 x+e^6 (x-x^2)}{e^6})}{20-40 x+20 x^2+e^3 (40 x-80 x^2+40 x^3)+e^6 (-20 x+60 x^2-60 x^3+20 x^4)} \, dx\)

Optimal. Leaf size=34 \[ \frac {1}{4} \left (2-\frac {x^2 \log ^2\left (x-\left (\frac {1}{e^3}+x\right )^2\right )}{-5+\frac {5}{x}}\right ) \]

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Rubi [C]  time = 13.17, antiderivative size = 3703, normalized size of antiderivative = 108.91, number of steps used = 156, number of rules used = 21, integrand size = 189, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6741, 6688, 12, 6728, 1628, 634, 618, 206, 628, 2528, 2523, 773, 2525, 800, 2524, 2418, 2394, 2393, 2391, 2390, 2301}

result too large to display

Antiderivative was successfully verified.

[In]

Int[((E^3*(-4*x^3 + 4*x^4) + E^6*(2*x^3 - 6*x^4 + 4*x^5))*Log[(-1 - 2*E^3*x + E^6*(x - x^2))/E^6] + (-3*x^2 +
2*x^3 + E^3*(-6*x^3 + 4*x^4) + E^6*(3*x^3 - 5*x^4 + 2*x^5))*Log[(-1 - 2*E^3*x + E^6*(x - x^2))/E^6]^2)/(20 - 4
0*x + 20*x^2 + E^3*(40*x - 80*x^2 + 40*x^3) + E^6*(-20*x + 60*x^2 - 60*x^3 + 20*x^4)),x]

[Out]

(2*x)/5 + ((2 - 3*E^3)*x)/(5*E^3) - ((2 - E^3)*x)/(5*E^3) - (Sqrt[-4 + E^3]*ArcTanh[(2 - E^3*(1 - 2*x))/(E^(3/
2)*Sqrt[-4 + E^3])])/(5*E^(3/2)) - ((2 - 3*E^3)*Sqrt[-4 + E^3]*ArcTanh[(2 - E^3*(1 - 2*x))/(E^(3/2)*Sqrt[-4 +
E^3])])/(10*E^(9/2)) + ((2 - E^3)*Sqrt[-4 + E^3]*ArcTanh[(2 - E^3*(1 - 2*x))/(E^(3/2)*Sqrt[-4 + E^3])])/(10*E^
(9/2)) - ((2 - E^3 - E^(3/2)*Sqrt[-4 + E^3])*Log[E^3*(2 - E^(3/2)*Sqrt[-4 + E^3] - E^3*(1 - 2*x))]^2)/(40*E^3)
- (E^3*(2 + E^3 + E^(3/2)*Sqrt[-4 + E^3])*Log[E^3*(2 - E^(3/2)*Sqrt[-4 + E^3] - E^3*(1 - 2*x))]^2)/(40*(1 + 2
*E^3)) + ((2 - 4*E^3 + E^6 + (E^(3/2)*(8 - 6*E^3 + E^6))/Sqrt[-4 + E^3])*Log[E^3*(2 - E^(3/2)*Sqrt[-4 + E^3] -
E^3*(1 - 2*x))]^2)/(40*E^6) - ((2 - 2*E^3 - 8*E^6 + 6*E^9 - E^12 + (E^(3/2)*(8 + 6*E^3 - 18*E^6 + 8*E^9 - E^1
2))/Sqrt[-4 + E^3])*Log[E^3*(2 - E^(3/2)*Sqrt[-4 + E^3] - E^3*(1 - 2*x))]^2)/(40*E^6*(1 + 2*E^3)) - (E^3*(2 +
E^3 - E^(3/2)*Sqrt[-4 + E^3])*Log[E^3*(2 + E^(3/2)*Sqrt[-4 + E^3] - E^3*(1 - 2*x))]^2)/(40*(1 + 2*E^3)) - ((2
- E^3 + E^(3/2)*Sqrt[-4 + E^3])*Log[E^3*(2 + E^(3/2)*Sqrt[-4 + E^3] - E^3*(1 - 2*x))]^2)/(40*E^3) + ((2 - 4*E^
3 + E^6 - (E^(3/2)*(8 - 6*E^3 + E^6))/Sqrt[-4 + E^3])*Log[E^3*(2 + E^(3/2)*Sqrt[-4 + E^3] - E^3*(1 - 2*x))]^2)
/(40*E^6) - ((2 - 2*E^3 - 8*E^6 + 6*E^9 - E^12 - (E^(3/2)*(8 + 6*E^3 - 18*E^6 + 8*E^9 - E^12))/Sqrt[-4 + E^3])
*Log[E^3*(2 + E^(3/2)*Sqrt[-4 + E^3] - E^3*(1 - 2*x))]^2)/(40*E^6*(1 + 2*E^3)) - ((2 - E^3 - E^(3/2)*Sqrt[-4 +
E^3])*Log[-1/2*(E^(3/2)*(1 - 2/E^3 - Sqrt[-4 + E^3]/E^(3/2) - 2*x))/Sqrt[-4 + E^3]]*Log[E^3*(2 - E^3 - E^(3/2
)*Sqrt[-4 + E^3]) + 2*E^6*x])/(20*E^3) - (E^3*(2 + E^3 + E^(3/2)*Sqrt[-4 + E^3])*Log[-1/2*(E^(3/2)*(1 - 2/E^3
- Sqrt[-4 + E^3]/E^(3/2) - 2*x))/Sqrt[-4 + E^3]]*Log[E^3*(2 - E^3 - E^(3/2)*Sqrt[-4 + E^3]) + 2*E^6*x])/(20*(1
+ 2*E^3)) + ((2 - 4*E^3 + E^6 + (E^(3/2)*(8 - 6*E^3 + E^6))/Sqrt[-4 + E^3])*Log[-1/2*(E^(3/2)*(1 - 2/E^3 - Sq
rt[-4 + E^3]/E^(3/2) - 2*x))/Sqrt[-4 + E^3]]*Log[E^3*(2 - E^3 - E^(3/2)*Sqrt[-4 + E^3]) + 2*E^6*x])/(20*E^6) -
((2 - 2*E^3 - 8*E^6 + 6*E^9 - E^12 + (E^(3/2)*(8 + 6*E^3 - 18*E^6 + 8*E^9 - E^12))/Sqrt[-4 + E^3])*Log[-1/2*(
E^(3/2)*(1 - 2/E^3 - Sqrt[-4 + E^3]/E^(3/2) - 2*x))/Sqrt[-4 + E^3]]*Log[E^3*(2 - E^3 - E^(3/2)*Sqrt[-4 + E^3])
+ 2*E^6*x])/(20*E^6*(1 + 2*E^3)) - (E^3*(2 + E^3 - E^(3/2)*Sqrt[-4 + E^3])*Log[(E^(3/2)*(1 - 2/E^3 + Sqrt[-4
+ E^3]/E^(3/2) - 2*x))/(2*Sqrt[-4 + E^3])]*Log[E^3*(2 - E^3 + E^(3/2)*Sqrt[-4 + E^3]) + 2*E^6*x])/(20*(1 + 2*E
^3)) - ((2 - E^3 + E^(3/2)*Sqrt[-4 + E^3])*Log[(E^(3/2)*(1 - 2/E^3 + Sqrt[-4 + E^3]/E^(3/2) - 2*x))/(2*Sqrt[-4
+ E^3])]*Log[E^3*(2 - E^3 + E^(3/2)*Sqrt[-4 + E^3]) + 2*E^6*x])/(20*E^3) + ((2 - 4*E^3 + E^6 - (E^(3/2)*(8 -
6*E^3 + E^6))/Sqrt[-4 + E^3])*Log[(E^(3/2)*(1 - 2/E^3 + Sqrt[-4 + E^3]/E^(3/2) - 2*x))/(2*Sqrt[-4 + E^3])]*Log
[E^3*(2 - E^3 + E^(3/2)*Sqrt[-4 + E^3]) + 2*E^6*x])/(20*E^6) - ((2 - 2*E^3 - 8*E^6 + 6*E^9 - E^12 - (E^(3/2)*(
8 + 6*E^3 - 18*E^6 + 8*E^9 - E^12))/Sqrt[-4 + E^3])*Log[(E^(3/2)*(1 - 2/E^3 + Sqrt[-4 + E^3]/E^(3/2) - 2*x))/(
2*Sqrt[-4 + E^3])]*Log[E^3*(2 - E^3 + E^(3/2)*Sqrt[-4 + E^3]) + 2*E^6*x])/(20*E^6*(1 + 2*E^3)) - (x*Log[-E^(-6
) + (1 - 2/E^3)*x - x^2])/5 - ((2 - 3*E^3)*x*Log[-E^(-6) + (1 - 2/E^3)*x - x^2])/(10*E^3) + ((2 - E^3)*x*Log[-
E^(-6) + (1 - 2/E^3)*x - x^2])/(10*E^3) + ((2 - E^3 - E^(3/2)*Sqrt[-4 + E^3])*Log[E^3*(2 - E^3 - E^(3/2)*Sqrt[
-4 + E^3]) + 2*E^6*x]*Log[-E^(-6) + (1 - 2/E^3)*x - x^2])/(20*E^3) + (E^3*(2 + E^3 + E^(3/2)*Sqrt[-4 + E^3])*L
og[E^3*(2 - E^3 - E^(3/2)*Sqrt[-4 + E^3]) + 2*E^6*x]*Log[-E^(-6) + (1 - 2/E^3)*x - x^2])/(20*(1 + 2*E^3)) - ((
2 - 4*E^3 + E^6 + (E^(3/2)*(8 - 6*E^3 + E^6))/Sqrt[-4 + E^3])*Log[E^3*(2 - E^3 - E^(3/2)*Sqrt[-4 + E^3]) + 2*E
^6*x]*Log[-E^(-6) + (1 - 2/E^3)*x - x^2])/(20*E^6) + ((2 - 2*E^3 - 8*E^6 + 6*E^9 - E^12 + (E^(3/2)*(8 + 6*E^3
- 18*E^6 + 8*E^9 - E^12))/Sqrt[-4 + E^3])*Log[E^3*(2 - E^3 - E^(3/2)*Sqrt[-4 + E^3]) + 2*E^6*x]*Log[-E^(-6) +
(1 - 2/E^3)*x - x^2])/(20*E^6*(1 + 2*E^3)) + (E^3*(2 + E^3 - E^(3/2)*Sqrt[-4 + E^3])*Log[E^3*(2 - E^3 + E^(3/2
)*Sqrt[-4 + E^3]) + 2*E^6*x]*Log[-E^(-6) + (1 - 2/E^3)*x - x^2])/(20*(1 + 2*E^3)) + ((2 - E^3 + E^(3/2)*Sqrt[-
4 + E^3])*Log[E^3*(2 - E^3 + E^(3/2)*Sqrt[-4 + E^3]) + 2*E^6*x]*Log[-E^(-6) + (1 - 2/E^3)*x - x^2])/(20*E^3) -
((2 - 4*E^3 + E^6 - (E^(3/2)*(8 - 6*E^3 + E^6))/Sqrt[-4 + E^3])*Log[E^3*(2 - E^3 + E^(3/2)*Sqrt[-4 + E^3]) +
2*E^6*x]*Log[-E^(-6) + (1 - 2/E^3)*x - x^2])/(20*E^6) + ((2 - 2*E^3 - 8*E^6 + 6*E^9 - E^12 - (E^(3/2)*(8 + 6*E
^3 - 18*E^6 + 8*E^9 - E^12))/Sqrt[-4 + E^3])*Log[E^3*(2 - E^3 + E^(3/2)*Sqrt[-4 + E^3]) + 2*E^6*x]*Log[-E^(-6)
+ (1 - 2/E^3)*x - x^2])/(20*E^6*(1 + 2*E^3)) - Log[-E^(-6) + (1 - 2/E^3)*x - x^2]^2/(20*(1 - x)) + (x*Log[-E^
(-6) + (1 - 2/E^3)*x - x^2]^2)/20 + (x^2*Log[-E^(-6) + (1 - 2/E^3)*x - x^2]^2)/20 - ((2 - E^3)*Log[1 + E^3*(2
- E^3)*x + E^6*x^2])/(10*E^3) - ((2 - 3*E^3)*(2 - E^3)*Log[1 + E^3*(2 - E^3)*x + E^6*x^2])/(20*E^6) + ((2 - E^
3)^2*Log[1 + E^3*(2 - E^3)*x + E^6*x^2])/(20*E^6) - ((2 - E^3 - E^(3/2)*Sqrt[-4 + E^3])*PolyLog[2, -1/2*(2 - E
^(3/2)*Sqrt[-4 + E^3] - E^3*(1 - 2*x))/(E^(3/2)*Sqrt[-4 + E^3])])/(20*E^3) - (E^3*(2 + E^3 + E^(3/2)*Sqrt[-4 +
E^3])*PolyLog[2, -1/2*(2 - E^(3/2)*Sqrt[-4 + E^3] - E^3*(1 - 2*x))/(E^(3/2)*Sqrt[-4 + E^3])])/(20*(1 + 2*E^3)
) + ((2 - 4*E^3 + E^6 + (E^(3/2)*(8 - 6*E^3 + E^6))/Sqrt[-4 + E^3])*PolyLog[2, -1/2*(2 - E^(3/2)*Sqrt[-4 + E^3
] - E^3*(1 - 2*x))/(E^(3/2)*Sqrt[-4 + E^3])])/(20*E^6) - ((2 - 2*E^3 - 8*E^6 + 6*E^9 - E^12 + (E^(3/2)*(8 + 6*
E^3 - 18*E^6 + 8*E^9 - E^12))/Sqrt[-4 + E^3])*PolyLog[2, -1/2*(2 - E^(3/2)*Sqrt[-4 + E^3] - E^3*(1 - 2*x))/(E^
(3/2)*Sqrt[-4 + E^3])])/(20*E^6*(1 + 2*E^3)) - (E^3*(2 + E^3 - E^(3/2)*Sqrt[-4 + E^3])*PolyLog[2, (1 + 2/(E^(3
/2)*Sqrt[-4 + E^3]))/2 - (E^(3/2)*(1 - 2*x))/(2*Sqrt[-4 + E^3])])/(20*(1 + 2*E^3)) - ((2 - E^3 + E^(3/2)*Sqrt[
-4 + E^3])*PolyLog[2, (1 + 2/(E^(3/2)*Sqrt[-4 + E^3]))/2 - (E^(3/2)*(1 - 2*x))/(2*Sqrt[-4 + E^3])])/(20*E^3) +
((2 - 4*E^3 + E^6 - (E^(3/2)*(8 - 6*E^3 + E^6))/Sqrt[-4 + E^3])*PolyLog[2, (1 + 2/(E^(3/2)*Sqrt[-4 + E^3]))/2
- (E^(3/2)*(1 - 2*x))/(2*Sqrt[-4 + E^3])])/(20*E^6) - ((2 - 2*E^3 - 8*E^6 + 6*E^9 - E^12 - (E^(3/2)*(8 + 6*E^
3 - 18*E^6 + 8*E^9 - E^12))/Sqrt[-4 + E^3])*PolyLog[2, (1 + 2/(E^(3/2)*Sqrt[-4 + E^3]))/2 - (E^(3/2)*(1 - 2*x)
)/(2*Sqrt[-4 + E^3])])/(20*E^6*(1 + 2*E^3))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 773

Int[(((d_.) + (e_.)*(x_))*((f_) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*g*x)/
c, x] + Dist[1/c, Int[(c*d*f - a*e*g + (c*e*f + c*d*g - b*e*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]

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 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2390

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

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2394

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

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

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 2524

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

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rule 6688

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

Rule 6728

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]

Rule 6741

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (e^3 \left (-4 x^3+4 x^4\right )+e^6 \left (2 x^3-6 x^4+4 x^5\right )\right ) \log \left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )+\left (-3 x^2+2 x^3+e^3 \left (-6 x^3+4 x^4\right )+e^6 \left (3 x^3-5 x^4+2 x^5\right )\right ) \log ^2\left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )}{20-20 \left (2-2 e^3+e^6\right ) x+20 \left (1-4 e^3+3 e^6\right ) x^2+20 e^3 \left (2-3 e^3\right ) x^3+20 e^6 x^4} \, dx\\ &=\int \frac {x^2 \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right ) \left (4 e^3 (-1+x) x+2 e^6 x \left (1-3 x+2 x^2\right )+\left (-3+\left (2-6 e^3+3 e^6\right ) x+\left (4 e^3-5 e^6\right ) x^2+2 e^6 x^3\right ) \log \left (-\frac {1}{e^6}+x-\frac {2 x}{e^3}-x^2\right )\right )}{20 (1-x)^2 \left (1+e^3 \left (2-e^3\right ) x+e^6 x^2\right )} \, dx\\ &=\frac {1}{20} \int \frac {x^2 \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right ) \left (4 e^3 (-1+x) x+2 e^6 x \left (1-3 x+2 x^2\right )+\left (-3+\left (2-6 e^3+3 e^6\right ) x+\left (4 e^3-5 e^6\right ) x^2+2 e^6 x^3\right ) \log \left (-\frac {1}{e^6}+x-\frac {2 x}{e^3}-x^2\right )\right )}{(1-x)^2 \left (1+e^3 \left (2-e^3\right ) x+e^6 x^2\right )} \, dx\\ &=\frac {1}{20} \int \left (\frac {2 e^3 x^3 \left (-2+e^3-2 e^3 x\right ) \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{(1-x) \left (1+e^3 \left (2-e^3\right ) x+e^6 x^2\right )}+\frac {x^2 (-3+2 x) \log ^2\left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{(1-x)^2}\right ) \, dx\\ &=\frac {1}{20} \int \frac {x^2 (-3+2 x) \log ^2\left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{(1-x)^2} \, dx+\frac {1}{10} e^3 \int \frac {x^3 \left (-2+e^3-2 e^3 x\right ) \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{(1-x) \left (1+e^3 \left (2-e^3\right ) x+e^6 x^2\right )} \, dx\\ &=\frac {1}{20} \int \left (\log ^2\left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )-\frac {\log ^2\left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{(-1+x)^2}+2 x \log ^2\left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )\right ) \, dx+\frac {1}{10} e^3 \int \left (\frac {\left (-2+3 e^3\right ) \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{e^6}+\frac {\left (2+e^3\right ) \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{\left (1+2 e^3\right ) (-1+x)}+\frac {2 x \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{e^3}+\frac {\left (2+e^3-4 e^6+e^9+e^3 \left (2-2 e^3-8 e^6+6 e^9-e^{12}\right ) x\right ) \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{e^6 \left (1+2 e^3\right ) \left (1+e^3 \left (2-e^3\right ) x+e^6 x^2\right )}\right ) \, dx\\ &=\frac {1}{20} \int \log ^2\left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right ) \, dx-\frac {1}{20} \int \frac {\log ^2\left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{(-1+x)^2} \, dx+\frac {1}{10} \int x \log ^2\left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right ) \, dx+\frac {1}{5} \int x \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right ) \, dx-\frac {\left (2-3 e^3\right ) \int \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right ) \, dx}{10 e^3}+\frac {\int \frac {\left (2+e^3-4 e^6+e^9+e^3 \left (2-2 e^3-8 e^6+6 e^9-e^{12}\right ) x\right ) \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{1+e^3 \left (2-e^3\right ) x+e^6 x^2} \, dx}{10 e^3 \left (1+2 e^3\right )}+\frac {\left (e^3 \left (2+e^3\right )\right ) \int \frac {\log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{-1+x} \, dx}{10 \left (1+2 e^3\right )}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 33, normalized size = 0.97 \begin {gather*} \frac {x^3 \log ^2\left (-\frac {1}{e^6}+x-\frac {2 x}{e^3}-x^2\right )}{20 (-1+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((E^3*(-4*x^3 + 4*x^4) + E^6*(2*x^3 - 6*x^4 + 4*x^5))*Log[(-1 - 2*E^3*x + E^6*(x - x^2))/E^6] + (-3*
x^2 + 2*x^3 + E^3*(-6*x^3 + 4*x^4) + E^6*(3*x^3 - 5*x^4 + 2*x^5))*Log[(-1 - 2*E^3*x + E^6*(x - x^2))/E^6]^2)/(
20 - 40*x + 20*x^2 + E^3*(40*x - 80*x^2 + 40*x^3) + E^6*(-20*x + 60*x^2 - 60*x^3 + 20*x^4)),x]

[Out]

(x^3*Log[-E^(-6) + x - (2*x)/E^3 - x^2]^2)/(20*(-1 + x))

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fricas [A]  time = 0.86, size = 34, normalized size = 1.00 \begin {gather*} \frac {x^{3} \log \left (-{\left ({\left (x^{2} - x\right )} e^{6} + 2 \, x e^{3} + 1\right )} e^{\left (-6\right )}\right )^{2}}{20 \, {\left (x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^5-5*x^4+3*x^3)*exp(3)^2+(4*x^4-6*x^3)*exp(3)+2*x^3-3*x^2)*log(((-x^2+x)*exp(3)^2-2*x*exp(3)-1
)/exp(3)^2)^2+((4*x^5-6*x^4+2*x^3)*exp(3)^2+(4*x^4-4*x^3)*exp(3))*log(((-x^2+x)*exp(3)^2-2*x*exp(3)-1)/exp(3)^
2))/((20*x^4-60*x^3+60*x^2-20*x)*exp(3)^2+(40*x^3-80*x^2+40*x)*exp(3)+20*x^2-40*x+20),x, algorithm="fricas")

[Out]

1/20*x^3*log(-((x^2 - x)*e^6 + 2*x*e^3 + 1)*e^(-6))^2/(x - 1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{3} - 3 \, x^{2} + {\left (2 \, x^{5} - 5 \, x^{4} + 3 \, x^{3}\right )} e^{6} + 2 \, {\left (2 \, x^{4} - 3 \, x^{3}\right )} e^{3}\right )} \log \left (-{\left ({\left (x^{2} - x\right )} e^{6} + 2 \, x e^{3} + 1\right )} e^{\left (-6\right )}\right )^{2} + 2 \, {\left ({\left (2 \, x^{5} - 3 \, x^{4} + x^{3}\right )} e^{6} + 2 \, {\left (x^{4} - x^{3}\right )} e^{3}\right )} \log \left (-{\left ({\left (x^{2} - x\right )} e^{6} + 2 \, x e^{3} + 1\right )} e^{\left (-6\right )}\right )}{20 \, {\left (x^{2} + {\left (x^{4} - 3 \, x^{3} + 3 \, x^{2} - x\right )} e^{6} + 2 \, {\left (x^{3} - 2 \, x^{2} + x\right )} e^{3} - 2 \, x + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^5-5*x^4+3*x^3)*exp(3)^2+(4*x^4-6*x^3)*exp(3)+2*x^3-3*x^2)*log(((-x^2+x)*exp(3)^2-2*x*exp(3)-1
)/exp(3)^2)^2+((4*x^5-6*x^4+2*x^3)*exp(3)^2+(4*x^4-4*x^3)*exp(3))*log(((-x^2+x)*exp(3)^2-2*x*exp(3)-1)/exp(3)^
2))/((20*x^4-60*x^3+60*x^2-20*x)*exp(3)^2+(40*x^3-80*x^2+40*x)*exp(3)+20*x^2-40*x+20),x, algorithm="giac")

[Out]

integrate(1/20*((2*x^3 - 3*x^2 + (2*x^5 - 5*x^4 + 3*x^3)*e^6 + 2*(2*x^4 - 3*x^3)*e^3)*log(-((x^2 - x)*e^6 + 2*
x*e^3 + 1)*e^(-6))^2 + 2*((2*x^5 - 3*x^4 + x^3)*e^6 + 2*(x^4 - x^3)*e^3)*log(-((x^2 - x)*e^6 + 2*x*e^3 + 1)*e^
(-6)))/(x^2 + (x^4 - 3*x^3 + 3*x^2 - x)*e^6 + 2*(x^3 - 2*x^2 + x)*e^3 - 2*x + 1), x)

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maple [A]  time = 0.34, size = 34, normalized size = 1.00

method	result	size
risch	\(\frac {\ln \left (\left (\left (-x^{2}+x \right ) {\mathrm e}^{6}-2 x \,{\mathrm e}^{3}-1\right ) {\mathrm e}^{-6}\right )^{2} x^{3}}{20 x -20}\)	\(34\)
norman	\(\frac {\ln \left (\left (\left (-x^{2}+x \right ) {\mathrm e}^{6}-2 x \,{\mathrm e}^{3}-1\right ) {\mathrm e}^{-6}\right )^{2} x^{3}}{20 x -20}\)	\(38\)
default	error in gcdex: invalid arguments\	N/A

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x^5-5*x^4+3*x^3)*exp(3)^2+(4*x^4-6*x^3)*exp(3)+2*x^3-3*x^2)*ln(((-x^2+x)*exp(3)^2-2*x*exp(3)-1)/exp(3
)^2)^2+((4*x^5-6*x^4+2*x^3)*exp(3)^2+(4*x^4-4*x^3)*exp(3))*ln(((-x^2+x)*exp(3)^2-2*x*exp(3)-1)/exp(3)^2))/((20
*x^4-60*x^3+60*x^2-20*x)*exp(3)^2+(40*x^3-80*x^2+40*x)*exp(3)+20*x^2-40*x+20),x,method=_RETURNVERBOSE)

[Out]

1/20*x^3*ln(((-x^2+x)*exp(6)-2*x*exp(3)-1)*exp(-6))^2/(x-1)

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maxima [B]  time = 0.59, size = 66, normalized size = 1.94 \begin {gather*} \frac {x^{3} \log \left (-x^{2} e^{6} + x {\left (e^{6} - 2 \, e^{3}\right )} - 1\right )^{2} - 12 \, x^{3} \log \left (-x^{2} e^{6} + x {\left (e^{6} - 2 \, e^{3}\right )} - 1\right ) + 36 \, x^{3} - 36 \, x + 36}{20 \, {\left (x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^5-5*x^4+3*x^3)*exp(3)^2+(4*x^4-6*x^3)*exp(3)+2*x^3-3*x^2)*log(((-x^2+x)*exp(3)^2-2*x*exp(3)-1
)/exp(3)^2)^2+((4*x^5-6*x^4+2*x^3)*exp(3)^2+(4*x^4-4*x^3)*exp(3))*log(((-x^2+x)*exp(3)^2-2*x*exp(3)-1)/exp(3)^
2))/((20*x^4-60*x^3+60*x^2-20*x)*exp(3)^2+(40*x^3-80*x^2+40*x)*exp(3)+20*x^2-40*x+20),x, algorithm="maxima")

[Out]

1/20*(x^3*log(-x^2*e^6 + x*(e^6 - 2*e^3) - 1)^2 - 12*x^3*log(-x^2*e^6 + x*(e^6 - 2*e^3) - 1) + 36*x^3 - 36*x +
36)/(x - 1)

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mupad [B]  time = 0.68, size = 30, normalized size = 0.88 \begin {gather*} \frac {x^3\,{\ln \left (x-{\mathrm {e}}^{-6}-2\,x\,{\mathrm {e}}^{-3}-x^2\right )}^2}{20\,\left (x-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(-exp(-6)*(2*x*exp(3) - exp(6)*(x - x^2) + 1))^2*(exp(3)*(6*x^3 - 4*x^4) - exp(6)*(3*x^3 - 5*x^4 + 2*
x^5) + 3*x^2 - 2*x^3) + log(-exp(-6)*(2*x*exp(3) - exp(6)*(x - x^2) + 1))*(exp(3)*(4*x^3 - 4*x^4) - exp(6)*(2*
x^3 - 6*x^4 + 4*x^5)))/(exp(3)*(40*x - 80*x^2 + 40*x^3) - 40*x - exp(6)*(20*x - 60*x^2 + 60*x^3 - 20*x^4) + 20
*x^2 + 20),x)

[Out]

(x^3*log(x - exp(-6) - 2*x*exp(-3) - x^2)^2)/(20*(x - 1))

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sympy [A]  time = 0.43, size = 31, normalized size = 0.91 \begin {gather*} \frac {x^{3} \log {\left (\frac {- 2 x e^{3} + \left (- x^{2} + x\right ) e^{6} - 1}{e^{6}} \right )}^{2}}{20 x - 20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x**5-5*x**4+3*x**3)*exp(3)**2+(4*x**4-6*x**3)*exp(3)+2*x**3-3*x**2)*ln(((-x**2+x)*exp(3)**2-2*x
*exp(3)-1)/exp(3)**2)**2+((4*x**5-6*x**4+2*x**3)*exp(3)**2+(4*x**4-4*x**3)*exp(3))*ln(((-x**2+x)*exp(3)**2-2*x
*exp(3)-1)/exp(3)**2))/((20*x**4-60*x**3+60*x**2-20*x)*exp(3)**2+(40*x**3-80*x**2+40*x)*exp(3)+20*x**2-40*x+20
),x)

[Out]

x**3*log((-2*x*exp(3) + (-x**2 + x)*exp(6) - 1)*exp(-6))**2/(20*x - 20)

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