3.95.72 $\int \frac{32+32e^3-32e^{\frac{e^3}{15}}-32x^2+(2+2e^3-2e^{\frac{e^3}{15}}-2x^2)\log (\frac{1+e^3-e^{\frac{e^3}{15}}-4x+x^2}{x})}{-x-e^{3}x+e^{\frac{e^3}{15}}x+4x^2-x^3}dx$

Optimal. Leaf size=28 $${(16+\log (-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x}+x))}^2$$

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Rubi [C]  time = 3.02, antiderivative size = 902, normalized size of antiderivative = 32.21, number of steps used = 46, number of rules used = 16, integrand size = 108, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.148, Rules used = {6, 1594, 6728, 1628, 628, 2528, 2524, 2357, 2301, 2317, 2391, 2418, 2394, 2315, 2390, 2393} $$\begin{array}{c}-{\log }^2(-2(-x-i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2))-{\log }^2(-2(-x+i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2))-{\log }^2(x)-32\log (x)-2\log (x)\log (x-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x})-2\log (-\frac{i(-x+i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2)}{2\sqrt{-3+e^3-e^{\frac{e^3}{15}}}})\log (2x-2(2-i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}))+2\log \left(\frac{x}{2-i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}}\right)\log (2x-2(2-i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}))+2\log (x-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x})\log (2x-2(2-i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}))-2\log \left(\frac{i(-x-i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2)}{2\sqrt{-3+e^3-e^{\frac{e^3}{15}}}}\right)\log (2x-2(2+i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}))+2\log \left(\frac{x}{2+i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}}\right)\log (2x-2(2+i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}))+2\log (x-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x})\log (2x-2(2+i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}))+2\log (x)\log (1-\frac{x}{2-i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}})+2\log (x)\log (1-\frac{x}{2+i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}})+32\log (x^2-4x-e^{\frac{e^3}{15}}+e^3+1)-2{\text{Li}}_2(-\frac{-ix-\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2i}{2\sqrt{-3+e^3-e^{\frac{e^3}{15}}}})-2{\text{Li}}_2\left(\frac{-ix+\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2i}{2\sqrt{-3+e^3-e^{\frac{e^3}{15}}}}\right)+2{\text{Li}}_2\left(\frac{x}{2-i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}}\right)+2{\text{Li}}_2\left(\frac{x}{2+i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}}\right)+2{\text{Li}}_2(1-\frac{x}{2-i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}})+2{\text{Li}}_2(1-\frac{x}{2+i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}})\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(32 + 32*E^3 - 32*E^(E^3/15) - 32*x^2 + (2 + 2*E^3 - 2*E^(E^3/15) - 2*x^2)*Log[(1 + E^3 - E^(E^3/15) - 4*x
+ x^2)/x])/(-x - E^3*x + E^(E^3/15)*x + 4*x^2 - x^3),x]

[Out]

-Log[-2*(2 - I*Sqrt[-3 + E^3 - E^(E^3/15)] - x)]^2 - Log[-2*(2 + I*Sqrt[-3 + E^3 - E^(E^3/15)] - x)]^2 - 32*Lo
g[x] - Log[x]^2 - 2*Log[x]*Log[-4 + (1 + E^3 - E^(E^3/15))/x + x] - 2*Log[((-1/2*I)*(2 + I*Sqrt[-3 + E^3 - E^(
E^3/15)] - x))/Sqrt[-3 + E^3 - E^(E^3/15)]]*Log[-2*(2 - I*Sqrt[-3 + E^3 - E^(E^3/15)]) + 2*x] + 2*Log[x/(2 - I
*Sqrt[-3 + E^3 - E^(E^3/15)])]*Log[-2*(2 - I*Sqrt[-3 + E^3 - E^(E^3/15)]) + 2*x] + 2*Log[-4 + (1 + E^3 - E^(E^
3/15))/x + x]*Log[-2*(2 - I*Sqrt[-3 + E^3 - E^(E^3/15)]) + 2*x] - 2*Log[((I/2)*(2 - I*Sqrt[-3 + E^3 - E^(E^3/1
5)] - x))/Sqrt[-3 + E^3 - E^(E^3/15)]]*Log[-2*(2 + I*Sqrt[-3 + E^3 - E^(E^3/15)]) + 2*x] + 2*Log[x/(2 + I*Sqrt
[-3 + E^3 - E^(E^3/15)])]*Log[-2*(2 + I*Sqrt[-3 + E^3 - E^(E^3/15)]) + 2*x] + 2*Log[-4 + (1 + E^3 - E^(E^3/15)
)/x + x]*Log[-2*(2 + I*Sqrt[-3 + E^3 - E^(E^3/15)]) + 2*x] + 2*Log[x]*Log[1 - x/(2 - I*Sqrt[-3 + E^3 - E^(E^3/
15)])] + 2*Log[x]*Log[1 - x/(2 + I*Sqrt[-3 + E^3 - E^(E^3/15)])] + 32*Log[1 + E^3 - E^(E^3/15) - 4*x + x^2] -
2*PolyLog[2, -1/2*(2*I - Sqrt[-3 + E^3 - E^(E^3/15)] - I*x)/Sqrt[-3 + E^3 - E^(E^3/15)]] - 2*PolyLog[2, (2*I +
Sqrt[-3 + E^3 - E^(E^3/15)] - I*x)/(2*Sqrt[-3 + E^3 - E^(E^3/15)])] + 2*PolyLog[2, x/(2 - I*Sqrt[-3 + E^3 - E
^(E^3/15)])] + 2*PolyLog[2, x/(2 + I*Sqrt[-3 + E^3 - E^(E^3/15)])] + 2*PolyLog[2, 1 - x/(2 - I*Sqrt[-3 + E^3 -
E^(E^3/15)])] + 2*PolyLog[2, 1 - x/(2 + I*Sqrt[-3 + E^3 - E^(E^3/15)])]

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

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 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

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 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2317

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

Rule 2357

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

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 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 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 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]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{32+32e^3-32e^{\frac{e^3}{15}}-32x^2+(2+2e^3-2e^{\frac{e^3}{15}}-2x^2)\log \left(\frac{1+e^3-e^{\frac{e^3}{15}}-4x+x^2}{x}\right)}{e^{\frac{e^3}{15}}x+(-1-e^3)x+4x^2-x^3}dx\\ & =\int \frac{32+32e^3-32e^{\frac{e^3}{15}}-32x^2+(2+2e^3-2e^{\frac{e^3}{15}}-2x^2)\log \left(\frac{1+e^3-e^{\frac{e^3}{15}}-4x+x^2}{x}\right)}{(-1-e^3+e^{\frac{e^3}{15}})x+4x^2-x^3}dx\\ & =\int \frac{32+32e^3-32e^{\frac{e^3}{15}}-32x^2+(2+2e^3-2e^{\frac{e^3}{15}}-2x^2)\log \left(\frac{1+e^3-e^{\frac{e^3}{15}}-4x+x^2}{x}\right)}{x(-1-e^3+e^{\frac{e^3}{15}}+4x-x^2)}dx\\ & =\int (\frac{32(-1-e^3+e^{\frac{e^3}{15}}+x^2)}{x(1+e^3-e^{\frac{e^3}{15}}-4x+x^2)}+\frac{2(-1-e^3+e^{\frac{e^3}{15}}+x^2)\log (-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x}+x)}{x(1+e^3-e^{\frac{e^3}{15}}-4x+x^2)})dx\\ & =2\int \frac{(-1-e^3+e^{\frac{e^3}{15}}+x^2)\log (-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x}+x)}{x(1+e^3-e^{\frac{e^3}{15}}-4x+x^2)}dx+32\int \frac{-1-e^3+e^{\frac{e^3}{15}}+x^2}{x(1+e^3-e^{\frac{e^3}{15}}-4x+x^2)}dx\\ & =2\int (-\frac{\log (-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x}+x)}{x}+\frac{2(-2+x)\log (-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x}+x)}{1+e^3-e^{\frac{e^3}{15}}-4x+x^2})dx+32\int (-\frac{1}{x}+\frac{2(-2+x)}{1+e^3-e^{\frac{e^3}{15}}-4x+x^2})dx\\ & =-32\log (x)-2\int \frac{\log (-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x}+x)}{x}dx+4\int \frac{(-2+x)\log (-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x}+x)}{1+e^3-e^{\frac{e^3}{15}}-4x+x^2}dx+64\int \frac{-2+x}{1+e^3-e^{\frac{e^3}{15}}-4x+x^2}dx\\ & =-32\log (x)-2\log (x)\log (-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x}+x)+32\log (1+e^3-e^{\frac{e^3}{15}}-4x+x^2)+2\int \frac{(1-\frac{1+e^3-e^{\frac{e^3}{15}}}{x^2})\log (x)}{-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x}+x}dx+4\int (\frac{\log (-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x}+x)}{-4-2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x}+\frac{\log (-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x}+x)}{-4+2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x})dx\\ & =-32\log (x)-2\log (x)\log (-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x}+x)+32\log (1+e^3-e^{\frac{e^3}{15}}-4x+x^2)+2\int (-\frac{\log (x)}{x}+\frac{2(-2+x)\log (x)}{1+e^3-e^{\frac{e^3}{15}}-4x+x^2})dx+4\int \frac{\log (-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x}+x)}{-4-2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x}dx+4\int \frac{\log (-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x}+x)}{-4+2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x}dx\\ & =-32\log (x)-2\log (x)\log (-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x}+x)+2\log (-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x}+x)\log (-2(2-i\sqrt{-3+e^3-e^{\frac{e^3}{15}}})+2x)+2\log (-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x}+x)\log (-2(2+i\sqrt{-3+e^3-e^{\frac{e^3}{15}}})+2x)+32\log (1+e^3-e^{\frac{e^3}{15}}-4x+x^2)-2\int \frac{\log (x)}{x}dx-2\int \frac{(1-\frac{1+e^3-e^{\frac{e^3}{15}}}{x^2})\log (-4-2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x)}{-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x}+x}dx-2\int \frac{(1-\frac{1+e^3-e^{\frac{e^3}{15}}}{x^2})\log (-4+2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x)}{-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x}+x}dx+4\int \frac{(-2+x)\log (x)}{1+e^3-e^{\frac{e^3}{15}}-4x+x^2}dx\\ & =-32\log (x)-{\log }^2(x)-2\log (x)\log (-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x}+x)+2\log (-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x}+x)\log (-2(2-i\sqrt{-3+e^3-e^{\frac{e^3}{15}}})+2x)+2\log (-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x}+x)\log (-2(2+i\sqrt{-3+e^3-e^{\frac{e^3}{15}}})+2x)+32\log (1+e^3-e^{\frac{e^3}{15}}-4x+x^2)-2\int (-\frac{\log (-4-2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x)}{x}+\frac{2(-2+x)\log (-4-2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x)}{1+e^3-e^{\frac{e^3}{15}}-4x+x^2})dx-2\int (-\frac{\log (-4+2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x)}{x}+\frac{2(-2+x)\log (-4+2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x)}{1+e^3-e^{\frac{e^3}{15}}-4x+x^2})dx+4\int (\frac{\log (x)}{-4-2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x}+\frac{\log (x)}{-4+2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x})dx\\ & =-32\log (x)-{\log }^2(x)-2\log (x)\log (-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x}+x)+2\log (-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x}+x)\log (-2(2-i\sqrt{-3+e^3-e^{\frac{e^3}{15}}})+2x)+2\log (-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x}+x)\log (-2(2+i\sqrt{-3+e^3-e^{\frac{e^3}{15}}})+2x)+32\log (1+e^3-e^{\frac{e^3}{15}}-4x+x^2)+2\int \frac{\log (-4-2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x)}{x}dx+2\int \frac{\log (-4+2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x)}{x}dx+4\int \frac{\log (x)}{-4-2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x}dx+4\int \frac{\log (x)}{-4+2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x}dx-4\int \frac{(-2+x)\log (-4-2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x)}{1+e^3-e^{\frac{e^3}{15}}-4x+x^2}dx-4\int \frac{(-2+x)\log (-4+2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x)}{1+e^3-e^{\frac{e^3}{15}}-4x+x^2}dx\\ & =-32\log (x)-{\log }^2(x)-2\log (x)\log (-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x}+x)+2\log \left(\frac{x}{2-i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}}\right)\log (-2(2-i\sqrt{-3+e^3-e^{\frac{e^3}{15}}})+2x)+2\log (-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x}+x)\log (-2(2-i\sqrt{-3+e^3-e^{\frac{e^3}{15}}})+2x)+2\log \left(\frac{x}{2+i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}}\right)\log (-2(2+i\sqrt{-3+e^3-e^{\frac{e^3}{15}}})+2x)+2\log (-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x}+x)\log (-2(2+i\sqrt{-3+e^3-e^{\frac{e^3}{15}}})+2x)+2\log (x)\log (1-\frac{x}{2-i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}})+2\log (x)\log (1-\frac{x}{2+i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}})+32\log (1+e^3-e^{\frac{e^3}{15}}-4x+x^2)-2\int \frac{\log (1+\frac{2x}{-4-2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}})}{x}dx-2\int \frac{\log (1+\frac{2x}{-4+2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}})}{x}dx-4\int \frac{\log \left(\frac{2x}{4-2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}}\right)}{-4+2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x}dx-4\int \frac{\log \left(\frac{2x}{4+2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}}\right)}{-4-2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x}dx-4\int (\frac{\log (-4-2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x)}{-4-2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x}+\frac{\log (-4-2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x)}{-4+2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x})dx-4\int (\frac{\log (-4+2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x)}{-4-2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x}+\frac{\log (-4+2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x)}{-4+2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x})dx\\ & =-32\log (x)-{\log }^2(x)-2\log (x)\log (-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x}+x)+2\log \left(\frac{x}{2-i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}}\right)\log (-2(2-i\sqrt{-3+e^3-e^{\frac{e^3}{15}}})+2x)+2\log (-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x}+x)\log (-2(2-i\sqrt{-3+e^3-e^{\frac{e^3}{15}}})+2x)+2\log \left(\frac{x}{2+i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}}\right)\log (-2(2+i\sqrt{-3+e^3-e^{\frac{e^3}{15}}})+2x)+2\log (-4+\frac{1+e^3-e^{\frac{e^3}{15}}}{x}+x)\log (-2(2+i\sqrt{-3+e^3-e^{\frac{e^3}{15}}})+2x)+2\log (x)\log (1-\frac{x}{2-i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}})+2\log (x)\log (1-\frac{x}{2+i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}})+32\log (1+e^3-e^{\frac{e^3}{15}}-4x+x^2)+2{\text{Li}}_2\left(\frac{x}{2-i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}}\right)+2{\text{Li}}_2\left(\frac{x}{2+i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}}\right)+2{\text{Li}}_2(1-\frac{x}{2-i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}})+2{\text{Li}}_2(1-\frac{x}{2+i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}})-4\int \frac{\log (-4-2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x)}{-4-2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x}dx-4\int \frac{\log (-4-2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x)}{-4+2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x}dx-4\int \frac{\log (-4+2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x)}{-4-2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x}dx-4\int \frac{\log (-4+2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x)}{-4+2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+2x}dx\\ & =\text{Rest of rules removed due to large latex content}\end{array}\end{array}$$

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Mathematica [C]  time = 0.37, size = 949, normalized size = 33.89 $$\begin{array}{c}2(\log \left(\frac{2i+\sqrt{-3+e^3-e^{\frac{e^3}{15}}}-ix}{2i+\sqrt{-3+e^3-e^{\frac{e^3}{15}}}}\right)\log (x)+\log \left(\frac{-2i+\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+ix}{-2i+\sqrt{-3+e^3-e^{\frac{e^3}{15}}}}\right)\log (x)-\frac{{\log }^2(x)}{2}-\log \left(\frac{2i+\sqrt{-3+e^3-e^{\frac{e^3}{15}}}-ix}{2\sqrt{-3+e^3-e^{\frac{e^3}{15}}}}\right)\log \left(2(-2-i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+x)\right)+\log \left(\frac{2x}{4+2i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}}\right)\log \left(2(-2-i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+x)\right)-\frac{1}{2}{\log }^2\left(2(-2-i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+x)\right)-\log \left(\frac{-2i+\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+ix}{2\sqrt{-3+e^3-e^{\frac{e^3}{15}}}}\right)\log \left(2(-2+i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+x)\right)+\log \left(\frac{ix}{2i+\sqrt{-3+e^3-e^{\frac{e^3}{15}}}}\right)\log \left(2(-2+i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+x)\right)-\frac{1}{2}{\log }^2\left(2(-2+i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+x)\right)-\log (x)(16+\log \left(\frac{1+e^3-e^{\frac{e^3}{15}}-4x+x^2}{x}\right))+\log \left(2(-2-i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+x)\right)(16+\log \left(\frac{1+e^3-e^{\frac{e^3}{15}}-4x+x^2}{x}\right))+\log \left(2(-2+i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+x)\right)(16+\log \left(\frac{1+e^3-e^{\frac{e^3}{15}}-4x+x^2}{x}\right))-{\text{Li}}_2\left(\frac{2i+\sqrt{-3+e^3-e^{\frac{e^3}{15}}}-ix}{2\sqrt{-3+e^3-e^{\frac{e^3}{15}}}}\right)+{\text{Li}}_2\left(\frac{2i+\sqrt{-3+e^3-e^{\frac{e^3}{15}}}-ix}{2i+\sqrt{-3+e^3-e^{\frac{e^3}{15}}}}\right)-{\text{Li}}_2\left(\frac{-2i+\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+ix}{2\sqrt{-3+e^3-e^{\frac{e^3}{15}}}}\right)+{\text{Li}}_2\left(\frac{-2i+\sqrt{-3+e^3-e^{\frac{e^3}{15}}}+ix}{-2i+\sqrt{-3+e^3-e^{\frac{e^3}{15}}}}\right)+{\text{Li}}_2\left(\frac{x}{2+i\sqrt{-3+e^3-e^{\frac{e^3}{15}}}}\right)+{\text{Li}}_2\left(\frac{ix}{2i+\sqrt{-3+e^3-e^{\frac{e^3}{15}}}}\right))\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(32 + 32*E^3 - 32*E^(E^3/15) - 32*x^2 + (2 + 2*E^3 - 2*E^(E^3/15) - 2*x^2)*Log[(1 + E^3 - E^(E^3/15)
- 4*x + x^2)/x])/(-x - E^3*x + E^(E^3/15)*x + 4*x^2 - x^3),x]

[Out]

2*(Log[(2*I + Sqrt[-3 + E^3 - E^(E^3/15)] - I*x)/(2*I + Sqrt[-3 + E^3 - E^(E^3/15)])]*Log[x] + Log[(-2*I + Sqr
t[-3 + E^3 - E^(E^3/15)] + I*x)/(-2*I + Sqrt[-3 + E^3 - E^(E^3/15)])]*Log[x] - Log[x]^2/2 - Log[(2*I + Sqrt[-3
+ E^3 - E^(E^3/15)] - I*x)/(2*Sqrt[-3 + E^3 - E^(E^3/15)])]*Log[2*(-2 - I*Sqrt[-3 + E^3 - E^(E^3/15)] + x)] +
Log[(2*x)/(4 + (2*I)*Sqrt[-3 + E^3 - E^(E^3/15)])]*Log[2*(-2 - I*Sqrt[-3 + E^3 - E^(E^3/15)] + x)] - Log[2*(-
2 - I*Sqrt[-3 + E^3 - E^(E^3/15)] + x)]^2/2 - Log[(-2*I + Sqrt[-3 + E^3 - E^(E^3/15)] + I*x)/(2*Sqrt[-3 + E^3
- E^(E^3/15)])]*Log[2*(-2 + I*Sqrt[-3 + E^3 - E^(E^3/15)] + x)] + Log[(I*x)/(2*I + Sqrt[-3 + E^3 - E^(E^3/15)]
)]*Log[2*(-2 + I*Sqrt[-3 + E^3 - E^(E^3/15)] + x)] - Log[2*(-2 + I*Sqrt[-3 + E^3 - E^(E^3/15)] + x)]^2/2 - Log
[x]*(16 + Log[(1 + E^3 - E^(E^3/15) - 4*x + x^2)/x]) + Log[2*(-2 - I*Sqrt[-3 + E^3 - E^(E^3/15)] + x)]*(16 + L
og[(1 + E^3 - E^(E^3/15) - 4*x + x^2)/x]) + Log[2*(-2 + I*Sqrt[-3 + E^3 - E^(E^3/15)] + x)]*(16 + Log[(1 + E^3
- E^(E^3/15) - 4*x + x^2)/x]) - PolyLog[2, (2*I + Sqrt[-3 + E^3 - E^(E^3/15)] - I*x)/(2*Sqrt[-3 + E^3 - E^(E^
3/15)])] + PolyLog[2, (2*I + Sqrt[-3 + E^3 - E^(E^3/15)] - I*x)/(2*I + Sqrt[-3 + E^3 - E^(E^3/15)])] - PolyLog
[2, (-2*I + Sqrt[-3 + E^3 - E^(E^3/15)] + I*x)/(2*Sqrt[-3 + E^3 - E^(E^3/15)])] + PolyLog[2, (-2*I + Sqrt[-3 +
E^3 - E^(E^3/15)] + I*x)/(-2*I + Sqrt[-3 + E^3 - E^(E^3/15)])] + PolyLog[2, x/(2 + I*Sqrt[-3 + E^3 - E^(E^3/1
5)])] + PolyLog[2, (I*x)/(2*I + Sqrt[-3 + E^3 - E^(E^3/15)])])

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fricas [B]  time = 0.60, size = 49, normalized size = 1.75 $$\begin{array}{c}\log {\left(\frac{x^2-4x+e^3-e^{\left(\frac{1}{15}{e}^3\right)}+1}{x}\right)}^2+32\log \left(\frac{x^2-4x+e^3-e^{\left(\frac{1}{15}{e}^3\right)}+1}{x}\right)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(1/15*exp(3))+2*exp(3)-2*x^2+2)*log((-exp(1/15*exp(3))+exp(3)+x^2-4*x+1)/x)-32*exp(1/15*exp(
3))+32*exp(3)-32*x^2+32)/(x*exp(1/15*exp(3))-x*exp(3)-x^3+4*x^2-x),x, algorithm="fricas")

[Out]

log((x^2 - 4*x + e^3 - e^(1/15*e^3) + 1)/x)^2 + 32*log((x^2 - 4*x + e^3 - e^(1/15*e^3) + 1)/x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}\int \frac{2(16x^2+(x^2-e^3+e^{\left(\frac{1}{15}{e}^3\right)}-1)\log \left(\frac{x^2-4x+e^3-e^{\left(\frac{1}{15}{e}^3\right)}+1}{x}\right)-16e^3+16e^{\left(\frac{1}{15}{e}^3\right)}-16)}{x^3-4x^2+x{e}^3-x{e}^{\left(\frac{1}{15}{e}^3\right)}+x}dx\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(1/15*exp(3))+2*exp(3)-2*x^2+2)*log((-exp(1/15*exp(3))+exp(3)+x^2-4*x+1)/x)-32*exp(1/15*exp(
3))+32*exp(3)-32*x^2+32)/(x*exp(1/15*exp(3))-x*exp(3)-x^3+4*x^2-x),x, algorithm="giac")

[Out]

integrate(2*(16*x^2 + (x^2 - e^3 + e^(1/15*e^3) - 1)*log((x^2 - 4*x + e^3 - e^(1/15*e^3) + 1)/x) - 16*e^3 + 16
*e^(1/15*e^3) - 16)/(x^3 - 4*x^2 + x*e^3 - x*e^(1/15*e^3) + x), x)

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maple [B]  time = 0.53, size = 50, normalized size = 1.79

method	result	size
norman	$\ln {\left(\frac{-{\mathrm{e}}^{\frac{{\mathrm{e}}^3}{15}}+{\mathrm{e}}^3+x^2-4x+1}{x}\right)}^2+32\ln \left(\frac{-{\mathrm{e}}^{\frac{{\mathrm{e}}^3}{15}}+{\mathrm{e}}^3+x^2-4x+1}{x}\right)$	$50$
default	error in gcdex: invalid arguments\	N/A

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*exp(1/15*exp(3))+2*exp(3)-2*x^2+2)*ln((-exp(1/15*exp(3))+exp(3)+x^2-4*x+1)/x)-32*exp(1/15*exp(3))+32*
exp(3)-32*x^2+32)/(x*exp(1/15*exp(3))-x*exp(3)-x^3+4*x^2-x),x,method=_RETURNVERBOSE)

[Out]

ln((-exp(1/15*exp(3))+exp(3)+x^2-4*x+1)/x)^2+32*ln((-exp(1/15*exp(3))+exp(3)+x^2-4*x+1)/x)

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maxima [B]  time = 3.55, size = 399, normalized size = 14.25 $$\begin{array}{c}16(\frac{\log (x^2-4x+e^3-e^{\left(\frac{1}{15}{e}^3\right)}+1)}{e^3-e^{\left(\frac{1}{15}{e}^3\right)}+1}-\frac{2\log (x)}{e^3-e^{\left(\frac{1}{15}{e}^3\right)}+1}-\frac{4\arctan \left(\frac{x-2}{\sqrt{e^3-e^{\left(\frac{1}{15}{e}^3\right)}-3}}\right)}{(e^3-e^{\left(\frac{1}{15}{e}^3\right)}+1)\sqrt{e^3-e^{\left(\frac{1}{15}{e}^3\right)}-3}})e^3-16(\frac{\log (x^2-4x+e^3-e^{\left(\frac{1}{15}{e}^3\right)}+1)}{e^3-e^{\left(\frac{1}{15}{e}^3\right)}+1}-\frac{2\log (x)}{e^3-e^{\left(\frac{1}{15}{e}^3\right)}+1}-\frac{4\arctan \left(\frac{x-2}{\sqrt{e^3-e^{\left(\frac{1}{15}{e}^3\right)}-3}}\right)}{(e^3-e^{\left(\frac{1}{15}{e}^3\right)}+1)\sqrt{e^3-e^{\left(\frac{1}{15}{e}^3\right)}-3}})e^{\left(\frac{1}{15}{e}^3\right)}+\log {(x^2-4x+e^3-e^{\left(\frac{1}{15}{e}^3\right)}+1)}^2-2\log (x^2-4x+e^3-e^{\left(\frac{1}{15}{e}^3\right)}+1)\log (x)+\log (x{)}^2+\frac{64\arctan \left(\frac{x-2}{\sqrt{e^3-e^{\left(\frac{1}{15}{e}^3\right)}-3}}\right)}{\sqrt{e^3-e^{\left(\frac{1}{15}{e}^3\right)}-3}}+\frac{16\log (x^2-4x+e^3-e^{\left(\frac{1}{15}{e}^3\right)}+1)}{e^3-e^{\left(\frac{1}{15}{e}^3\right)}+1}-\frac{32\log (x)}{e^3-e^{\left(\frac{1}{15}{e}^3\right)}+1}-\frac{64\arctan \left(\frac{x-2}{\sqrt{e^3-e^{\left(\frac{1}{15}{e}^3\right)}-3}}\right)}{(e^3-e^{\left(\frac{1}{15}{e}^3\right)}+1)\sqrt{e^3-e^{\left(\frac{1}{15}{e}^3\right)}-3}}+16\log (x^2-4x+e^3-e^{\left(\frac{1}{15}{e}^3\right)}+1)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(1/15*exp(3))+2*exp(3)-2*x^2+2)*log((-exp(1/15*exp(3))+exp(3)+x^2-4*x+1)/x)-32*exp(1/15*exp(
3))+32*exp(3)-32*x^2+32)/(x*exp(1/15*exp(3))-x*exp(3)-x^3+4*x^2-x),x, algorithm="maxima")

[Out]

16*(log(x^2 - 4*x + e^3 - e^(1/15*e^3) + 1)/(e^3 - e^(1/15*e^3) + 1) - 2*log(x)/(e^3 - e^(1/15*e^3) + 1) - 4*a
rctan((x - 2)/sqrt(e^3 - e^(1/15*e^3) - 3))/((e^3 - e^(1/15*e^3) + 1)*sqrt(e^3 - e^(1/15*e^3) - 3)))*e^3 - 16*
(log(x^2 - 4*x + e^3 - e^(1/15*e^3) + 1)/(e^3 - e^(1/15*e^3) + 1) - 2*log(x)/(e^3 - e^(1/15*e^3) + 1) - 4*arct
an((x - 2)/sqrt(e^3 - e^(1/15*e^3) - 3))/((e^3 - e^(1/15*e^3) + 1)*sqrt(e^3 - e^(1/15*e^3) - 3)))*e^(1/15*e^3)
+ log(x^2 - 4*x + e^3 - e^(1/15*e^3) + 1)^2 - 2*log(x^2 - 4*x + e^3 - e^(1/15*e^3) + 1)*log(x) + log(x)^2 + 6
4*arctan((x - 2)/sqrt(e^3 - e^(1/15*e^3) - 3))/sqrt(e^3 - e^(1/15*e^3) - 3) + 16*log(x^2 - 4*x + e^3 - e^(1/15
*e^3) + 1)/(e^3 - e^(1/15*e^3) + 1) - 32*log(x)/(e^3 - e^(1/15*e^3) + 1) - 64*arctan((x - 2)/sqrt(e^3 - e^(1/1
5*e^3) - 3))/((e^3 - e^(1/15*e^3) + 1)*sqrt(e^3 - e^(1/15*e^3) - 3)) + 16*log(x^2 - 4*x + e^3 - e^(1/15*e^3) +
1)

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mupad [B]  time = 33.35, size = 49, normalized size = 1.75 $$\begin{array}{c}{\ln \left(\frac{x^2-4x+{\mathrm{e}}^3-{\left({\mathrm{e}}^{{\mathrm{e}}^3}\right)}^{1/15}+1}{x}\right)}^2+32\ln (x^2-4x-{\mathrm{e}}^{\frac{{\mathrm{e}}^3}{15}}+{\mathrm{e}}^3+1)-32\ln (x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((32*exp(exp(3)/15) - 32*exp(3) + 32*x^2 + log((exp(3) - exp(exp(3)/15) - 4*x + x^2 + 1)/x)*(2*exp(exp(3)/1
5) - 2*exp(3) + 2*x^2 - 2) - 32)/(x - x*exp(exp(3)/15) + x*exp(3) - 4*x^2 + x^3),x)

[Out]

32*log(exp(3) - exp(exp(3)/15) - 4*x + x^2 + 1) - 32*log(x) + log((exp(3) - 4*x - exp(exp(3))^(1/15) + x^2 + 1
)/x)^2

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sympy [B]  time = 49.07, size = 49, normalized size = 1.75 $$\begin{array}{c}-32\log (x)+\log {\left(\frac{x^2-4x-e^{\frac{e^3}{15}}+1+e^3}{x}\right)}^2+32\log (x^2-4x-e^{\frac{e^3}{15}}+1+e^3)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(1/15*exp(3))+2*exp(3)-2*x**2+2)*ln((-exp(1/15*exp(3))+exp(3)+x**2-4*x+1)/x)-32*exp(1/15*exp
(3))+32*exp(3)-32*x**2+32)/(x*exp(1/15*exp(3))-x*exp(3)-x**3+4*x**2-x),x)

[Out]

-32*log(x) + log((x**2 - 4*x - exp(exp(3)/15) + 1 + exp(3))/x)**2 + 32*log(x**2 - 4*x - exp(exp(3)/15) + 1 + e
xp(3))

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