\(\int \frac {e^5 (54 e x+e^2 (36+6 x^3))+(-18 e x+e^2 (-12-2 x^3)) \log (\frac {e x^2+x^3}{-3+x^3})}{-3 x^2+x^5+e (-3 x+x^4)} \, dx\) [782]

Optimal result

Integrand size = 77, antiderivative size = 26 \[ \int \frac {e^5 \left (54 e x+e^2 \left (36+6 x^3\right )\right )+\left (-18 e x+e^2 \left (-12-2 x^3\right )\right ) \log \left (\frac {e x^2+x^3}{-3+x^3}\right )}{-3 x^2+x^5+e \left (-3 x+x^4\right )} \, dx=e \left (3 e^5-\log \left (\frac {e+x}{-\frac {3}{x^2}+x}\right )\right )^2 \]

[Out]

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

Rubi [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 2.06 (sec) , antiderivative size = 1214, normalized size of antiderivative = 46.69, number of steps used = 86, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.221, Rules used = {6820, 12, 2608, 2604, 2404, 2338, 2354, 2438, 2375, 2465, 2439, 2437, 266, 2463, 2441, 2440, 2352} \[ \int \frac {e^5 \left (54 e x+e^2 \left (36+6 x^3\right )\right )+\left (-18 e x+e^2 \left (-12-2 x^3\right )\right ) \log \left (\frac {e x^2+x^3}{-3+x^3}\right )}{-3 x^2+x^5+e \left (-3 x+x^4\right )} \, dx =\text {Too large to display} \]

[In]

Int[(E^5*(54*E*x + E^2*(36 + 6*x^3)) + (-18*E*x + E^2*(-12 - 2*x^3))*Log[(E*x^2 + x^3)/(-3 + x^3)])/(-3*x^2 +
x^5 + E*(-3*x + x^4)),x]

[Out]

-(E*Log[-(-3)^(1/3) - x]^2) - E*Log[3^(1/3) - x]^2 - 2*E*Log[-(-3)^(1/3) - x]*Log[(3^(1/3) - x)/((-3)^(1/3) +
3^(1/3))] - 2*E*Log[(3^(1/3) - x)/(3^(1/3)*(1 - (-1)^(2/3)))]*Log[(-1)^(2/3)*3^(1/3) - x] - E*Log[(-1)^(2/3)*3
^(1/3) - x]^2 - 2*E*Log[-(-3)^(1/3) - x]*Log[-(((-1)^(2/3)*((-1)^(2/3)*3^(1/3) - x))/((-3)^(1/3) + 3^(1/3)))]
- 4*E*Log[x] + (4*E*Log[3]*Log[x])/3 - 4*E*Log[x]^2 + 4*E*Log[(-1)^(2/3)*3^(1/3) - x]*Log[-((-1/3)^(1/3)*x)] +
4*E*Log[-(-3)^(1/3) - x]*Log[((-1)^(2/3)*x)/3^(1/3)] - 2*E*Log[(-1)^(2/3)*3^(1/3) - x]*Log[((-I)*((-3)^(1/3)
+ x))/3^(5/6)] - 2*E*Log[3^(1/3) - x]*Log[((-3)^(1/3) + x)/((-3)^(1/3) + 3^(1/3))] + 2*E*Log[(3^(1/3) - x)/(3^
(1/3) + E)]*Log[E + x] + 2*E*Log[((-3)^(1/3) + x)/((-3)^(1/3) - E)]*Log[E + x] - E*Log[E + x]^2 + 2*E*Log[-(-3
)^(1/3) - x]*Log[-((E + x)/((-3)^(1/3) - E))] + 2*E*Log[3^(1/3) - x]*Log[(E + x)/(3^(1/3) + E)] + 2*E*Log[(-1)
^(2/3)*3^(1/3) - x]*Log[(E + x)/((-1)^(2/3)*3^(1/3) + E)] - 2*E*Log[3^(1/3) - x]*Log[-(((-1)^(2/3)*(3^(1/3) +
(-1)^(1/3)*x))/(3^(1/3)*(1 - (-1)^(2/3))))] + 2*E*Log[E + x]*Log[((-1)^(2/3)*(3^(1/3) + (-1)^(1/3)*x))/((-1)^(
2/3)*3^(1/3) + E)] - 4*E*Log[x]*Log[1 + x/E] + 2*E*Log[-(-3)^(1/3) - x]*(3*E^5 - Log[-((x^2*(E + x))/(3 - x^3)
)]) + 2*E*Log[3^(1/3) - x]*(3*E^5 - Log[-((x^2*(E + x))/(3 - x^3))]) + 2*E*Log[(-1)^(2/3)*3^(1/3) - x]*(3*E^5
- Log[-((x^2*(E + x))/(3 - x^3))]) - 4*E*Log[x]*(3*E^5 - Log[-((x^2*(E + x))/(3 - x^3))]) - 2*E*Log[E + x]*(3*
E^5 - Log[-((x^2*(E + x))/(3 - x^3))]) + 4*E*Log[x]*Log[1 - x^3/3] - 2*E*PolyLog[2, (3^(1/3) - x)/((-3)^(1/3)
+ 3^(1/3))] - 2*E*PolyLog[2, (2*(3^(1/3) - x))/(3^(1/3)*(3 - I*Sqrt[3]))] + 2*E*PolyLog[2, (3^(1/3) - x)/(3^(1
/3) + E)] - 2*E*PolyLog[2, ((-1)^(1/6)*3^(1/3) + I*x)/3^(5/6)] - 2*E*PolyLog[2, (3^(1/3)*(I + Sqrt[3]) + (2*I)
*x)/(3^(1/3)*(3*I + Sqrt[3]))] - 4*E*PolyLog[2, x/3^(1/3)] + (4*E*PolyLog[2, x^3/3])/3 - 2*E*PolyLog[2, ((-3)^
(1/3) + x)/((-3)^(1/3) + 3^(1/3))] + 2*E*PolyLog[2, ((-3)^(1/3) + x)/((-3)^(1/3) - E)] + 2*E*PolyLog[2, -((E +
x)/((-3)^(1/3) - E))] + 2*E*PolyLog[2, (E + x)/(3^(1/3) + E)] + 2*E*PolyLog[2, (E + x)/((-1)^(2/3)*3^(1/3) +
E)] + 2*E*PolyLog[2, ((-1)^(2/3)*(3^(1/3) + (-1)^(1/3)*x))/((-1)^(2/3)*3^(1/3) + E)] - 2*E*PolyLog[2, (3^(1/3)
- (-1)^(2/3)*x)/((-3)^(1/3) + 3^(1/3))] + 4*E*PolyLog[2, 1 + (-1/3)^(1/3)*x] + 4*E*PolyLog[2, 1 - ((-1)^(2/3)
*x)/3^(1/3)]

Rule 12

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

Rule 266

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

Rule 2338

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 2352

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

Rule 2354

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 2375

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^(r_)), x_Symbol] :> Si
mp[f^m*Log[1 + e*(x^r/d)]*((a + b*Log[c*x^n])^p/(e*r)), x] - Dist[b*f^m*n*(p/(e*r)), Int[Log[1 + e*(x^r/d)]*((
a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] &
& (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n]

Rule 2404

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 2437

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 2438

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 2439

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

Rule 2440

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 2441

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 2463

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2465

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 2604

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 2608

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 6820

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {2 e \left (6 e+9 x+e x^3\right ) \left (-3 e^5+\log \left (\frac {x^2 (e+x)}{-3+x^3}\right )\right )}{x (e+x) \left (3-x^3\right )} \, dx \\ & = (2 e) \int \frac {\left (6 e+9 x+e x^3\right ) \left (-3 e^5+\log \left (\frac {x^2 (e+x)}{-3+x^3}\right )\right )}{x (e+x) \left (3-x^3\right )} \, dx \\ & = (2 e) \int \left (\frac {2 \left (-3 e^5+\log \left (\frac {x^2 (e+x)}{-3+x^3}\right )\right )}{x}+\frac {-3 e^5+\log \left (\frac {x^2 (e+x)}{-3+x^3}\right )}{e+x}-\frac {3 x^2 \left (-3 e^5+\log \left (\frac {x^2 (e+x)}{-3+x^3}\right )\right )}{-3+x^3}\right ) \, dx \\ & = (2 e) \int \frac {-3 e^5+\log \left (\frac {x^2 (e+x)}{-3+x^3}\right )}{e+x} \, dx+(4 e) \int \frac {-3 e^5+\log \left (\frac {x^2 (e+x)}{-3+x^3}\right )}{x} \, dx-(6 e) \int \frac {x^2 \left (-3 e^5+\log \left (\frac {x^2 (e+x)}{-3+x^3}\right )\right )}{-3+x^3} \, dx \\ & = -4 e \log (x) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )-2 e \log (e+x) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )-(2 e) \int \frac {\left (-3+x^3\right ) \left (-\frac {3 x^4 (e+x)}{\left (-3+x^3\right )^2}+\frac {x^2}{-3+x^3}+\frac {2 x (e+x)}{-3+x^3}\right ) \log (e+x)}{x^2 (e+x)} \, dx-(4 e) \int \frac {\left (-3+x^3\right ) \left (-\frac {3 x^4 (e+x)}{\left (-3+x^3\right )^2}+\frac {x^2}{-3+x^3}+\frac {2 x (e+x)}{-3+x^3}\right ) \log (x)}{x^2 (e+x)} \, dx-(6 e) \int \left (-\frac {-3 e^5+\log \left (\frac {x^2 (e+x)}{-3+x^3}\right )}{3 \left (-\sqrt [3]{-3}-x\right )}-\frac {-3 e^5+\log \left (\frac {x^2 (e+x)}{-3+x^3}\right )}{3 \left (\sqrt [3]{3}-x\right )}-\frac {-3 e^5+\log \left (\frac {x^2 (e+x)}{-3+x^3}\right )}{3 \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}\right ) \, dx \\ & = -4 e \log (x) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )-2 e \log (e+x) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )-(2 e) \int \left (\frac {2 \log (e+x)}{x}+\frac {\log (e+x)}{e+x}-\frac {3 x^2 \log (e+x)}{-3+x^3}\right ) \, dx+(2 e) \int \frac {-3 e^5+\log \left (\frac {x^2 (e+x)}{-3+x^3}\right )}{-\sqrt [3]{-3}-x} \, dx+(2 e) \int \frac {-3 e^5+\log \left (\frac {x^2 (e+x)}{-3+x^3}\right )}{\sqrt [3]{3}-x} \, dx+(2 e) \int \frac {-3 e^5+\log \left (\frac {x^2 (e+x)}{-3+x^3}\right )}{(-1)^{2/3} \sqrt [3]{3}-x} \, dx-(4 e) \int \left (\frac {2 \log (x)}{x}+\frac {\log (x)}{e+x}-\frac {3 x^2 \log (x)}{-3+x^3}\right ) \, dx \\ & = 2 e \log \left (-\sqrt [3]{-3}-x\right ) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )+2 e \log \left (\sqrt [3]{3}-x\right ) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )+2 e \log \left ((-1)^{2/3} \sqrt [3]{3}-x\right ) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )-4 e \log (x) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )-2 e \log (e+x) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )+(2 e) \int \frac {\left (-3+x^3\right ) \left (-\frac {3 x^4 (e+x)}{\left (-3+x^3\right )^2}+\frac {x^2}{-3+x^3}+\frac {2 x (e+x)}{-3+x^3}\right ) \log \left (-\sqrt [3]{-3}-x\right )}{x^2 (e+x)} \, dx+(2 e) \int \frac {\left (-3+x^3\right ) \left (-\frac {3 x^4 (e+x)}{\left (-3+x^3\right )^2}+\frac {x^2}{-3+x^3}+\frac {2 x (e+x)}{-3+x^3}\right ) \log \left (\sqrt [3]{3}-x\right )}{x^2 (e+x)} \, dx+(2 e) \int \frac {\left (-3+x^3\right ) \left (-\frac {3 x^4 (e+x)}{\left (-3+x^3\right )^2}+\frac {x^2}{-3+x^3}+\frac {2 x (e+x)}{-3+x^3}\right ) \log \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}{x^2 (e+x)} \, dx-(2 e) \int \frac {\log (e+x)}{e+x} \, dx-(4 e) \int \frac {\log (x)}{e+x} \, dx-(4 e) \int \frac {\log (e+x)}{x} \, dx+(6 e) \int \frac {x^2 \log (e+x)}{-3+x^3} \, dx-(8 e) \int \frac {\log (x)}{x} \, dx+(12 e) \int \frac {x^2 \log (x)}{-3+x^3} \, dx \\ & = -4 e \log (x)-4 e \log ^2(x)-4 e \log (x) \log \left (1+\frac {x}{e}\right )+2 e \log \left (-\sqrt [3]{-3}-x\right ) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )+2 e \log \left (\sqrt [3]{3}-x\right ) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )+2 e \log \left ((-1)^{2/3} \sqrt [3]{3}-x\right ) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )-4 e \log (x) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )-2 e \log (e+x) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )+4 e \log (x) \log \left (1-\frac {x^3}{3}\right )+(2 e) \int \left (\frac {2 \log \left (-\sqrt [3]{-3}-x\right )}{x}+\frac {\log \left (-\sqrt [3]{-3}-x\right )}{e+x}-\frac {3 x^2 \log \left (-\sqrt [3]{-3}-x\right )}{-3+x^3}\right ) \, dx+(2 e) \int \left (\frac {2 \log \left (\sqrt [3]{3}-x\right )}{x}+\frac {\log \left (\sqrt [3]{3}-x\right )}{e+x}-\frac {3 x^2 \log \left (\sqrt [3]{3}-x\right )}{-3+x^3}\right ) \, dx+(2 e) \int \left (\frac {2 \log \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}{x}+\frac {\log \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}{e+x}-\frac {3 x^2 \log \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}{-3+x^3}\right ) \, dx-(2 e) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,e+x\right )-(4 e) \int \frac {\log \left (1-\frac {x^3}{3}\right )}{x} \, dx+(6 e) \int \left (-\frac {\log (e+x)}{3 \left (-\sqrt [3]{-3}-x\right )}-\frac {\log (e+x)}{3 \left (\sqrt [3]{3}-x\right )}-\frac {\log (e+x)}{3 \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}\right ) \, dx \\ & = -4 e \log (x)-4 e \log ^2(x)-e \log ^2(e+x)-4 e \log (x) \log \left (1+\frac {x}{e}\right )+2 e \log \left (-\sqrt [3]{-3}-x\right ) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )+2 e \log \left (\sqrt [3]{3}-x\right ) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )+2 e \log \left ((-1)^{2/3} \sqrt [3]{3}-x\right ) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )-4 e \log (x) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )-2 e \log (e+x) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )+4 e \log (x) \log \left (1-\frac {x^3}{3}\right )+\frac {4}{3} e \operatorname {PolyLog}\left (2,\frac {x^3}{3}\right )+(2 e) \int \frac {\log \left (-\sqrt [3]{-3}-x\right )}{e+x} \, dx+(2 e) \int \frac {\log \left (\sqrt [3]{3}-x\right )}{e+x} \, dx+(2 e) \int \frac {\log \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}{e+x} \, dx-(2 e) \int \frac {\log (e+x)}{-\sqrt [3]{-3}-x} \, dx-(2 e) \int \frac {\log (e+x)}{\sqrt [3]{3}-x} \, dx-(2 e) \int \frac {\log (e+x)}{(-1)^{2/3} \sqrt [3]{3}-x} \, dx+(4 e) \int \frac {\log \left (-\sqrt [3]{-3}-x\right )}{x} \, dx+(4 e) \int \frac {\log \left (\sqrt [3]{3}-x\right )}{x} \, dx+(4 e) \int \frac {\log \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}{x} \, dx-(6 e) \int \frac {x^2 \log \left (-\sqrt [3]{-3}-x\right )}{-3+x^3} \, dx-(6 e) \int \frac {x^2 \log \left (\sqrt [3]{3}-x\right )}{-3+x^3} \, dx-(6 e) \int \frac {x^2 \log \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}{-3+x^3} \, dx \\ & = -4 e \log (x)+\frac {4}{3} e \log (3) \log (x)-4 e \log ^2(x)+4 e \log \left ((-1)^{2/3} \sqrt [3]{3}-x\right ) \log \left (-\sqrt [3]{-\frac {1}{3}} x\right )+4 e \log \left (-\sqrt [3]{-3}-x\right ) \log \left (\frac {(-1)^{2/3} x}{\sqrt [3]{3}}\right )+2 e \log \left (\frac {\sqrt [3]{3}-x}{\sqrt [3]{3}+e}\right ) \log (e+x)+2 e \log \left (\frac {\sqrt [3]{-3}+x}{\sqrt [3]{-3}-e}\right ) \log (e+x)-e \log ^2(e+x)+2 e \log \left (-\sqrt [3]{-3}-x\right ) \log \left (-\frac {e+x}{\sqrt [3]{-3}-e}\right )+2 e \log \left (\sqrt [3]{3}-x\right ) \log \left (\frac {e+x}{\sqrt [3]{3}+e}\right )+2 e \log \left ((-1)^{2/3} \sqrt [3]{3}-x\right ) \log \left (\frac {e+x}{(-1)^{2/3} \sqrt [3]{3}+e}\right )+2 e \log (e+x) \log \left (\frac {(-1)^{2/3} \left (\sqrt [3]{3}+\sqrt [3]{-1} x\right )}{(-1)^{2/3} \sqrt [3]{3}+e}\right )-4 e \log (x) \log \left (1+\frac {x}{e}\right )+2 e \log \left (-\sqrt [3]{-3}-x\right ) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )+2 e \log \left (\sqrt [3]{3}-x\right ) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )+2 e \log \left ((-1)^{2/3} \sqrt [3]{3}-x\right ) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )-4 e \log (x) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )-2 e \log (e+x) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )+4 e \log (x) \log \left (1-\frac {x^3}{3}\right )+\frac {4}{3} e \operatorname {PolyLog}\left (2,\frac {x^3}{3}\right )-(2 e) \int \frac {\log \left (\frac {-\sqrt [3]{-3}-x}{-\sqrt [3]{-3}+e}\right )}{e+x} \, dx-(2 e) \int \frac {\log \left (\frac {\sqrt [3]{3}-x}{\sqrt [3]{3}+e}\right )}{e+x} \, dx-(2 e) \int \frac {\log \left (\frac {(-1)^{2/3} \sqrt [3]{3}-x}{(-1)^{2/3} \sqrt [3]{3}+e}\right )}{e+x} \, dx+(2 e) \int \frac {\log \left (\frac {-e-x}{\sqrt [3]{-3}-e}\right )}{-\sqrt [3]{-3}-x} \, dx+(2 e) \int \frac {\log \left (\frac {-e-x}{-\sqrt [3]{3}-e}\right )}{\sqrt [3]{3}-x} \, dx+(2 e) \int \frac {\log \left (\frac {-e-x}{-(-1)^{2/3} \sqrt [3]{3}-e}\right )}{(-1)^{2/3} \sqrt [3]{3}-x} \, dx+(4 e) \int \frac {\log \left (-\sqrt [3]{-\frac {1}{3}} x\right )}{(-1)^{2/3} \sqrt [3]{3}-x} \, dx+(4 e) \int \frac {\log \left (\frac {(-1)^{2/3} x}{\sqrt [3]{3}}\right )}{-\sqrt [3]{-3}-x} \, dx+(4 e) \int \frac {\log \left (1-\frac {x}{\sqrt [3]{3}}\right )}{x} \, dx-(6 e) \int \left (-\frac {\log \left (-\sqrt [3]{-3}-x\right )}{3 \left (-\sqrt [3]{-3}-x\right )}-\frac {\log \left (-\sqrt [3]{-3}-x\right )}{3 \left (\sqrt [3]{3}-x\right )}-\frac {\log \left (-\sqrt [3]{-3}-x\right )}{3 \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}\right ) \, dx-(6 e) \int \left (-\frac {\log \left (\sqrt [3]{3}-x\right )}{3 \left (-\sqrt [3]{-3}-x\right )}-\frac {\log \left (\sqrt [3]{3}-x\right )}{3 \left (\sqrt [3]{3}-x\right )}-\frac {\log \left (\sqrt [3]{3}-x\right )}{3 \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}\right ) \, dx-(6 e) \int \left (-\frac {\log \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}{3 \left (-\sqrt [3]{-3}-x\right )}-\frac {\log \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}{3 \left (\sqrt [3]{3}-x\right )}-\frac {\log \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}{3 \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}\right ) \, dx \\ & = -4 e \log (x)+\frac {4}{3} e \log (3) \log (x)-4 e \log ^2(x)+4 e \log \left ((-1)^{2/3} \sqrt [3]{3}-x\right ) \log \left (-\sqrt [3]{-\frac {1}{3}} x\right )+4 e \log \left (-\sqrt [3]{-3}-x\right ) \log \left (\frac {(-1)^{2/3} x}{\sqrt [3]{3}}\right )+2 e \log \left (\frac {\sqrt [3]{3}-x}{\sqrt [3]{3}+e}\right ) \log (e+x)+2 e \log \left (\frac {\sqrt [3]{-3}+x}{\sqrt [3]{-3}-e}\right ) \log (e+x)-e \log ^2(e+x)+2 e \log \left (-\sqrt [3]{-3}-x\right ) \log \left (-\frac {e+x}{\sqrt [3]{-3}-e}\right )+2 e \log \left (\sqrt [3]{3}-x\right ) \log \left (\frac {e+x}{\sqrt [3]{3}+e}\right )+2 e \log \left ((-1)^{2/3} \sqrt [3]{3}-x\right ) \log \left (\frac {e+x}{(-1)^{2/3} \sqrt [3]{3}+e}\right )+2 e \log (e+x) \log \left (\frac {(-1)^{2/3} \left (\sqrt [3]{3}+\sqrt [3]{-1} x\right )}{(-1)^{2/3} \sqrt [3]{3}+e}\right )-4 e \log (x) \log \left (1+\frac {x}{e}\right )+2 e \log \left (-\sqrt [3]{-3}-x\right ) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )+2 e \log \left (\sqrt [3]{3}-x\right ) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )+2 e \log \left ((-1)^{2/3} \sqrt [3]{3}-x\right ) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )-4 e \log (x) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )-2 e \log (e+x) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )+4 e \log (x) \log \left (1-\frac {x^3}{3}\right )-4 e \operatorname {PolyLog}\left (2,\frac {x}{\sqrt [3]{3}}\right )+\frac {4}{3} e \operatorname {PolyLog}\left (2,\frac {x^3}{3}\right )+4 e \operatorname {PolyLog}\left (2,1+\sqrt [3]{-\frac {1}{3}} x\right )+4 e \operatorname {PolyLog}\left (2,1-\frac {(-1)^{2/3} x}{\sqrt [3]{3}}\right )+(2 e) \int \frac {\log \left (-\sqrt [3]{-3}-x\right )}{-\sqrt [3]{-3}-x} \, dx+(2 e) \int \frac {\log \left (-\sqrt [3]{-3}-x\right )}{\sqrt [3]{3}-x} \, dx+(2 e) \int \frac {\log \left (-\sqrt [3]{-3}-x\right )}{(-1)^{2/3} \sqrt [3]{3}-x} \, dx+(2 e) \int \frac {\log \left (\sqrt [3]{3}-x\right )}{-\sqrt [3]{-3}-x} \, dx+(2 e) \int \frac {\log \left (\sqrt [3]{3}-x\right )}{\sqrt [3]{3}-x} \, dx+(2 e) \int \frac {\log \left (\sqrt [3]{3}-x\right )}{(-1)^{2/3} \sqrt [3]{3}-x} \, dx+(2 e) \int \frac {\log \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}{-\sqrt [3]{-3}-x} \, dx+(2 e) \int \frac {\log \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}{\sqrt [3]{3}-x} \, dx+(2 e) \int \frac {\log \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}{(-1)^{2/3} \sqrt [3]{3}-x} \, dx-(2 e) \text {Subst}\left (\int \frac {\log \left (1+\frac {x}{\sqrt [3]{-3}-e}\right )}{x} \, dx,x,-\sqrt [3]{-3}-x\right )-(2 e) \text {Subst}\left (\int \frac {\log \left (1+\frac {x}{-\sqrt [3]{3}-e}\right )}{x} \, dx,x,\sqrt [3]{3}-x\right )-(2 e) \text {Subst}\left (\int \frac {\log \left (1+\frac {x}{-(-1)^{2/3} \sqrt [3]{3}-e}\right )}{x} \, dx,x,(-1)^{2/3} \sqrt [3]{3}-x\right )-(2 e) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{-\sqrt [3]{-3}+e}\right )}{x} \, dx,x,e+x\right )-(2 e) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{\sqrt [3]{3}+e}\right )}{x} \, dx,x,e+x\right )-(2 e) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{(-1)^{2/3} \sqrt [3]{3}+e}\right )}{x} \, dx,x,e+x\right ) \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 1.12 (sec) , antiderivative size = 1186, normalized size of antiderivative = 45.62 \[ \int \frac {e^5 \left (54 e x+e^2 \left (36+6 x^3\right )\right )+\left (-18 e x+e^2 \left (-12-2 x^3\right )\right ) \log \left (\frac {e x^2+x^3}{-3+x^3}\right )}{-3 x^2+x^5+e \left (-3 x+x^4\right )} \, dx =\text {Too large to display} \]

[In]

Integrate[(E^5*(54*E*x + E^2*(36 + 6*x^3)) + (-18*E*x + E^2*(-12 - 2*x^3))*Log[(E*x^2 + x^3)/(-3 + x^3)])/(-3*
x^2 + x^5 + E*(-3*x + x^4)),x]

[Out]

2*E*(-1/2*Log[-(-3)^(1/3) - x]^2 + (2*Log[3]*Log[3^(1/3) - x])/3 - Log[3^(1/3) - x]^2/2 - Log[-(-3)^(1/3) - x]
*Log[(3^(1/3) - x)/((-3)^(1/3) + 3^(1/3))] - Log[(-1)^(2/3)*3^(1/3) - x]^2/2 + Log[(3^(1/3) - x)/(3^(1/3) + E)
]*Log[-E - x] + Log[((-1)^(2/3)*3^(1/3) - x)/((-1)^(2/3)*3^(1/3) + E)]*Log[-E - x] - Log[-E - x]^2/2 - 2*Log[-
E - x]*(-1 + Log[-x]) - 6*E^5*Log[x] + (2*Log[3]*Log[x])/3 - 2*Log[x]^2 + 2*Log[(-1)^(2/3)*3^(1/3) - x]*Log[-(
(-1/3)^(1/3)*x)] + 2*Log[-(-3)^(1/3) - x]*Log[((-1)^(2/3)*x)/3^(1/3)] - Log[(-1)^(2/3)*3^(1/3) - x]*Log[((-I)*
((-3)^(1/3) + x))/3^(5/6)] - Log[3^(1/3) - x]*Log[((-3)^(1/3) + x)/((-3)^(1/3) + 3^(1/3))] + Log[-E - x]*Log[(
(-3)^(1/3) + x)/((-3)^(1/3) - E)] - 2*Log[x]*(-1 + Log[E + x]) + Log[3^(1/3) + E]*Log[E + x] + Log[-(-3)^(1/3)
- x]*Log[-((E + x)/((-3)^(1/3) - E))] + Log[(-1)^(2/3)*3^(1/3) - x]*Log[(E + x)/((-1)^(2/3)*3^(1/3) + E)] + 2
*Log[x]*Log[(3 - (-3)^(2/3)*x)/3] - Log[-(-3)^(1/3) - x]*Log[((-3)^(1/3) + (-1)^(2/3)*x)/((-3)^(1/3) + 3^(1/3)
)] + 2*Log[x]*Log[1 + (-1/3)^(1/3)*x] - Log[3^(1/3) - x]*Log[(I + Sqrt[3] + ((2*I)*x)/3^(1/3))/(3*I + Sqrt[3])
] - Log[(-1)^(2/3)*3^(1/3) - x]*Log[(((-2*I)/3)*(-3 + 3^(2/3)*x))/(3*I + Sqrt[3])] + Log[-(-3)^(1/3) - x]*(3*E
^5 - Log[(x^2*(E + x))/(-3 + x^3)]) + Log[3^(1/3) - x]*(3*E^5 - Log[(x^2*(E + x))/(-3 + x^3)]) + Log[(-1)^(2/3
)*3^(1/3) - x]*(3*E^5 - Log[(x^2*(E + x))/(-3 + x^3)]) + 2*Log[x]*Log[(x^2*(E + x))/(-3 + x^3)] + Log[-E - x]*
(-3*E^5 + Log[(x^2*(E + x))/(-3 + x^3)]) - PolyLog[2, (3^(1/3) - x)/((-3)^(1/3) + 3^(1/3))] + PolyLog[2, ((-1)
^(2/3)*3^(1/3) - x)/((-1)^(2/3)*3^(1/3) + E)] + 2*PolyLog[2, -((-1/3)^(1/3)*x)] - 2*PolyLog[2, x/3^(1/3)] + 2*
PolyLog[2, ((-1)^(2/3)*x)/3^(1/3)] - 2*PolyLog[2, -(x/E)] - PolyLog[2, ((-3)^(1/3) + x)/((-3)^(1/3) + 3^(1/3))
] + PolyLog[2, ((-3)^(1/3) + x)/((-3)^(1/3) - E)] + PolyLog[2, -((E + x)/((-3)^(1/3) - E))] - 2*PolyLog[2, (E
+ x)/E] + PolyLog[2, (E + x)/((-1)^(2/3)*3^(1/3) + E)] + 2*PolyLog[2, (3 - (-3)^(2/3)*x)/3] - PolyLog[2, (3^(1
/3) - (-1)^(2/3)*x)/((-3)^(1/3) + 3^(1/3))] - PolyLog[2, ((-3)^(1/3) + (-1)^(2/3)*x)/((-3)^(1/3) + 3^(1/3))] +
2*PolyLog[2, 1 + (-1/3)^(1/3)*x] - 2*PolyLog[2, 1 - x/3^(1/3)] - PolyLog[2, (I + Sqrt[3] + ((2*I)*x)/3^(1/3))
/(3*I + Sqrt[3])] - PolyLog[2, (((-2*I)/3)*(-3 + 3^(2/3)*x))/(3*I + Sqrt[3])])

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.96

method	result	size
norman	\(-6 \,{\mathrm e} \,{\mathrm e}^{5} \ln \left (\frac {x^{2} {\mathrm e}+x^{3}}{x^{3}-3}\right )+{\mathrm e} \ln \left (\frac {x^{2} {\mathrm e}+x^{3}}{x^{3}-3}\right )^{2}\)	\(51\)
risch	\(\text {Expression too large to display}\)	\(1571\)
default	\(\text {Expression too large to display}\)	\(1579\)
parts	\(\text {Expression too large to display}\)	\(1579\)

[In]

int((((-2*x^3-12)*exp(1)^2-18*x*exp(1))*ln((x^2*exp(1)+x^3)/(x^3-3))+((6*x^3+36)*exp(1)^2+54*x*exp(1))*exp(5))
/((x^4-3*x)*exp(1)+x^5-3*x^2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.85 \[ \int \frac {e^5 \left (54 e x+e^2 \left (36+6 x^3\right )\right )+\left (-18 e x+e^2 \left (-12-2 x^3\right )\right ) \log \left (\frac {e x^2+x^3}{-3+x^3}\right )}{-3 x^2+x^5+e \left (-3 x+x^4\right )} \, dx=e \log \left (\frac {x^{3} + x^{2} e}{x^{3} - 3}\right )^{2} - 6 \, e^{6} \log \left (\frac {x^{3} + x^{2} e}{x^{3} - 3}\right ) \]

[In]

integrate((((-2*x^3-12)*exp(1)^2-18*x*exp(1))*log((x^2*exp(1)+x^3)/(x^3-3))+((6*x^3+36)*exp(1)^2+54*x*exp(1))*
exp(5))/((x^4-3*x)*exp(1)+x^5-3*x^2),x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (22) = 44\).

Time = 1.23 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.04 \[ \int \frac {e^5 \left (54 e x+e^2 \left (36+6 x^3\right )\right )+\left (-18 e x+e^2 \left (-12-2 x^3\right )\right ) \log \left (\frac {e x^2+x^3}{-3+x^3}\right )}{-3 x^2+x^5+e \left (-3 x+x^4\right )} \, dx=- 12 e^{6} \log {\left (x \right )} + e \log {\left (\frac {x^{3} + e x^{2}}{x^{3} - 3} \right )}^{2} - 6 e^{6} \log {\left (x + e \right )} + 6 e^{6} \log {\left (x^{3} - 3 \right )} \]

[In]

integrate((((-2*x**3-12)*exp(1)**2-18*x*exp(1))*ln((x**2*exp(1)+x**3)/(x**3-3))+((6*x**3+36)*exp(1)**2+54*x*ex
p(1))*exp(5))/((x**4-3*x)*exp(1)+x**5-3*x**2),x)

[Out]

-12*exp(6)*log(x) + E*log((x**3 + E*x**2)/(x**3 - 3))**2 - 6*exp(6)*log(x + E) + 6*exp(6)*log(x**3 - 3)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 513 vs. \(2 (27) = 54\).

Time = 2.93 (sec) , antiderivative size = 513, normalized size of antiderivative = 19.73 \[ \int \frac {e^5 \left (54 e x+e^2 \left (36+6 x^3\right )\right )+\left (-18 e x+e^2 \left (-12-2 x^3\right )\right ) \log \left (\frac {e x^2+x^3}{-3+x^3}\right )}{-3 x^2+x^5+e \left (-3 x+x^4\right )} \, dx=e \log \left (x^{3} - 3\right )^{2} + e \log \left (x + e\right )^{2} + 4 \, e \log \left (x + e\right ) \log \left (x\right ) + 4 \, e \log \left (x\right )^{2} - 2 \, {\left (6 \, e^{\left (-1\right )} \log \left (x\right ) - \frac {6 \cdot 3^{\frac {1}{6}} {\left (3^{\frac {1}{3}} e + 3^{\frac {2}{3}}\right )} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x + 3^{\frac {1}{3}}\right )}\right )}{3^{\frac {2}{3}} e^{3} + 3 \cdot 3^{\frac {2}{3}}} - \frac {{\left (2 \cdot 3^{\frac {2}{3}} e^{2} - 3 \cdot 3^{\frac {1}{3}} + 3 \, e\right )} \log \left (x^{2} + 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right )}{3^{\frac {2}{3}} e^{3} + 3 \cdot 3^{\frac {2}{3}}} - \frac {2 \, {\left (3^{\frac {2}{3}} e^{2} + 3 \cdot 3^{\frac {1}{3}} - 3 \, e\right )} \log \left (x - 3^{\frac {1}{3}}\right )}{3^{\frac {2}{3}} e^{3} + 3 \cdot 3^{\frac {2}{3}}} - \frac {18 \, \log \left (x + e\right )}{e^{4} + 3 \, e}\right )} e^{7} + {\left (\frac {6 \cdot 3^{\frac {1}{6}} {\left (3^{\frac {1}{3}} e + 3^{\frac {2}{3}}\right )} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x + 3^{\frac {1}{3}}\right )}\right )}{3^{\frac {2}{3}} e^{3} + 3 \cdot 3^{\frac {2}{3}}} + \frac {{\left (2 \cdot 3^{\frac {2}{3}} e^{2} - 3 \cdot 3^{\frac {1}{3}} + 3 \, e\right )} \log \left (x^{2} + 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right )}{3^{\frac {2}{3}} e^{3} + 3 \cdot 3^{\frac {2}{3}}} + \frac {2 \, {\left (3^{\frac {2}{3}} e^{2} + 3 \cdot 3^{\frac {1}{3}} - 3 \, e\right )} \log \left (x - 3^{\frac {1}{3}}\right )}{3^{\frac {2}{3}} e^{3} + 3 \cdot 3^{\frac {2}{3}}} - \frac {6 \, e^{2} \log \left (x + e\right )}{e^{3} + 3}\right )} e^{7} - 9 \, {\left (\frac {2 \cdot 3^{\frac {1}{6}} {\left (3^{\frac {2}{3}} e + 3^{\frac {1}{3}} e^{2}\right )} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x + 3^{\frac {1}{3}}\right )}\right )}{3^{\frac {2}{3}} e^{3} + 3 \cdot 3^{\frac {2}{3}}} - \frac {{\left (3^{\frac {1}{3}} e + 2 \cdot 3^{\frac {2}{3}} - e^{2}\right )} \log \left (x^{2} + 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right )}{3^{\frac {2}{3}} e^{3} + 3 \cdot 3^{\frac {2}{3}}} + \frac {2 \, {\left (3^{\frac {1}{3}} e - 3^{\frac {2}{3}} - e^{2}\right )} \log \left (x - 3^{\frac {1}{3}}\right )}{3^{\frac {2}{3}} e^{3} + 3 \cdot 3^{\frac {2}{3}}} + \frac {6 \, \log \left (x + e\right )}{e^{3} + 3}\right )} e^{6} - 2 \, {\left (e \log \left (x + e\right ) + 2 \, e \log \left (x\right )\right )} \log \left (x^{3} - 3\right ) \]

[In]

integrate((((-2*x^3-12)*exp(1)^2-18*x*exp(1))*log((x^2*exp(1)+x^3)/(x^3-3))+((6*x^3+36)*exp(1)^2+54*x*exp(1))*
exp(5))/((x^4-3*x)*exp(1)+x^5-3*x^2),x, algorithm="maxima")

[Out]

e*log(x^3 - 3)^2 + e*log(x + e)^2 + 4*e*log(x + e)*log(x) + 4*e*log(x)^2 - 2*(6*e^(-1)*log(x) - 6*3^(1/6)*(3^(
1/3)*e + 3^(2/3))*arctan(1/3*3^(1/6)*(2*x + 3^(1/3)))/(3^(2/3)*e^3 + 3*3^(2/3)) - (2*3^(2/3)*e^2 - 3*3^(1/3) +
3*e)*log(x^2 + 3^(1/3)*x + 3^(2/3))/(3^(2/3)*e^3 + 3*3^(2/3)) - 2*(3^(2/3)*e^2 + 3*3^(1/3) - 3*e)*log(x - 3^(
1/3))/(3^(2/3)*e^3 + 3*3^(2/3)) - 18*log(x + e)/(e^4 + 3*e))*e^7 + (6*3^(1/6)*(3^(1/3)*e + 3^(2/3))*arctan(1/3
*3^(1/6)*(2*x + 3^(1/3)))/(3^(2/3)*e^3 + 3*3^(2/3)) + (2*3^(2/3)*e^2 - 3*3^(1/3) + 3*e)*log(x^2 + 3^(1/3)*x +
3^(2/3))/(3^(2/3)*e^3 + 3*3^(2/3)) + 2*(3^(2/3)*e^2 + 3*3^(1/3) - 3*e)*log(x - 3^(1/3))/(3^(2/3)*e^3 + 3*3^(2/
3)) - 6*e^2*log(x + e)/(e^3 + 3))*e^7 - 9*(2*3^(1/6)*(3^(2/3)*e + 3^(1/3)*e^2)*arctan(1/3*3^(1/6)*(2*x + 3^(1/
3)))/(3^(2/3)*e^3 + 3*3^(2/3)) - (3^(1/3)*e + 2*3^(2/3) - e^2)*log(x^2 + 3^(1/3)*x + 3^(2/3))/(3^(2/3)*e^3 + 3
*3^(2/3)) + 2*(3^(1/3)*e - 3^(2/3) - e^2)*log(x - 3^(1/3))/(3^(2/3)*e^3 + 3*3^(2/3)) + 6*log(x + e)/(e^3 + 3))
*e^6 - 2*(e*log(x + e) + 2*e*log(x))*log(x^3 - 3)

Giac [F]

\[ \int \frac {e^5 \left (54 e x+e^2 \left (36+6 x^3\right )\right )+\left (-18 e x+e^2 \left (-12-2 x^3\right )\right ) \log \left (\frac {e x^2+x^3}{-3+x^3}\right )}{-3 x^2+x^5+e \left (-3 x+x^4\right )} \, dx=\int { \frac {2 \, {\left (3 \, {\left ({\left (x^{3} + 6\right )} e^{2} + 9 \, x e\right )} e^{5} - {\left ({\left (x^{3} + 6\right )} e^{2} + 9 \, x e\right )} \log \left (\frac {x^{3} + x^{2} e}{x^{3} - 3}\right )\right )}}{x^{5} - 3 \, x^{2} + {\left (x^{4} - 3 \, x\right )} e} \,d x } \]

[In]

integrate((((-2*x^3-12)*exp(1)^2-18*x*exp(1))*log((x^2*exp(1)+x^3)/(x^3-3))+((6*x^3+36)*exp(1)^2+54*x*exp(1))*
exp(5))/((x^4-3*x)*exp(1)+x^5-3*x^2),x, algorithm="giac")

[Out]

integrate(2*(3*((x^3 + 6)*e^2 + 9*x*e)*e^5 - ((x^3 + 6)*e^2 + 9*x*e)*log((x^3 + x^2*e)/(x^3 - 3)))/(x^5 - 3*x^
2 + (x^4 - 3*x)*e), x)

Mupad [B] (verification not implemented)

Time = 9.06 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.92 \[ \int \frac {e^5 \left (54 e x+e^2 \left (36+6 x^3\right )\right )+\left (-18 e x+e^2 \left (-12-2 x^3\right )\right ) \log \left (\frac {e x^2+x^3}{-3+x^3}\right )}{-3 x^2+x^5+e \left (-3 x+x^4\right )} \, dx=\mathrm {e}\,{\ln \left (\frac {x^3+\mathrm {e}\,x^2}{x^3-3}\right )}^2+6\,{\mathrm {e}}^6\,\ln \left (x^3-3\right )-6\,{\mathrm {e}}^6\,\ln \left (x+\mathrm {e}\right )-12\,{\mathrm {e}}^6\,\ln \left (x\right ) \]

[In]

int((log((x^2*exp(1) + x^3)/(x^3 - 3))*(18*x*exp(1) + exp(2)*(2*x^3 + 12)) - exp(5)*(54*x*exp(1) + exp(2)*(6*x
^3 + 36)))/(exp(1)*(3*x - x^4) + 3*x^2 - x^5),x)

[Out]

log((x^2*exp(1) + x^3)/(x^3 - 3))^2*exp(1) + 6*exp(6)*log(x^3 - 3) - 6*exp(6)*log(x + exp(1)) - 12*exp(6)*log(
x)