3.43.60 \(\int \frac {-5 \log (\log (3))+e^{e^x} \log (\log (3))+24 e^{e^x+x} \log (25-10 e^{e^x}+e^{2 e^x}) \log (\log (3))+8 e^{e^x+x} \log ^3(25-10 e^{e^x}+e^{2 e^x}) \log (\log (3))}{-5+e^{e^x}} \, dx\) [4260]

Optimal. Leaf size=26 \[ 5+\left (x+\left (3+\log ^2\left (\left (5-e^{e^x}\right )^2\right )\right )^2\right ) \log (\log (3)) \]

[Out]

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

________________________________________________________________________________________

Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 1.18, antiderivative size = 318, normalized size of antiderivative = 12.23, number of steps used = 26, number of rules used = 17, integrand size = 85, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2320, 12, 6874, 36, 31, 29, 2458, 2379, 2421, 6724, 2438, 2430, 14, 2441, 2352, 2443, 2481} \begin {gather*} 48 \log (\log (3)) \log ^2\left (\left (e^{e^x}-5\right )^2\right ) \text {PolyLog}\left (2,\frac {5}{5-e^{e^x}}\right )+48 \log (\log (3)) \log ^2\left (\left (e^{e^x}-5\right )^2\right ) \text {PolyLog}\left (2,1-\frac {e^{e^x}}{5}\right )+192 \log (\log (3)) \log \left (\left (e^{e^x}-5\right )^2\right ) \text {PolyLog}\left (3,\frac {5}{5-e^{e^x}}\right )-192 \log (\log (3)) \log \left (\left (e^{e^x}-5\right )^2\right ) \text {PolyLog}\left (3,1-\frac {e^{e^x}}{5}\right )+48 \log (\log (3)) \text {PolyLog}\left (2,\frac {5}{5-e^{e^x}}\right )+48 \log (\log (3)) \text {PolyLog}\left (2,1-\frac {e^{e^x}}{5}\right )+384 \log (\log (3)) \text {PolyLog}\left (4,\frac {5}{5-e^{e^x}}\right )+384 \log (\log (3)) \text {PolyLog}\left (4,1-\frac {e^{e^x}}{5}\right )+8 \log (\log (3)) \log \left (\frac {e^{e^x}}{5}\right ) \log ^3\left (\left (e^{e^x}-5\right )^2\right )-8 \log (\log (3)) \log \left (1-\frac {5}{5-e^{e^x}}\right ) \log ^3\left (\left (e^{e^x}-5\right )^2\right )+24 \log (\log (3)) \log \left (\frac {e^{e^x}}{5}\right ) \log \left (\left (e^{e^x}-5\right )^2\right )-24 \log (\log (3)) \log \left (1-\frac {5}{5-e^{e^x}}\right ) \log \left (\left (e^{e^x}-5\right )^2\right )+x \log (\log (3)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-5*Log[Log[3]] + E^E^x*Log[Log[3]] + 24*E^(E^x + x)*Log[25 - 10*E^E^x + E^(2*E^x)]*Log[Log[3]] + 8*E^(E^x
+ x)*Log[25 - 10*E^E^x + E^(2*E^x)]^3*Log[Log[3]])/(-5 + E^E^x),x]

[Out]

x*Log[Log[3]] + 24*Log[E^E^x/5]*Log[(-5 + E^E^x)^2]*Log[Log[3]] + 8*Log[E^E^x/5]*Log[(-5 + E^E^x)^2]^3*Log[Log
[3]] - 24*Log[(-5 + E^E^x)^2]*Log[1 - 5/(5 - E^E^x)]*Log[Log[3]] - 8*Log[(-5 + E^E^x)^2]^3*Log[1 - 5/(5 - E^E^
x)]*Log[Log[3]] + 48*Log[Log[3]]*PolyLog[2, 5/(5 - E^E^x)] + 48*Log[(-5 + E^E^x)^2]^2*Log[Log[3]]*PolyLog[2, 5
/(5 - E^E^x)] + 48*Log[Log[3]]*PolyLog[2, 1 - E^E^x/5] + 48*Log[(-5 + E^E^x)^2]^2*Log[Log[3]]*PolyLog[2, 1 - E
^E^x/5] + 192*Log[(-5 + E^E^x)^2]*Log[Log[3]]*PolyLog[3, 5/(5 - E^E^x)] - 192*Log[(-5 + E^E^x)^2]*Log[Log[3]]*
PolyLog[3, 1 - E^E^x/5] + 384*Log[Log[3]]*PolyLog[4, 5/(5 - E^E^x)] + 384*Log[Log[3]]*PolyLog[4, 1 - E^E^x/5]

Rule 12

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 36

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

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[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 2379

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

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c*x^n])^p/m), x] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L
og[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2430

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[PolyLo
g[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q), x] - Dist[b*n*(p/q), Int[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^(
p - 1)/x), x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 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 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 2443

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

Rule 2458

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2481

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + j*(x/e))^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

Rule 6724

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

Rule 6874

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\text {Subst}\left (\int \frac {\left (5-e^x-24 e^x x \log \left (\left (-5+e^x\right )^2\right )-8 e^x x \log ^3\left (\left (-5+e^x\right )^2\right )\right ) \log (\log (3))}{\left (5-e^x\right ) x} \, dx,x,e^x\right )\\ &=\log (\log (3)) \text {Subst}\left (\int \frac {5-e^x-24 e^x x \log \left (\left (-5+e^x\right )^2\right )-8 e^x x \log ^3\left (\left (-5+e^x\right )^2\right )}{\left (5-e^x\right ) x} \, dx,x,e^x\right )\\ &=\log (\log (3)) \text {Subst}\left (\int \left (\frac {40 \log \left (\left (-5+e^x\right )^2\right ) \left (3+\log ^2\left (\left (-5+e^x\right )^2\right )\right )}{-5+e^x}+\frac {1+24 x \log \left (\left (-5+e^x\right )^2\right )+8 x \log ^3\left (\left (-5+e^x\right )^2\right )}{x}\right ) \, dx,x,e^x\right )\\ &=\log (\log (3)) \text {Subst}\left (\int \frac {1+24 x \log \left (\left (-5+e^x\right )^2\right )+8 x \log ^3\left (\left (-5+e^x\right )^2\right )}{x} \, dx,x,e^x\right )+(40 \log (\log (3))) \text {Subst}\left (\int \frac {\log \left (\left (-5+e^x\right )^2\right ) \left (3+\log ^2\left (\left (-5+e^x\right )^2\right )\right )}{-5+e^x} \, dx,x,e^x\right )\\ &=\log (\log (3)) \text {Subst}\left (\int \left (\frac {1}{x}+24 \log \left (\left (-5+e^x\right )^2\right )+8 \log ^3\left (\left (-5+e^x\right )^2\right )\right ) \, dx,x,e^x\right )+(40 \log (\log (3))) \text {Subst}\left (\int \frac {\log \left ((-5+x)^2\right ) \left (3+\log ^2\left ((-5+x)^2\right )\right )}{(-5+x) x} \, dx,x,e^{e^x}\right )\\ &=x \log (\log (3))+(8 \log (\log (3))) \text {Subst}\left (\int \log ^3\left (\left (-5+e^x\right )^2\right ) \, dx,x,e^x\right )+(24 \log (\log (3))) \text {Subst}\left (\int \log \left (\left (-5+e^x\right )^2\right ) \, dx,x,e^x\right )+(40 \log (\log (3))) \text {Subst}\left (\int \left (\frac {3 \log \left ((-5+x)^2\right )}{(-5+x) x}+\frac {\log ^3\left ((-5+x)^2\right )}{(-5+x) x}\right ) \, dx,x,e^{e^x}\right )\\ &=x \log (\log (3))+(8 \log (\log (3))) \text {Subst}\left (\int \frac {\log ^3\left ((-5+x)^2\right )}{x} \, dx,x,e^{e^x}\right )+(24 \log (\log (3))) \text {Subst}\left (\int \frac {\log \left ((-5+x)^2\right )}{x} \, dx,x,e^{e^x}\right )+(40 \log (\log (3))) \text {Subst}\left (\int \frac {\log ^3\left ((-5+x)^2\right )}{(-5+x) x} \, dx,x,e^{e^x}\right )+(120 \log (\log (3))) \text {Subst}\left (\int \frac {\log \left ((-5+x)^2\right )}{(-5+x) x} \, dx,x,e^{e^x}\right )\\ &=x \log (\log (3))+24 \log \left (\frac {e^{e^x}}{5}\right ) \log \left (\left (-5+e^{e^x}\right )^2\right ) \log (\log (3))+8 \log \left (\frac {e^{e^x}}{5}\right ) \log ^3\left (\left (-5+e^{e^x}\right )^2\right ) \log (\log (3))+(40 \log (\log (3))) \text {Subst}\left (\int \frac {\log ^3\left (x^2\right )}{x (5+x)} \, dx,x,-5+e^{e^x}\right )-(48 \log (\log (3))) \text {Subst}\left (\int \frac {\log \left (\frac {x}{5}\right )}{-5+x} \, dx,x,e^{e^x}\right )-(48 \log (\log (3))) \text {Subst}\left (\int \frac {\log ^2\left ((-5+x)^2\right ) \log \left (\frac {x}{5}\right )}{-5+x} \, dx,x,e^{e^x}\right )+(120 \log (\log (3))) \text {Subst}\left (\int \frac {\log \left (x^2\right )}{x (5+x)} \, dx,x,-5+e^{e^x}\right )\\ &=x \log (\log (3))+24 \log \left (\frac {e^{e^x}}{5}\right ) \log \left (\left (-5+e^{e^x}\right )^2\right ) \log (\log (3))+8 \log \left (\frac {e^{e^x}}{5}\right ) \log ^3\left (\left (-5+e^{e^x}\right )^2\right ) \log (\log (3))+48 \log (\log (3)) \text {Li}_2\left (1-\frac {e^{e^x}}{5}\right )+(8 \log (\log (3))) \text {Subst}\left (\int \frac {\log ^3\left (x^2\right )}{x} \, dx,x,-5+e^{e^x}\right )-(8 \log (\log (3))) \text {Subst}\left (\int \frac {\log ^3\left (x^2\right )}{5+x} \, dx,x,-5+e^{e^x}\right )+(24 \log (\log (3))) \text {Subst}\left (\int \frac {\log \left (x^2\right )}{x} \, dx,x,-5+e^{e^x}\right )-(24 \log (\log (3))) \text {Subst}\left (\int \frac {\log \left (x^2\right )}{5+x} \, dx,x,-5+e^{e^x}\right )-(48 \log (\log (3))) \text {Subst}\left (\int \frac {\log ^2\left (x^2\right ) \log \left (\frac {5+x}{5}\right )}{x} \, dx,x,-5+e^{e^x}\right )\\ &=x \log (\log (3))+24 \log \left (\frac {e^{e^x}}{5}\right ) \log \left (\left (-5+e^{e^x}\right )^2\right ) \log (\log (3))+6 \log ^2\left (\left (-5+e^{e^x}\right )^2\right ) \log (\log (3))+8 \log \left (\frac {e^{e^x}}{5}\right ) \log ^3\left (\left (-5+e^{e^x}\right )^2\right ) \log (\log (3))-24 \log \left (\left (-5+e^{e^x}\right )^2\right ) \log \left (1+\frac {1}{5} \left (-5+e^{e^x}\right )\right ) \log (\log (3))-8 \log ^3\left (\left (-5+e^{e^x}\right )^2\right ) \log \left (1+\frac {1}{5} \left (-5+e^{e^x}\right )\right ) \log (\log (3))+48 \log ^2\left (\left (-5+e^{e^x}\right )^2\right ) \log (\log (3)) \text {Li}_2\left (\frac {1}{5} \left (5-e^{e^x}\right )\right )+48 \log (\log (3)) \text {Li}_2\left (1-\frac {e^{e^x}}{5}\right )+(4 \log (\log (3))) \text {Subst}\left (\int x^3 \, dx,x,\log \left (\left (-5+e^{e^x}\right )^2\right )\right )+(48 \log (\log (3))) \text {Subst}\left (\int \frac {\log \left (1+\frac {x}{5}\right )}{x} \, dx,x,-5+e^{e^x}\right )+(48 \log (\log (3))) \text {Subst}\left (\int \frac {\log \left (1+\frac {x}{5}\right ) \log ^2\left (x^2\right )}{x} \, dx,x,-5+e^{e^x}\right )-(192 \log (\log (3))) \text {Subst}\left (\int \frac {\log \left (x^2\right ) \text {Li}_2\left (-\frac {x}{5}\right )}{x} \, dx,x,-5+e^{e^x}\right )\\ &=x \log (\log (3))+24 \log \left (\frac {e^{e^x}}{5}\right ) \log \left (\left (-5+e^{e^x}\right )^2\right ) \log (\log (3))+6 \log ^2\left (\left (-5+e^{e^x}\right )^2\right ) \log (\log (3))+8 \log \left (\frac {e^{e^x}}{5}\right ) \log ^3\left (\left (-5+e^{e^x}\right )^2\right ) \log (\log (3))+\log ^4\left (\left (-5+e^{e^x}\right )^2\right ) \log (\log (3))-24 \log \left (\left (-5+e^{e^x}\right )^2\right ) \log \left (1+\frac {1}{5} \left (-5+e^{e^x}\right )\right ) \log (\log (3))-8 \log ^3\left (\left (-5+e^{e^x}\right )^2\right ) \log \left (1+\frac {1}{5} \left (-5+e^{e^x}\right )\right ) \log (\log (3))-48 \log (\log (3)) \text {Li}_2\left (\frac {1}{5} \left (5-e^{e^x}\right )\right )+48 \log (\log (3)) \text {Li}_2\left (1-\frac {e^{e^x}}{5}\right )-192 \log \left (\left (-5+e^{e^x}\right )^2\right ) \log (\log (3)) \text {Li}_3\left (\frac {1}{5} \left (5-e^{e^x}\right )\right )+(192 \log (\log (3))) \text {Subst}\left (\int \frac {\log \left (x^2\right ) \text {Li}_2\left (-\frac {x}{5}\right )}{x} \, dx,x,-5+e^{e^x}\right )+(384 \log (\log (3))) \text {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {x}{5}\right )}{x} \, dx,x,-5+e^{e^x}\right )\\ &=x \log (\log (3))+24 \log \left (\frac {e^{e^x}}{5}\right ) \log \left (\left (-5+e^{e^x}\right )^2\right ) \log (\log (3))+6 \log ^2\left (\left (-5+e^{e^x}\right )^2\right ) \log (\log (3))+8 \log \left (\frac {e^{e^x}}{5}\right ) \log ^3\left (\left (-5+e^{e^x}\right )^2\right ) \log (\log (3))+\log ^4\left (\left (-5+e^{e^x}\right )^2\right ) \log (\log (3))-24 \log \left (\left (-5+e^{e^x}\right )^2\right ) \log \left (1+\frac {1}{5} \left (-5+e^{e^x}\right )\right ) \log (\log (3))-8 \log ^3\left (\left (-5+e^{e^x}\right )^2\right ) \log \left (1+\frac {1}{5} \left (-5+e^{e^x}\right )\right ) \log (\log (3))-48 \log (\log (3)) \text {Li}_2\left (\frac {1}{5} \left (5-e^{e^x}\right )\right )+48 \log (\log (3)) \text {Li}_2\left (1-\frac {e^{e^x}}{5}\right )+384 \log (\log (3)) \text {Li}_4\left (\frac {1}{5} \left (5-e^{e^x}\right )\right )-(384 \log (\log (3))) \text {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {x}{5}\right )}{x} \, dx,x,-5+e^{e^x}\right )\\ &=x \log (\log (3))+24 \log \left (\frac {e^{e^x}}{5}\right ) \log \left (\left (-5+e^{e^x}\right )^2\right ) \log (\log (3))+6 \log ^2\left (\left (-5+e^{e^x}\right )^2\right ) \log (\log (3))+8 \log \left (\frac {e^{e^x}}{5}\right ) \log ^3\left (\left (-5+e^{e^x}\right )^2\right ) \log (\log (3))+\log ^4\left (\left (-5+e^{e^x}\right )^2\right ) \log (\log (3))-24 \log \left (\left (-5+e^{e^x}\right )^2\right ) \log \left (1+\frac {1}{5} \left (-5+e^{e^x}\right )\right ) \log (\log (3))-8 \log ^3\left (\left (-5+e^{e^x}\right )^2\right ) \log \left (1+\frac {1}{5} \left (-5+e^{e^x}\right )\right ) \log (\log (3))-48 \log (\log (3)) \text {Li}_2\left (\frac {1}{5} \left (5-e^{e^x}\right )\right )+48 \log (\log (3)) \text {Li}_2\left (1-\frac {e^{e^x}}{5}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.07, size = 32, normalized size = 1.23 \begin {gather*} \left (x+6 \log ^2\left (\left (-5+e^{e^x}\right )^2\right )+\log ^4\left (\left (-5+e^{e^x}\right )^2\right )\right ) \log (\log (3)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5*Log[Log[3]] + E^E^x*Log[Log[3]] + 24*E^(E^x + x)*Log[25 - 10*E^E^x + E^(2*E^x)]*Log[Log[3]] + 8*
E^(E^x + x)*Log[25 - 10*E^E^x + E^(2*E^x)]^3*Log[Log[3]])/(-5 + E^E^x),x]

[Out]

(x + 6*Log[(-5 + E^E^x)^2]^2 + Log[(-5 + E^E^x)^2]^4)*Log[Log[3]]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(154\) vs. \(2(24)=48\).
time = 1.30, size = 155, normalized size = 5.96

method	result	size
derivativedivides	\(24 \ln \left (\ln \left (3\right )\right ) \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2} \left (\ln \left (\left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right )-2 \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )\right )^{2}+24 \ln \left (\ln \left (3\right )\right ) \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}+8 \ln \left (\ln \left (3\right )\right ) \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right ) \left (\ln \left (\left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right )-2 \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )\right )^{3}+24 \ln \left (\ln \left (3\right )\right ) \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right ) \left (\ln \left (\left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right )-2 \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )\right )+16 \ln \left (\ln \left (3\right )\right ) \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{4}+32 \ln \left (\ln \left (3\right )\right ) \left (\ln \left (\left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right )-2 \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )\right ) \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{3}+\ln \left (\ln \left (3\right )\right ) \ln \left ({\mathrm e}^{x}\right )\)	\(155\)
default	\(24 \ln \left (\ln \left (3\right )\right ) \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2} \left (\ln \left (\left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right )-2 \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )\right )^{2}+24 \ln \left (\ln \left (3\right )\right ) \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}+8 \ln \left (\ln \left (3\right )\right ) \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right ) \left (\ln \left (\left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right )-2 \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )\right )^{3}+24 \ln \left (\ln \left (3\right )\right ) \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right ) \left (\ln \left (\left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right )-2 \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )\right )+16 \ln \left (\ln \left (3\right )\right ) \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{4}+32 \ln \left (\ln \left (3\right )\right ) \left (\ln \left (\left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right )-2 \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )\right ) \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{3}+\ln \left (\ln \left (3\right )\right ) \ln \left ({\mathrm e}^{x}\right )\)	\(155\)
risch	\(\text {Expression too large to display}\)	\(611\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*exp(x)*ln(ln(3))*exp(exp(x))*ln(exp(exp(x))^2-10*exp(exp(x))+25)^3+24*exp(x)*ln(ln(3))*exp(exp(x))*ln(e
xp(exp(x))^2-10*exp(exp(x))+25)+ln(ln(3))*exp(exp(x))-5*ln(ln(3)))/(exp(exp(x))-5),x,method=_RETURNVERBOSE)

[Out]

24*ln(ln(3))*ln(exp(exp(x))-5)^2*(ln((exp(exp(x))-5)^2)-2*ln(exp(exp(x))-5))^2+24*ln(ln(3))*ln(exp(exp(x))-5)^
2+8*ln(ln(3))*ln(exp(exp(x))-5)*(ln((exp(exp(x))-5)^2)-2*ln(exp(exp(x))-5))^3+24*ln(ln(3))*ln(exp(exp(x))-5)*(
ln((exp(exp(x))-5)^2)-2*ln(exp(exp(x))-5))+16*ln(ln(3))*ln(exp(exp(x))-5)^4+32*ln(ln(3))*(ln((exp(exp(x))-5)^2
)-2*ln(exp(exp(x))-5))*ln(exp(exp(x))-5)^3+ln(ln(3))*ln(exp(x))

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Maxima [A]
time = 0.54, size = 32, normalized size = 1.23 \begin {gather*} 16 \, \log \left (e^{\left (e^{x}\right )} - 5\right )^{4} \log \left (\log \left (3\right )\right ) + 24 \, \log \left (e^{\left (e^{x}\right )} - 5\right )^{2} \log \left (\log \left (3\right )\right ) + x \log \left (\log \left (3\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*exp(x)*log(log(3))*exp(exp(x))*log(exp(exp(x))^2-10*exp(exp(x))+25)^3+24*exp(x)*log(log(3))*exp(e
xp(x))*log(exp(exp(x))^2-10*exp(exp(x))+25)+log(log(3))*exp(exp(x))-5*log(log(3)))/(exp(exp(x))-5),x, algorith
m="maxima")

[Out]

16*log(e^(e^x) - 5)^4*log(log(3)) + 24*log(e^(e^x) - 5)^2*log(log(3)) + x*log(log(3))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (22) = 44\).
time = 0.37, size = 81, normalized size = 3.12 \begin {gather*} \log \left ({\left (25 \, e^{\left (2 \, x\right )} + e^{\left (2 \, x + 2 \, e^{x}\right )} - 10 \, e^{\left (2 \, x + e^{x}\right )}\right )} e^{\left (-2 \, x\right )}\right )^{4} \log \left (\log \left (3\right )\right ) + 6 \, \log \left ({\left (25 \, e^{\left (2 \, x\right )} + e^{\left (2 \, x + 2 \, e^{x}\right )} - 10 \, e^{\left (2 \, x + e^{x}\right )}\right )} e^{\left (-2 \, x\right )}\right )^{2} \log \left (\log \left (3\right )\right ) + x \log \left (\log \left (3\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*exp(x)*log(log(3))*exp(exp(x))*log(exp(exp(x))^2-10*exp(exp(x))+25)^3+24*exp(x)*log(log(3))*exp(e
xp(x))*log(exp(exp(x))^2-10*exp(exp(x))+25)+log(log(3))*exp(exp(x))-5*log(log(3)))/(exp(exp(x))-5),x, algorith
m="fricas")

[Out]

log((25*e^(2*x) + e^(2*x + 2*e^x) - 10*e^(2*x + e^x))*e^(-2*x))^4*log(log(3)) + 6*log((25*e^(2*x) + e^(2*x + 2
*e^x) - 10*e^(2*x + e^x))*e^(-2*x))^2*log(log(3)) + x*log(log(3))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (22) = 44\).
time = 0.15, size = 54, normalized size = 2.08 \begin {gather*} x \log {\left (\log {\left (3 \right )} \right )} + \log {\left (e^{2 e^{x}} - 10 e^{e^{x}} + 25 \right )}^{4} \log {\left (\log {\left (3 \right )} \right )} + 6 \log {\left (e^{2 e^{x}} - 10 e^{e^{x}} + 25 \right )}^{2} \log {\left (\log {\left (3 \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*exp(x)*ln(ln(3))*exp(exp(x))*ln(exp(exp(x))**2-10*exp(exp(x))+25)**3+24*exp(x)*ln(ln(3))*exp(exp(
x))*ln(exp(exp(x))**2-10*exp(exp(x))+25)+ln(ln(3))*exp(exp(x))-5*ln(ln(3)))/(exp(exp(x))-5),x)

[Out]

x*log(log(3)) + log(exp(2*exp(x)) - 10*exp(exp(x)) + 25)**4*log(log(3)) + 6*log(exp(2*exp(x)) - 10*exp(exp(x))
+ 25)**2*log(log(3))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*exp(x)*log(log(3))*exp(exp(x))*log(exp(exp(x))^2-10*exp(exp(x))+25)^3+24*exp(x)*log(log(3))*exp(e
xp(x))*log(exp(exp(x))^2-10*exp(exp(x))+25)+log(log(3))*exp(exp(x))-5*log(log(3)))/(exp(exp(x))-5),x, algorith
m="giac")

[Out]

integrate((8*e^(x + e^x)*log(e^(2*e^x) - 10*e^(e^x) + 25)^3*log(log(3)) + 24*e^(x + e^x)*log(e^(2*e^x) - 10*e^
(e^x) + 25)*log(log(3)) + e^(e^x)*log(log(3)) - 5*log(log(3)))/(e^(e^x) - 5), x)

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Mupad [B]
time = 3.37, size = 38, normalized size = 1.46 \begin {gather*} \ln \left (\ln \left (3\right )\right )\,\left ({\ln \left ({\mathrm {e}}^{2\,{\mathrm {e}}^x}-10\,{\mathrm {e}}^{{\mathrm {e}}^x}+25\right )}^4+6\,{\ln \left ({\mathrm {e}}^{2\,{\mathrm {e}}^x}-10\,{\mathrm {e}}^{{\mathrm {e}}^x}+25\right )}^2+x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(x))*log(log(3)) - 5*log(log(3)) + 24*log(exp(2*exp(x)) - 10*exp(exp(x)) + 25)*exp(exp(x))*exp(x)*
log(log(3)) + 8*log(exp(2*exp(x)) - 10*exp(exp(x)) + 25)^3*exp(exp(x))*exp(x)*log(log(3)))/(exp(exp(x)) - 5),x
)

[Out]

log(log(3))*(x + 6*log(exp(2*exp(x)) - 10*exp(exp(x)) + 25)^2 + log(exp(2*exp(x)) - 10*exp(exp(x)) + 25)^4)

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