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

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

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Rubi [A]  time = 1.12, antiderivative size = 39, normalized size of antiderivative = 1.50, number of steps used = 31, number of rules used = 21, integrand size = 85, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.247, Rules used = {2282, 12, 6742, 36, 31, 29, 2411, 2344, 2302, 30, 2317, 2374, 6589, 2301, 2391, 2383, 14, 2394, 2315, 2396, 2433} \begin {gather*} \log (\log (3)) \log ^4\left (\left (e^{e^x}-5\right )^2\right )+6 \log (\log (3)) \log ^2\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]] + 6*Log[(-5 + E^E^x)^2]^2*Log[Log[3]] + Log[(-5 + E^E^x)^2]^4*Log[Log[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 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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 2282

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 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 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, 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 2344

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

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(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*Log
[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 2383

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[(PolyL
og[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 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 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 2396

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 2411

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 2433

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 6589

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 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\operatorname {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)) \operatorname {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)) \operatorname {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)) \operatorname {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))) \operatorname {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)) \operatorname {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))) \operatorname {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))) \operatorname {Subst}\left (\int \log ^3\left (\left (-5+e^x\right )^2\right ) \, dx,x,e^x\right )+(24 \log (\log (3))) \operatorname {Subst}\left (\int \log \left (\left (-5+e^x\right )^2\right ) \, dx,x,e^x\right )+(40 \log (\log (3))) \operatorname {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))) \operatorname {Subst}\left (\int \frac {\log ^3\left ((-5+x)^2\right )}{x} \, dx,x,e^{e^x}\right )+(24 \log (\log (3))) \operatorname {Subst}\left (\int \frac {\log \left ((-5+x)^2\right )}{x} \, dx,x,e^{e^x}\right )+(40 \log (\log (3))) \operatorname {Subst}\left (\int \frac {\log ^3\left ((-5+x)^2\right )}{(-5+x) x} \, dx,x,e^{e^x}\right )+(120 \log (\log (3))) \operatorname {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))) \operatorname {Subst}\left (\int \frac {\log ^3\left (x^2\right )}{x (5+x)} \, dx,x,-5+e^{e^x}\right )-(48 \log (\log (3))) \operatorname {Subst}\left (\int \frac {\log \left (\frac {x}{5}\right )}{-5+x} \, dx,x,e^{e^x}\right )-(48 \log (\log (3))) \operatorname {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))) \operatorname {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))) \operatorname {Subst}\left (\int \frac {\log ^3\left (x^2\right )}{x} \, dx,x,-5+e^{e^x}\right )-(8 \log (\log (3))) \operatorname {Subst}\left (\int \frac {\log ^3\left (x^2\right )}{5+x} \, dx,x,-5+e^{e^x}\right )+(24 \log (\log (3))) \operatorname {Subst}\left (\int \frac {\log \left (x^2\right )}{x} \, dx,x,-5+e^{e^x}\right )-(24 \log (\log (3))) \operatorname {Subst}\left (\int \frac {\log \left (x^2\right )}{5+x} \, dx,x,-5+e^{e^x}\right )-(48 \log (\log (3))) \operatorname {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))) \operatorname {Subst}\left (\int x^3 \, dx,x,\log \left (\left (-5+e^{e^x}\right )^2\right )\right )+(48 \log (\log (3))) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{5}\right )}{x} \, dx,x,-5+e^{e^x}\right )+(48 \log (\log (3))) \operatorname {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))) \operatorname {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))) \operatorname {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))) \operatorname {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))) \operatorname {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 = 0.16, size = 22, normalized size = 0.85 \begin {gather*} \left (x+\left (3+\log ^2\left (\left (-5+e^{e^x}\right )^2\right )\right )^2\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 + (3 + Log[(-5 + E^E^x)^2]^2)^2)*Log[Log[3]]

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fricas [B]  time = 0.58, 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 \relax (3)\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 \relax (3)\right ) + x \log \left (\log \relax (3)\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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {8 \, e^{\left (x + e^{x}\right )} \log \left (e^{\left (2 \, e^{x}\right )} - 10 \, e^{\left (e^{x}\right )} + 25\right )^{3} \log \left (\log \relax (3)\right ) + 24 \, e^{\left (x + e^{x}\right )} \log \left (e^{\left (2 \, e^{x}\right )} - 10 \, e^{\left (e^{x}\right )} + 25\right ) \log \left (\log \relax (3)\right ) + e^{\left (e^{x}\right )} \log \left (\log \relax (3)\right ) - 5 \, \log \left (\log \relax (3)\right )}{e^{\left (e^{x}\right )} - 5}\,{d x} \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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maple [B]  time = 0.25, size = 155, normalized size = 5.96

method	result	size
derivativedivides	\(24 \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2} \ln \left (\ln \relax (3)\right ) \left (\ln \left (\left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right )-2 \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )\right )^{2}+24 \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2} \ln \left (\ln \relax (3)\right )+8 \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right ) \ln \left (\ln \relax (3)\right ) \left (\ln \left (\left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right )-2 \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )\right )^{3}+24 \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right ) \ln \left (\ln \relax (3)\right ) \left (\ln \left (\left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right )-2 \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )\right )+16 \ln \left (\ln \relax (3)\right ) \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{4}+32 \ln \left (\ln \relax (3)\right ) \left (\ln \left (\left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right )-2 \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )\right ) \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{3}+\ln \left (\ln \relax (3)\right ) \ln \left ({\mathrm e}^{x}\right )\)	\(155\)
default	\(24 \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2} \ln \left (\ln \relax (3)\right ) \left (\ln \left (\left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right )-2 \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )\right )^{2}+24 \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2} \ln \left (\ln \relax (3)\right )+8 \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right ) \ln \left (\ln \relax (3)\right ) \left (\ln \left (\left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right )-2 \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )\right )^{3}+24 \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right ) \ln \left (\ln \relax (3)\right ) \left (\ln \left (\left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right )-2 \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )\right )+16 \ln \left (\ln \relax (3)\right ) \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{4}+32 \ln \left (\ln \relax (3)\right ) \left (\ln \left (\left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right )-2 \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )\right ) \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{3}+\ln \left (\ln \relax (3)\right ) \ln \left ({\mathrm e}^{x}\right )\)	\(155\)
risch	\(16 \ln \left (\ln \relax (3)\right ) \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{4}-16 i \ln \left (\ln \relax (3)\right ) \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right ) \left (\mathrm {csgn}\left (i \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )\right )^{2}-2 \,\mathrm {csgn}\left (i \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )\right )+\mathrm {csgn}\left (i \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right )^{2}\right ) \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{3}+\left (-6 \pi ^{2} \ln \left (\ln \relax (3)\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )\right )^{4} \mathrm {csgn}\left (i \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right )^{2}+24 \pi ^{2} \ln \left (\ln \relax (3)\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )\right )^{3} \mathrm {csgn}\left (i \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right )^{3}-36 \pi ^{2} \ln \left (\ln \relax (3)\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )\right )^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right )^{4}+24 \pi ^{2} \ln \left (\ln \relax (3)\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right )^{5}-6 \pi ^{2} \ln \left (\ln \relax (3)\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right )^{6}+24 \ln \left (\ln \relax (3)\right )\right ) \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}+i \pi ^{3} \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right ) \ln \left (\ln \relax (3)\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )\right )^{6} \mathrm {csgn}\left (i \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right )^{3}-6 i \pi ^{3} \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right ) \ln \left (\ln \relax (3)\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )\right )^{5} \mathrm {csgn}\left (i \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right )^{4}+15 i \pi ^{3} \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right ) \ln \left (\ln \relax (3)\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )\right )^{4} \mathrm {csgn}\left (i \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right )^{5}-20 i \pi ^{3} \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right ) \ln \left (\ln \relax (3)\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )\right )^{3} \mathrm {csgn}\left (i \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right )^{6}+15 i \pi ^{3} \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right ) \ln \left (\ln \relax (3)\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )\right )^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right )^{7}-6 i \pi ^{3} \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right ) \ln \left (\ln \relax (3)\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right )^{8}+i \pi ^{3} \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right ) \ln \left (\ln \relax (3)\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right )^{9}-12 i \pi \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right ) \ln \left (\ln \relax (3)\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )\right )^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right )+24 i \pi \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right ) \ln \left (\ln \relax (3)\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right )^{2}-12 i \pi \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right ) \ln \left (\ln \relax (3)\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{{\mathrm e}^{x}}-5\right )^{2}\right )^{3}+\ln \left (\ln \relax (3)\right ) x\)	\(611\)

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(exp(exp(x))-5)^2*ln(ln(3))*(ln((exp(exp(x))-5)^2)-2*ln(exp(exp(x))-5))^2+24*ln(exp(exp(x))-5)^2*ln(ln(3)
)+8*ln(exp(exp(x))-5)*ln(ln(3))*(ln((exp(exp(x))-5)^2)-2*ln(exp(exp(x))-5))^3+24*ln(exp(exp(x))-5)*ln(ln(3))*(
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 = 0.47, size = 32, normalized size = 1.23 \begin {gather*} 16 \, \log \left (e^{\left (e^{x}\right )} - 5\right )^{4} \log \left (\log \relax (3)\right ) + 24 \, \log \left (e^{\left (e^{x}\right )} - 5\right )^{2} \log \left (\log \relax (3)\right ) + x \log \left (\log \relax (3)\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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mupad [B]  time = 3.37, size = 38, normalized size = 1.46 \begin {gather*} \ln \left (\ln \relax (3)\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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sympy [B]  time = 0.42, size = 54, normalized size = 2.08 \begin {gather*} x \log {\left (\log {\relax (3 )} \right )} + \log {\left (e^{2 e^{x}} - 10 e^{e^{x}} + 25 \right )}^{4} \log {\left (\log {\relax (3 )} \right )} + 6 \log {\left (e^{2 e^{x}} - 10 e^{e^{x}} + 25 \right )}^{2} \log {\left (\log {\relax (3 )} \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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