3.62.22 \(\int \frac {-6 e^8 x \log ^4(\frac {5}{3})+e^{12} \log ^6(\frac {5}{3}) (-6 x+6 x \log (x))}{-1+3 e^4 \log ^2(\frac {5}{3}) \log (x)-3 e^8 \log ^4(\frac {5}{3}) \log ^2(x)+e^{12} \log ^6(\frac {5}{3}) \log ^3(x)} \, dx\) [6122]

Optimal. Leaf size=22 \[ \frac {3 x^2}{\left (\frac {1}{e^4 \log ^2\left (\frac {5}{3}\right )}-\log (x)\right )^2} \]

[Out]

3/(exp(4-2*ln(exp(4)*ln(5/3)))-ln(x))^2*x^2

________________________________________________________________________________________

Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 0.52, antiderivative size = 398, normalized size of antiderivative = 18.09, number of steps used = 12, number of rules used = 7, integrand size = 79, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {6820, 12, 2343, 2346, 2209, 2413, 6617} \begin {gather*} 6 e^{\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}} \text {ExpIntegralEi}\left (-\frac {2 \left (1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{e^4 \log ^2\left (\frac {5}{3}\right )}\right )+24 e^{\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}-4} \left (\frac {1}{\log ^2\left (\frac {5}{3}\right )}-e^4 \log (x)\right ) \text {ExpIntegralEi}\left (2 \log (x)-\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}\right )-\frac {6 e^{\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}-4} \left (-2 e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)+2-e^4 \log ^2\left (\frac {5}{3}\right )\right ) \text {ExpIntegralEi}\left (-\frac {2 \left (1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{e^4 \log ^2\left (\frac {5}{3}\right )}\right )}{\log ^2\left (\frac {5}{3}\right )}-\frac {12 e^{\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}-4} \left (-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)+1+e^4 \log ^2\left (\frac {5}{3}\right )\right ) \text {ExpIntegralEi}\left (-\frac {2 \left (1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{e^4 \log ^2\left (\frac {5}{3}\right )}\right )}{\log ^2\left (\frac {5}{3}\right )}+12 x^2-\frac {6 x^2 \left (-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)+1+e^4 \log ^2\left (\frac {5}{3}\right )\right )}{1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)}+\frac {3 e^4 x^2 \log ^2\left (\frac {5}{3}\right ) \left (-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)+1+e^4 \log ^2\left (\frac {5}{3}\right )\right )}{\left (1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )^2}-\frac {3 x^2 \left (-2 e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)+2-e^4 \log ^2\left (\frac {5}{3}\right )\right )}{1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-6*E^8*x*Log[5/3]^4 + E^12*Log[5/3]^6*(-6*x + 6*x*Log[x]))/(-1 + 3*E^4*Log[5/3]^2*Log[x] - 3*E^8*Log[5/3]
^4*Log[x]^2 + E^12*Log[5/3]^6*Log[x]^3),x]

[Out]

12*x^2 + 6*E^(2/(E^4*Log[5/3]^2))*ExpIntegralEi[(-2*(1 - E^4*Log[5/3]^2*Log[x]))/(E^4*Log[5/3]^2)] + 24*E^(-4
+ 2/(E^4*Log[5/3]^2))*ExpIntegralEi[-2/(E^4*Log[5/3]^2) + 2*Log[x]]*(Log[5/3]^(-2) - E^4*Log[x]) - (6*E^(-4 +
2/(E^4*Log[5/3]^2))*ExpIntegralEi[(-2*(1 - E^4*Log[5/3]^2*Log[x]))/(E^4*Log[5/3]^2)]*(2 - E^4*Log[5/3]^2 - 2*E
^4*Log[5/3]^2*Log[x]))/Log[5/3]^2 - (3*x^2*(2 - E^4*Log[5/3]^2 - 2*E^4*Log[5/3]^2*Log[x]))/(1 - E^4*Log[5/3]^2
*Log[x]) - (12*E^(-4 + 2/(E^4*Log[5/3]^2))*ExpIntegralEi[(-2*(1 - E^4*Log[5/3]^2*Log[x]))/(E^4*Log[5/3]^2)]*(1
+ E^4*Log[5/3]^2 - E^4*Log[5/3]^2*Log[x]))/Log[5/3]^2 + (3*E^4*x^2*Log[5/3]^2*(1 + E^4*Log[5/3]^2 - E^4*Log[5
/3]^2*Log[x]))/(1 - E^4*Log[5/3]^2*Log[x])^2 - (6*x^2*(1 + E^4*Log[5/3]^2 - E^4*Log[5/3]^2*Log[x]))/(1 - E^4*L
og[5/3]^2*Log[x])

Rule 12

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

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2343

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

Rule 2346

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[E^((m + 1)*x)*(a
+ b*x)^p, x], x, Log[c*x]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[m]

Rule 2413

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_.)]*(e_.))*((g_.)*(x_))^(m_.), x_Sy
mbol] :> With[{u = IntHide[(g*x)^m*(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - Dist[e*r, Int[Sim
plifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, r}, x] &&  !(EqQ[p, 1] && EqQ[a, 0] &&
NeQ[d, 0])

Rule 6617

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

Rule 6820

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {6 e^8 x \log ^4\left (\frac {5}{3}\right ) \left (1+e^4 \log ^2\left (\frac {5}{3}\right )-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{\left (1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )^3} \, dx\\ &=\left (6 e^8 \log ^4\left (\frac {5}{3}\right )\right ) \int \frac {x \left (1+e^4 \log ^2\left (\frac {5}{3}\right )-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{\left (1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )^3} \, dx\\ &=-\frac {12 e^{-4+\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}} \text {Ei}\left (-\frac {2 \left (1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{e^4 \log ^2\left (\frac {5}{3}\right )}\right ) \left (1+e^4 \log ^2\left (\frac {5}{3}\right )-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{\log ^2\left (\frac {5}{3}\right )}+\frac {3 e^4 x^2 \log ^2\left (\frac {5}{3}\right ) \left (1+e^4 \log ^2\left (\frac {5}{3}\right )-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{\left (1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )^2}-\frac {6 x^2 \left (1+e^4 \log ^2\left (\frac {5}{3}\right )-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)}+\left (6 e^{12} \log ^6\left (\frac {5}{3}\right )\right ) \int \left (-\frac {2 e^{-12+\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}} \text {Ei}\left (-\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}+2 \log (x)\right )}{x \log ^6\left (\frac {5}{3}\right )}+\frac {x \left (-2 \left (1-\frac {1}{2} e^4 \log ^2\left (\frac {5}{3}\right )\right )+2 e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{2 e^8 \log ^4\left (\frac {5}{3}\right ) \left (-1+e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )^2}\right ) \, dx\\ &=-\frac {12 e^{-4+\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}} \text {Ei}\left (-\frac {2 \left (1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{e^4 \log ^2\left (\frac {5}{3}\right )}\right ) \left (1+e^4 \log ^2\left (\frac {5}{3}\right )-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{\log ^2\left (\frac {5}{3}\right )}+\frac {3 e^4 x^2 \log ^2\left (\frac {5}{3}\right ) \left (1+e^4 \log ^2\left (\frac {5}{3}\right )-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{\left (1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )^2}-\frac {6 x^2 \left (1+e^4 \log ^2\left (\frac {5}{3}\right )-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)}-\left (12 e^{\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}}\right ) \int \frac {\text {Ei}\left (-\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}+2 \log (x)\right )}{x} \, dx+\left (3 e^4 \log ^2\left (\frac {5}{3}\right )\right ) \int \frac {x \left (-2 \left (1-\frac {1}{2} e^4 \log ^2\left (\frac {5}{3}\right )\right )+2 e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{\left (-1+e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )^2} \, dx\\ &=-\frac {6 e^{-4+\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}} \text {Ei}\left (-\frac {2 \left (1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{e^4 \log ^2\left (\frac {5}{3}\right )}\right ) \left (2-e^4 \log ^2\left (\frac {5}{3}\right )-2 e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{\log ^2\left (\frac {5}{3}\right )}-\frac {3 x^2 \left (2-e^4 \log ^2\left (\frac {5}{3}\right )-2 e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)}-\frac {12 e^{-4+\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}} \text {Ei}\left (-\frac {2 \left (1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{e^4 \log ^2\left (\frac {5}{3}\right )}\right ) \left (1+e^4 \log ^2\left (\frac {5}{3}\right )-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{\log ^2\left (\frac {5}{3}\right )}+\frac {3 e^4 x^2 \log ^2\left (\frac {5}{3}\right ) \left (1+e^4 \log ^2\left (\frac {5}{3}\right )-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{\left (1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )^2}-\frac {6 x^2 \left (1+e^4 \log ^2\left (\frac {5}{3}\right )-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)}-\left (12 e^{\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}}\right ) \text {Subst}\left (\int \text {Ei}\left (2 x-\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}\right ) \, dx,x,\log (x)\right )-\left (6 e^8 \log ^4\left (\frac {5}{3}\right )\right ) \int \left (\frac {2 e^{-8+\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}} \text {Ei}\left (-\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}+2 \log (x)\right )}{x \log ^4\left (\frac {5}{3}\right )}+\frac {x}{e^4 \log ^2\left (\frac {5}{3}\right ) \left (1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}\right ) \, dx\\ &=6 x^2+12 e^{-4+\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}} \text {Ei}\left (-\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}+2 \log (x)\right ) \left (\frac {1}{\log ^2\left (\frac {5}{3}\right )}-e^4 \log (x)\right )-\frac {6 e^{-4+\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}} \text {Ei}\left (-\frac {2 \left (1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{e^4 \log ^2\left (\frac {5}{3}\right )}\right ) \left (2-e^4 \log ^2\left (\frac {5}{3}\right )-2 e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{\log ^2\left (\frac {5}{3}\right )}-\frac {3 x^2 \left (2-e^4 \log ^2\left (\frac {5}{3}\right )-2 e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)}-\frac {12 e^{-4+\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}} \text {Ei}\left (-\frac {2 \left (1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{e^4 \log ^2\left (\frac {5}{3}\right )}\right ) \left (1+e^4 \log ^2\left (\frac {5}{3}\right )-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{\log ^2\left (\frac {5}{3}\right )}+\frac {3 e^4 x^2 \log ^2\left (\frac {5}{3}\right ) \left (1+e^4 \log ^2\left (\frac {5}{3}\right )-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{\left (1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )^2}-\frac {6 x^2 \left (1+e^4 \log ^2\left (\frac {5}{3}\right )-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)}-\left (12 e^{\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}}\right ) \int \frac {\text {Ei}\left (-\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}+2 \log (x)\right )}{x} \, dx-\left (6 e^4 \log ^2\left (\frac {5}{3}\right )\right ) \int \frac {x}{1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)} \, dx\\ &=6 x^2+12 e^{-4+\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}} \text {Ei}\left (-\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}+2 \log (x)\right ) \left (\frac {1}{\log ^2\left (\frac {5}{3}\right )}-e^4 \log (x)\right )-\frac {6 e^{-4+\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}} \text {Ei}\left (-\frac {2 \left (1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{e^4 \log ^2\left (\frac {5}{3}\right )}\right ) \left (2-e^4 \log ^2\left (\frac {5}{3}\right )-2 e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{\log ^2\left (\frac {5}{3}\right )}-\frac {3 x^2 \left (2-e^4 \log ^2\left (\frac {5}{3}\right )-2 e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)}-\frac {12 e^{-4+\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}} \text {Ei}\left (-\frac {2 \left (1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{e^4 \log ^2\left (\frac {5}{3}\right )}\right ) \left (1+e^4 \log ^2\left (\frac {5}{3}\right )-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{\log ^2\left (\frac {5}{3}\right )}+\frac {3 e^4 x^2 \log ^2\left (\frac {5}{3}\right ) \left (1+e^4 \log ^2\left (\frac {5}{3}\right )-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{\left (1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )^2}-\frac {6 x^2 \left (1+e^4 \log ^2\left (\frac {5}{3}\right )-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)}-\left (12 e^{\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}}\right ) \text {Subst}\left (\int \text {Ei}\left (2 x-\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}\right ) \, dx,x,\log (x)\right )-\left (6 e^4 \log ^2\left (\frac {5}{3}\right )\right ) \text {Subst}\left (\int \frac {e^{2 x}}{1-e^4 x \log ^2\left (\frac {5}{3}\right )} \, dx,x,\log (x)\right )\\ &=12 x^2+6 e^{\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}} \text {Ei}\left (-\frac {2 \left (1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{e^4 \log ^2\left (\frac {5}{3}\right )}\right )+24 e^{-4+\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}} \text {Ei}\left (-\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}+2 \log (x)\right ) \left (\frac {1}{\log ^2\left (\frac {5}{3}\right )}-e^4 \log (x)\right )-\frac {6 e^{-4+\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}} \text {Ei}\left (-\frac {2 \left (1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{e^4 \log ^2\left (\frac {5}{3}\right )}\right ) \left (2-e^4 \log ^2\left (\frac {5}{3}\right )-2 e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{\log ^2\left (\frac {5}{3}\right )}-\frac {3 x^2 \left (2-e^4 \log ^2\left (\frac {5}{3}\right )-2 e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)}-\frac {12 e^{-4+\frac {2}{e^4 \log ^2\left (\frac {5}{3}\right )}} \text {Ei}\left (-\frac {2 \left (1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{e^4 \log ^2\left (\frac {5}{3}\right )}\right ) \left (1+e^4 \log ^2\left (\frac {5}{3}\right )-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{\log ^2\left (\frac {5}{3}\right )}+\frac {3 e^4 x^2 \log ^2\left (\frac {5}{3}\right ) \left (1+e^4 \log ^2\left (\frac {5}{3}\right )-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{\left (1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )^2}-\frac {6 x^2 \left (1+e^4 \log ^2\left (\frac {5}{3}\right )-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )}{1-e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]
time = 0.04, size = 30, normalized size = 1.36 \begin {gather*} \frac {3 e^8 x^2 \log ^4\left (\frac {5}{3}\right )}{\left (-1+e^4 \log ^2\left (\frac {5}{3}\right ) \log (x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6*E^8*x*Log[5/3]^4 + E^12*Log[5/3]^6*(-6*x + 6*x*Log[x]))/(-1 + 3*E^4*Log[5/3]^2*Log[x] - 3*E^8*Lo
g[5/3]^4*Log[x]^2 + E^12*Log[5/3]^6*Log[x]^3),x]

[Out]

(3*E^8*x^2*Log[5/3]^4)/(-1 + E^4*Log[5/3]^2*Log[x])^2

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(154\) vs. \(2(23)=46\).
time = 1.30, size = 155, normalized size = 7.05

method	result	size
norman	\(\frac {3 \,{\mathrm e}^{8} \left (\ln \left (5\right )^{4}-4 \ln \left (5\right )^{3} \ln \left (3\right )+6 \ln \left (3\right )^{2} \ln \left (5\right )^{2}-4 \ln \left (3\right )^{3} \ln \left (5\right )+\ln \left (3\right )^{4}\right ) x^{2}}{\left (\ln \left (x \right ) {\mathrm e}^{4} \ln \left (\frac {5}{3}\right )^{2}-1\right )^{2}}\)	\(58\)
risch	\(\frac {3 \left (\ln \left (5\right )^{4}-4 \ln \left (5\right )^{3} \ln \left (3\right )+6 \ln \left (3\right )^{2} \ln \left (5\right )^{2}-4 \ln \left (3\right )^{3} \ln \left (5\right )+\ln \left (3\right )^{4}\right ) x^{2} {\mathrm e}^{8}}{\left (\ln \left (x \right ) \ln \left (5\right )^{2} {\mathrm e}^{4}-2 \ln \left (x \right ) {\mathrm e}^{4} \ln \left (3\right ) \ln \left (5\right )+\ln \left (x \right ) {\mathrm e}^{4} \ln \left (3\right )^{2}-1\right )^{2}}\)	\(75\)
default	\(\frac {3 \,{\mathrm e}^{8} x^{2} \left (\ln \left (5\right )^{4}-4 \ln \left (5\right )^{3} \ln \left (3\right )+6 \ln \left (3\right )^{2} \ln \left (5\right )^{2}-4 \ln \left (3\right )^{3} \ln \left (5\right )+\ln \left (3\right )^{4}\right )}{\ln \left (x \right )^{2} \ln \left (5\right )^{4} {\mathrm e}^{8}-4 \ln \left (x \right )^{2} \ln \left (5\right )^{3} {\mathrm e}^{8} \ln \left (3\right )+6 \ln \left (x \right )^{2} \ln \left (5\right )^{2} {\mathrm e}^{8} \ln \left (3\right )^{2}-4 \ln \left (x \right )^{2} {\mathrm e}^{8} \ln \left (3\right )^{3} \ln \left (5\right )+\ln \left (x \right )^{2} {\mathrm e}^{8} \ln \left (3\right )^{4}-2 \ln \left (x \right ) \ln \left (5\right )^{2} {\mathrm e}^{4}+4 \ln \left (x \right ) {\mathrm e}^{4} \ln \left (3\right ) \ln \left (5\right )-2 \ln \left (x \right ) {\mathrm e}^{4} \ln \left (3\right )^{2}+1}\)	\(155\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((6*x*ln(x)-6*x)*exp(2*ln(exp(4)*ln(5/3))-4)^3-6*x*exp(2*ln(exp(4)*ln(5/3))-4)^2)/(ln(x)^3*exp(2*ln(exp(4)
*ln(5/3))-4)^3-3*ln(x)^2*exp(2*ln(exp(4)*ln(5/3))-4)^2+3*ln(x)*exp(2*ln(exp(4)*ln(5/3))-4)-1),x,method=_RETURN
VERBOSE)

[Out]

3*exp(4)^2*x^2*(ln(5)^4-4*ln(5)^3*ln(3)+6*ln(3)^2*ln(5)^2-4*ln(3)^3*ln(5)+ln(3)^4)/(ln(x)^2*ln(5)^4*exp(4)^2-4
*ln(x)^2*ln(5)^3*exp(4)^2*ln(3)+6*ln(x)^2*ln(5)^2*exp(4)^2*ln(3)^2-4*ln(x)^2*exp(4)^2*ln(3)^3*ln(5)+ln(x)^2*ex
p(4)^2*ln(3)^4-2*ln(x)*ln(5)^2*exp(4)+4*ln(x)*exp(4)*ln(3)*ln(5)-2*ln(x)*exp(4)*ln(3)^2+1)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(4*((log(5)^2 - 2*log(5)*log(3) + log(3)^2)*e^4 + 1)*integrate(x/((log(5)^8 - 8*log(5)^7*log(3) + 28*log(5)^
6*log(3)^2 - 56*log(5)^5*log(3)^3 + 70*log(5)^4*log(3)^4 - 56*log(5)^3*log(3)^5 + 28*log(5)^2*log(3)^6 - 8*log
(5)*log(3)^7 + log(3)^8)*e^16*log(x) - (log(5)^6 - 6*log(5)^5*log(3) + 15*log(5)^4*log(3)^2 - 20*log(5)^3*log(
3)^3 + 15*log(5)^2*log(3)^4 - 6*log(5)*log(3)^5 + log(3)^6)*e^12), x) - (2*((log(5)^4 - 4*log(5)^3*log(3) + 6*
log(5)^2*log(3)^2 - 4*log(5)*log(3)^3 + log(3)^4)*e^8 + (log(5)^2 - 2*log(5)*log(3) + log(3)^2)*e^4)*x^2*log(x
) - ((log(5)^2 - 2*log(5)*log(3) + log(3)^2)*e^4 + 2)*x^2)/((log(5)^10 - 10*log(5)^9*log(3) + 45*log(5)^8*log(
3)^2 - 120*log(5)^7*log(3)^3 + 210*log(5)^6*log(3)^4 - 252*log(5)^5*log(3)^5 + 210*log(5)^4*log(3)^6 - 120*log
(5)^3*log(3)^7 + 45*log(5)^2*log(3)^8 - 10*log(5)*log(3)^9 + log(3)^10)*e^20*log(x)^2 - 2*(log(5)^8 - 8*log(5)
^7*log(3) + 28*log(5)^6*log(3)^2 - 56*log(5)^5*log(3)^3 + 70*log(5)^4*log(3)^4 - 56*log(5)^3*log(3)^5 + 28*log
(5)^2*log(3)^6 - 8*log(5)*log(3)^7 + log(3)^8)*e^16*log(x) + (log(5)^6 - 6*log(5)^5*log(3) + 15*log(5)^4*log(3
)^2 - 20*log(5)^3*log(3)^3 + 15*log(5)^2*log(3)^4 - 6*log(5)*log(3)^5 + log(3)^6)*e^12))*e^12*log(5/3)^6 + 6*e
^(2*e^(-4)/log(5/3)^2 + 8)*exp_integral_e(3, -2*(e^4*log(5/3)^2*log(x) - 1)*e^(-4)/log(5/3)^2)*log(5/3)^4/(e^4
*log(5/3)^2*log(x) - 1)^2 + 6*e^(2*e^(-4)/log(5/3)^2 + 4)*exp_integral_e(3, -2*(e^4*log(5/3)^2*log(x) - 1)*e^(
-4)/log(5/3)^2)*log(5/3)^2/(e^4*log(5/3)^2*log(x) - 1)^2

________________________________________________________________________________________

Fricas [A]
time = 0.38, size = 36, normalized size = 1.64 \begin {gather*} \frac {3 \, x^{2} e^{8} \log \left (\frac {5}{3}\right )^{4}}{e^{8} \log \left (\frac {5}{3}\right )^{4} \log \left (x\right )^{2} - 2 \, e^{4} \log \left (\frac {5}{3}\right )^{2} \log \left (x\right ) + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x^2*e^8*log(5/3)^4/(e^8*log(5/3)^4*log(x)^2 - 2*e^4*log(5/3)^2*log(x) + 1)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (22) = 44\).
time = 0.51, size = 180, normalized size = 8.18 \begin {gather*} \frac {- 12 x^{2} e^{8} \log {\left (3 \right )} \log {\left (5 \right )}^{3} - 12 x^{2} e^{8} \log {\left (3 \right )}^{3} \log {\left (5 \right )} + 3 x^{2} e^{8} \log {\left (3 \right )}^{4} + 3 x^{2} e^{8} \log {\left (5 \right )}^{4} + 18 x^{2} e^{8} \log {\left (3 \right )}^{2} \log {\left (5 \right )}^{2}}{\left (- 4 e^{8} \log {\left (3 \right )} \log {\left (5 \right )}^{3} - 4 e^{8} \log {\left (3 \right )}^{3} \log {\left (5 \right )} + e^{8} \log {\left (3 \right )}^{4} + e^{8} \log {\left (5 \right )}^{4} + 6 e^{8} \log {\left (3 \right )}^{2} \log {\left (5 \right )}^{2}\right ) \log {\left (x \right )}^{2} + \left (- 2 e^{4} \log {\left (5 \right )}^{2} - 2 e^{4} \log {\left (3 \right )}^{2} + 4 e^{4} \log {\left (3 \right )} \log {\left (5 \right )}\right ) \log {\left (x \right )} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-12*x**2*exp(8)*log(3)*log(5)**3 - 12*x**2*exp(8)*log(3)**3*log(5) + 3*x**2*exp(8)*log(3)**4 + 3*x**2*exp(8)*
log(5)**4 + 18*x**2*exp(8)*log(3)**2*log(5)**2)/((-4*exp(8)*log(3)*log(5)**3 - 4*exp(8)*log(3)**3*log(5) + exp
(8)*log(3)**4 + exp(8)*log(5)**4 + 6*exp(8)*log(3)**2*log(5)**2)*log(x)**2 + (-2*exp(4)*log(5)**2 - 2*exp(4)*l
og(3)**2 + 4*exp(4)*log(3)*log(5))*log(x) + 1)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 564 vs. \(2 (23) = 46\).
time = 0.59, size = 564, normalized size = 25.64 \begin {gather*} \frac {3 \, x^{2} e^{8} \log \left (5\right )^{4}}{e^{8} \log \left (5\right )^{4} \log \left (x\right )^{2} - 4 \, e^{8} \log \left (5\right )^{3} \log \left (3\right ) \log \left (x\right )^{2} + 6 \, e^{8} \log \left (5\right )^{2} \log \left (3\right )^{2} \log \left (x\right )^{2} - 4 \, e^{8} \log \left (5\right ) \log \left (3\right )^{3} \log \left (x\right )^{2} + e^{8} \log \left (3\right )^{4} \log \left (x\right )^{2} - 2 \, e^{4} \log \left (5\right )^{2} \log \left (x\right ) + 4 \, e^{4} \log \left (5\right ) \log \left (3\right ) \log \left (x\right ) - 2 \, e^{4} \log \left (3\right )^{2} \log \left (x\right ) + 1} - \frac {12 \, x^{2} e^{8} \log \left (5\right )^{3} \log \left (3\right )}{e^{8} \log \left (5\right )^{4} \log \left (x\right )^{2} - 4 \, e^{8} \log \left (5\right )^{3} \log \left (3\right ) \log \left (x\right )^{2} + 6 \, e^{8} \log \left (5\right )^{2} \log \left (3\right )^{2} \log \left (x\right )^{2} - 4 \, e^{8} \log \left (5\right ) \log \left (3\right )^{3} \log \left (x\right )^{2} + e^{8} \log \left (3\right )^{4} \log \left (x\right )^{2} - 2 \, e^{4} \log \left (5\right )^{2} \log \left (x\right ) + 4 \, e^{4} \log \left (5\right ) \log \left (3\right ) \log \left (x\right ) - 2 \, e^{4} \log \left (3\right )^{2} \log \left (x\right ) + 1} + \frac {18 \, x^{2} e^{8} \log \left (5\right )^{2} \log \left (3\right )^{2}}{e^{8} \log \left (5\right )^{4} \log \left (x\right )^{2} - 4 \, e^{8} \log \left (5\right )^{3} \log \left (3\right ) \log \left (x\right )^{2} + 6 \, e^{8} \log \left (5\right )^{2} \log \left (3\right )^{2} \log \left (x\right )^{2} - 4 \, e^{8} \log \left (5\right ) \log \left (3\right )^{3} \log \left (x\right )^{2} + e^{8} \log \left (3\right )^{4} \log \left (x\right )^{2} - 2 \, e^{4} \log \left (5\right )^{2} \log \left (x\right ) + 4 \, e^{4} \log \left (5\right ) \log \left (3\right ) \log \left (x\right ) - 2 \, e^{4} \log \left (3\right )^{2} \log \left (x\right ) + 1} - \frac {12 \, x^{2} e^{8} \log \left (5\right ) \log \left (3\right )^{3}}{e^{8} \log \left (5\right )^{4} \log \left (x\right )^{2} - 4 \, e^{8} \log \left (5\right )^{3} \log \left (3\right ) \log \left (x\right )^{2} + 6 \, e^{8} \log \left (5\right )^{2} \log \left (3\right )^{2} \log \left (x\right )^{2} - 4 \, e^{8} \log \left (5\right ) \log \left (3\right )^{3} \log \left (x\right )^{2} + e^{8} \log \left (3\right )^{4} \log \left (x\right )^{2} - 2 \, e^{4} \log \left (5\right )^{2} \log \left (x\right ) + 4 \, e^{4} \log \left (5\right ) \log \left (3\right ) \log \left (x\right ) - 2 \, e^{4} \log \left (3\right )^{2} \log \left (x\right ) + 1} + \frac {3 \, x^{2} e^{8} \log \left (3\right )^{4}}{e^{8} \log \left (5\right )^{4} \log \left (x\right )^{2} - 4 \, e^{8} \log \left (5\right )^{3} \log \left (3\right ) \log \left (x\right )^{2} + 6 \, e^{8} \log \left (5\right )^{2} \log \left (3\right )^{2} \log \left (x\right )^{2} - 4 \, e^{8} \log \left (5\right ) \log \left (3\right )^{3} \log \left (x\right )^{2} + e^{8} \log \left (3\right )^{4} \log \left (x\right )^{2} - 2 \, e^{4} \log \left (5\right )^{2} \log \left (x\right ) + 4 \, e^{4} \log \left (5\right ) \log \left (3\right ) \log \left (x\right ) - 2 \, e^{4} \log \left (3\right )^{2} \log \left (x\right ) + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x^2*e^8*log(5)^4/(e^8*log(5)^4*log(x)^2 - 4*e^8*log(5)^3*log(3)*log(x)^2 + 6*e^8*log(5)^2*log(3)^2*log(x)^2
- 4*e^8*log(5)*log(3)^3*log(x)^2 + e^8*log(3)^4*log(x)^2 - 2*e^4*log(5)^2*log(x) + 4*e^4*log(5)*log(3)*log(x)
- 2*e^4*log(3)^2*log(x) + 1) - 12*x^2*e^8*log(5)^3*log(3)/(e^8*log(5)^4*log(x)^2 - 4*e^8*log(5)^3*log(3)*log(x
)^2 + 6*e^8*log(5)^2*log(3)^2*log(x)^2 - 4*e^8*log(5)*log(3)^3*log(x)^2 + e^8*log(3)^4*log(x)^2 - 2*e^4*log(5)
^2*log(x) + 4*e^4*log(5)*log(3)*log(x) - 2*e^4*log(3)^2*log(x) + 1) + 18*x^2*e^8*log(5)^2*log(3)^2/(e^8*log(5)
^4*log(x)^2 - 4*e^8*log(5)^3*log(3)*log(x)^2 + 6*e^8*log(5)^2*log(3)^2*log(x)^2 - 4*e^8*log(5)*log(3)^3*log(x)
^2 + e^8*log(3)^4*log(x)^2 - 2*e^4*log(5)^2*log(x) + 4*e^4*log(5)*log(3)*log(x) - 2*e^4*log(3)^2*log(x) + 1) -
12*x^2*e^8*log(5)*log(3)^3/(e^8*log(5)^4*log(x)^2 - 4*e^8*log(5)^3*log(3)*log(x)^2 + 6*e^8*log(5)^2*log(3)^2*
log(x)^2 - 4*e^8*log(5)*log(3)^3*log(x)^2 + e^8*log(3)^4*log(x)^2 - 2*e^4*log(5)^2*log(x) + 4*e^4*log(5)*log(3
)*log(x) - 2*e^4*log(3)^2*log(x) + 1) + 3*x^2*e^8*log(3)^4/(e^8*log(5)^4*log(x)^2 - 4*e^8*log(5)^3*log(3)*log(
x)^2 + 6*e^8*log(5)^2*log(3)^2*log(x)^2 - 4*e^8*log(5)*log(3)^3*log(x)^2 + e^8*log(3)^4*log(x)^2 - 2*e^4*log(5
)^2*log(x) + 4*e^4*log(5)*log(3)*log(x) - 2*e^4*log(3)^2*log(x) + 1)

________________________________________________________________________________________

Mupad [B]
time = 5.65, size = 102, normalized size = 4.64 \begin {gather*} \frac {3\,x^2\,{\mathrm {e}}^{24}\,{\left (\ln \left (3\right )-\ln \left (5\right )\right )}^4}{\left ({\mathrm {e}}^{24}\,{\ln \left (3\right )}^4+{\mathrm {e}}^{24}\,{\ln \left (5\right )}^4+6\,{\mathrm {e}}^{24}\,{\ln \left (3\right )}^2\,{\ln \left (5\right )}^2-4\,{\mathrm {e}}^{24}\,\ln \left (3\right )\,{\ln \left (5\right )}^3-4\,{\mathrm {e}}^{24}\,{\ln \left (3\right )}^3\,\ln \left (5\right )\right )\,{\ln \left (x\right )}^2+\left (4\,{\mathrm {e}}^{20}\,\ln \left (3\right )\,\ln \left (5\right )-2\,{\mathrm {e}}^{20}\,{\ln \left (5\right )}^2-2\,{\mathrm {e}}^{20}\,{\ln \left (3\right )}^2\right )\,\ln \left (x\right )+{\mathrm {e}}^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(6*log(exp(4)*log(5/3)) - 12)*(6*x - 6*x*log(x)) + 6*x*exp(4*log(exp(4)*log(5/3)) - 8))/(3*exp(2*log(
exp(4)*log(5/3)) - 4)*log(x) - 3*exp(4*log(exp(4)*log(5/3)) - 8)*log(x)^2 + exp(6*log(exp(4)*log(5/3)) - 12)*l
og(x)^3 - 1),x)

[Out]

(3*x^2*exp(24)*(log(3) - log(5))^4)/(exp(16) - log(x)*(2*exp(20)*log(3)^2 + 2*exp(20)*log(5)^2 - 4*exp(20)*log
(3)*log(5)) + log(x)^2*(exp(24)*log(3)^4 + exp(24)*log(5)^4 + 6*exp(24)*log(3)^2*log(5)^2 - 4*exp(24)*log(3)*l
og(5)^3 - 4*exp(24)*log(3)^3*log(5)))

________________________________________________________________________________________