\(\int \frac {(8+2 e^2-2 x) \log (5)+(-54-3 e^4+24 x-3 x^2+e^2 (-24+6 x)-3 \log (4)) \log (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)) \log ^2(\log (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)))}{(18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)) \log (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)) \log ^2(\log (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)))} \, dx\) [8517]

Optimal result

Integrand size = 168, antiderivative size = 26 \[ \int \frac {\left (8+2 e^2-2 x\right ) \log (5)+\left (-54-3 e^4+24 x-3 x^2+e^2 (-24+6 x)-3 \log (4)\right ) \log \left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right ) \log ^2\left (\log \left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right )\right )}{\left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right ) \log \left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right ) \log ^2\left (\log \left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right )\right )} \, dx=2-3 x+\frac {\log (5)}{\log \left (\log \left (2+\left (4+e^2-x\right )^2+\log (4)\right )\right )} \]

[Out]

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(265\) vs. \(2(26)=52\).

Time = 0.66 (sec) , antiderivative size = 265, normalized size of antiderivative = 10.19, number of steps used = 19, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {6820, 6860, 717, 648, 632, 210, 642, 6818} \[ \int \frac {\left (8+2 e^2-2 x\right ) \log (5)+\left (-54-3 e^4+24 x-3 x^2+e^2 (-24+6 x)-3 \log (4)\right ) \log \left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right ) \log ^2\left (\log \left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right )\right )}{\left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right ) \log \left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right ) \log ^2\left (\log \left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right )\right )} \, dx=-\frac {48 \left (4+e^2\right ) \arctan \left (\frac {2 \left (-x+e^2+4\right )}{\sqrt {8+\log (256)}}\right )}{\sqrt {8+\log (256)}}-\frac {12 e^4 \arctan \left (\frac {2 \left (-x+e^2+4\right )}{\sqrt {8+\log (256)}}\right )}{\sqrt {8+\log (256)}}+\frac {6 \left (18+e^4+\log (4)\right ) \arctan \left (\frac {2 \left (-x+e^2+4\right )}{\sqrt {8+\log (256)}}\right )}{\sqrt {8+\log (256)}}+\frac {6 \left (14+8 e^2+e^4-\log (4)\right ) \arctan \left (\frac {2 \left (-x+e^2+4\right )}{\sqrt {8+\log (256)}}\right )}{\sqrt {8+\log (256)}}-3 \left (4+e^2\right ) \log \left (x^2-2 \left (4+e^2\right ) x+e^4+8 e^2+18+\log (4)\right )+3 e^2 \log \left (x^2-2 \left (4+e^2\right ) x+e^4+8 e^2+18+\log (4)\right )+12 \log \left (x^2-2 \left (4+e^2\right ) x+e^4+8 e^2+18+\log (4)\right )+\frac {\log (5)}{\log \left (\log \left (x^2-2 \left (4+e^2\right ) x+e^4+8 e^2+18+\log (4)\right )\right )}-3 x \]

[In]

Int[((8 + 2*E^2 - 2*x)*Log[5] + (-54 - 3*E^4 + 24*x - 3*x^2 + E^2*(-24 + 6*x) - 3*Log[4])*Log[18 + E^4 + E^2*(
8 - 2*x) - 8*x + x^2 + Log[4]]*Log[Log[18 + E^4 + E^2*(8 - 2*x) - 8*x + x^2 + Log[4]]]^2)/((18 + E^4 + E^2*(8
- 2*x) - 8*x + x^2 + Log[4])*Log[18 + E^4 + E^2*(8 - 2*x) - 8*x + x^2 + Log[4]]*Log[Log[18 + E^4 + E^2*(8 - 2*
x) - 8*x + x^2 + Log[4]]]^2),x]

[Out]

-3*x - (12*E^4*ArcTan[(2*(4 + E^2 - x))/Sqrt[8 + Log[256]]])/Sqrt[8 + Log[256]] - (48*(4 + E^2)*ArcTan[(2*(4 +
E^2 - x))/Sqrt[8 + Log[256]]])/Sqrt[8 + Log[256]] + (6*ArcTan[(2*(4 + E^2 - x))/Sqrt[8 + Log[256]]]*(14 + 8*E
^2 + E^4 - Log[4]))/Sqrt[8 + Log[256]] + (6*ArcTan[(2*(4 + E^2 - x))/Sqrt[8 + Log[256]]]*(18 + E^4 + Log[4]))/
Sqrt[8 + Log[256]] + 12*Log[18 + 8*E^2 + E^4 - 2*(4 + E^2)*x + x^2 + Log[4]] + 3*E^2*Log[18 + 8*E^2 + E^4 - 2*
(4 + E^2)*x + x^2 + Log[4]] - 3*(4 + E^2)*Log[18 + 8*E^2 + E^4 - 2*(4 + E^2)*x + x^2 + Log[4]] + Log[5]/Log[Lo
g[18 + 8*E^2 + E^4 - 2*(4 + E^2)*x + x^2 + Log[4]]]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

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

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 717

Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[e*((d + e*x)^(m - 1)/(c*(
m - 1))), x] + Dist[1/c, Int[(d + e*x)^(m - 2)*(Simp[c*d^2 - a*e^2 + e*(2*c*d - b*e)*x, x]/(a + b*x + c*x^2)),
x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*
e, 0] && GtQ[m, 1]

Rule 6818

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*(y^(m + 1)/(m + 1)), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6820

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

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \frac {6 e^2 (-4+x)+24 x-3 x^2-54 \left (1+\frac {1}{18} \left (e^4+\log (4)\right )\right )+\frac {2 \left (4+e^2-x\right ) \log (5)}{\log \left (18+e^4-2 e^2 (-4+x)-8 x+x^2+\log (4)\right ) \log ^2\left (\log \left (18+e^4-2 e^2 (-4+x)-8 x+x^2+\log (4)\right )\right )}}{18+8 e^2+e^4-2 \left (4+e^2\right ) x+x^2+\log (4)} \, dx \\ & = \int \left (\frac {3 x^2}{-18-8 e^2-e^4+2 \left (4+e^2\right ) x-x^2-\log (4)}+\frac {6 e^2 (-4+x)}{18+8 e^2+e^4-2 \left (4+e^2\right ) x+x^2+\log (4)}+\frac {24 x}{18+8 e^2+e^4-2 \left (4+e^2\right ) x+x^2+\log (4)}+\frac {3 \left (-18-e^4-\log (4)\right )}{18+8 e^2+e^4-2 \left (4+e^2\right ) x+x^2+\log (4)}+\frac {2 \left (4+e^2-x\right ) \log (5)}{\left (18+8 e^2+e^4-2 \left (4+e^2\right ) x+x^2+\log (4)\right ) \log \left (18+8 e^2+e^4-2 \left (4+e^2\right ) x+x^2+\log (4)\right ) \log ^2\left (\log \left (18+8 e^2+e^4-2 \left (4+e^2\right ) x+x^2+\log (4)\right )\right )}\right ) \, dx \\ & = 3 \int \frac {x^2}{-18-8 e^2-e^4+2 \left (4+e^2\right ) x-x^2-\log (4)} \, dx+24 \int \frac {x}{18+8 e^2+e^4-2 \left (4+e^2\right ) x+x^2+\log (4)} \, dx+\left (6 e^2\right ) \int \frac {-4+x}{18+8 e^2+e^4-2 \left (4+e^2\right ) x+x^2+\log (4)} \, dx-\left (3 \left (18+e^4+\log (4)\right )\right ) \int \frac {1}{18+8 e^2+e^4-2 \left (4+e^2\right ) x+x^2+\log (4)} \, dx+(2 \log (5)) \int \frac {4+e^2-x}{\left (18+8 e^2+e^4-2 \left (4+e^2\right ) x+x^2+\log (4)\right ) \log \left (18+8 e^2+e^4-2 \left (4+e^2\right ) x+x^2+\log (4)\right ) \log ^2\left (\log \left (18+8 e^2+e^4-2 \left (4+e^2\right ) x+x^2+\log (4)\right )\right )} \, dx \\ & = -3 x+\frac {\log (5)}{\log \left (\log \left (18+8 e^2+e^4-2 \left (4+e^2\right ) x+x^2+\log (4)\right )\right )}-3 \int \frac {18+8 e^2+e^4-2 \left (4+e^2\right ) x+\log (4)}{-18-8 e^2-e^4+2 \left (4+e^2\right ) x-x^2-\log (4)} \, dx+12 \int \frac {-2 \left (4+e^2\right )+2 x}{18+8 e^2+e^4-2 \left (4+e^2\right ) x+x^2+\log (4)} \, dx+\left (3 e^2\right ) \int \frac {-2 \left (4+e^2\right )+2 x}{18+8 e^2+e^4-2 \left (4+e^2\right ) x+x^2+\log (4)} \, dx+\left (6 e^4\right ) \int \frac {1}{18+8 e^2+e^4-2 \left (4+e^2\right ) x+x^2+\log (4)} \, dx+\left (24 \left (4+e^2\right )\right ) \int \frac {1}{18+8 e^2+e^4-2 \left (4+e^2\right ) x+x^2+\log (4)} \, dx+\left (6 \left (18+e^4+\log (4)\right )\right ) \text {Subst}\left (\int \frac {1}{-x^2-4 (2+\log (4))} \, dx,x,-2 \left (4+e^2\right )+2 x\right ) \\ & = -3 x+\frac {6 \arctan \left (\frac {2 \left (4+e^2-x\right )}{\sqrt {8+\log (256)}}\right ) \left (18+e^4+\log (4)\right )}{\sqrt {8+\log (256)}}+12 \log \left (18+8 e^2+e^4-2 \left (4+e^2\right ) x+x^2+\log (4)\right )+3 e^2 \log \left (18+8 e^2+e^4-2 \left (4+e^2\right ) x+x^2+\log (4)\right )+\frac {\log (5)}{\log \left (\log \left (18+8 e^2+e^4-2 \left (4+e^2\right ) x+x^2+\log (4)\right )\right )}-\left (12 e^4\right ) \text {Subst}\left (\int \frac {1}{-x^2-4 (2+\log (4))} \, dx,x,-2 \left (4+e^2\right )+2 x\right )-\left (3 \left (4+e^2\right )\right ) \int \frac {2 \left (4+e^2\right )-2 x}{-18-8 e^2-e^4+2 \left (4+e^2\right ) x-x^2-\log (4)} \, dx-\left (48 \left (4+e^2\right )\right ) \text {Subst}\left (\int \frac {1}{-x^2-4 (2+\log (4))} \, dx,x,-2 \left (4+e^2\right )+2 x\right )+\left (3 \left (14+8 e^2+e^4-\log (4)\right )\right ) \int \frac {1}{-18-8 e^2-e^4+2 \left (4+e^2\right ) x-x^2-\log (4)} \, dx \\ & = -3 x-\frac {12 e^4 \arctan \left (\frac {2 \left (4+e^2-x\right )}{\sqrt {8+\log (256)}}\right )}{\sqrt {8+\log (256)}}-\frac {48 \left (4+e^2\right ) \arctan \left (\frac {2 \left (4+e^2-x\right )}{\sqrt {8+\log (256)}}\right )}{\sqrt {8+\log (256)}}+\frac {6 \arctan \left (\frac {2 \left (4+e^2-x\right )}{\sqrt {8+\log (256)}}\right ) \left (18+e^4+\log (4)\right )}{\sqrt {8+\log (256)}}+12 \log \left (18+8 e^2+e^4-2 \left (4+e^2\right ) x+x^2+\log (4)\right )+3 e^2 \log \left (18+8 e^2+e^4-2 \left (4+e^2\right ) x+x^2+\log (4)\right )-3 \left (4+e^2\right ) \log \left (18+8 e^2+e^4-2 \left (4+e^2\right ) x+x^2+\log (4)\right )+\frac {\log (5)}{\log \left (\log \left (18+8 e^2+e^4-2 \left (4+e^2\right ) x+x^2+\log (4)\right )\right )}-\left (6 \left (14+8 e^2+e^4-\log (4)\right )\right ) \text {Subst}\left (\int \frac {1}{-x^2-4 (2+\log (4))} \, dx,x,2 \left (4+e^2\right )-2 x\right ) \\ & = -3 x-\frac {12 e^4 \arctan \left (\frac {2 \left (4+e^2-x\right )}{\sqrt {8+\log (256)}}\right )}{\sqrt {8+\log (256)}}-\frac {48 \left (4+e^2\right ) \arctan \left (\frac {2 \left (4+e^2-x\right )}{\sqrt {8+\log (256)}}\right )}{\sqrt {8+\log (256)}}+\frac {6 \arctan \left (\frac {2 \left (4+e^2-x\right )}{\sqrt {8+\log (256)}}\right ) \left (14+8 e^2+e^4-\log (4)\right )}{\sqrt {8+\log (256)}}+\frac {6 \arctan \left (\frac {2 \left (4+e^2-x\right )}{\sqrt {8+\log (256)}}\right ) \left (18+e^4+\log (4)\right )}{\sqrt {8+\log (256)}}+12 \log \left (18+8 e^2+e^4-2 \left (4+e^2\right ) x+x^2+\log (4)\right )+3 e^2 \log \left (18+8 e^2+e^4-2 \left (4+e^2\right ) x+x^2+\log (4)\right )-3 \left (4+e^2\right ) \log \left (18+8 e^2+e^4-2 \left (4+e^2\right ) x+x^2+\log (4)\right )+\frac {\log (5)}{\log \left (\log \left (18+8 e^2+e^4-2 \left (4+e^2\right ) x+x^2+\log (4)\right )\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {\left (8+2 e^2-2 x\right ) \log (5)+\left (-54-3 e^4+24 x-3 x^2+e^2 (-24+6 x)-3 \log (4)\right ) \log \left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right ) \log ^2\left (\log \left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right )\right )}{\left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right ) \log \left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right ) \log ^2\left (\log \left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right )\right )} \, dx=-3 x+\frac {\log (5)}{\log \left (\log \left (18+e^4-2 e^2 (-4+x)-8 x+x^2+\log (4)\right )\right )} \]

[In]

Integrate[((8 + 2*E^2 - 2*x)*Log[5] + (-54 - 3*E^4 + 24*x - 3*x^2 + E^2*(-24 + 6*x) - 3*Log[4])*Log[18 + E^4 +
E^2*(8 - 2*x) - 8*x + x^2 + Log[4]]*Log[Log[18 + E^4 + E^2*(8 - 2*x) - 8*x + x^2 + Log[4]]]^2)/((18 + E^4 + E
^2*(8 - 2*x) - 8*x + x^2 + Log[4])*Log[18 + E^4 + E^2*(8 - 2*x) - 8*x + x^2 + Log[4]]*Log[Log[18 + E^4 + E^2*(
8 - 2*x) - 8*x + x^2 + Log[4]]]^2),x]

[Out]

-3*x + Log[5]/Log[Log[18 + E^4 - 2*E^2*(-4 + x) - 8*x + x^2 + Log[4]]]

Maple [A] (verified)

Time = 2.94 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31

method	result	size
risch	\(-3 x +\frac {\ln \left (5\right )}{\ln \left (\ln \left (2 \ln \left (2\right )+{\mathrm e}^{4}+\left (-2 x +8\right ) {\mathrm e}^{2}+x^{2}-8 x +18\right )\right )}\)	\(34\)
default	\(-3 x +\frac {\ln \left (5\right )}{\ln \left (\ln \left (-2 \,{\mathrm e}^{2} x +x^{2}+8 \,{\mathrm e}^{2}+{\mathrm e}^{4}+2 \ln \left (2\right )-8 x +18\right )\right )}\)	\(37\)
parts	\(-3 x +\frac {\ln \left (5\right )}{\ln \left (\ln \left (-2 \,{\mathrm e}^{2} x +x^{2}+8 \,{\mathrm e}^{2}+{\mathrm e}^{4}+2 \ln \left (2\right )-8 x +18\right )\right )}\)	\(37\)
parallelrisch	\(-\frac {12 \,{\mathrm e}^{2} \ln \left (\ln \left (2 \ln \left (2\right )+{\mathrm e}^{4}+\left (-2 x +8\right ) {\mathrm e}^{2}+x^{2}-8 x +18\right )\right )+3 x \ln \left (\ln \left (2 \ln \left (2\right )+{\mathrm e}^{4}+\left (-2 x +8\right ) {\mathrm e}^{2}+x^{2}-8 x +18\right )\right )-\ln \left (5\right )+48 \ln \left (\ln \left (2 \ln \left (2\right )+{\mathrm e}^{4}+\left (-2 x +8\right ) {\mathrm e}^{2}+x^{2}-8 x +18\right )\right )}{\ln \left (\ln \left (2 \ln \left (2\right )+{\mathrm e}^{4}+\left (-2 x +8\right ) {\mathrm e}^{2}+x^{2}-8 x +18\right )\right )}\)	\(123\)

[In]

int(((-6*ln(2)-3*exp(2)^2+(6*x-24)*exp(2)-3*x^2+24*x-54)*ln(2*ln(2)+exp(2)^2+(-2*x+8)*exp(2)+x^2-8*x+18)*ln(ln
(2*ln(2)+exp(2)^2+(-2*x+8)*exp(2)+x^2-8*x+18))^2+(2*exp(2)-2*x+8)*ln(5))/(2*ln(2)+exp(2)^2+(-2*x+8)*exp(2)+x^2
-8*x+18)/ln(2*ln(2)+exp(2)^2+(-2*x+8)*exp(2)+x^2-8*x+18)/ln(ln(2*ln(2)+exp(2)^2+(-2*x+8)*exp(2)+x^2-8*x+18))^2
,x,method=_RETURNVERBOSE)

[Out]

-3*x+ln(5)/ln(ln(2*ln(2)+exp(4)+(-2*x+8)*exp(2)+x^2-8*x+18))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (27) = 54\).

Time = 0.26 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.23 \[ \int \frac {\left (8+2 e^2-2 x\right ) \log (5)+\left (-54-3 e^4+24 x-3 x^2+e^2 (-24+6 x)-3 \log (4)\right ) \log \left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right ) \log ^2\left (\log \left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right )\right )}{\left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right ) \log \left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right ) \log ^2\left (\log \left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right )\right )} \, dx=-\frac {3 \, x \log \left (\log \left (x^{2} - 2 \, {\left (x - 4\right )} e^{2} - 8 \, x + e^{4} + 2 \, \log \left (2\right ) + 18\right )\right ) - \log \left (5\right )}{\log \left (\log \left (x^{2} - 2 \, {\left (x - 4\right )} e^{2} - 8 \, x + e^{4} + 2 \, \log \left (2\right ) + 18\right )\right )} \]

[In]

integrate(((-6*log(2)-3*exp(2)^2+(6*x-24)*exp(2)-3*x^2+24*x-54)*log(2*log(2)+exp(2)^2+(-2*x+8)*exp(2)+x^2-8*x+
18)*log(log(2*log(2)+exp(2)^2+(-2*x+8)*exp(2)+x^2-8*x+18))^2+(2*exp(2)-2*x+8)*log(5))/(2*log(2)+exp(2)^2+(-2*x
+8)*exp(2)+x^2-8*x+18)/log(2*log(2)+exp(2)^2+(-2*x+8)*exp(2)+x^2-8*x+18)/log(log(2*log(2)+exp(2)^2+(-2*x+8)*ex
p(2)+x^2-8*x+18))^2,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int \frac {\left (8+2 e^2-2 x\right ) \log (5)+\left (-54-3 e^4+24 x-3 x^2+e^2 (-24+6 x)-3 \log (4)\right ) \log \left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right ) \log ^2\left (\log \left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right )\right )}{\left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right ) \log \left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right ) \log ^2\left (\log \left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right )\right )} \, dx=- 3 x + \frac {\log {\left (5 \right )}}{\log {\left (\log {\left (x^{2} - 8 x + \left (8 - 2 x\right ) e^{2} + 2 \log {\left (2 \right )} + 18 + e^{4} \right )} \right )}} \]

[In]

integrate(((-6*ln(2)-3*exp(2)**2+(6*x-24)*exp(2)-3*x**2+24*x-54)*ln(2*ln(2)+exp(2)**2+(-2*x+8)*exp(2)+x**2-8*x
+18)*ln(ln(2*ln(2)+exp(2)**2+(-2*x+8)*exp(2)+x**2-8*x+18))**2+(2*exp(2)-2*x+8)*ln(5))/(2*ln(2)+exp(2)**2+(-2*x
+8)*exp(2)+x**2-8*x+18)/ln(2*ln(2)+exp(2)**2+(-2*x+8)*exp(2)+x**2-8*x+18)/ln(ln(2*ln(2)+exp(2)**2+(-2*x+8)*exp
(2)+x**2-8*x+18))**2,x)

[Out]

-3*x + log(5)/log(log(x**2 - 8*x + (8 - 2*x)*exp(2) + 2*log(2) + 18 + exp(4)))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (27) = 54\).

Time = 0.34 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.31 \[ \int \frac {\left (8+2 e^2-2 x\right ) \log (5)+\left (-54-3 e^4+24 x-3 x^2+e^2 (-24+6 x)-3 \log (4)\right ) \log \left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right ) \log ^2\left (\log \left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right )\right )}{\left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right ) \log \left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right ) \log ^2\left (\log \left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right )\right )} \, dx=-\frac {3 \, x \log \left (\log \left (x^{2} - 2 \, x {\left (e^{2} + 4\right )} + e^{4} + 8 \, e^{2} + 2 \, \log \left (2\right ) + 18\right )\right ) - \log \left (5\right )}{\log \left (\log \left (x^{2} - 2 \, x {\left (e^{2} + 4\right )} + e^{4} + 8 \, e^{2} + 2 \, \log \left (2\right ) + 18\right )\right )} \]

[In]

integrate(((-6*log(2)-3*exp(2)^2+(6*x-24)*exp(2)-3*x^2+24*x-54)*log(2*log(2)+exp(2)^2+(-2*x+8)*exp(2)+x^2-8*x+
18)*log(log(2*log(2)+exp(2)^2+(-2*x+8)*exp(2)+x^2-8*x+18))^2+(2*exp(2)-2*x+8)*log(5))/(2*log(2)+exp(2)^2+(-2*x
+8)*exp(2)+x^2-8*x+18)/log(2*log(2)+exp(2)^2+(-2*x+8)*exp(2)+x^2-8*x+18)/log(log(2*log(2)+exp(2)^2+(-2*x+8)*ex
p(2)+x^2-8*x+18))^2,x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (27) = 54\).

Time = 1.74 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.38 \[ \int \frac {\left (8+2 e^2-2 x\right ) \log (5)+\left (-54-3 e^4+24 x-3 x^2+e^2 (-24+6 x)-3 \log (4)\right ) \log \left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right ) \log ^2\left (\log \left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right )\right )}{\left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right ) \log \left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right ) \log ^2\left (\log \left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right )\right )} \, dx=-\frac {3 \, x \log \left (\log \left (x^{2} - 2 \, x e^{2} - 8 \, x + e^{4} + 8 \, e^{2} + 2 \, \log \left (2\right ) + 18\right )\right ) - \log \left (5\right )}{\log \left (\log \left (x^{2} - 2 \, x e^{2} - 8 \, x + e^{4} + 8 \, e^{2} + 2 \, \log \left (2\right ) + 18\right )\right )} \]

[In]

integrate(((-6*log(2)-3*exp(2)^2+(6*x-24)*exp(2)-3*x^2+24*x-54)*log(2*log(2)+exp(2)^2+(-2*x+8)*exp(2)+x^2-8*x+
18)*log(log(2*log(2)+exp(2)^2+(-2*x+8)*exp(2)+x^2-8*x+18))^2+(2*exp(2)-2*x+8)*log(5))/(2*log(2)+exp(2)^2+(-2*x
+8)*exp(2)+x^2-8*x+18)/log(2*log(2)+exp(2)^2+(-2*x+8)*exp(2)+x^2-8*x+18)/log(log(2*log(2)+exp(2)^2+(-2*x+8)*ex
p(2)+x^2-8*x+18))^2,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 13.44 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int \frac {\left (8+2 e^2-2 x\right ) \log (5)+\left (-54-3 e^4+24 x-3 x^2+e^2 (-24+6 x)-3 \log (4)\right ) \log \left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right ) \log ^2\left (\log \left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right )\right )}{\left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right ) \log \left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right ) \log ^2\left (\log \left (18+e^4+e^2 (8-2 x)-8 x+x^2+\log (4)\right )\right )} \, dx=\frac {\ln \left (5\right )}{\ln \left (\ln \left (8\,{\mathrm {e}}^2-8\,x+{\mathrm {e}}^4+2\,\ln \left (2\right )-2\,x\,{\mathrm {e}}^2+x^2+18\right )\right )}-3\,x \]

[In]

int((log(5)*(2*exp(2) - 2*x + 8) - log(log(exp(4) - 8*x + 2*log(2) + x^2 - exp(2)*(2*x - 8) + 18))^2*log(exp(4
) - 8*x + 2*log(2) + x^2 - exp(2)*(2*x - 8) + 18)*(3*exp(4) - 24*x + 6*log(2) + 3*x^2 - exp(2)*(6*x - 24) + 54
))/(log(log(exp(4) - 8*x + 2*log(2) + x^2 - exp(2)*(2*x - 8) + 18))^2*log(exp(4) - 8*x + 2*log(2) + x^2 - exp(
2)*(2*x - 8) + 18)*(exp(4) - 8*x + 2*log(2) + x^2 - exp(2)*(2*x - 8) + 18)),x)

[Out]

log(5)/log(log(8*exp(2) - 8*x + exp(4) + 2*log(2) - 2*x*exp(2) + x^2 + 18)) - 3*x