Integrand size = 455, antiderivative size = 28 \[ \int \frac {-x+18 e^{2 x} x+e^x \left (-54 x-18 e^4 x+18 x \log (5)\right )+\left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right ) \log \left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right )}{80 x^2+54 e^4 x^2+9 e^8 x^2+9 e^{2 x} x^2-x^3+\left (-54 x^2-18 e^4 x^2\right ) \log (5)+9 x^2 \log ^2(5)+e^x \left (-54 x^2-18 e^4 x^2+18 x^2 \log (5)\right )+\left (4000 x+2700 e^4 x+450 e^8 x+450 e^{2 x} x-50 x^2+\left (-2700 x-900 e^4 x\right ) \log (5)+450 x \log ^2(5)+e^x \left (-2700 x-900 e^4 x+900 x \log (5)\right )\right ) \log \left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right )+\left (50000+33750 e^4+5625 e^8+5625 e^{2 x}-625 x+\left (-33750-11250 e^4\right ) \log (5)+5625 \log ^2(5)+e^x \left (-33750-11250 e^4+11250 \log (5)\right )\right ) \log ^2\left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right )} \, dx=\frac {1}{25+\frac {x}{\log \left (1+x-9 \left (-3-e^4+e^x+\log (5)\right )^2\right )}} \] Output:
1/(x/ln(x-3*(exp(x)-exp(4)+ln(5)-3)*(3*exp(x)-3*exp(4)+3*ln(5)-9)+1)+25)
Leaf count is larger than twice the leaf count of optimal. \(61\) vs. \(2(28)=56\).
Time = 0.14 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.18 \[ \int \frac {-x+18 e^{2 x} x+e^x \left (-54 x-18 e^4 x+18 x \log (5)\right )+\left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right ) \log \left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right )}{80 x^2+54 e^4 x^2+9 e^8 x^2+9 e^{2 x} x^2-x^3+\left (-54 x^2-18 e^4 x^2\right ) \log (5)+9 x^2 \log ^2(5)+e^x \left (-54 x^2-18 e^4 x^2+18 x^2 \log (5)\right )+\left (4000 x+2700 e^4 x+450 e^8 x+450 e^{2 x} x-50 x^2+\left (-2700 x-900 e^4 x\right ) \log (5)+450 x \log ^2(5)+e^x \left (-2700 x-900 e^4 x+900 x \log (5)\right )\right ) \log \left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right )+\left (50000+33750 e^4+5625 e^8+5625 e^{2 x}-625 x+\left (-33750-11250 e^4\right ) \log (5)+5625 \log ^2(5)+e^x \left (-33750-11250 e^4+11250 \log (5)\right )\right ) \log ^2\left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right )} \, dx=-\frac {x}{25 \left (x+25 \log \left (-80-54 e^4-9 e^8-9 e^{2 x}+x+18 e^x \left (3+e^4-\log (5)\right )+18 \left (3+e^4\right ) \log (5)-9 \log ^2(5)\right )\right )} \] Input:
Integrate[(-x + 18*E^(2*x)*x + E^x*(-54*x - 18*E^4*x + 18*x*Log[5]) + (-80 - 54*E^4 - 9*E^8 - 9*E^(2*x) + x + E^x*(54 + 18*E^4 - 18*Log[5]) + (54 + 18*E^4)*Log[5] - 9*Log[5]^2)*Log[-80 - 54*E^4 - 9*E^8 - 9*E^(2*x) + x + E^ x*(54 + 18*E^4 - 18*Log[5]) + (54 + 18*E^4)*Log[5] - 9*Log[5]^2])/(80*x^2 + 54*E^4*x^2 + 9*E^8*x^2 + 9*E^(2*x)*x^2 - x^3 + (-54*x^2 - 18*E^4*x^2)*Lo g[5] + 9*x^2*Log[5]^2 + E^x*(-54*x^2 - 18*E^4*x^2 + 18*x^2*Log[5]) + (4000 *x + 2700*E^4*x + 450*E^8*x + 450*E^(2*x)*x - 50*x^2 + (-2700*x - 900*E^4* x)*Log[5] + 450*x*Log[5]^2 + E^x*(-2700*x - 900*E^4*x + 900*x*Log[5]))*Log [-80 - 54*E^4 - 9*E^8 - 9*E^(2*x) + x + E^x*(54 + 18*E^4 - 18*Log[5]) + (5 4 + 18*E^4)*Log[5] - 9*Log[5]^2] + (50000 + 33750*E^4 + 5625*E^8 + 5625*E^ (2*x) - 625*x + (-33750 - 11250*E^4)*Log[5] + 5625*Log[5]^2 + E^x*(-33750 - 11250*E^4 + 11250*Log[5]))*Log[-80 - 54*E^4 - 9*E^8 - 9*E^(2*x) + x + E^ x*(54 + 18*E^4 - 18*Log[5]) + (54 + 18*E^4)*Log[5] - 9*Log[5]^2]^2),x]
Output:
-1/25*x/(x + 25*Log[-80 - 54*E^4 - 9*E^8 - 9*E^(2*x) + x + 18*E^x*(3 + E^4 - Log[5]) + 18*(3 + E^4)*Log[5] - 9*Log[5]^2])
Time = 2.54 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.013, Rules used = {6, 6, 6, 7239, 7262, 17}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {18 e^{2 x} x-x+\left (x-9 e^{2 x}+e^x \left (54+18 e^4-18 \log (5)\right )-9 e^8-54 e^4-80-9 \log ^2(5)+\left (54+18 e^4\right ) \log (5)\right ) \log \left (x-9 e^{2 x}+e^x \left (54+18 e^4-18 \log (5)\right )-9 e^8-54 e^4-80-9 \log ^2(5)+\left (54+18 e^4\right ) \log (5)\right )+e^x \left (-18 e^4 x-54 x+18 x \log (5)\right )}{-x^3+9 e^{2 x} x^2+9 e^8 x^2+54 e^4 x^2+80 x^2+9 x^2 \log ^2(5)+\left (-50 x^2+450 e^{2 x} x+450 e^8 x+2700 e^4 x+4000 x+450 x \log ^2(5)+e^x \left (-900 e^4 x-2700 x+900 x \log (5)\right )+\left (-900 e^4 x-2700 x\right ) \log (5)\right ) \log \left (x-9 e^{2 x}+e^x \left (54+18 e^4-18 \log (5)\right )-9 e^8-54 e^4-80-9 \log ^2(5)+\left (54+18 e^4\right ) \log (5)\right )+e^x \left (-18 e^4 x^2-54 x^2+18 x^2 \log (5)\right )+\left (-18 e^4 x^2-54 x^2\right ) \log (5)+\left (-625 x+5625 e^{2 x}+e^x \left (-33750-11250 e^4+11250 \log (5)\right )+5625 e^8+33750 e^4+50000+5625 \log ^2(5)+\left (-33750-11250 e^4\right ) \log (5)\right ) \log ^2\left (x-9 e^{2 x}+e^x \left (54+18 e^4-18 \log (5)\right )-9 e^8-54 e^4-80-9 \log ^2(5)+\left (54+18 e^4\right ) \log (5)\right )} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {18 e^{2 x} x-x+\left (x-9 e^{2 x}+e^x \left (54+18 e^4-18 \log (5)\right )-9 e^8-54 e^4-80-9 \log ^2(5)+\left (54+18 e^4\right ) \log (5)\right ) \log \left (x-9 e^{2 x}+e^x \left (54+18 e^4-18 \log (5)\right )-9 e^8-54 e^4-80-9 \log ^2(5)+\left (54+18 e^4\right ) \log (5)\right )+e^x \left (-18 e^4 x-54 x+18 x \log (5)\right )}{-x^3+9 e^{2 x} x^2+\left (80+54 e^4\right ) x^2+9 e^8 x^2+9 x^2 \log ^2(5)+\left (-50 x^2+450 e^{2 x} x+450 e^8 x+2700 e^4 x+4000 x+450 x \log ^2(5)+e^x \left (-900 e^4 x-2700 x+900 x \log (5)\right )+\left (-900 e^4 x-2700 x\right ) \log (5)\right ) \log \left (x-9 e^{2 x}+e^x \left (54+18 e^4-18 \log (5)\right )-9 e^8-54 e^4-80-9 \log ^2(5)+\left (54+18 e^4\right ) \log (5)\right )+e^x \left (-18 e^4 x^2-54 x^2+18 x^2 \log (5)\right )+\left (-18 e^4 x^2-54 x^2\right ) \log (5)+\left (-625 x+5625 e^{2 x}+e^x \left (-33750-11250 e^4+11250 \log (5)\right )+5625 e^8+33750 e^4+50000+5625 \log ^2(5)+\left (-33750-11250 e^4\right ) \log (5)\right ) \log ^2\left (x-9 e^{2 x}+e^x \left (54+18 e^4-18 \log (5)\right )-9 e^8-54 e^4-80-9 \log ^2(5)+\left (54+18 e^4\right ) \log (5)\right )}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {18 e^{2 x} x-x+\left (x-9 e^{2 x}+e^x \left (54+18 e^4-18 \log (5)\right )-9 e^8-54 e^4-80-9 \log ^2(5)+\left (54+18 e^4\right ) \log (5)\right ) \log \left (x-9 e^{2 x}+e^x \left (54+18 e^4-18 \log (5)\right )-9 e^8-54 e^4-80-9 \log ^2(5)+\left (54+18 e^4\right ) \log (5)\right )+e^x \left (-18 e^4 x-54 x+18 x \log (5)\right )}{-x^3+9 e^{2 x} x^2+\left (80+54 e^4+9 e^8\right ) x^2+9 x^2 \log ^2(5)+\left (-50 x^2+450 e^{2 x} x+450 e^8 x+2700 e^4 x+4000 x+450 x \log ^2(5)+e^x \left (-900 e^4 x-2700 x+900 x \log (5)\right )+\left (-900 e^4 x-2700 x\right ) \log (5)\right ) \log \left (x-9 e^{2 x}+e^x \left (54+18 e^4-18 \log (5)\right )-9 e^8-54 e^4-80-9 \log ^2(5)+\left (54+18 e^4\right ) \log (5)\right )+e^x \left (-18 e^4 x^2-54 x^2+18 x^2 \log (5)\right )+\left (-18 e^4 x^2-54 x^2\right ) \log (5)+\left (-625 x+5625 e^{2 x}+e^x \left (-33750-11250 e^4+11250 \log (5)\right )+5625 e^8+33750 e^4+50000+5625 \log ^2(5)+\left (-33750-11250 e^4\right ) \log (5)\right ) \log ^2\left (x-9 e^{2 x}+e^x \left (54+18 e^4-18 \log (5)\right )-9 e^8-54 e^4-80-9 \log ^2(5)+\left (54+18 e^4\right ) \log (5)\right )}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {18 e^{2 x} x-x+\left (x-9 e^{2 x}+e^x \left (54+18 e^4-18 \log (5)\right )-9 e^8-54 e^4-80-9 \log ^2(5)+\left (54+18 e^4\right ) \log (5)\right ) \log \left (x-9 e^{2 x}+e^x \left (54+18 e^4-18 \log (5)\right )-9 e^8-54 e^4-80-9 \log ^2(5)+\left (54+18 e^4\right ) \log (5)\right )+e^x \left (-18 e^4 x-54 x+18 x \log (5)\right )}{-x^3+9 e^{2 x} x^2+x^2 \left (80+54 e^4+9 e^8+9 \log ^2(5)\right )+\left (-50 x^2+450 e^{2 x} x+450 e^8 x+2700 e^4 x+4000 x+450 x \log ^2(5)+e^x \left (-900 e^4 x-2700 x+900 x \log (5)\right )+\left (-900 e^4 x-2700 x\right ) \log (5)\right ) \log \left (x-9 e^{2 x}+e^x \left (54+18 e^4-18 \log (5)\right )-9 e^8-54 e^4-80-9 \log ^2(5)+\left (54+18 e^4\right ) \log (5)\right )+e^x \left (-18 e^4 x^2-54 x^2+18 x^2 \log (5)\right )+\left (-18 e^4 x^2-54 x^2\right ) \log (5)+\left (-625 x+5625 e^{2 x}+e^x \left (-33750-11250 e^4+11250 \log (5)\right )+5625 e^8+33750 e^4+50000+5625 \log ^2(5)+\left (-33750-11250 e^4\right ) \log (5)\right ) \log ^2\left (x-9 e^{2 x}+e^x \left (54+18 e^4-18 \log (5)\right )-9 e^8-54 e^4-80-9 \log ^2(5)+\left (54+18 e^4\right ) \log (5)\right )}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (x-9 e^{2 x}+18 e^{x+4}-18 e^x (\log (5)-3)-9 e^8-80-9 \log ^2(5)+54 \log (5)+18 e^4 (\log (5)-3)\right ) \log \left (x-9 e^{2 x}+18 e^x \left (3+e^4-\log (5)\right )-9 e^8-54 e^4-80-9 \log ^2(5)+18 \left (3+e^4\right ) \log (5)\right )+x \left (18 e^{2 x}-18 e^{x+4}+18 e^x (\log (5)-3)-1\right )}{\left (-x+9 e^{2 x}-18 e^{x+4} \left (1+\frac {3-\log (5)}{e^4}\right )+80 \left (1+\frac {9}{80} \left (e^8-2 e^4 (\log (5)-3)+(\log (5)-6) \log (5)\right )\right )\right ) \left (x+25 \log \left (x-9 e^{2 x}+18 e^x \left (3+e^4-\log (5)\right )-9 e^8-54 e^4-80-9 \log ^2(5)+18 \left (3+e^4\right ) \log (5)\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7262 |
| \(\displaystyle -\int \frac {1}{\left (\frac {x}{\log \left (x-9 e^{2 x}+18 e^x \left (3+e^4-\log (5)\right )-\left (8+3 e^4-3 \log (5)\right ) \left (10+3 e^4-3 \log (5)\right )\right )}+25\right )^2}d\frac {x}{\log \left (x-9 e^{2 x}+18 e^x \left (3+e^4-\log (5)\right )-\left (8+3 e^4-3 \log (5)\right ) \left (10+3 e^4-3 \log (5)\right )\right )}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {1}{\frac {x}{\log \left (x-9 e^{2 x}+18 e^x \left (3+e^4-\log (5)\right )-\left (8+3 e^4-3 \log (5)\right ) \left (10+3 e^4-3 \log (5)\right )\right )}+25}\) |
Input:
Int[(-x + 18*E^(2*x)*x + E^x*(-54*x - 18*E^4*x + 18*x*Log[5]) + (-80 - 54* E^4 - 9*E^8 - 9*E^(2*x) + x + E^x*(54 + 18*E^4 - 18*Log[5]) + (54 + 18*E^4 )*Log[5] - 9*Log[5]^2)*Log[-80 - 54*E^4 - 9*E^8 - 9*E^(2*x) + x + E^x*(54 + 18*E^4 - 18*Log[5]) + (54 + 18*E^4)*Log[5] - 9*Log[5]^2])/(80*x^2 + 54*E ^4*x^2 + 9*E^8*x^2 + 9*E^(2*x)*x^2 - x^3 + (-54*x^2 - 18*E^4*x^2)*Log[5] + 9*x^2*Log[5]^2 + E^x*(-54*x^2 - 18*E^4*x^2 + 18*x^2*Log[5]) + (4000*x + 2 700*E^4*x + 450*E^8*x + 450*E^(2*x)*x - 50*x^2 + (-2700*x - 900*E^4*x)*Log [5] + 450*x*Log[5]^2 + E^x*(-2700*x - 900*E^4*x + 900*x*Log[5]))*Log[-80 - 54*E^4 - 9*E^8 - 9*E^(2*x) + x + E^x*(54 + 18*E^4 - 18*Log[5]) + (54 + 18 *E^4)*Log[5] - 9*Log[5]^2] + (50000 + 33750*E^4 + 5625*E^8 + 5625*E^(2*x) - 625*x + (-33750 - 11250*E^4)*Log[5] + 5625*Log[5]^2 + E^x*(-33750 - 1125 0*E^4 + 11250*Log[5]))*Log[-80 - 54*E^4 - 9*E^8 - 9*E^(2*x) + x + E^x*(54 + 18*E^4 - 18*Log[5]) + (54 + 18*E^4)*Log[5] - 9*Log[5]^2]^2),x]
Output:
(25 + x/Log[-9*E^(2*x) + x - (8 + 3*E^4 - 3*Log[5])*(10 + 3*E^4 - 3*Log[5] ) + 18*E^x*(3 + E^4 - Log[5])])^(-1)
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_)*((a_.)*(v_)^(p_.) + (b_.)*(w_)^(q_.))^(m_.), x_Symbol] :> With[{c = Simplify[u/(p*w*D[v, x] - q*v*D[w, x])]}, Simp[c*p Subst[Int[(b + a*x^p )^m, x], x, v*w^(m*q + 1)], x] /; FreeQ[c, x]] /; FreeQ[{a, b, m, p, q}, x] && EqQ[p + q*(m*p + 1), 0] && IntegerQ[p] && IntegerQ[m]
Time = 2.96 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.00
| method | result | size |
| risch | \(-\frac {x}{25 \left (x +25 \ln \left (-9 \,{\mathrm e}^{2 x}+\left (-18 \ln \left (5\right )+18 \,{\mathrm e}^{4}+54\right ) {\mathrm e}^{x}-9 \ln \left (5\right )^{2}+\left (18 \,{\mathrm e}^{4}+54\right ) \ln \left (5\right )-9 \,{\mathrm e}^{8}-54 \,{\mathrm e}^{4}+x -80\right )\right )}\) | \(56\) |
| parallelrisch | \(\frac {\ln \left (-9 \,{\mathrm e}^{2 x}+\left (-18 \ln \left (5\right )+18 \,{\mathrm e}^{4}+54\right ) {\mathrm e}^{x}-9 \ln \left (5\right )^{2}+\left (18 \,{\mathrm e}^{4}+54\right ) \ln \left (5\right )-9 \,{\mathrm e}^{8}-54 \,{\mathrm e}^{4}+x -80\right )}{x +25 \ln \left (-9 \,{\mathrm e}^{2 x}+\left (-18 \ln \left (5\right )+18 \,{\mathrm e}^{4}+54\right ) {\mathrm e}^{x}-9 \ln \left (5\right )^{2}+\left (18 \,{\mathrm e}^{4}+54\right ) \ln \left (5\right )-9 \,{\mathrm e}^{8}-54 \,{\mathrm e}^{4}+x -80\right )}\) | \(104\) |
Input:
int(((-9*exp(x)^2+(-18*ln(5)+18*exp(4)+54)*exp(x)-9*ln(5)^2+(18*exp(4)+54) *ln(5)-9*exp(4)^2-54*exp(4)+x-80)*ln(-9*exp(x)^2+(-18*ln(5)+18*exp(4)+54)* exp(x)-9*ln(5)^2+(18*exp(4)+54)*ln(5)-9*exp(4)^2-54*exp(4)+x-80)+18*x*exp( x)^2+(18*x*ln(5)-18*x*exp(4)-54*x)*exp(x)-x)/((5625*exp(x)^2+(11250*ln(5)- 11250*exp(4)-33750)*exp(x)+5625*ln(5)^2+(-11250*exp(4)-33750)*ln(5)+5625*e xp(4)^2+33750*exp(4)-625*x+50000)*ln(-9*exp(x)^2+(-18*ln(5)+18*exp(4)+54)* exp(x)-9*ln(5)^2+(18*exp(4)+54)*ln(5)-9*exp(4)^2-54*exp(4)+x-80)^2+(450*x* exp(x)^2+(900*x*ln(5)-900*x*exp(4)-2700*x)*exp(x)+450*x*ln(5)^2+(-900*x*ex p(4)-2700*x)*ln(5)+450*x*exp(4)^2+2700*x*exp(4)-50*x^2+4000*x)*ln(-9*exp(x )^2+(-18*ln(5)+18*exp(4)+54)*exp(x)-9*ln(5)^2+(18*exp(4)+54)*ln(5)-9*exp(4 )^2-54*exp(4)+x-80)+9*exp(x)^2*x^2+(18*x^2*ln(5)-18*x^2*exp(4)-54*x^2)*exp (x)+9*x^2*ln(5)^2+(-18*x^2*exp(4)-54*x^2)*ln(5)+9*x^2*exp(4)^2+54*x^2*exp( 4)-x^3+80*x^2),x,method=_RETURNVERBOSE)
Output:
-1/25*x/(x+25*ln(-9*exp(2*x)+(-18*ln(5)+18*exp(4)+54)*exp(x)-9*ln(5)^2+(18 *exp(4)+54)*ln(5)-9*exp(8)-54*exp(4)+x-80))
Time = 0.09 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.89 \[ \int \frac {-x+18 e^{2 x} x+e^x \left (-54 x-18 e^4 x+18 x \log (5)\right )+\left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right ) \log \left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right )}{80 x^2+54 e^4 x^2+9 e^8 x^2+9 e^{2 x} x^2-x^3+\left (-54 x^2-18 e^4 x^2\right ) \log (5)+9 x^2 \log ^2(5)+e^x \left (-54 x^2-18 e^4 x^2+18 x^2 \log (5)\right )+\left (4000 x+2700 e^4 x+450 e^8 x+450 e^{2 x} x-50 x^2+\left (-2700 x-900 e^4 x\right ) \log (5)+450 x \log ^2(5)+e^x \left (-2700 x-900 e^4 x+900 x \log (5)\right )\right ) \log \left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right )+\left (50000+33750 e^4+5625 e^8+5625 e^{2 x}-625 x+\left (-33750-11250 e^4\right ) \log (5)+5625 \log ^2(5)+e^x \left (-33750-11250 e^4+11250 \log (5)\right )\right ) \log ^2\left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right )} \, dx=-\frac {x}{25 \, {\left (x + 25 \, \log \left (18 \, {\left (e^{4} - \log \left (5\right ) + 3\right )} e^{x} + 18 \, {\left (e^{4} + 3\right )} \log \left (5\right ) - 9 \, \log \left (5\right )^{2} + x - 9 \, e^{8} - 54 \, e^{4} - 9 \, e^{\left (2 \, x\right )} - 80\right )\right )}} \] Input:
integrate(((-9*exp(x)^2+(-18*log(5)+18*exp(4)+54)*exp(x)-9*log(5)^2+(18*ex p(4)+54)*log(5)-9*exp(4)^2-54*exp(4)+x-80)*log(-9*exp(x)^2+(-18*log(5)+18* exp(4)+54)*exp(x)-9*log(5)^2+(18*exp(4)+54)*log(5)-9*exp(4)^2-54*exp(4)+x- 80)+18*x*exp(x)^2+(18*x*log(5)-18*x*exp(4)-54*x)*exp(x)-x)/((5625*exp(x)^2 +(11250*log(5)-11250*exp(4)-33750)*exp(x)+5625*log(5)^2+(-11250*exp(4)-337 50)*log(5)+5625*exp(4)^2+33750*exp(4)-625*x+50000)*log(-9*exp(x)^2+(-18*lo g(5)+18*exp(4)+54)*exp(x)-9*log(5)^2+(18*exp(4)+54)*log(5)-9*exp(4)^2-54*e xp(4)+x-80)^2+(450*x*exp(x)^2+(900*x*log(5)-900*x*exp(4)-2700*x)*exp(x)+45 0*x*log(5)^2+(-900*x*exp(4)-2700*x)*log(5)+450*x*exp(4)^2+2700*x*exp(4)-50 *x^2+4000*x)*log(-9*exp(x)^2+(-18*log(5)+18*exp(4)+54)*exp(x)-9*log(5)^2+( 18*exp(4)+54)*log(5)-9*exp(4)^2-54*exp(4)+x-80)+9*exp(x)^2*x^2+(18*x^2*log (5)-18*x^2*exp(4)-54*x^2)*exp(x)+9*x^2*log(5)^2+(-18*x^2*exp(4)-54*x^2)*lo g(5)+9*x^2*exp(4)^2+54*x^2*exp(4)-x^3+80*x^2),x, algorithm="fricas")
Output:
-1/25*x/(x + 25*log(18*(e^4 - log(5) + 3)*e^x + 18*(e^4 + 3)*log(5) - 9*lo g(5)^2 + x - 9*e^8 - 54*e^4 - 9*e^(2*x) - 80))
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (27) = 54\).
Time = 0.45 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.18 \[ \int \frac {-x+18 e^{2 x} x+e^x \left (-54 x-18 e^4 x+18 x \log (5)\right )+\left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right ) \log \left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right )}{80 x^2+54 e^4 x^2+9 e^8 x^2+9 e^{2 x} x^2-x^3+\left (-54 x^2-18 e^4 x^2\right ) \log (5)+9 x^2 \log ^2(5)+e^x \left (-54 x^2-18 e^4 x^2+18 x^2 \log (5)\right )+\left (4000 x+2700 e^4 x+450 e^8 x+450 e^{2 x} x-50 x^2+\left (-2700 x-900 e^4 x\right ) \log (5)+450 x \log ^2(5)+e^x \left (-2700 x-900 e^4 x+900 x \log (5)\right )\right ) \log \left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right )+\left (50000+33750 e^4+5625 e^8+5625 e^{2 x}-625 x+\left (-33750-11250 e^4\right ) \log (5)+5625 \log ^2(5)+e^x \left (-33750-11250 e^4+11250 \log (5)\right )\right ) \log ^2\left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right )} \, dx=- \frac {x}{25 x + 625 \log {\left (x - 9 e^{2 x} + \left (- 18 \log {\left (5 \right )} + 54 + 18 e^{4}\right ) e^{x} - 9 e^{8} - 54 e^{4} - 80 - 9 \log {\left (5 \right )}^{2} + \left (54 + 18 e^{4}\right ) \log {\left (5 \right )} \right )}} \] Input:
integrate(((-9*exp(x)**2+(-18*ln(5)+18*exp(4)+54)*exp(x)-9*ln(5)**2+(18*ex p(4)+54)*ln(5)-9*exp(4)**2-54*exp(4)+x-80)*ln(-9*exp(x)**2+(-18*ln(5)+18*e xp(4)+54)*exp(x)-9*ln(5)**2+(18*exp(4)+54)*ln(5)-9*exp(4)**2-54*exp(4)+x-8 0)+18*x*exp(x)**2+(18*x*ln(5)-18*x*exp(4)-54*x)*exp(x)-x)/((5625*exp(x)**2 +(11250*ln(5)-11250*exp(4)-33750)*exp(x)+5625*ln(5)**2+(-11250*exp(4)-3375 0)*ln(5)+5625*exp(4)**2+33750*exp(4)-625*x+50000)*ln(-9*exp(x)**2+(-18*ln( 5)+18*exp(4)+54)*exp(x)-9*ln(5)**2+(18*exp(4)+54)*ln(5)-9*exp(4)**2-54*exp (4)+x-80)**2+(450*x*exp(x)**2+(900*x*ln(5)-900*x*exp(4)-2700*x)*exp(x)+450 *x*ln(5)**2+(-900*x*exp(4)-2700*x)*ln(5)+450*x*exp(4)**2+2700*x*exp(4)-50* x**2+4000*x)*ln(-9*exp(x)**2+(-18*ln(5)+18*exp(4)+54)*exp(x)-9*ln(5)**2+(1 8*exp(4)+54)*ln(5)-9*exp(4)**2-54*exp(4)+x-80)+9*exp(x)**2*x**2+(18*x**2*l n(5)-18*x**2*exp(4)-54*x**2)*exp(x)+9*x**2*ln(5)**2+(-18*x**2*exp(4)-54*x* *2)*ln(5)+9*x**2*exp(4)**2+54*x**2*exp(4)-x**3+80*x**2),x)
Output:
-x/(25*x + 625*log(x - 9*exp(2*x) + (-18*log(5) + 54 + 18*exp(4))*exp(x) - 9*exp(8) - 54*exp(4) - 80 - 9*log(5)**2 + (54 + 18*exp(4))*log(5)))
Time = 3.99 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.89 \[ \int \frac {-x+18 e^{2 x} x+e^x \left (-54 x-18 e^4 x+18 x \log (5)\right )+\left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right ) \log \left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right )}{80 x^2+54 e^4 x^2+9 e^8 x^2+9 e^{2 x} x^2-x^3+\left (-54 x^2-18 e^4 x^2\right ) \log (5)+9 x^2 \log ^2(5)+e^x \left (-54 x^2-18 e^4 x^2+18 x^2 \log (5)\right )+\left (4000 x+2700 e^4 x+450 e^8 x+450 e^{2 x} x-50 x^2+\left (-2700 x-900 e^4 x\right ) \log (5)+450 x \log ^2(5)+e^x \left (-2700 x-900 e^4 x+900 x \log (5)\right )\right ) \log \left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right )+\left (50000+33750 e^4+5625 e^8+5625 e^{2 x}-625 x+\left (-33750-11250 e^4\right ) \log (5)+5625 \log ^2(5)+e^x \left (-33750-11250 e^4+11250 \log (5)\right )\right ) \log ^2\left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right )} \, dx=-\frac {x}{25 \, {\left (x + 25 \, \log \left (18 \, {\left (e^{4} - \log \left (5\right ) + 3\right )} e^{x} + 18 \, {\left (e^{4} + 3\right )} \log \left (5\right ) - 9 \, \log \left (5\right )^{2} + x - 9 \, e^{8} - 54 \, e^{4} - 9 \, e^{\left (2 \, x\right )} - 80\right )\right )}} \] Input:
integrate(((-9*exp(x)^2+(-18*log(5)+18*exp(4)+54)*exp(x)-9*log(5)^2+(18*ex p(4)+54)*log(5)-9*exp(4)^2-54*exp(4)+x-80)*log(-9*exp(x)^2+(-18*log(5)+18* exp(4)+54)*exp(x)-9*log(5)^2+(18*exp(4)+54)*log(5)-9*exp(4)^2-54*exp(4)+x- 80)+18*x*exp(x)^2+(18*x*log(5)-18*x*exp(4)-54*x)*exp(x)-x)/((5625*exp(x)^2 +(11250*log(5)-11250*exp(4)-33750)*exp(x)+5625*log(5)^2+(-11250*exp(4)-337 50)*log(5)+5625*exp(4)^2+33750*exp(4)-625*x+50000)*log(-9*exp(x)^2+(-18*lo g(5)+18*exp(4)+54)*exp(x)-9*log(5)^2+(18*exp(4)+54)*log(5)-9*exp(4)^2-54*e xp(4)+x-80)^2+(450*x*exp(x)^2+(900*x*log(5)-900*x*exp(4)-2700*x)*exp(x)+45 0*x*log(5)^2+(-900*x*exp(4)-2700*x)*log(5)+450*x*exp(4)^2+2700*x*exp(4)-50 *x^2+4000*x)*log(-9*exp(x)^2+(-18*log(5)+18*exp(4)+54)*exp(x)-9*log(5)^2+( 18*exp(4)+54)*log(5)-9*exp(4)^2-54*exp(4)+x-80)+9*exp(x)^2*x^2+(18*x^2*log (5)-18*x^2*exp(4)-54*x^2)*exp(x)+9*x^2*log(5)^2+(-18*x^2*exp(4)-54*x^2)*lo g(5)+9*x^2*exp(4)^2+54*x^2*exp(4)-x^3+80*x^2),x, algorithm="maxima")
Output:
-1/25*x/(x + 25*log(18*(e^4 - log(5) + 3)*e^x + 18*(e^4 + 3)*log(5) - 9*lo g(5)^2 + x - 9*e^8 - 54*e^4 - 9*e^(2*x) - 80))
Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (28) = 56\).
Time = 56.32 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.11 \[ \int \frac {-x+18 e^{2 x} x+e^x \left (-54 x-18 e^4 x+18 x \log (5)\right )+\left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right ) \log \left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right )}{80 x^2+54 e^4 x^2+9 e^8 x^2+9 e^{2 x} x^2-x^3+\left (-54 x^2-18 e^4 x^2\right ) \log (5)+9 x^2 \log ^2(5)+e^x \left (-54 x^2-18 e^4 x^2+18 x^2 \log (5)\right )+\left (4000 x+2700 e^4 x+450 e^8 x+450 e^{2 x} x-50 x^2+\left (-2700 x-900 e^4 x\right ) \log (5)+450 x \log ^2(5)+e^x \left (-2700 x-900 e^4 x+900 x \log (5)\right )\right ) \log \left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right )+\left (50000+33750 e^4+5625 e^8+5625 e^{2 x}-625 x+\left (-33750-11250 e^4\right ) \log (5)+5625 \log ^2(5)+e^x \left (-33750-11250 e^4+11250 \log (5)\right )\right ) \log ^2\left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right )} \, dx=-\frac {x}{25 \, {\left (x + 25 \, \log \left (18 \, e^{4} \log \left (5\right ) - 18 \, e^{x} \log \left (5\right ) - 9 \, \log \left (5\right )^{2} + x - 9 \, e^{8} - 54 \, e^{4} - 9 \, e^{\left (2 \, x\right )} + 18 \, e^{\left (x + 4\right )} + 54 \, e^{x} + 54 \, \log \left (5\right ) - 80\right )\right )}} \] Input:
integrate(((-9*exp(x)^2+(-18*log(5)+18*exp(4)+54)*exp(x)-9*log(5)^2+(18*ex p(4)+54)*log(5)-9*exp(4)^2-54*exp(4)+x-80)*log(-9*exp(x)^2+(-18*log(5)+18* exp(4)+54)*exp(x)-9*log(5)^2+(18*exp(4)+54)*log(5)-9*exp(4)^2-54*exp(4)+x- 80)+18*x*exp(x)^2+(18*x*log(5)-18*x*exp(4)-54*x)*exp(x)-x)/((5625*exp(x)^2 +(11250*log(5)-11250*exp(4)-33750)*exp(x)+5625*log(5)^2+(-11250*exp(4)-337 50)*log(5)+5625*exp(4)^2+33750*exp(4)-625*x+50000)*log(-9*exp(x)^2+(-18*lo g(5)+18*exp(4)+54)*exp(x)-9*log(5)^2+(18*exp(4)+54)*log(5)-9*exp(4)^2-54*e xp(4)+x-80)^2+(450*x*exp(x)^2+(900*x*log(5)-900*x*exp(4)-2700*x)*exp(x)+45 0*x*log(5)^2+(-900*x*exp(4)-2700*x)*log(5)+450*x*exp(4)^2+2700*x*exp(4)-50 *x^2+4000*x)*log(-9*exp(x)^2+(-18*log(5)+18*exp(4)+54)*exp(x)-9*log(5)^2+( 18*exp(4)+54)*log(5)-9*exp(4)^2-54*exp(4)+x-80)+9*exp(x)^2*x^2+(18*x^2*log (5)-18*x^2*exp(4)-54*x^2)*exp(x)+9*x^2*log(5)^2+(-18*x^2*exp(4)-54*x^2)*lo g(5)+9*x^2*exp(4)^2+54*x^2*exp(4)-x^3+80*x^2),x, algorithm="giac")
Output:
-1/25*x/(x + 25*log(18*e^4*log(5) - 18*e^x*log(5) - 9*log(5)^2 + x - 9*e^8 - 54*e^4 - 9*e^(2*x) + 18*e^(x + 4) + 54*e^x + 54*log(5) - 80))
Timed out. \[ \int \frac {-x+18 e^{2 x} x+e^x \left (-54 x-18 e^4 x+18 x \log (5)\right )+\left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right ) \log \left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right )}{80 x^2+54 e^4 x^2+9 e^8 x^2+9 e^{2 x} x^2-x^3+\left (-54 x^2-18 e^4 x^2\right ) \log (5)+9 x^2 \log ^2(5)+e^x \left (-54 x^2-18 e^4 x^2+18 x^2 \log (5)\right )+\left (4000 x+2700 e^4 x+450 e^8 x+450 e^{2 x} x-50 x^2+\left (-2700 x-900 e^4 x\right ) \log (5)+450 x \log ^2(5)+e^x \left (-2700 x-900 e^4 x+900 x \log (5)\right )\right ) \log \left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right )+\left (50000+33750 e^4+5625 e^8+5625 e^{2 x}-625 x+\left (-33750-11250 e^4\right ) \log (5)+5625 \log ^2(5)+e^x \left (-33750-11250 e^4+11250 \log (5)\right )\right ) \log ^2\left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right )} \, dx=\int -\frac {x-18\,x\,{\mathrm {e}}^{2\,x}+\ln \left (x-9\,{\mathrm {e}}^{2\,x}-54\,{\mathrm {e}}^4-9\,{\mathrm {e}}^8+{\mathrm {e}}^x\,\left (18\,{\mathrm {e}}^4-18\,\ln \left (5\right )+54\right )-9\,{\ln \left (5\right )}^2+\ln \left (5\right )\,\left (18\,{\mathrm {e}}^4+54\right )-80\right )\,\left (9\,{\mathrm {e}}^{2\,x}-x+54\,{\mathrm {e}}^4+9\,{\mathrm {e}}^8-{\mathrm {e}}^x\,\left (18\,{\mathrm {e}}^4-18\,\ln \left (5\right )+54\right )+9\,{\ln \left (5\right )}^2-\ln \left (5\right )\,\left (18\,{\mathrm {e}}^4+54\right )+80\right )+{\mathrm {e}}^x\,\left (54\,x+18\,x\,{\mathrm {e}}^4-18\,x\,\ln \left (5\right )\right )}{9\,x^2\,{\ln \left (5\right )}^2-{\mathrm {e}}^x\,\left (18\,x^2\,{\mathrm {e}}^4-18\,x^2\,\ln \left (5\right )+54\,x^2\right )+{\ln \left (x-9\,{\mathrm {e}}^{2\,x}-54\,{\mathrm {e}}^4-9\,{\mathrm {e}}^8+{\mathrm {e}}^x\,\left (18\,{\mathrm {e}}^4-18\,\ln \left (5\right )+54\right )-9\,{\ln \left (5\right )}^2+\ln \left (5\right )\,\left (18\,{\mathrm {e}}^4+54\right )-80\right )}^2\,\left (5625\,{\mathrm {e}}^{2\,x}-625\,x+33750\,{\mathrm {e}}^4+5625\,{\mathrm {e}}^8-{\mathrm {e}}^x\,\left (11250\,{\mathrm {e}}^4-11250\,\ln \left (5\right )+33750\right )+5625\,{\ln \left (5\right )}^2-\ln \left (5\right )\,\left (11250\,{\mathrm {e}}^4+33750\right )+50000\right )+9\,x^2\,{\mathrm {e}}^{2\,x}+54\,x^2\,{\mathrm {e}}^4+9\,x^2\,{\mathrm {e}}^8+80\,x^2-x^3+\ln \left (x-9\,{\mathrm {e}}^{2\,x}-54\,{\mathrm {e}}^4-9\,{\mathrm {e}}^8+{\mathrm {e}}^x\,\left (18\,{\mathrm {e}}^4-18\,\ln \left (5\right )+54\right )-9\,{\ln \left (5\right )}^2+\ln \left (5\right )\,\left (18\,{\mathrm {e}}^4+54\right )-80\right )\,\left (4000\,x+450\,x\,{\mathrm {e}}^{2\,x}+2700\,x\,{\mathrm {e}}^4+450\,x\,{\mathrm {e}}^8-\ln \left (5\right )\,\left (2700\,x+900\,x\,{\mathrm {e}}^4\right )+450\,x\,{\ln \left (5\right )}^2-50\,x^2-{\mathrm {e}}^x\,\left (2700\,x+900\,x\,{\mathrm {e}}^4-900\,x\,\ln \left (5\right )\right )\right )-\ln \left (5\right )\,\left (18\,x^2\,{\mathrm {e}}^4+54\,x^2\right )} \,d x \] Input:
int(-(x - 18*x*exp(2*x) + log(x - 9*exp(2*x) - 54*exp(4) - 9*exp(8) + exp( x)*(18*exp(4) - 18*log(5) + 54) - 9*log(5)^2 + log(5)*(18*exp(4) + 54) - 8 0)*(9*exp(2*x) - x + 54*exp(4) + 9*exp(8) - exp(x)*(18*exp(4) - 18*log(5) + 54) + 9*log(5)^2 - log(5)*(18*exp(4) + 54) + 80) + exp(x)*(54*x + 18*x*e xp(4) - 18*x*log(5)))/(9*x^2*log(5)^2 - exp(x)*(18*x^2*exp(4) - 18*x^2*log (5) + 54*x^2) + log(x - 9*exp(2*x) - 54*exp(4) - 9*exp(8) + exp(x)*(18*exp (4) - 18*log(5) + 54) - 9*log(5)^2 + log(5)*(18*exp(4) + 54) - 80)^2*(5625 *exp(2*x) - 625*x + 33750*exp(4) + 5625*exp(8) - exp(x)*(11250*exp(4) - 11 250*log(5) + 33750) + 5625*log(5)^2 - log(5)*(11250*exp(4) + 33750) + 5000 0) + 9*x^2*exp(2*x) + 54*x^2*exp(4) + 9*x^2*exp(8) + 80*x^2 - x^3 + log(x - 9*exp(2*x) - 54*exp(4) - 9*exp(8) + exp(x)*(18*exp(4) - 18*log(5) + 54) - 9*log(5)^2 + log(5)*(18*exp(4) + 54) - 80)*(4000*x + 450*x*exp(2*x) + 27 00*x*exp(4) + 450*x*exp(8) - log(5)*(2700*x + 900*x*exp(4)) + 450*x*log(5) ^2 - 50*x^2 - exp(x)*(2700*x + 900*x*exp(4) - 900*x*log(5))) - log(5)*(18* x^2*exp(4) + 54*x^2)),x)
Output:
int(-(x - 18*x*exp(2*x) + log(x - 9*exp(2*x) - 54*exp(4) - 9*exp(8) + exp( x)*(18*exp(4) - 18*log(5) + 54) - 9*log(5)^2 + log(5)*(18*exp(4) + 54) - 8 0)*(9*exp(2*x) - x + 54*exp(4) + 9*exp(8) - exp(x)*(18*exp(4) - 18*log(5) + 54) + 9*log(5)^2 - log(5)*(18*exp(4) + 54) + 80) + exp(x)*(54*x + 18*x*e xp(4) - 18*x*log(5)))/(9*x^2*log(5)^2 - exp(x)*(18*x^2*exp(4) - 18*x^2*log (5) + 54*x^2) + log(x - 9*exp(2*x) - 54*exp(4) - 9*exp(8) + exp(x)*(18*exp (4) - 18*log(5) + 54) - 9*log(5)^2 + log(5)*(18*exp(4) + 54) - 80)^2*(5625 *exp(2*x) - 625*x + 33750*exp(4) + 5625*exp(8) - exp(x)*(11250*exp(4) - 11 250*log(5) + 33750) + 5625*log(5)^2 - log(5)*(11250*exp(4) + 33750) + 5000 0) + 9*x^2*exp(2*x) + 54*x^2*exp(4) + 9*x^2*exp(8) + 80*x^2 - x^3 + log(x - 9*exp(2*x) - 54*exp(4) - 9*exp(8) + exp(x)*(18*exp(4) - 18*log(5) + 54) - 9*log(5)^2 + log(5)*(18*exp(4) + 54) - 80)*(4000*x + 450*x*exp(2*x) + 27 00*x*exp(4) + 450*x*exp(8) - log(5)*(2700*x + 900*x*exp(4)) + 450*x*log(5) ^2 - 50*x^2 - exp(x)*(2700*x + 900*x*exp(4) - 900*x*log(5))) - log(5)*(18* x^2*exp(4) + 54*x^2)), x)
Time = 0.27 (sec) , antiderivative size = 123, normalized size of antiderivative = 4.39 \[ \int \frac {-x+18 e^{2 x} x+e^x \left (-54 x-18 e^4 x+18 x \log (5)\right )+\left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right ) \log \left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right )}{80 x^2+54 e^4 x^2+9 e^8 x^2+9 e^{2 x} x^2-x^3+\left (-54 x^2-18 e^4 x^2\right ) \log (5)+9 x^2 \log ^2(5)+e^x \left (-54 x^2-18 e^4 x^2+18 x^2 \log (5)\right )+\left (4000 x+2700 e^4 x+450 e^8 x+450 e^{2 x} x-50 x^2+\left (-2700 x-900 e^4 x\right ) \log (5)+450 x \log ^2(5)+e^x \left (-2700 x-900 e^4 x+900 x \log (5)\right )\right ) \log \left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right )+\left (50000+33750 e^4+5625 e^8+5625 e^{2 x}-625 x+\left (-33750-11250 e^4\right ) \log (5)+5625 \log ^2(5)+e^x \left (-33750-11250 e^4+11250 \log (5)\right )\right ) \log ^2\left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right )} \, dx=\frac {\mathrm {log}\left (-9 e^{2 x}-18 e^{x} \mathrm {log}\left (5\right )+18 e^{x} e^{4}+54 e^{x}-9 \mathrm {log}\left (5\right )^{2}+18 \,\mathrm {log}\left (5\right ) e^{4}+54 \,\mathrm {log}\left (5\right )-9 e^{8}-54 e^{4}+x -80\right )}{25 \,\mathrm {log}\left (-9 e^{2 x}-18 e^{x} \mathrm {log}\left (5\right )+18 e^{x} e^{4}+54 e^{x}-9 \mathrm {log}\left (5\right )^{2}+18 \,\mathrm {log}\left (5\right ) e^{4}+54 \,\mathrm {log}\left (5\right )-9 e^{8}-54 e^{4}+x -80\right )+x} \] Input:
int(((-9*exp(x)^2+(-18*log(5)+18*exp(4)+54)*exp(x)-9*log(5)^2+(18*exp(4)+5 4)*log(5)-9*exp(4)^2-54*exp(4)+x-80)*log(-9*exp(x)^2+(-18*log(5)+18*exp(4) +54)*exp(x)-9*log(5)^2+(18*exp(4)+54)*log(5)-9*exp(4)^2-54*exp(4)+x-80)+18 *x*exp(x)^2+(18*x*log(5)-18*x*exp(4)-54*x)*exp(x)-x)/((5625*exp(x)^2+(1125 0*log(5)-11250*exp(4)-33750)*exp(x)+5625*log(5)^2+(-11250*exp(4)-33750)*lo g(5)+5625*exp(4)^2+33750*exp(4)-625*x+50000)*log(-9*exp(x)^2+(-18*log(5)+1 8*exp(4)+54)*exp(x)-9*log(5)^2+(18*exp(4)+54)*log(5)-9*exp(4)^2-54*exp(4)+ x-80)^2+(450*x*exp(x)^2+(900*x*log(5)-900*x*exp(4)-2700*x)*exp(x)+450*x*lo g(5)^2+(-900*x*exp(4)-2700*x)*log(5)+450*x*exp(4)^2+2700*x*exp(4)-50*x^2+4 000*x)*log(-9*exp(x)^2+(-18*log(5)+18*exp(4)+54)*exp(x)-9*log(5)^2+(18*exp (4)+54)*log(5)-9*exp(4)^2-54*exp(4)+x-80)+9*exp(x)^2*x^2+(18*x^2*log(5)-18 *x^2*exp(4)-54*x^2)*exp(x)+9*x^2*log(5)^2+(-18*x^2*exp(4)-54*x^2)*log(5)+9 *x^2*exp(4)^2+54*x^2*exp(4)-x^3+80*x^2),x)
Output:
log( - 9*e**(2*x) - 18*e**x*log(5) + 18*e**x*e**4 + 54*e**x - 9*log(5)**2 + 18*log(5)*e**4 + 54*log(5) - 9*e**8 - 54*e**4 + x - 80)/(25*log( - 9*e** (2*x) - 18*e**x*log(5) + 18*e**x*e**4 + 54*e**x - 9*log(5)**2 + 18*log(5)* e**4 + 54*log(5) - 9*e**8 - 54*e**4 + x - 80) + x)