Integrand size = 113, antiderivative size = 22 \[ \int \frac {-16 x^7+7 x^8+e^x \left (20 x^4-16 x^5+x^6+x^7\right )+\left (8 x^7+e^x \left (-20 x^4+6 x^5+2 x^6\right )\right ) \log (3+\log (2))+e^x \left (5 x^4+x^5\right ) \log ^2(3+\log (2))}{4-4 x+x^2+(-4+2 x) \log (3+\log (2))+\log ^2(3+\log (2))} \, dx=x^5 \left (e^x+\frac {x^3}{-2+x+\log (3+\log (2))}\right ) \]
Leaf count is larger than twice the leaf count of optimal. \(56\) vs. \(2(22)=44\).
Time = 0.09 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.55 \[ \int \frac {-16 x^7+7 x^8+e^x \left (20 x^4-16 x^5+x^6+x^7\right )+\left (8 x^7+e^x \left (-20 x^4+6 x^5+2 x^6\right )\right ) \log (3+\log (2))+e^x \left (5 x^4+x^5\right ) \log ^2(3+\log (2))}{4-4 x+x^2+(-4+2 x) \log (3+\log (2))+\log ^2(3+\log (2))} \, dx=\frac {e^x x^6+x^8+e^x x^5 (-2+\log (3+\log (2)))+x (-2+\log (3+\log (2)))^7+(-2+\log (3+\log (2)))^8}{-2+x+\log (3+\log (2))} \]
Integrate[(-16*x^7 + 7*x^8 + E^x*(20*x^4 - 16*x^5 + x^6 + x^7) + (8*x^7 + E^x*(-20*x^4 + 6*x^5 + 2*x^6))*Log[3 + Log[2]] + E^x*(5*x^4 + x^5)*Log[3 + Log[2]]^2)/(4 - 4*x + x^2 + (-4 + 2*x)*Log[3 + Log[2]] + Log[3 + Log[2]]^ 2),x]
(E^x*x^6 + x^8 + E^x*x^5*(-2 + Log[3 + Log[2]]) + x*(-2 + Log[3 + Log[2]]) ^7 + (-2 + Log[3 + Log[2]])^8)/(-2 + x + Log[3 + Log[2]])
Time = 0.50 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {2007, 7239, 2009}
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.
| \(\displaystyle \int \frac {7 x^8-16 x^7+e^x \left (x^5+5 x^4\right ) \log ^2(3+\log (2))+e^x \left (x^7+x^6-16 x^5+20 x^4\right )+\left (8 x^7+e^x \left (2 x^6+6 x^5-20 x^4\right )\right ) \log (3+\log (2))}{x^2-4 x+(2 x-4) \log (3+\log (2))+4+\log ^2(3+\log (2))} \, dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {7 x^8-16 x^7+e^x \left (x^5+5 x^4\right ) \log ^2(3+\log (2))+e^x \left (x^7+x^6-16 x^5+20 x^4\right )+\left (8 x^7+e^x \left (2 x^6+6 x^5-20 x^4\right )\right ) \log (3+\log (2))}{(x-2+\log (3+\log (2)))^2}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \left (\frac {x^7 (7 x+8 (\log (3+\log (2))-2))}{(x-2+\log (3+\log (2)))^2}+e^x (x+5) x^4\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle e^x x^5-\frac {x^8}{-x+2-\log (3+\log (2))}\) |
Int[(-16*x^7 + 7*x^8 + E^x*(20*x^4 - 16*x^5 + x^6 + x^7) + (8*x^7 + E^x*(- 20*x^4 + 6*x^5 + 2*x^6))*Log[3 + Log[2]] + E^x*(5*x^4 + x^5)*Log[3 + Log[2 ]]^2)/(4 - 4*x + x^2 + (-4 + 2*x)*Log[3 + Log[2]] + Log[3 + Log[2]]^2),x]
3.2.18.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.89 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59
| method | result | size |
| norman | \(\frac {x^{8}+x^{6} {\mathrm e}^{x}+\left (\ln \left (3+\ln \left (2\right )\right )-2\right ) x^{5} {\mathrm e}^{x}}{x -2+\ln \left (3+\ln \left (2\right )\right )}\) | \(35\) |
| parallelrisch | \(\frac {x^{8}+\ln \left (3+\ln \left (2\right )\right ) x^{5} {\mathrm e}^{x}+x^{6} {\mathrm e}^{x}-2 x^{5} {\mathrm e}^{x}}{x -2+\ln \left (3+\ln \left (2\right )\right )}\) | \(40\) |
| parts | \(64 x +x^{5} {\mathrm e}^{x}+x^{7}+8 x^{4}+16 x^{3}+32 x^{2}+2 x^{6}+4 x^{5}+6 \ln \left (3+\ln \left (2\right )\right )^{2} x^{4}-4 x^{5} \ln \left (3+\ln \left (2\right )\right )+60 \ln \left (3+\ln \left (2\right )\right )^{4} x -40 \ln \left (3+\ln \left (2\right )\right )^{3} x^{2}-160 \ln \left (3+\ln \left (2\right )\right )^{3} x +80 \ln \left (3+\ln \left (2\right )\right )^{2} x^{2}+240 \ln \left (3+\ln \left (2\right )\right )^{2} x -80 \ln \left (3+\ln \left (2\right )\right ) x^{2}-192 \ln \left (3+\ln \left (2\right )\right ) x -12 \ln \left (3+\ln \left (2\right )\right )^{5} x +10 \ln \left (3+\ln \left (2\right )\right )^{4} x^{2}-8 \ln \left (3+\ln \left (2\right )\right )^{3} x^{3}+\ln \left (3+\ln \left (2\right )\right )^{6} x -\ln \left (3+\ln \left (2\right )\right )^{3} x^{4}-x^{6} \ln \left (3+\ln \left (2\right )\right )+\ln \left (3+\ln \left (2\right )\right )^{2} x^{5}+\ln \left (3+\ln \left (2\right )\right )^{4} x^{3}-\ln \left (3+\ln \left (2\right )\right )^{5} x^{2}+24 \ln \left (3+\ln \left (2\right )\right )^{2} x^{3}-12 \ln \left (3+\ln \left (2\right )\right ) x^{4}-32 \ln \left (3+\ln \left (2\right )\right ) x^{3}-\frac {-\ln \left (3+\ln \left (2\right )\right )^{8}+16 \ln \left (3+\ln \left (2\right )\right )^{7}-112 \ln \left (3+\ln \left (2\right )\right )^{6}+448 \ln \left (3+\ln \left (2\right )\right )^{5}-1120 \ln \left (3+\ln \left (2\right )\right )^{4}+1792 \ln \left (3+\ln \left (2\right )\right )^{3}-1792 \ln \left (3+\ln \left (2\right )\right )^{2}+1024 \ln \left (3+\ln \left (2\right )\right )-256}{x -2+\ln \left (3+\ln \left (2\right )\right )}\) | \(348\) |
| risch | \(64 x +x^{5} {\mathrm e}^{x}+x^{7}+8 x^{4}+16 x^{3}+32 x^{2}+2 x^{6}+4 x^{5}+6 \ln \left (3+\ln \left (2\right )\right )^{2} x^{4}-4 x^{5} \ln \left (3+\ln \left (2\right )\right )+60 \ln \left (3+\ln \left (2\right )\right )^{4} x -40 \ln \left (3+\ln \left (2\right )\right )^{3} x^{2}-160 \ln \left (3+\ln \left (2\right )\right )^{3} x +80 \ln \left (3+\ln \left (2\right )\right )^{2} x^{2}+240 \ln \left (3+\ln \left (2\right )\right )^{2} x -80 \ln \left (3+\ln \left (2\right )\right ) x^{2}-192 \ln \left (3+\ln \left (2\right )\right ) x -12 \ln \left (3+\ln \left (2\right )\right )^{5} x +10 \ln \left (3+\ln \left (2\right )\right )^{4} x^{2}-8 \ln \left (3+\ln \left (2\right )\right )^{3} x^{3}+\frac {\ln \left (3+\ln \left (2\right )\right )^{8}}{x -2+\ln \left (3+\ln \left (2\right )\right )}-\frac {16 \ln \left (3+\ln \left (2\right )\right )^{7}}{x -2+\ln \left (3+\ln \left (2\right )\right )}+\frac {112 \ln \left (3+\ln \left (2\right )\right )^{6}}{x -2+\ln \left (3+\ln \left (2\right )\right )}-\frac {448 \ln \left (3+\ln \left (2\right )\right )^{5}}{x -2+\ln \left (3+\ln \left (2\right )\right )}+\frac {1120 \ln \left (3+\ln \left (2\right )\right )^{4}}{x -2+\ln \left (3+\ln \left (2\right )\right )}-\frac {1792 \ln \left (3+\ln \left (2\right )\right )^{3}}{x -2+\ln \left (3+\ln \left (2\right )\right )}+\frac {1792 \ln \left (3+\ln \left (2\right )\right )^{2}}{x -2+\ln \left (3+\ln \left (2\right )\right )}-\frac {1024 \ln \left (3+\ln \left (2\right )\right )}{x -2+\ln \left (3+\ln \left (2\right )\right )}+\ln \left (3+\ln \left (2\right )\right )^{6} x -\ln \left (3+\ln \left (2\right )\right )^{3} x^{4}-x^{6} \ln \left (3+\ln \left (2\right )\right )+\ln \left (3+\ln \left (2\right )\right )^{2} x^{5}+\ln \left (3+\ln \left (2\right )\right )^{4} x^{3}-\ln \left (3+\ln \left (2\right )\right )^{5} x^{2}+24 \ln \left (3+\ln \left (2\right )\right )^{2} x^{3}-12 \ln \left (3+\ln \left (2\right )\right ) x^{4}-32 \ln \left (3+\ln \left (2\right )\right ) x^{3}+\frac {256}{x -2+\ln \left (3+\ln \left (2\right )\right )}\) | \(425\) |
| default | \(\text {Expression too large to display}\) | \(3101\) |
int(((x^5+5*x^4)*exp(x)*ln(3+ln(2))^2+((2*x^6+6*x^5-20*x^4)*exp(x)+8*x^7)* ln(3+ln(2))+(x^7+x^6-16*x^5+20*x^4)*exp(x)+7*x^8-16*x^7)/(ln(3+ln(2))^2+(2 *x-4)*ln(3+ln(2))+x^2-4*x+4),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (21) = 42\).
Time = 0.28 (sec) , antiderivative size = 132, normalized size of antiderivative = 6.00 \[ \int \frac {-16 x^7+7 x^8+e^x \left (20 x^4-16 x^5+x^6+x^7\right )+\left (8 x^7+e^x \left (-20 x^4+6 x^5+2 x^6\right )\right ) \log (3+\log (2))+e^x \left (5 x^4+x^5\right ) \log ^2(3+\log (2))}{4-4 x+x^2+(-4+2 x) \log (3+\log (2))+\log ^2(3+\log (2))} \, dx=\frac {x^{8} + {\left (x - 16\right )} \log \left (\log \left (2\right ) + 3\right )^{7} + \log \left (\log \left (2\right ) + 3\right )^{8} - 14 \, {\left (x - 8\right )} \log \left (\log \left (2\right ) + 3\right )^{6} + 28 \, {\left (3 \, x - 16\right )} \log \left (\log \left (2\right ) + 3\right )^{5} - 280 \, {\left (x - 4\right )} \log \left (\log \left (2\right ) + 3\right )^{4} + 112 \, {\left (5 \, x - 16\right )} \log \left (\log \left (2\right ) + 3\right )^{3} - 224 \, {\left (3 \, x - 8\right )} \log \left (\log \left (2\right ) + 3\right )^{2} + {\left (x^{6} - 2 \, x^{5}\right )} e^{x} + {\left (x^{5} e^{x} + 448 \, x - 1024\right )} \log \left (\log \left (2\right ) + 3\right ) - 128 \, x + 256}{x + \log \left (\log \left (2\right ) + 3\right ) - 2} \]
integrate(((x^5+5*x^4)*exp(x)*log(3+log(2))^2+((2*x^6+6*x^5-20*x^4)*exp(x) +8*x^7)*log(3+log(2))+(x^7+x^6-16*x^5+20*x^4)*exp(x)+7*x^8-16*x^7)/(log(3+ log(2))^2+(2*x-4)*log(3+log(2))+x^2-4*x+4),x, algorithm=\
(x^8 + (x - 16)*log(log(2) + 3)^7 + log(log(2) + 3)^8 - 14*(x - 8)*log(log (2) + 3)^6 + 28*(3*x - 16)*log(log(2) + 3)^5 - 280*(x - 4)*log(log(2) + 3) ^4 + 112*(5*x - 16)*log(log(2) + 3)^3 - 224*(3*x - 8)*log(log(2) + 3)^2 + (x^6 - 2*x^5)*e^x + (x^5*e^x + 448*x - 1024)*log(log(2) + 3) - 128*x + 256 )/(x + log(log(2) + 3) - 2)
Leaf count of result is larger than twice the leaf count of optimal. 321 vs. \(2 (19) = 38\).
Time = 0.51 (sec) , antiderivative size = 321, normalized size of antiderivative = 14.59 \[ \int \frac {-16 x^7+7 x^8+e^x \left (20 x^4-16 x^5+x^6+x^7\right )+\left (8 x^7+e^x \left (-20 x^4+6 x^5+2 x^6\right )\right ) \log (3+\log (2))+e^x \left (5 x^4+x^5\right ) \log ^2(3+\log (2))}{4-4 x+x^2+(-4+2 x) \log (3+\log (2))+\log ^2(3+\log (2))} \, dx=x^{7} + x^{6} \cdot \left (2 - \log {\left (\log {\left (2 \right )} + 3 \right )}\right ) + x^{5} e^{x} + x^{5} \left (- 4 \log {\left (\log {\left (2 \right )} + 3 \right )} + \log {\left (\log {\left (2 \right )} + 3 \right )}^{2} + 4\right ) + x^{4} \left (- 12 \log {\left (\log {\left (2 \right )} + 3 \right )} - \log {\left (\log {\left (2 \right )} + 3 \right )}^{3} + 8 + 6 \log {\left (\log {\left (2 \right )} + 3 \right )}^{2}\right ) + x^{3} \left (- 32 \log {\left (\log {\left (2 \right )} + 3 \right )} - 8 \log {\left (\log {\left (2 \right )} + 3 \right )}^{3} + \log {\left (\log {\left (2 \right )} + 3 \right )}^{4} + 16 + 24 \log {\left (\log {\left (2 \right )} + 3 \right )}^{2}\right ) + x^{2} \left (- 80 \log {\left (\log {\left (2 \right )} + 3 \right )} - 40 \log {\left (\log {\left (2 \right )} + 3 \right )}^{3} - \log {\left (\log {\left (2 \right )} + 3 \right )}^{5} + 10 \log {\left (\log {\left (2 \right )} + 3 \right )}^{4} + 32 + 80 \log {\left (\log {\left (2 \right )} + 3 \right )}^{2}\right ) + x \left (- 160 \log {\left (\log {\left (2 \right )} + 3 \right )}^{3} - 192 \log {\left (\log {\left (2 \right )} + 3 \right )} - 12 \log {\left (\log {\left (2 \right )} + 3 \right )}^{5} + \log {\left (\log {\left (2 \right )} + 3 \right )}^{6} + 64 + 60 \log {\left (\log {\left (2 \right )} + 3 \right )}^{4} + 240 \log {\left (\log {\left (2 \right )} + 3 \right )}^{2}\right ) + \frac {- 1792 \log {\left (\log {\left (2 \right )} + 3 \right )}^{3} - 448 \log {\left (\log {\left (2 \right )} + 3 \right )}^{5} - 1024 \log {\left (\log {\left (2 \right )} + 3 \right )} - 16 \log {\left (\log {\left (2 \right )} + 3 \right )}^{7} + \log {\left (\log {\left (2 \right )} + 3 \right )}^{8} + 256 + 112 \log {\left (\log {\left (2 \right )} + 3 \right )}^{6} + 1792 \log {\left (\log {\left (2 \right )} + 3 \right )}^{2} + 1120 \log {\left (\log {\left (2 \right )} + 3 \right )}^{4}}{x - 2 + \log {\left (\log {\left (2 \right )} + 3 \right )}} \]
integrate(((x**5+5*x**4)*exp(x)*ln(3+ln(2))**2+((2*x**6+6*x**5-20*x**4)*ex p(x)+8*x**7)*ln(3+ln(2))+(x**7+x**6-16*x**5+20*x**4)*exp(x)+7*x**8-16*x**7 )/(ln(3+ln(2))**2+(2*x-4)*ln(3+ln(2))+x**2-4*x+4),x)
x**7 + x**6*(2 - log(log(2) + 3)) + x**5*exp(x) + x**5*(-4*log(log(2) + 3) + log(log(2) + 3)**2 + 4) + x**4*(-12*log(log(2) + 3) - log(log(2) + 3)** 3 + 8 + 6*log(log(2) + 3)**2) + x**3*(-32*log(log(2) + 3) - 8*log(log(2) + 3)**3 + log(log(2) + 3)**4 + 16 + 24*log(log(2) + 3)**2) + x**2*(-80*log( log(2) + 3) - 40*log(log(2) + 3)**3 - log(log(2) + 3)**5 + 10*log(log(2) + 3)**4 + 32 + 80*log(log(2) + 3)**2) + x*(-160*log(log(2) + 3)**3 - 192*lo g(log(2) + 3) - 12*log(log(2) + 3)**5 + log(log(2) + 3)**6 + 64 + 60*log(l og(2) + 3)**4 + 240*log(log(2) + 3)**2) + (-1792*log(log(2) + 3)**3 - 448* log(log(2) + 3)**5 - 1024*log(log(2) + 3) - 16*log(log(2) + 3)**7 + log(lo g(2) + 3)**8 + 256 + 112*log(log(2) + 3)**6 + 1792*log(log(2) + 3)**2 + 11 20*log(log(2) + 3)**4)/(x - 2 + log(log(2) + 3))
Leaf count of result is larger than twice the leaf count of optimal. 955 vs. \(2 (21) = 42\).
Time = 0.33 (sec) , antiderivative size = 955, normalized size of antiderivative = 43.41 \[ \int \frac {-16 x^7+7 x^8+e^x \left (20 x^4-16 x^5+x^6+x^7\right )+\left (8 x^7+e^x \left (-20 x^4+6 x^5+2 x^6\right )\right ) \log (3+\log (2))+e^x \left (5 x^4+x^5\right ) \log ^2(3+\log (2))}{4-4 x+x^2+(-4+2 x) \log (3+\log (2))+\log ^2(3+\log (2))} \, dx=\text {Too large to display} \]
integrate(((x^5+5*x^4)*exp(x)*log(3+log(2))^2+((2*x^6+6*x^5-20*x^4)*exp(x) +8*x^7)*log(3+log(2))+(x^7+x^6-16*x^5+20*x^4)*exp(x)+7*x^8-16*x^7)/(log(3+ log(2))^2+(2*x-4)*log(3+log(2))+x^2-4*x+4),x, algorithm=\
x^7 - 7/3*x^6*(log(log(2) + 3) - 2) + 21/5*(log(log(2) + 3)^2 - 4*log(log( 2) + 3) + 4)*x^5 - 8/3*x^6 + 32/5*x^5*(log(log(2) + 3) - 2) + x^5*e^x - 7* (log(log(2) + 3)^3 - 6*log(log(2) + 3)^2 + 12*log(log(2) + 3) - 8)*x^4 - 1 2*(log(log(2) + 3)^2 - 4*log(log(2) + 3) + 4)*x^4 + 35/3*(log(log(2) + 3)^ 4 - 8*log(log(2) + 3)^3 + 24*log(log(2) + 3)^2 - 32*log(log(2) + 3) + 16)* x^3 + 64/3*(log(log(2) + 3)^3 - 6*log(log(2) + 3)^2 + 12*log(log(2) + 3) - 8)*x^3 - 21*(log(log(2) + 3)^5 - 10*log(log(2) + 3)^4 + 40*log(log(2) + 3 )^3 - 80*log(log(2) + 3)^2 + 80*log(log(2) + 3) - 32)*x^2 - 40*(log(log(2) + 3)^4 - 8*log(log(2) + 3)^3 + 24*log(log(2) + 3)^2 - 32*log(log(2) + 3) + 16)*x^2 + 49*(log(log(2) + 3)^6 - 12*log(log(2) + 3)^5 + 60*log(log(2) + 3)^4 - 160*log(log(2) + 3)^3 + 240*log(log(2) + 3)^2 - 192*log(log(2) + 3 ) + 64)*x + 96*(log(log(2) + 3)^5 - 10*log(log(2) + 3)^4 + 40*log(log(2) + 3)^3 - 80*log(log(2) + 3)^2 + 80*log(log(2) + 3) - 32)*x - 56*(log(log(2) + 3)^7 - 14*log(log(2) + 3)^6 + 84*log(log(2) + 3)^5 - 280*log(log(2) + 3 )^4 + 560*log(log(2) + 3)^3 - 672*log(log(2) + 3)^2 + 448*log(log(2) + 3) - 128)*log(x + log(log(2) + 3) - 2) - 112*(log(log(2) + 3)^6 - 12*log(log( 2) + 3)^5 + 60*log(log(2) + 3)^4 - 160*log(log(2) + 3)^3 + 240*log(log(2) + 3)^2 - 192*log(log(2) + 3) + 64)*log(x + log(log(2) + 3) - 2) + 2/15*(10 *x^6 - 24*x^5*(log(log(2) + 3) - 2) + 45*(log(log(2) + 3)^2 - 4*log(log(2) + 3) + 4)*x^4 - 80*(log(log(2) + 3)^3 - 6*log(log(2) + 3)^2 + 12*log(l...
Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (21) = 42\).
Time = 0.26 (sec) , antiderivative size = 178, normalized size of antiderivative = 8.09 \[ \int \frac {-16 x^7+7 x^8+e^x \left (20 x^4-16 x^5+x^6+x^7\right )+\left (8 x^7+e^x \left (-20 x^4+6 x^5+2 x^6\right )\right ) \log (3+\log (2))+e^x \left (5 x^4+x^5\right ) \log ^2(3+\log (2))}{4-4 x+x^2+(-4+2 x) \log (3+\log (2))+\log ^2(3+\log (2))} \, dx=\frac {x^{8} + x \log \left (\log \left (2\right ) + 3\right )^{7} + \log \left (\log \left (2\right ) + 3\right )^{8} + x^{6} e^{x} + x^{5} e^{x} \log \left (\log \left (2\right ) + 3\right ) - 14 \, x \log \left (\log \left (2\right ) + 3\right )^{6} - 16 \, \log \left (\log \left (2\right ) + 3\right )^{7} - 2 \, x^{5} e^{x} + 84 \, x \log \left (\log \left (2\right ) + 3\right )^{5} + 112 \, \log \left (\log \left (2\right ) + 3\right )^{6} - 280 \, x \log \left (\log \left (2\right ) + 3\right )^{4} - 448 \, \log \left (\log \left (2\right ) + 3\right )^{5} + 560 \, x \log \left (\log \left (2\right ) + 3\right )^{3} + 1120 \, \log \left (\log \left (2\right ) + 3\right )^{4} - 672 \, x \log \left (\log \left (2\right ) + 3\right )^{2} - 1792 \, \log \left (\log \left (2\right ) + 3\right )^{3} + 448 \, x \log \left (\log \left (2\right ) + 3\right ) + 1792 \, \log \left (\log \left (2\right ) + 3\right )^{2} - 128 \, x - 1024 \, \log \left (\log \left (2\right ) + 3\right ) + 256}{x + \log \left (\log \left (2\right ) + 3\right ) - 2} \]
integrate(((x^5+5*x^4)*exp(x)*log(3+log(2))^2+((2*x^6+6*x^5-20*x^4)*exp(x) +8*x^7)*log(3+log(2))+(x^7+x^6-16*x^5+20*x^4)*exp(x)+7*x^8-16*x^7)/(log(3+ log(2))^2+(2*x-4)*log(3+log(2))+x^2-4*x+4),x, algorithm=\
(x^8 + x*log(log(2) + 3)^7 + log(log(2) + 3)^8 + x^6*e^x + x^5*e^x*log(log (2) + 3) - 14*x*log(log(2) + 3)^6 - 16*log(log(2) + 3)^7 - 2*x^5*e^x + 84* x*log(log(2) + 3)^5 + 112*log(log(2) + 3)^6 - 280*x*log(log(2) + 3)^4 - 44 8*log(log(2) + 3)^5 + 560*x*log(log(2) + 3)^3 + 1120*log(log(2) + 3)^4 - 6 72*x*log(log(2) + 3)^2 - 1792*log(log(2) + 3)^3 + 448*x*log(log(2) + 3) + 1792*log(log(2) + 3)^2 - 128*x - 1024*log(log(2) + 3) + 256)/(x + log(log( 2) + 3) - 2)
Time = 12.18 (sec) , antiderivative size = 1697, normalized size of antiderivative = 77.14 \[ \int \frac {-16 x^7+7 x^8+e^x \left (20 x^4-16 x^5+x^6+x^7\right )+\left (8 x^7+e^x \left (-20 x^4+6 x^5+2 x^6\right )\right ) \log (3+\log (2))+e^x \left (5 x^4+x^5\right ) \log ^2(3+\log (2))}{4-4 x+x^2+(-4+2 x) \log (3+\log (2))+\log ^2(3+\log (2))} \, dx=\text {Too large to display} \]
int((log(log(2) + 3)*(exp(x)*(6*x^5 - 20*x^4 + 2*x^6) + 8*x^7) + exp(x)*(2 0*x^4 - 16*x^5 + x^6 + x^7) - 16*x^7 + 7*x^8 + log(log(2) + 3)^2*exp(x)*(5 *x^4 + x^5))/(log(log(2) + 3)^2 - 4*x + log(log(2) + 3)*(2*x - 4) + x^2 + 4),x)
x^5*exp(x) + x*((((log((log(2) + 3)^2) - 4)*((log((log(2) + 3)^2) - 4)*(8* log(log(2) + 3) - 7*log((log(2) + 3)^2) + 12) - 28*log(log(2) + 3) + 7*log (log(2) + 3)^2 + 28) - (log(log(2) + 3)^2 - 4*log(log(2) + 3) + 4)*(8*log( log(2) + 3) - 7*log((log(2) + 3)^2) + 12))*(log(log(2) + 3)^2 - 4*log(log( 2) + 3) + 4) - (((log((log(2) + 3)^2) - 4)*((log((log(2) + 3)^2) - 4)*(8*l og(log(2) + 3) - 7*log((log(2) + 3)^2) + 12) - 28*log(log(2) + 3) + 7*log( log(2) + 3)^2 + 28) - (log(log(2) + 3)^2 - 4*log(log(2) + 3) + 4)*(8*log(l og(2) + 3) - 7*log((log(2) + 3)^2) + 12))*(log((log(2) + 3)^2) - 4) - (log (log(2) + 3)^2 - 4*log(log(2) + 3) + 4)*((log((log(2) + 3)^2) - 4)*(8*log( log(2) + 3) - 7*log((log(2) + 3)^2) + 12) - 28*log(log(2) + 3) + 7*log(log (2) + 3)^2 + 28))*(log((log(2) + 3)^2) - 4))*(log((log(2) + 3)^2) - 4) + ( ((log((log(2) + 3)^2) - 4)*((log((log(2) + 3)^2) - 4)*(8*log(log(2) + 3) - 7*log((log(2) + 3)^2) + 12) - 28*log(log(2) + 3) + 7*log(log(2) + 3)^2 + 28) - (log(log(2) + 3)^2 - 4*log(log(2) + 3) + 4)*(8*log(log(2) + 3) - 7*l og((log(2) + 3)^2) + 12))*(log((log(2) + 3)^2) - 4) - (log(log(2) + 3)^2 - 4*log(log(2) + 3) + 4)*((log((log(2) + 3)^2) - 4)*(8*log(log(2) + 3) - 7* log((log(2) + 3)^2) + 12) - 28*log(log(2) + 3) + 7*log(log(2) + 3)^2 + 28) )*(log(log(2) + 3)^2 - 4*log(log(2) + 3) + 4)) + x^6*((4*log(log(2) + 3))/ 3 - (7*log((log(2) + 3)^2))/6 + 2) - x^2*((((log((log(2) + 3)^2) - 4)*((lo g((log(2) + 3)^2) - 4)*(8*log(log(2) + 3) - 7*log((log(2) + 3)^2) + 12)...