Integrand size = 398, antiderivative size = 34 \[ \int \frac {e^x \left (-4+111 x-48 x^2-125 x^3+3 x^4+111 x^5-59 x^6+9 x^7\right )+\left (e^x \left (x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8\right )+e^x \left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right ) \log \left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right )}{\left (x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8+\left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right ) \log ^2\left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right )} \, dx=\frac {e^x}{\log \left (-x+\log \left (x+\frac {1}{9} \left (2+\frac {3}{-1-2 x+x^2}\right )^2\right )\right )} \]
Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.44 \[ \int \frac {e^x \left (-4+111 x-48 x^2-125 x^3+3 x^4+111 x^5-59 x^6+9 x^7\right )+\left (e^x \left (x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8\right )+e^x \left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right ) \log \left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right )}{\left (x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8+\left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right ) \log ^2\left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right )} \, dx=\frac {e^x}{\log \left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9 \left (-1-2 x+x^2\right )^2}\right )\right )} \]
Integrate[(E^x*(-4 + 111*x - 48*x^2 - 125*x^3 + 3*x^4 + 111*x^5 - 59*x^6 + 9*x^7) + (E^x*(x + 3*x^2 + 57*x^3 + 113*x^4 - 84*x^5 - 57*x^6 + 50*x^7 - 9*x^8) + E^x*(-1 - 3*x - 57*x^2 - 113*x^3 + 84*x^4 + 57*x^5 - 50*x^6 + 9*x ^7)*Log[(1 + x + 56*x^2 + 2*x^3 - 32*x^4 + 9*x^5)/(9 + 36*x + 18*x^2 - 36* x^3 + 9*x^4)])*Log[-x + Log[(1 + x + 56*x^2 + 2*x^3 - 32*x^4 + 9*x^5)/(9 + 36*x + 18*x^2 - 36*x^3 + 9*x^4)]])/((x + 3*x^2 + 57*x^3 + 113*x^4 - 84*x^ 5 - 57*x^6 + 50*x^7 - 9*x^8 + (-1 - 3*x - 57*x^2 - 113*x^3 + 84*x^4 + 57*x ^5 - 50*x^6 + 9*x^7)*Log[(1 + x + 56*x^2 + 2*x^3 - 32*x^4 + 9*x^5)/(9 + 36 *x + 18*x^2 - 36*x^3 + 9*x^4)])*Log[-x + Log[(1 + x + 56*x^2 + 2*x^3 - 32* x^4 + 9*x^5)/(9 + 36*x + 18*x^2 - 36*x^3 + 9*x^4)]]^2),x]
Leaf count is larger than twice the leaf count of optimal. \(123\) vs. \(2(34)=68\).
Time = 13.45 (sec) , antiderivative size = 123, normalized size of antiderivative = 3.62, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.010, Rules used = {7292, 2463, 7239, 2726}
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.
| \(\displaystyle \int \frac {e^x \left (9 x^7-59 x^6+111 x^5+3 x^4-125 x^3-48 x^2+111 x-4\right )+\left (e^x \left (9 x^7-50 x^6+57 x^5+84 x^4-113 x^3-57 x^2-3 x-1\right ) \log \left (\frac {9 x^5-32 x^4+2 x^3+56 x^2+x+1}{9 x^4-36 x^3+18 x^2+36 x+9}\right )+e^x \left (-9 x^8+50 x^7-57 x^6-84 x^5+113 x^4+57 x^3+3 x^2+x\right )\right ) \log \left (\log \left (\frac {9 x^5-32 x^4+2 x^3+56 x^2+x+1}{9 x^4-36 x^3+18 x^2+36 x+9}\right )-x\right )}{\left (-9 x^8+50 x^7-57 x^6-84 x^5+113 x^4+57 x^3+3 x^2+\left (9 x^7-50 x^6+57 x^5+84 x^4-113 x^3-57 x^2-3 x-1\right ) \log \left (\frac {9 x^5-32 x^4+2 x^3+56 x^2+x+1}{9 x^4-36 x^3+18 x^2+36 x+9}\right )+x\right ) \log ^2\left (\log \left (\frac {9 x^5-32 x^4+2 x^3+56 x^2+x+1}{9 x^4-36 x^3+18 x^2+36 x+9}\right )-x\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^x \left (9 x^7-59 x^6+111 x^5+3 x^4-125 x^3-48 x^2+111 x-4\right )+\left (e^x \left (9 x^7-50 x^6+57 x^5+84 x^4-113 x^3-57 x^2-3 x-1\right ) \log \left (\frac {9 x^5-32 x^4+2 x^3+56 x^2+x+1}{9 x^4-36 x^3+18 x^2+36 x+9}\right )+e^x \left (-9 x^8+50 x^7-57 x^6-84 x^5+113 x^4+57 x^3+3 x^2+x\right )\right ) \log \left (\log \left (\frac {9 x^5-32 x^4+2 x^3+56 x^2+x+1}{9 x^4-36 x^3+18 x^2+36 x+9}\right )-x\right )}{\left (-9 x^7+50 x^6-57 x^5-84 x^4+113 x^3+57 x^2+3 x+1\right ) \left (x-\log \left (\frac {9 x^5-32 x^4+2 x^3+56 x^2+x+1}{9 \left (x^2-2 x-1\right )^2}\right )\right ) \log ^2\left (\log \left (\frac {9 x^5-32 x^4+2 x^3+56 x^2+x+1}{9 x^4-36 x^3+18 x^2+36 x+9}\right )-x\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {\left (9 x^3-14 x^2-17 x+8\right ) \left (e^x \left (9 x^7-59 x^6+111 x^5+3 x^4-125 x^3-48 x^2+111 x-4\right )+\left (e^x \left (9 x^7-50 x^6+57 x^5+84 x^4-113 x^3-57 x^2-3 x-1\right ) \log \left (\frac {9 x^5-32 x^4+2 x^3+56 x^2+x+1}{9 x^4-36 x^3+18 x^2+36 x+9}\right )+e^x \left (-9 x^8+50 x^7-57 x^6-84 x^5+113 x^4+57 x^3+3 x^2+x\right )\right ) \log \left (\log \left (\frac {9 x^5-32 x^4+2 x^3+56 x^2+x+1}{9 x^4-36 x^3+18 x^2+36 x+9}\right )-x\right )\right )}{9 \left (9 x^5-32 x^4+2 x^3+56 x^2+x+1\right ) \left (x-\log \left (\frac {9 x^5-32 x^4+2 x^3+56 x^2+x+1}{9 \left (x^2-2 x-1\right )^2}\right )\right ) \log ^2\left (\log \left (\frac {9 x^5-32 x^4+2 x^3+56 x^2+x+1}{9 x^4-36 x^3+18 x^2+36 x+9}\right )-x\right )}-\frac {e^x \left (9 x^7-59 x^6+111 x^5+3 x^4-125 x^3-48 x^2+111 x-4\right )+\left (e^x \left (9 x^7-50 x^6+57 x^5+84 x^4-113 x^3-57 x^2-3 x-1\right ) \log \left (\frac {9 x^5-32 x^4+2 x^3+56 x^2+x+1}{9 x^4-36 x^3+18 x^2+36 x+9}\right )+e^x \left (-9 x^8+50 x^7-57 x^6-84 x^5+113 x^4+57 x^3+3 x^2+x\right )\right ) \log \left (\log \left (\frac {9 x^5-32 x^4+2 x^3+56 x^2+x+1}{9 x^4-36 x^3+18 x^2+36 x+9}\right )-x\right )}{9 \left (x^2-2 x-1\right ) \left (x-\log \left (\frac {9 x^5-32 x^4+2 x^3+56 x^2+x+1}{9 \left (x^2-2 x-1\right )^2}\right )\right ) \log ^2\left (\log \left (\frac {9 x^5-32 x^4+2 x^3+56 x^2+x+1}{9 x^4-36 x^3+18 x^2+36 x+9}\right )-x\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^x \left (9 x^7-59 x^6+111 x^5+3 x^4-125 x^3-48 x^2-\left (9 x^7-50 x^6+57 x^5+84 x^4-113 x^3-57 x^2-3 x-1\right ) \left (x-\log \left (\frac {9 x^5-32 x^4+2 x^3+56 x^2+x+1}{9 \left (x^2-2 x-1\right )^2}\right )\right ) \log \left (\log \left (\frac {9 x^5-32 x^4+2 x^3+56 x^2+x+1}{9 \left (x^2-2 x-1\right )^2}\right )-x\right )+111 x-4\right )}{\left (-x^2+2 x+1\right ) \left (9 x^5-32 x^4+2 x^3+56 x^2+x+1\right ) \left (x-\log \left (\frac {9 x^5-32 x^4+2 x^3+56 x^2+x+1}{9 \left (x^2-2 x-1\right )^2}\right )\right ) \log ^2\left (\log \left (\frac {9 x^5-32 x^4+2 x^3+56 x^2+x+1}{9 \left (x^2-2 x-1\right )^2}\right )-x\right )}dx\) |
| \(\Big \downarrow \) 2726 |
| \(\displaystyle \frac {e^x \left (-9 x^7+50 x^6-57 x^5-84 x^4+113 x^3+57 x^2+3 x+1\right )}{\left (-x^2+2 x+1\right ) \left (9 x^5-32 x^4+2 x^3+56 x^2+x+1\right ) \log \left (\log \left (\frac {9 x^5-32 x^4+2 x^3+56 x^2+x+1}{9 \left (-x^2+2 x+1\right )^2}\right )-x\right )}\) |
Int[(E^x*(-4 + 111*x - 48*x^2 - 125*x^3 + 3*x^4 + 111*x^5 - 59*x^6 + 9*x^7 ) + (E^x*(x + 3*x^2 + 57*x^3 + 113*x^4 - 84*x^5 - 57*x^6 + 50*x^7 - 9*x^8) + E^x*(-1 - 3*x - 57*x^2 - 113*x^3 + 84*x^4 + 57*x^5 - 50*x^6 + 9*x^7)*Lo g[(1 + x + 56*x^2 + 2*x^3 - 32*x^4 + 9*x^5)/(9 + 36*x + 18*x^2 - 36*x^3 + 9*x^4)])*Log[-x + Log[(1 + x + 56*x^2 + 2*x^3 - 32*x^4 + 9*x^5)/(9 + 36*x + 18*x^2 - 36*x^3 + 9*x^4)]])/((x + 3*x^2 + 57*x^3 + 113*x^4 - 84*x^5 - 57 *x^6 + 50*x^7 - 9*x^8 + (-1 - 3*x - 57*x^2 - 113*x^3 + 84*x^4 + 57*x^5 - 5 0*x^6 + 9*x^7)*Log[(1 + x + 56*x^2 + 2*x^3 - 32*x^4 + 9*x^5)/(9 + 36*x + 1 8*x^2 - 36*x^3 + 9*x^4)])*Log[-x + Log[(1 + x + 56*x^2 + 2*x^3 - 32*x^4 + 9*x^5)/(9 + 36*x + 18*x^2 - 36*x^3 + 9*x^4)]]^2),x]
(E^x*(1 + 3*x + 57*x^2 + 113*x^3 - 84*x^4 - 57*x^5 + 50*x^6 - 9*x^7))/((1 + 2*x - x^2)*(1 + x + 56*x^2 + 2*x^3 - 32*x^4 + 9*x^5)*Log[-x + Log[(1 + x + 56*x^2 + 2*x^3 - 32*x^4 + 9*x^5)/(9*(1 + 2*x - x^2)^2)]])
3.20.10.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z, x], w*y]] /; FreeQ[F, x]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.04 (sec) , antiderivative size = 257, normalized size of antiderivative = 7.56
\[\frac {{\mathrm e}^{x}}{\ln \left (\ln \left (x^{5}-\frac {32}{9} x^{4}+\frac {2}{9} x^{3}+\frac {56}{9} x^{2}+\frac {1}{9} x +\frac {1}{9}\right )-2 \ln \left (x^{2}-2 x -1\right )+\frac {i \pi \,\operatorname {csgn}\left (i \left (x^{2}-2 x -1\right )^{2}\right ) {\left (-\operatorname {csgn}\left (i \left (x^{2}-2 x -1\right )^{2}\right )+\operatorname {csgn}\left (i \left (x^{2}-2 x -1\right )\right )\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (x^{5}-\frac {32}{9} x^{4}+\frac {2}{9} x^{3}+\frac {56}{9} x^{2}+\frac {1}{9} x +\frac {1}{9}\right )}{\left (x^{2}-2 x -1\right )^{2}}\right ) \left (-\operatorname {csgn}\left (\frac {i \left (x^{5}-\frac {32}{9} x^{4}+\frac {2}{9} x^{3}+\frac {56}{9} x^{2}+\frac {1}{9} x +\frac {1}{9}\right )}{\left (x^{2}-2 x -1\right )^{2}}\right )+\operatorname {csgn}\left (i \left (x^{5}-\frac {32}{9} x^{4}+\frac {2}{9} x^{3}+\frac {56}{9} x^{2}+\frac {1}{9} x +\frac {1}{9}\right )\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \left (x^{5}-\frac {32}{9} x^{4}+\frac {2}{9} x^{3}+\frac {56}{9} x^{2}+\frac {1}{9} x +\frac {1}{9}\right )}{\left (x^{2}-2 x -1\right )^{2}}\right )+\operatorname {csgn}\left (\frac {i}{\left (x^{2}-2 x -1\right )^{2}}\right )\right )}{2}-x \right )}\]
int((((9*x^7-50*x^6+57*x^5+84*x^4-113*x^3-57*x^2-3*x-1)*exp(x)*ln((9*x^5-3 2*x^4+2*x^3+56*x^2+x+1)/(9*x^4-36*x^3+18*x^2+36*x+9))+(-9*x^8+50*x^7-57*x^ 6-84*x^5+113*x^4+57*x^3+3*x^2+x)*exp(x))*ln(ln((9*x^5-32*x^4+2*x^3+56*x^2+ x+1)/(9*x^4-36*x^3+18*x^2+36*x+9))-x)+(9*x^7-59*x^6+111*x^5+3*x^4-125*x^3- 48*x^2+111*x-4)*exp(x))/((9*x^7-50*x^6+57*x^5+84*x^4-113*x^3-57*x^2-3*x-1) *ln((9*x^5-32*x^4+2*x^3+56*x^2+x+1)/(9*x^4-36*x^3+18*x^2+36*x+9))-9*x^8+50 *x^7-57*x^6-84*x^5+113*x^4+57*x^3+3*x^2+x)/ln(ln((9*x^5-32*x^4+2*x^3+56*x^ 2+x+1)/(9*x^4-36*x^3+18*x^2+36*x+9))-x)^2,x)
exp(x)/ln(ln(x^5-32/9*x^4+2/9*x^3+56/9*x^2+1/9*x+1/9)-2*ln(x^2-2*x-1)+1/2* I*Pi*csgn(I*(x^2-2*x-1)^2)*(-csgn(I*(x^2-2*x-1)^2)+csgn(I*(x^2-2*x-1)))^2- 1/2*I*Pi*csgn(I*(x^5-32/9*x^4+2/9*x^3+56/9*x^2+1/9*x+1/9)/(x^2-2*x-1)^2)*( -csgn(I*(x^5-32/9*x^4+2/9*x^3+56/9*x^2+1/9*x+1/9)/(x^2-2*x-1)^2)+csgn(I*(x ^5-32/9*x^4+2/9*x^3+56/9*x^2+1/9*x+1/9)))*(-csgn(I*(x^5-32/9*x^4+2/9*x^3+5 6/9*x^2+1/9*x+1/9)/(x^2-2*x-1)^2)+csgn(I/(x^2-2*x-1)^2))-x)
Time = 0.25 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.65 \[ \int \frac {e^x \left (-4+111 x-48 x^2-125 x^3+3 x^4+111 x^5-59 x^6+9 x^7\right )+\left (e^x \left (x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8\right )+e^x \left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right ) \log \left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right )}{\left (x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8+\left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right ) \log ^2\left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right )} \, dx=\frac {e^{x}}{\log \left (-x + \log \left (\frac {9 \, x^{5} - 32 \, x^{4} + 2 \, x^{3} + 56 \, x^{2} + x + 1}{9 \, {\left (x^{4} - 4 \, x^{3} + 2 \, x^{2} + 4 \, x + 1\right )}}\right )\right )} \]
integrate((((9*x^7-50*x^6+57*x^5+84*x^4-113*x^3-57*x^2-3*x-1)*exp(x)*log(( 9*x^5-32*x^4+2*x^3+56*x^2+x+1)/(9*x^4-36*x^3+18*x^2+36*x+9))+(-9*x^8+50*x^ 7-57*x^6-84*x^5+113*x^4+57*x^3+3*x^2+x)*exp(x))*log(log((9*x^5-32*x^4+2*x^ 3+56*x^2+x+1)/(9*x^4-36*x^3+18*x^2+36*x+9))-x)+(9*x^7-59*x^6+111*x^5+3*x^4 -125*x^3-48*x^2+111*x-4)*exp(x))/((9*x^7-50*x^6+57*x^5+84*x^4-113*x^3-57*x ^2-3*x-1)*log((9*x^5-32*x^4+2*x^3+56*x^2+x+1)/(9*x^4-36*x^3+18*x^2+36*x+9) )-9*x^8+50*x^7-57*x^6-84*x^5+113*x^4+57*x^3+3*x^2+x)/log(log((9*x^5-32*x^4 +2*x^3+56*x^2+x+1)/(9*x^4-36*x^3+18*x^2+36*x+9))-x)^2,x, algorithm=\
e^x/log(-x + log(1/9*(9*x^5 - 32*x^4 + 2*x^3 + 56*x^2 + x + 1)/(x^4 - 4*x^ 3 + 2*x^2 + 4*x + 1)))
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (24) = 48\).
Time = 4.48 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.50 \[ \int \frac {e^x \left (-4+111 x-48 x^2-125 x^3+3 x^4+111 x^5-59 x^6+9 x^7\right )+\left (e^x \left (x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8\right )+e^x \left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right ) \log \left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right )}{\left (x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8+\left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right ) \log ^2\left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right )} \, dx=\frac {e^{x}}{\log {\left (- x + \log {\left (\frac {9 x^{5} - 32 x^{4} + 2 x^{3} + 56 x^{2} + x + 1}{9 x^{4} - 36 x^{3} + 18 x^{2} + 36 x + 9} \right )} \right )}} \]
integrate((((9*x**7-50*x**6+57*x**5+84*x**4-113*x**3-57*x**2-3*x-1)*exp(x) *ln((9*x**5-32*x**4+2*x**3+56*x**2+x+1)/(9*x**4-36*x**3+18*x**2+36*x+9))+( -9*x**8+50*x**7-57*x**6-84*x**5+113*x**4+57*x**3+3*x**2+x)*exp(x))*ln(ln(( 9*x**5-32*x**4+2*x**3+56*x**2+x+1)/(9*x**4-36*x**3+18*x**2+36*x+9))-x)+(9* x**7-59*x**6+111*x**5+3*x**4-125*x**3-48*x**2+111*x-4)*exp(x))/((9*x**7-50 *x**6+57*x**5+84*x**4-113*x**3-57*x**2-3*x-1)*ln((9*x**5-32*x**4+2*x**3+56 *x**2+x+1)/(9*x**4-36*x**3+18*x**2+36*x+9))-9*x**8+50*x**7-57*x**6-84*x**5 +113*x**4+57*x**3+3*x**2+x)/ln(ln((9*x**5-32*x**4+2*x**3+56*x**2+x+1)/(9*x **4-36*x**3+18*x**2+36*x+9))-x)**2,x)
exp(x)/log(-x + log((9*x**5 - 32*x**4 + 2*x**3 + 56*x**2 + x + 1)/(9*x**4 - 36*x**3 + 18*x**2 + 36*x + 9)))
Time = 0.50 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.44 \[ \int \frac {e^x \left (-4+111 x-48 x^2-125 x^3+3 x^4+111 x^5-59 x^6+9 x^7\right )+\left (e^x \left (x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8\right )+e^x \left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right ) \log \left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right )}{\left (x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8+\left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right ) \log ^2\left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right )} \, dx=\frac {e^{x}}{\log \left (-x - 2 \, \log \left (3\right ) + \log \left (9 \, x^{5} - 32 \, x^{4} + 2 \, x^{3} + 56 \, x^{2} + x + 1\right ) - 2 \, \log \left (x^{2} - 2 \, x - 1\right )\right )} \]
integrate((((9*x^7-50*x^6+57*x^5+84*x^4-113*x^3-57*x^2-3*x-1)*exp(x)*log(( 9*x^5-32*x^4+2*x^3+56*x^2+x+1)/(9*x^4-36*x^3+18*x^2+36*x+9))+(-9*x^8+50*x^ 7-57*x^6-84*x^5+113*x^4+57*x^3+3*x^2+x)*exp(x))*log(log((9*x^5-32*x^4+2*x^ 3+56*x^2+x+1)/(9*x^4-36*x^3+18*x^2+36*x+9))-x)+(9*x^7-59*x^6+111*x^5+3*x^4 -125*x^3-48*x^2+111*x-4)*exp(x))/((9*x^7-50*x^6+57*x^5+84*x^4-113*x^3-57*x ^2-3*x-1)*log((9*x^5-32*x^4+2*x^3+56*x^2+x+1)/(9*x^4-36*x^3+18*x^2+36*x+9) )-9*x^8+50*x^7-57*x^6-84*x^5+113*x^4+57*x^3+3*x^2+x)/log(log((9*x^5-32*x^4 +2*x^3+56*x^2+x+1)/(9*x^4-36*x^3+18*x^2+36*x+9))-x)^2,x, algorithm=\
e^x/log(-x - 2*log(3) + log(9*x^5 - 32*x^4 + 2*x^3 + 56*x^2 + x + 1) - 2*l og(x^2 - 2*x - 1))
Leaf count of result is larger than twice the leaf count of optimal. 262 vs. \(2 (31) = 62\).
Time = 7.03 (sec) , antiderivative size = 262, normalized size of antiderivative = 7.71 \[ \int \frac {e^x \left (-4+111 x-48 x^2-125 x^3+3 x^4+111 x^5-59 x^6+9 x^7\right )+\left (e^x \left (x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8\right )+e^x \left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right ) \log \left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right )}{\left (x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8+\left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right ) \log ^2\left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right )} \, dx=\frac {x e^{x} - e^{x} \log \left (\frac {9 \, x^{5} - 32 \, x^{4} + 2 \, x^{3} + 56 \, x^{2} + x + 1}{9 \, {\left (x^{4} - 4 \, x^{3} + 2 \, x^{2} + 4 \, x + 1\right )}}\right )}{x \log \left (-x + \log \left (\frac {9 \, x^{5} - 32 \, x^{4} + 2 \, x^{3} + 56 \, x^{2} + x + 1}{9 \, {\left (x^{4} - 4 \, x^{3} + 2 \, x^{2} + 4 \, x + 1\right )}}\right )\right ) - \log \left (9 \, x^{5} - 32 \, x^{4} + 2 \, x^{3} + 56 \, x^{2} + x + 1\right ) \log \left (-x + \log \left (\frac {9 \, x^{5} - 32 \, x^{4} + 2 \, x^{3} + 56 \, x^{2} + x + 1}{9 \, {\left (x^{4} - 4 \, x^{3} + 2 \, x^{2} + 4 \, x + 1\right )}}\right )\right ) + \log \left (9 \, x^{4} - 36 \, x^{3} + 18 \, x^{2} + 36 \, x + 9\right ) \log \left (-x + \log \left (\frac {9 \, x^{5} - 32 \, x^{4} + 2 \, x^{3} + 56 \, x^{2} + x + 1}{9 \, {\left (x^{4} - 4 \, x^{3} + 2 \, x^{2} + 4 \, x + 1\right )}}\right )\right )} \]
integrate((((9*x^7-50*x^6+57*x^5+84*x^4-113*x^3-57*x^2-3*x-1)*exp(x)*log(( 9*x^5-32*x^4+2*x^3+56*x^2+x+1)/(9*x^4-36*x^3+18*x^2+36*x+9))+(-9*x^8+50*x^ 7-57*x^6-84*x^5+113*x^4+57*x^3+3*x^2+x)*exp(x))*log(log((9*x^5-32*x^4+2*x^ 3+56*x^2+x+1)/(9*x^4-36*x^3+18*x^2+36*x+9))-x)+(9*x^7-59*x^6+111*x^5+3*x^4 -125*x^3-48*x^2+111*x-4)*exp(x))/((9*x^7-50*x^6+57*x^5+84*x^4-113*x^3-57*x ^2-3*x-1)*log((9*x^5-32*x^4+2*x^3+56*x^2+x+1)/(9*x^4-36*x^3+18*x^2+36*x+9) )-9*x^8+50*x^7-57*x^6-84*x^5+113*x^4+57*x^3+3*x^2+x)/log(log((9*x^5-32*x^4 +2*x^3+56*x^2+x+1)/(9*x^4-36*x^3+18*x^2+36*x+9))-x)^2,x, algorithm=\
(x*e^x - e^x*log(1/9*(9*x^5 - 32*x^4 + 2*x^3 + 56*x^2 + x + 1)/(x^4 - 4*x^ 3 + 2*x^2 + 4*x + 1)))/(x*log(-x + log(1/9*(9*x^5 - 32*x^4 + 2*x^3 + 56*x^ 2 + x + 1)/(x^4 - 4*x^3 + 2*x^2 + 4*x + 1))) - log(9*x^5 - 32*x^4 + 2*x^3 + 56*x^2 + x + 1)*log(-x + log(1/9*(9*x^5 - 32*x^4 + 2*x^3 + 56*x^2 + x + 1)/(x^4 - 4*x^3 + 2*x^2 + 4*x + 1))) + log(9*x^4 - 36*x^3 + 18*x^2 + 36*x + 9)*log(-x + log(1/9*(9*x^5 - 32*x^4 + 2*x^3 + 56*x^2 + x + 1)/(x^4 - 4*x ^3 + 2*x^2 + 4*x + 1))))
Time = 12.24 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.68 \[ \int \frac {e^x \left (-4+111 x-48 x^2-125 x^3+3 x^4+111 x^5-59 x^6+9 x^7\right )+\left (e^x \left (x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8\right )+e^x \left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right ) \log \left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right )}{\left (x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8+\left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right ) \log ^2\left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right )} \, dx=\frac {{\mathrm {e}}^x}{\ln \left (\ln \left (\frac {9\,x^5-32\,x^4+2\,x^3+56\,x^2+x+1}{9\,x^4-36\,x^3+18\,x^2+36\,x+9}\right )-x\right )} \]
int((log(log((x + 56*x^2 + 2*x^3 - 32*x^4 + 9*x^5 + 1)/(36*x + 18*x^2 - 36 *x^3 + 9*x^4 + 9)) - x)*(exp(x)*(x + 3*x^2 + 57*x^3 + 113*x^4 - 84*x^5 - 5 7*x^6 + 50*x^7 - 9*x^8) - exp(x)*log((x + 56*x^2 + 2*x^3 - 32*x^4 + 9*x^5 + 1)/(36*x + 18*x^2 - 36*x^3 + 9*x^4 + 9))*(3*x + 57*x^2 + 113*x^3 - 84*x^ 4 - 57*x^5 + 50*x^6 - 9*x^7 + 1)) + exp(x)*(111*x - 48*x^2 - 125*x^3 + 3*x ^4 + 111*x^5 - 59*x^6 + 9*x^7 - 4))/(log(log((x + 56*x^2 + 2*x^3 - 32*x^4 + 9*x^5 + 1)/(36*x + 18*x^2 - 36*x^3 + 9*x^4 + 9)) - x)^2*(x - log((x + 56 *x^2 + 2*x^3 - 32*x^4 + 9*x^5 + 1)/(36*x + 18*x^2 - 36*x^3 + 9*x^4 + 9))*( 3*x + 57*x^2 + 113*x^3 - 84*x^4 - 57*x^5 + 50*x^6 - 9*x^7 + 1) + 3*x^2 + 5 7*x^3 + 113*x^4 - 84*x^5 - 57*x^6 + 50*x^7 - 9*x^8)),x)