Integrand size = 325, antiderivative size = 28 \[ \int \frac {-36 x^4-162 e^4 x^4+81 x^4 \log (3)+e^{4 x} \left (-162 e^4+81 \log (3)\right )+e^{3 x} \left (648 e^4 x-324 x \log (3)\right )+e^{2 x} \left (-108 x^2-972 e^4 x^2+54 x^3+18 x^4+486 x^2 \log (3)\right )+e^x \left (144 x^3+648 e^4 x^3-54 x^4-18 x^5-324 x^3 \log (3)\right )}{324 e^8 x^4-36 e^4 x^5+x^6+\left (-324 e^4 x^4+18 x^5\right ) \log (3)+81 x^4 \log ^2(3)+e^{4 x} \left (324 e^8-324 e^4 \log (3)+81 \log ^2(3)\right )+e^{3 x} \left (-1296 e^8 x+1296 e^4 x \log (3)-324 x \log ^2(3)\right )+e^{2 x} \left (1944 e^8 x^2-36 e^4 x^3+\left (-1944 e^4 x^2+18 x^3\right ) \log (3)+486 x^2 \log ^2(3)\right )+e^x \left (-1296 e^8 x^3+72 e^4 x^4+\left (1296 e^4 x^3-36 x^4\right ) \log (3)-324 x^3 \log ^2(3)\right )} \, dx=\frac {4+x}{-2 e^4+\frac {x}{\left (3-\frac {3 e^x}{x}\right )^2}+\log (3)} \]
Leaf count is larger than twice the leaf count of optimal. \(79\) vs. \(2(28)=56\).
Time = 0.22 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.82 \[ \int \frac {-36 x^4-162 e^4 x^4+81 x^4 \log (3)+e^{4 x} \left (-162 e^4+81 \log (3)\right )+e^{3 x} \left (648 e^4 x-324 x \log (3)\right )+e^{2 x} \left (-108 x^2-972 e^4 x^2+54 x^3+18 x^4+486 x^2 \log (3)\right )+e^x \left (144 x^3+648 e^4 x^3-54 x^4-18 x^5-324 x^3 \log (3)\right )}{324 e^8 x^4-36 e^4 x^5+x^6+\left (-324 e^4 x^4+18 x^5\right ) \log (3)+81 x^4 \log ^2(3)+e^{4 x} \left (324 e^8-324 e^4 \log (3)+81 \log ^2(3)\right )+e^{3 x} \left (-1296 e^8 x+1296 e^4 x \log (3)-324 x \log ^2(3)\right )+e^{2 x} \left (1944 e^8 x^2-36 e^4 x^3+\left (-1944 e^4 x^2+18 x^3\right ) \log (3)+486 x^2 \log ^2(3)\right )+e^x \left (-1296 e^8 x^3+72 e^4 x^4+\left (1296 e^4 x^3-36 x^4\right ) \log (3)-324 x^3 \log ^2(3)\right )} \, dx=-\frac {x-\frac {x^3 (4+x)}{-18 e^{4+2 x}+36 e^{4+x} x-18 e^4 x^2+9 e^{2 x} \log (3)-18 e^x x \log (3)+x^2 (x+9 \log (3))}}{2 e^4-\log (3)} \]
Integrate[(-36*x^4 - 162*E^4*x^4 + 81*x^4*Log[3] + E^(4*x)*(-162*E^4 + 81* Log[3]) + E^(3*x)*(648*E^4*x - 324*x*Log[3]) + E^(2*x)*(-108*x^2 - 972*E^4 *x^2 + 54*x^3 + 18*x^4 + 486*x^2*Log[3]) + E^x*(144*x^3 + 648*E^4*x^3 - 54 *x^4 - 18*x^5 - 324*x^3*Log[3]))/(324*E^8*x^4 - 36*E^4*x^5 + x^6 + (-324*E ^4*x^4 + 18*x^5)*Log[3] + 81*x^4*Log[3]^2 + E^(4*x)*(324*E^8 - 324*E^4*Log [3] + 81*Log[3]^2) + E^(3*x)*(-1296*E^8*x + 1296*E^4*x*Log[3] - 324*x*Log[ 3]^2) + E^(2*x)*(1944*E^8*x^2 - 36*E^4*x^3 + (-1944*E^4*x^2 + 18*x^3)*Log[ 3] + 486*x^2*Log[3]^2) + E^x*(-1296*E^8*x^3 + 72*E^4*x^4 + (1296*E^4*x^3 - 36*x^4)*Log[3] - 324*x^3*Log[3]^2)),x]
-((x - (x^3*(4 + x))/(-18*E^(4 + 2*x) + 36*E^(4 + x)*x - 18*E^4*x^2 + 9*E^ (2*x)*Log[3] - 18*E^x*x*Log[3] + x^2*(x + 9*Log[3])))/(2*E^4 - Log[3]))
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.
| \(\displaystyle \int \frac {-162 e^4 x^4-36 x^4+81 x^4 \log (3)+e^x \left (-18 x^5-54 x^4+648 e^4 x^3+144 x^3-324 x^3 \log (3)\right )+e^{2 x} \left (18 x^4+54 x^3-972 e^4 x^2-108 x^2+486 x^2 \log (3)\right )+e^{3 x} \left (648 e^4 x-324 x \log (3)\right )+e^{4 x} \left (81 \log (3)-162 e^4\right )}{x^6-36 e^4 x^5+324 e^8 x^4+81 x^4 \log ^2(3)+\left (18 x^5-324 e^4 x^4\right ) \log (3)+e^x \left (72 e^4 x^4-1296 e^8 x^3-324 x^3 \log ^2(3)+\left (1296 e^4 x^3-36 x^4\right ) \log (3)\right )+e^{2 x} \left (-36 e^4 x^3+1944 e^8 x^2+486 x^2 \log ^2(3)+\left (18 x^3-1944 e^4 x^2\right ) \log (3)\right )+e^{3 x} \left (-1296 e^8 x-324 x \log ^2(3)+1296 e^4 x \log (3)\right )+e^{4 x} \left (324 e^8+81 \log ^2(3)-324 e^4 \log (3)\right )} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {\left (-36-162 e^4\right ) x^4+81 x^4 \log (3)+e^x \left (-18 x^5-54 x^4+648 e^4 x^3+144 x^3-324 x^3 \log (3)\right )+e^{2 x} \left (18 x^4+54 x^3-972 e^4 x^2-108 x^2+486 x^2 \log (3)\right )+e^{3 x} \left (648 e^4 x-324 x \log (3)\right )+e^{4 x} \left (81 \log (3)-162 e^4\right )}{x^6-36 e^4 x^5+324 e^8 x^4+81 x^4 \log ^2(3)+\left (18 x^5-324 e^4 x^4\right ) \log (3)+e^x \left (72 e^4 x^4-1296 e^8 x^3-324 x^3 \log ^2(3)+\left (1296 e^4 x^3-36 x^4\right ) \log (3)\right )+e^{2 x} \left (-36 e^4 x^3+1944 e^8 x^2+486 x^2 \log ^2(3)+\left (18 x^3-1944 e^4 x^2\right ) \log (3)\right )+e^{3 x} \left (-1296 e^8 x-324 x \log ^2(3)+1296 e^4 x \log (3)\right )+e^{4 x} \left (324 e^8+81 \log ^2(3)-324 e^4 \log (3)\right )}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {x^4 \left (-36-162 e^4+81 \log (3)\right )+e^x \left (-18 x^5-54 x^4+648 e^4 x^3+144 x^3-324 x^3 \log (3)\right )+e^{2 x} \left (18 x^4+54 x^3-972 e^4 x^2-108 x^2+486 x^2 \log (3)\right )+e^{3 x} \left (648 e^4 x-324 x \log (3)\right )+e^{4 x} \left (81 \log (3)-162 e^4\right )}{x^6-36 e^4 x^5+324 e^8 x^4+81 x^4 \log ^2(3)+\left (18 x^5-324 e^4 x^4\right ) \log (3)+e^x \left (72 e^4 x^4-1296 e^8 x^3-324 x^3 \log ^2(3)+\left (1296 e^4 x^3-36 x^4\right ) \log (3)\right )+e^{2 x} \left (-36 e^4 x^3+1944 e^8 x^2+486 x^2 \log ^2(3)+\left (18 x^3-1944 e^4 x^2\right ) \log (3)\right )+e^{3 x} \left (-1296 e^8 x-324 x \log ^2(3)+1296 e^4 x \log (3)\right )+e^{4 x} \left (324 e^8+81 \log ^2(3)-324 e^4 \log (3)\right )}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {x^4 \left (-36-162 e^4+81 \log (3)\right )+e^x \left (-18 x^5-54 x^4+648 e^4 x^3+144 x^3-324 x^3 \log (3)\right )+e^{2 x} \left (18 x^4+54 x^3-972 e^4 x^2-108 x^2+486 x^2 \log (3)\right )+e^{3 x} \left (648 e^4 x-324 x \log (3)\right )+e^{4 x} \left (81 \log (3)-162 e^4\right )}{x^6-36 e^4 x^5+x^4 \left (324 e^8+81 \log ^2(3)\right )+\left (18 x^5-324 e^4 x^4\right ) \log (3)+e^x \left (72 e^4 x^4-1296 e^8 x^3-324 x^3 \log ^2(3)+\left (1296 e^4 x^3-36 x^4\right ) \log (3)\right )+e^{2 x} \left (-36 e^4 x^3+1944 e^8 x^2+486 x^2 \log ^2(3)+\left (18 x^3-1944 e^4 x^2\right ) \log (3)\right )+e^{3 x} \left (-1296 e^8 x-324 x \log ^2(3)+1296 e^4 x \log (3)\right )+e^{4 x} \left (324 e^8+81 \log ^2(3)-324 e^4 \log (3)\right )}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {9 \left (e^x-x\right ) \left (18 e^4 x^3 \left (1+\frac {4-9 \log (3)}{18 e^4}\right )-54 e^{x+4} x^2+e^x x^2 \left (2 x^2+6 x-12+27 \log (3)\right )+54 e^{2 x+4} x \left (1-\frac {\log (3)}{2 e^4}\right )-18 e^{3 x+4} \left (1-\frac {\log (3)}{2 e^4}\right )\right )}{\left (18 e^4 x^2-\left (x^2 (x+9 \log (3))\right )-36 e^{x+4} x \left (1-\frac {\log (3)}{2 e^4}\right )+18 e^{2 x+4} \left (1-\frac {\log (3)}{2 e^4}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 9 \int -\frac {\left (e^x-x\right ) \left (-\left (\left (4+18 e^4-9 \log (3)\right ) x^3\right )+54 e^{x+4} x^2-e^x \left (2 x^2+6 x-3 (4-9 \log (3))\right ) x^2-27 e^{2 x+4} \left (2-\frac {\log (3)}{e^4}\right ) x+9 e^{3 x+4} \left (2-\frac {\log (3)}{e^4}\right )\right )}{\left (-\left ((x+9 \log (3)) x^2\right )+18 e^4 x^2-18 e^{x+4} \left (2-\frac {\log (3)}{e^4}\right ) x+9 e^{2 x+4} \left (2-\frac {\log (3)}{e^4}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -9 \int \frac {\left (e^x-x\right ) \left (-\left (\left (4+18 e^4-9 \log (3)\right ) x^3\right )+54 e^{x+4} x^2-e^x \left (2 x^2+6 x-3 (4-9 \log (3))\right ) x^2-27 e^{2 x+4} \left (2-\frac {\log (3)}{e^4}\right ) x+9 e^{3 x+4} \left (2-\frac {\log (3)}{e^4}\right )\right )}{\left (-\left ((x+9 \log (3)) x^2\right )+18 e^4 x^2-18 e^{x+4} \left (2-\frac {\log (3)}{e^4}\right ) x+9 e^{2 x+4} \left (2-\frac {\log (3)}{e^4}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -9 \int \left (\frac {(x+4) \left (-2 x^3+3 \left (1+12 e^4-\log (729)\right ) x^2-36 e^{x+4} \left (1-\frac {\log (3)}{2 e^4}\right ) x-36 e^4 \left (1-\frac {\log (3)}{2 e^4}\right ) x+36 e^{x+4} \left (1-\frac {\log (3)}{2 e^4}\right )\right ) x^3}{9 \left (2 e^4-\log (3)\right ) \left (x^3-18 e^4 \left (1-\frac {\log (3)}{2 e^4}\right ) x^2+36 e^{x+4} \left (1-\frac {\log (3)}{2 e^4}\right ) x-18 e^{2 x+4} \left (1-\frac {\log (3)}{2 e^4}\right )\right )^2}+\frac {2 \left (x^2+2 x-6\right ) x^2}{9 \left (2 e^4-\log (3)\right ) \left (x^3-18 e^4 \left (1-\frac {\log (3)}{2 e^4}\right ) x^2+36 e^{x+4} \left (1-\frac {\log (3)}{2 e^4}\right ) x-18 e^{2 x+4} \left (1-\frac {\log (3)}{2 e^4}\right )\right )}+\frac {1}{9 \left (2 e^4-\log (3)\right )}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle -9 \int \left (\frac {(x+4) \left (-2 x^3+3 \left (1+12 e^4-\log (729)\right ) x^2-36 e^{x+4} \left (1-\frac {\log (3)}{2 e^4}\right ) x-36 e^4 \left (1-\frac {\log (3)}{2 e^4}\right ) x+36 e^{x+4} \left (1-\frac {\log (3)}{2 e^4}\right )\right ) x^3}{9 \left (2 e^4-\log (3)\right ) \left (x^3-18 e^4 \left (1-\frac {\log (3)}{2 e^4}\right ) x^2+36 e^{x+4} \left (1-\frac {\log (3)}{2 e^4}\right ) x-18 e^{2 x+4} \left (1-\frac {\log (3)}{2 e^4}\right )\right )^2}+\frac {2 \left (x^2+2 x-6\right ) x^2}{9 \left (2 e^4-\log (3)\right ) \left (x^3-18 e^4 \left (1-\frac {\log (3)}{2 e^4}\right ) x^2+36 e^{x+4} \left (1-\frac {\log (3)}{2 e^4}\right ) x-18 e^{2 x+4} \left (1-\frac {\log (3)}{2 e^4}\right )\right )}+\frac {1}{9 \left (2 e^4-\log (3)\right )}\right )dx\) |
Int[(-36*x^4 - 162*E^4*x^4 + 81*x^4*Log[3] + E^(4*x)*(-162*E^4 + 81*Log[3] ) + E^(3*x)*(648*E^4*x - 324*x*Log[3]) + E^(2*x)*(-108*x^2 - 972*E^4*x^2 + 54*x^3 + 18*x^4 + 486*x^2*Log[3]) + E^x*(144*x^3 + 648*E^4*x^3 - 54*x^4 - 18*x^5 - 324*x^3*Log[3]))/(324*E^8*x^4 - 36*E^4*x^5 + x^6 + (-324*E^4*x^4 + 18*x^5)*Log[3] + 81*x^4*Log[3]^2 + E^(4*x)*(324*E^8 - 324*E^4*Log[3] + 81*Log[3]^2) + E^(3*x)*(-1296*E^8*x + 1296*E^4*x*Log[3] - 324*x*Log[3]^2) + E^(2*x)*(1944*E^8*x^2 - 36*E^4*x^3 + (-1944*E^4*x^2 + 18*x^3)*Log[3] + 4 86*x^2*Log[3]^2) + E^x*(-1296*E^8*x^3 + 72*E^4*x^4 + (1296*E^4*x^3 - 36*x^ 4)*Log[3] - 324*x^3*Log[3]^2)),x]
3.30.37.3.1 Defintions of rubi rules used
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[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(86\) vs. \(2(26)=52\).
Time = 0.38 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.11
| method | result | size |
| risch | \(-\frac {x}{-\ln \left (3\right )+2 \,{\mathrm e}^{4}}-\frac {x^{3} \left (4+x \right )}{\left (-\ln \left (3\right )+2 \,{\mathrm e}^{4}\right ) \left (18 \,{\mathrm e}^{4+2 x}-36 x \,{\mathrm e}^{4+x}+18 x^{2} {\mathrm e}^{4}-9 \ln \left (3\right ) {\mathrm e}^{2 x}+18 x \ln \left (3\right ) {\mathrm e}^{x}-9 x^{2} \ln \left (3\right )-x^{3}\right )}\) | \(87\) |
| parallelrisch | \(-\frac {162 x^{2} {\mathrm e}^{4}-324 x \,{\mathrm e}^{4} {\mathrm e}^{x}+162 \,{\mathrm e}^{4} {\mathrm e}^{2 x}-81 x^{2} \ln \left (3\right )+162 x \ln \left (3\right ) {\mathrm e}^{x}-81 \ln \left (3\right ) {\mathrm e}^{2 x}-18 \,{\mathrm e}^{x} x^{2}+9 x \,{\mathrm e}^{2 x}+36 x^{2}-72 \,{\mathrm e}^{x} x +36 \,{\mathrm e}^{2 x}}{18 \,{\mathrm e}^{4} {\mathrm e}^{2 x}-36 x \,{\mathrm e}^{4} {\mathrm e}^{x}+18 x^{2} {\mathrm e}^{4}-9 \ln \left (3\right ) {\mathrm e}^{2 x}+18 x \ln \left (3\right ) {\mathrm e}^{x}-9 x^{2} \ln \left (3\right )-x^{3}}\) | \(130\) |
int(((81*ln(3)-162*exp(4))*exp(x)^4+(-324*x*ln(3)+648*x*exp(4))*exp(x)^3+( 486*x^2*ln(3)-972*x^2*exp(4)+18*x^4+54*x^3-108*x^2)*exp(x)^2+(-324*x^3*ln( 3)+648*x^3*exp(4)-18*x^5-54*x^4+144*x^3)*exp(x)+81*x^4*ln(3)-162*x^4*exp(4 )-36*x^4)/((81*ln(3)^2-324*exp(4)*ln(3)+324*exp(4)^2)*exp(x)^4+(-324*x*ln( 3)^2+1296*exp(4)*x*ln(3)-1296*x*exp(4)^2)*exp(x)^3+(486*x^2*ln(3)^2+(-1944 *x^2*exp(4)+18*x^3)*ln(3)+1944*x^2*exp(4)^2-36*x^3*exp(4))*exp(x)^2+(-324* x^3*ln(3)^2+(1296*x^3*exp(4)-36*x^4)*ln(3)-1296*x^3*exp(4)^2+72*x^4*exp(4) )*exp(x)+81*x^4*ln(3)^2+(-324*x^4*exp(4)+18*x^5)*ln(3)+324*x^4*exp(4)^2-36 *x^5*exp(4)+x^6),x,method=_RETURNVERBOSE)
-x/(-ln(3)+2*exp(4))-x^3*(4+x)/(-ln(3)+2*exp(4))/(18*exp(4+2*x)-36*x*exp(4 +x)+18*x^2*exp(4)-9*ln(3)*exp(2*x)+18*x*ln(3)*exp(x)-9*x^2*ln(3)-x^3)
Leaf count of result is larger than twice the leaf count of optimal. 145 vs. \(2 (28) = 56\).
Time = 0.27 (sec) , antiderivative size = 145, normalized size of antiderivative = 5.18 \[ \int \frac {-36 x^4-162 e^4 x^4+81 x^4 \log (3)+e^{4 x} \left (-162 e^4+81 \log (3)\right )+e^{3 x} \left (648 e^4 x-324 x \log (3)\right )+e^{2 x} \left (-108 x^2-972 e^4 x^2+54 x^3+18 x^4+486 x^2 \log (3)\right )+e^x \left (144 x^3+648 e^4 x^3-54 x^4-18 x^5-324 x^3 \log (3)\right )}{324 e^8 x^4-36 e^4 x^5+x^6+\left (-324 e^4 x^4+18 x^5\right ) \log (3)+81 x^4 \log ^2(3)+e^{4 x} \left (324 e^8-324 e^4 \log (3)+81 \log ^2(3)\right )+e^{3 x} \left (-1296 e^8 x+1296 e^4 x \log (3)-324 x \log ^2(3)\right )+e^{2 x} \left (1944 e^8 x^2-36 e^4 x^3+\left (-1944 e^4 x^2+18 x^3\right ) \log (3)+486 x^2 \log ^2(3)\right )+e^x \left (-1296 e^8 x^3+72 e^4 x^4+\left (1296 e^4 x^3-36 x^4\right ) \log (3)-324 x^3 \log ^2(3)\right )} \, dx=\frac {18 \, x^{3} e^{4} - 9 \, x^{3} \log \left (3\right ) + 4 \, x^{3} + 9 \, {\left (2 \, x e^{4} - x \log \left (3\right )\right )} e^{\left (2 \, x\right )} - 18 \, {\left (2 \, x^{2} e^{4} - x^{2} \log \left (3\right )\right )} e^{x}}{2 \, x^{3} e^{4} - 9 \, x^{2} \log \left (3\right )^{2} - 36 \, x^{2} e^{8} + 9 \, {\left (4 \, e^{4} \log \left (3\right ) - \log \left (3\right )^{2} - 4 \, e^{8}\right )} e^{\left (2 \, x\right )} - 18 \, {\left (4 \, x e^{4} \log \left (3\right ) - x \log \left (3\right )^{2} - 4 \, x e^{8}\right )} e^{x} - {\left (x^{3} - 36 \, x^{2} e^{4}\right )} \log \left (3\right )} \]
integrate(((81*log(3)-162*exp(4))*exp(x)^4+(-324*x*log(3)+648*x*exp(4))*ex p(x)^3+(486*x^2*log(3)-972*x^2*exp(4)+18*x^4+54*x^3-108*x^2)*exp(x)^2+(-32 4*x^3*log(3)+648*x^3*exp(4)-18*x^5-54*x^4+144*x^3)*exp(x)+81*x^4*log(3)-16 2*x^4*exp(4)-36*x^4)/((81*log(3)^2-324*exp(4)*log(3)+324*exp(4)^2)*exp(x)^ 4+(-324*x*log(3)^2+1296*exp(4)*x*log(3)-1296*x*exp(4)^2)*exp(x)^3+(486*x^2 *log(3)^2+(-1944*x^2*exp(4)+18*x^3)*log(3)+1944*x^2*exp(4)^2-36*x^3*exp(4) )*exp(x)^2+(-324*x^3*log(3)^2+(1296*x^3*exp(4)-36*x^4)*log(3)-1296*x^3*exp (4)^2+72*x^4*exp(4))*exp(x)+81*x^4*log(3)^2+(-324*x^4*exp(4)+18*x^5)*log(3 )+324*x^4*exp(4)^2-36*x^5*exp(4)+x^6),x, algorithm=\
(18*x^3*e^4 - 9*x^3*log(3) + 4*x^3 + 9*(2*x*e^4 - x*log(3))*e^(2*x) - 18*( 2*x^2*e^4 - x^2*log(3))*e^x)/(2*x^3*e^4 - 9*x^2*log(3)^2 - 36*x^2*e^8 + 9* (4*e^4*log(3) - log(3)^2 - 4*e^8)*e^(2*x) - 18*(4*x*e^4*log(3) - x*log(3)^ 2 - 4*x*e^8)*e^x - (x^3 - 36*x^2*e^4)*log(3))
Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (22) = 44\).
Time = 0.31 (sec) , antiderivative size = 119, normalized size of antiderivative = 4.25 \[ \int \frac {-36 x^4-162 e^4 x^4+81 x^4 \log (3)+e^{4 x} \left (-162 e^4+81 \log (3)\right )+e^{3 x} \left (648 e^4 x-324 x \log (3)\right )+e^{2 x} \left (-108 x^2-972 e^4 x^2+54 x^3+18 x^4+486 x^2 \log (3)\right )+e^x \left (144 x^3+648 e^4 x^3-54 x^4-18 x^5-324 x^3 \log (3)\right )}{324 e^8 x^4-36 e^4 x^5+x^6+\left (-324 e^4 x^4+18 x^5\right ) \log (3)+81 x^4 \log ^2(3)+e^{4 x} \left (324 e^8-324 e^4 \log (3)+81 \log ^2(3)\right )+e^{3 x} \left (-1296 e^8 x+1296 e^4 x \log (3)-324 x \log ^2(3)\right )+e^{2 x} \left (1944 e^8 x^2-36 e^4 x^3+\left (-1944 e^4 x^2+18 x^3\right ) \log (3)+486 x^2 \log ^2(3)\right )+e^x \left (-1296 e^8 x^3+72 e^4 x^4+\left (1296 e^4 x^3-36 x^4\right ) \log (3)-324 x^3 \log ^2(3)\right )} \, dx=- \frac {x}{- \log {\left (3 \right )} + 2 e^{4}} + \frac {- x^{4} - 4 x^{3}}{- 2 x^{3} e^{4} + x^{3} \log {\left (3 \right )} - 36 x^{2} e^{4} \log {\left (3 \right )} + 9 x^{2} \log {\left (3 \right )}^{2} + 36 x^{2} e^{8} + \left (- 72 x e^{8} - 18 x \log {\left (3 \right )}^{2} + 72 x e^{4} \log {\left (3 \right )}\right ) e^{x} + \left (- 36 e^{4} \log {\left (3 \right )} + 9 \log {\left (3 \right )}^{2} + 36 e^{8}\right ) e^{2 x}} \]
integrate(((81*ln(3)-162*exp(4))*exp(x)**4+(-324*x*ln(3)+648*x*exp(4))*exp (x)**3+(486*x**2*ln(3)-972*x**2*exp(4)+18*x**4+54*x**3-108*x**2)*exp(x)**2 +(-324*x**3*ln(3)+648*x**3*exp(4)-18*x**5-54*x**4+144*x**3)*exp(x)+81*x**4 *ln(3)-162*x**4*exp(4)-36*x**4)/((81*ln(3)**2-324*exp(4)*ln(3)+324*exp(4)* *2)*exp(x)**4+(-324*x*ln(3)**2+1296*exp(4)*x*ln(3)-1296*x*exp(4)**2)*exp(x )**3+(486*x**2*ln(3)**2+(-1944*x**2*exp(4)+18*x**3)*ln(3)+1944*x**2*exp(4) **2-36*x**3*exp(4))*exp(x)**2+(-324*x**3*ln(3)**2+(1296*x**3*exp(4)-36*x** 4)*ln(3)-1296*x**3*exp(4)**2+72*x**4*exp(4))*exp(x)+81*x**4*ln(3)**2+(-324 *x**4*exp(4)+18*x**5)*ln(3)+324*x**4*exp(4)**2-36*x**5*exp(4)+x**6),x)
-x/(-log(3) + 2*exp(4)) + (-x**4 - 4*x**3)/(-2*x**3*exp(4) + x**3*log(3) - 36*x**2*exp(4)*log(3) + 9*x**2*log(3)**2 + 36*x**2*exp(8) + (-72*x*exp(8) - 18*x*log(3)**2 + 72*x*exp(4)*log(3))*exp(x) + (-36*exp(4)*log(3) + 9*lo g(3)**2 + 36*exp(8))*exp(2*x))
Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (28) = 56\).
Time = 0.50 (sec) , antiderivative size = 131, normalized size of antiderivative = 4.68 \[ \int \frac {-36 x^4-162 e^4 x^4+81 x^4 \log (3)+e^{4 x} \left (-162 e^4+81 \log (3)\right )+e^{3 x} \left (648 e^4 x-324 x \log (3)\right )+e^{2 x} \left (-108 x^2-972 e^4 x^2+54 x^3+18 x^4+486 x^2 \log (3)\right )+e^x \left (144 x^3+648 e^4 x^3-54 x^4-18 x^5-324 x^3 \log (3)\right )}{324 e^8 x^4-36 e^4 x^5+x^6+\left (-324 e^4 x^4+18 x^5\right ) \log (3)+81 x^4 \log ^2(3)+e^{4 x} \left (324 e^8-324 e^4 \log (3)+81 \log ^2(3)\right )+e^{3 x} \left (-1296 e^8 x+1296 e^4 x \log (3)-324 x \log ^2(3)\right )+e^{2 x} \left (1944 e^8 x^2-36 e^4 x^3+\left (-1944 e^4 x^2+18 x^3\right ) \log (3)+486 x^2 \log ^2(3)\right )+e^x \left (-1296 e^8 x^3+72 e^4 x^4+\left (1296 e^4 x^3-36 x^4\right ) \log (3)-324 x^3 \log ^2(3)\right )} \, dx=\frac {x^{3} {\left (18 \, e^{4} - 9 \, \log \left (3\right ) + 4\right )} - 18 \, x^{2} {\left (2 \, e^{4} - \log \left (3\right )\right )} e^{x} + 9 \, x {\left (2 \, e^{4} - \log \left (3\right )\right )} e^{\left (2 \, x\right )}}{x^{3} {\left (2 \, e^{4} - \log \left (3\right )\right )} + 9 \, {\left (4 \, e^{4} \log \left (3\right ) - \log \left (3\right )^{2} - 4 \, e^{8}\right )} x^{2} - 18 \, {\left (4 \, e^{4} \log \left (3\right ) - \log \left (3\right )^{2} - 4 \, e^{8}\right )} x e^{x} + 9 \, {\left (4 \, e^{4} \log \left (3\right ) - \log \left (3\right )^{2} - 4 \, e^{8}\right )} e^{\left (2 \, x\right )}} \]
integrate(((81*log(3)-162*exp(4))*exp(x)^4+(-324*x*log(3)+648*x*exp(4))*ex p(x)^3+(486*x^2*log(3)-972*x^2*exp(4)+18*x^4+54*x^3-108*x^2)*exp(x)^2+(-32 4*x^3*log(3)+648*x^3*exp(4)-18*x^5-54*x^4+144*x^3)*exp(x)+81*x^4*log(3)-16 2*x^4*exp(4)-36*x^4)/((81*log(3)^2-324*exp(4)*log(3)+324*exp(4)^2)*exp(x)^ 4+(-324*x*log(3)^2+1296*exp(4)*x*log(3)-1296*x*exp(4)^2)*exp(x)^3+(486*x^2 *log(3)^2+(-1944*x^2*exp(4)+18*x^3)*log(3)+1944*x^2*exp(4)^2-36*x^3*exp(4) )*exp(x)^2+(-324*x^3*log(3)^2+(1296*x^3*exp(4)-36*x^4)*log(3)-1296*x^3*exp (4)^2+72*x^4*exp(4))*exp(x)+81*x^4*log(3)^2+(-324*x^4*exp(4)+18*x^5)*log(3 )+324*x^4*exp(4)^2-36*x^5*exp(4)+x^6),x, algorithm=\
(x^3*(18*e^4 - 9*log(3) + 4) - 18*x^2*(2*e^4 - log(3))*e^x + 9*x*(2*e^4 - log(3))*e^(2*x))/(x^3*(2*e^4 - log(3)) + 9*(4*e^4*log(3) - log(3)^2 - 4*e^ 8)*x^2 - 18*(4*e^4*log(3) - log(3)^2 - 4*e^8)*x*e^x + 9*(4*e^4*log(3) - lo g(3)^2 - 4*e^8)*e^(2*x))
Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (28) = 56\).
Time = 3.54 (sec) , antiderivative size = 155, normalized size of antiderivative = 5.54 \[ \int \frac {-36 x^4-162 e^4 x^4+81 x^4 \log (3)+e^{4 x} \left (-162 e^4+81 \log (3)\right )+e^{3 x} \left (648 e^4 x-324 x \log (3)\right )+e^{2 x} \left (-108 x^2-972 e^4 x^2+54 x^3+18 x^4+486 x^2 \log (3)\right )+e^x \left (144 x^3+648 e^4 x^3-54 x^4-18 x^5-324 x^3 \log (3)\right )}{324 e^8 x^4-36 e^4 x^5+x^6+\left (-324 e^4 x^4+18 x^5\right ) \log (3)+81 x^4 \log ^2(3)+e^{4 x} \left (324 e^8-324 e^4 \log (3)+81 \log ^2(3)\right )+e^{3 x} \left (-1296 e^8 x+1296 e^4 x \log (3)-324 x \log ^2(3)\right )+e^{2 x} \left (1944 e^8 x^2-36 e^4 x^3+\left (-1944 e^4 x^2+18 x^3\right ) \log (3)+486 x^2 \log ^2(3)\right )+e^x \left (-1296 e^8 x^3+72 e^4 x^4+\left (1296 e^4 x^3-36 x^4\right ) \log (3)-324 x^3 \log ^2(3)\right )} \, dx=\frac {x^{4} + 18 \, x^{3} e^{4} - 9 \, x^{3} \log \left (3\right ) + 18 \, x^{2} e^{x} \log \left (3\right ) + 8 \, x^{3} - 36 \, x^{2} e^{\left (x + 4\right )} - 9 \, x e^{\left (2 \, x\right )} \log \left (3\right ) + 18 \, x e^{\left (2 \, x + 4\right )}}{2 \, x^{3} e^{4} - x^{3} \log \left (3\right ) + 36 \, x^{2} e^{4} \log \left (3\right ) - 9 \, x^{2} \log \left (3\right )^{2} + 18 \, x e^{x} \log \left (3\right )^{2} - 36 \, x^{2} e^{8} - 72 \, x e^{\left (x + 4\right )} \log \left (3\right ) - 9 \, e^{\left (2 \, x\right )} \log \left (3\right )^{2} + 72 \, x e^{\left (x + 8\right )} + 36 \, e^{\left (2 \, x + 4\right )} \log \left (3\right ) - 36 \, e^{\left (2 \, x + 8\right )}} \]
integrate(((81*log(3)-162*exp(4))*exp(x)^4+(-324*x*log(3)+648*x*exp(4))*ex p(x)^3+(486*x^2*log(3)-972*x^2*exp(4)+18*x^4+54*x^3-108*x^2)*exp(x)^2+(-32 4*x^3*log(3)+648*x^3*exp(4)-18*x^5-54*x^4+144*x^3)*exp(x)+81*x^4*log(3)-16 2*x^4*exp(4)-36*x^4)/((81*log(3)^2-324*exp(4)*log(3)+324*exp(4)^2)*exp(x)^ 4+(-324*x*log(3)^2+1296*exp(4)*x*log(3)-1296*x*exp(4)^2)*exp(x)^3+(486*x^2 *log(3)^2+(-1944*x^2*exp(4)+18*x^3)*log(3)+1944*x^2*exp(4)^2-36*x^3*exp(4) )*exp(x)^2+(-324*x^3*log(3)^2+(1296*x^3*exp(4)-36*x^4)*log(3)-1296*x^3*exp (4)^2+72*x^4*exp(4))*exp(x)+81*x^4*log(3)^2+(-324*x^4*exp(4)+18*x^5)*log(3 )+324*x^4*exp(4)^2-36*x^5*exp(4)+x^6),x, algorithm=\
(x^4 + 18*x^3*e^4 - 9*x^3*log(3) + 18*x^2*e^x*log(3) + 8*x^3 - 36*x^2*e^(x + 4) - 9*x*e^(2*x)*log(3) + 18*x*e^(2*x + 4))/(2*x^3*e^4 - x^3*log(3) + 3 6*x^2*e^4*log(3) - 9*x^2*log(3)^2 + 18*x*e^x*log(3)^2 - 36*x^2*e^8 - 72*x* e^(x + 4)*log(3) - 9*e^(2*x)*log(3)^2 + 72*x*e^(x + 8) + 36*e^(2*x + 4)*lo g(3) - 36*e^(2*x + 8))
Timed out. \[ \int \frac {-36 x^4-162 e^4 x^4+81 x^4 \log (3)+e^{4 x} \left (-162 e^4+81 \log (3)\right )+e^{3 x} \left (648 e^4 x-324 x \log (3)\right )+e^{2 x} \left (-108 x^2-972 e^4 x^2+54 x^3+18 x^4+486 x^2 \log (3)\right )+e^x \left (144 x^3+648 e^4 x^3-54 x^4-18 x^5-324 x^3 \log (3)\right )}{324 e^8 x^4-36 e^4 x^5+x^6+\left (-324 e^4 x^4+18 x^5\right ) \log (3)+81 x^4 \log ^2(3)+e^{4 x} \left (324 e^8-324 e^4 \log (3)+81 \log ^2(3)\right )+e^{3 x} \left (-1296 e^8 x+1296 e^4 x \log (3)-324 x \log ^2(3)\right )+e^{2 x} \left (1944 e^8 x^2-36 e^4 x^3+\left (-1944 e^4 x^2+18 x^3\right ) \log (3)+486 x^2 \log ^2(3)\right )+e^x \left (-1296 e^8 x^3+72 e^4 x^4+\left (1296 e^4 x^3-36 x^4\right ) \log (3)-324 x^3 \log ^2(3)\right )} \, dx=\int -\frac {{\mathrm {e}}^x\,\left (324\,x^3\,\ln \left (3\right )-648\,x^3\,{\mathrm {e}}^4-144\,x^3+54\,x^4+18\,x^5\right )-{\mathrm {e}}^{3\,x}\,\left (648\,x\,{\mathrm {e}}^4-324\,x\,\ln \left (3\right )\right )+{\mathrm {e}}^{4\,x}\,\left (162\,{\mathrm {e}}^4-81\,\ln \left (3\right )\right )+162\,x^4\,{\mathrm {e}}^4-81\,x^4\,\ln \left (3\right )+36\,x^4-{\mathrm {e}}^{2\,x}\,\left (486\,x^2\,\ln \left (3\right )-972\,x^2\,{\mathrm {e}}^4-108\,x^2+54\,x^3+18\,x^4\right )}{81\,x^4\,{\ln \left (3\right )}^2+{\mathrm {e}}^{4\,x}\,\left (324\,{\mathrm {e}}^8-324\,{\mathrm {e}}^4\,\ln \left (3\right )+81\,{\ln \left (3\right )}^2\right )+{\mathrm {e}}^{2\,x}\,\left (486\,x^2\,{\ln \left (3\right )}^2-36\,x^3\,{\mathrm {e}}^4+1944\,x^2\,{\mathrm {e}}^8-\ln \left (3\right )\,\left (1944\,x^2\,{\mathrm {e}}^4-18\,x^3\right )\right )-{\mathrm {e}}^{3\,x}\,\left (1296\,x\,{\mathrm {e}}^8+324\,x\,{\ln \left (3\right )}^2-1296\,x\,{\mathrm {e}}^4\,\ln \left (3\right )\right )-36\,x^5\,{\mathrm {e}}^4+324\,x^4\,{\mathrm {e}}^8-{\mathrm {e}}^x\,\left (324\,x^3\,{\ln \left (3\right )}^2-72\,x^4\,{\mathrm {e}}^4+1296\,x^3\,{\mathrm {e}}^8-\ln \left (3\right )\,\left (1296\,x^3\,{\mathrm {e}}^4-36\,x^4\right )\right )+x^6-\ln \left (3\right )\,\left (324\,x^4\,{\mathrm {e}}^4-18\,x^5\right )} \,d x \]
int(-(exp(x)*(324*x^3*log(3) - 648*x^3*exp(4) - 144*x^3 + 54*x^4 + 18*x^5) - exp(3*x)*(648*x*exp(4) - 324*x*log(3)) + exp(4*x)*(162*exp(4) - 81*log( 3)) + 162*x^4*exp(4) - 81*x^4*log(3) + 36*x^4 - exp(2*x)*(486*x^2*log(3) - 972*x^2*exp(4) - 108*x^2 + 54*x^3 + 18*x^4))/(81*x^4*log(3)^2 + exp(4*x)* (324*exp(8) - 324*exp(4)*log(3) + 81*log(3)^2) + exp(2*x)*(486*x^2*log(3)^ 2 - 36*x^3*exp(4) + 1944*x^2*exp(8) - log(3)*(1944*x^2*exp(4) - 18*x^3)) - exp(3*x)*(1296*x*exp(8) + 324*x*log(3)^2 - 1296*x*exp(4)*log(3)) - 36*x^5 *exp(4) + 324*x^4*exp(8) - exp(x)*(324*x^3*log(3)^2 - 72*x^4*exp(4) + 1296 *x^3*exp(8) - log(3)*(1296*x^3*exp(4) - 36*x^4)) + x^6 - log(3)*(324*x^4*e xp(4) - 18*x^5)),x)
int(-(exp(x)*(324*x^3*log(3) - 648*x^3*exp(4) - 144*x^3 + 54*x^4 + 18*x^5) - exp(3*x)*(648*x*exp(4) - 324*x*log(3)) + exp(4*x)*(162*exp(4) - 81*log( 3)) + 162*x^4*exp(4) - 81*x^4*log(3) + 36*x^4 - exp(2*x)*(486*x^2*log(3) - 972*x^2*exp(4) - 108*x^2 + 54*x^3 + 18*x^4))/(81*x^4*log(3)^2 + exp(4*x)* (324*exp(8) - 324*exp(4)*log(3) + 81*log(3)^2) + exp(2*x)*(486*x^2*log(3)^ 2 - 36*x^3*exp(4) + 1944*x^2*exp(8) - log(3)*(1944*x^2*exp(4) - 18*x^3)) - exp(3*x)*(1296*x*exp(8) + 324*x*log(3)^2 - 1296*x*exp(4)*log(3)) - 36*x^5 *exp(4) + 324*x^4*exp(8) - exp(x)*(324*x^3*log(3)^2 - 72*x^4*exp(4) + 1296 *x^3*exp(8) - log(3)*(1296*x^3*exp(4) - 36*x^4)) + x^6 - log(3)*(324*x^4*e xp(4) - 18*x^5)), x)