Integrand size = 164, antiderivative size = 32 \[ \int \frac {1}{5} e^{e^{\frac {1}{25} \left (3600 x^2+1200 x^3+100 x^4+e^6 \left (144+48 x+4 x^2\right )+e^3 \left (1440 x+480 x^2+40 x^3\right )\right )}} \left (-25+e^{\frac {1}{25} \left (3600 x^2+1200 x^3+100 x^4+e^6 \left (144+48 x+4 x^2\right )+e^3 \left (1440 x+480 x^2+40 x^3\right )\right )} \left (14400 x-2800 x^3-400 x^4+e^6 \left (96-32 x-8 x^2\right )+e^3 \left (2880+480 x-720 x^2-120 x^3\right )\right )\right ) \, dx=2+5 e^{e^{(6+x)^2 \left (\frac {2 e^3}{5}+2 x\right )^2}} (2-x) \]
Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {1}{5} e^{e^{\frac {1}{25} \left (3600 x^2+1200 x^3+100 x^4+e^6 \left (144+48 x+4 x^2\right )+e^3 \left (1440 x+480 x^2+40 x^3\right )\right )}} \left (-25+e^{\frac {1}{25} \left (3600 x^2+1200 x^3+100 x^4+e^6 \left (144+48 x+4 x^2\right )+e^3 \left (1440 x+480 x^2+40 x^3\right )\right )} \left (14400 x-2800 x^3-400 x^4+e^6 \left (96-32 x-8 x^2\right )+e^3 \left (2880+480 x-720 x^2-120 x^3\right )\right )\right ) \, dx=-5 e^{e^{\frac {4}{25} (6+x)^2 \left (e^3+5 x\right )^2}} (-2+x) \]
Integrate[(E^E^((3600*x^2 + 1200*x^3 + 100*x^4 + E^6*(144 + 48*x + 4*x^2) + E^3*(1440*x + 480*x^2 + 40*x^3))/25)*(-25 + E^((3600*x^2 + 1200*x^3 + 10 0*x^4 + E^6*(144 + 48*x + 4*x^2) + E^3*(1440*x + 480*x^2 + 40*x^3))/25)*(1 4400*x - 2800*x^3 - 400*x^4 + E^6*(96 - 32*x - 8*x^2) + E^3*(2880 + 480*x - 720*x^2 - 120*x^3))))/5,x]
Leaf count is larger than twice the leaf count of optimal. \(139\) vs. \(2(32)=64\).
Time = 0.90 (sec) , antiderivative size = 139, normalized size of antiderivative = 4.34, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {27, 25, 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.
| \(\displaystyle \int \frac {1}{5} \exp \left (\exp \left (\frac {1}{25} \left (100 x^4+1200 x^3+3600 x^2+e^6 \left (4 x^2+48 x+144\right )+e^3 \left (40 x^3+480 x^2+1440 x\right )\right )\right )\right ) \left (\left (-400 x^4-2800 x^3+e^6 \left (-8 x^2-32 x+96\right )+e^3 \left (-120 x^3-720 x^2+480 x+2880\right )+14400 x\right ) \exp \left (\frac {1}{25} \left (100 x^4+1200 x^3+3600 x^2+e^6 \left (4 x^2+48 x+144\right )+e^3 \left (40 x^3+480 x^2+1440 x\right )\right )\right )-25\right ) \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int -\exp \left (\exp \left (\frac {4}{25} \left (25 x^4+300 x^3+900 x^2+e^6 \left (x^2+12 x+36\right )+10 e^3 \left (x^3+12 x^2+36 x\right )\right )\right )\right ) \left (25-8 \exp \left (\frac {4}{25} \left (25 x^4+300 x^3+900 x^2+e^6 \left (x^2+12 x+36\right )+10 e^3 \left (x^3+12 x^2+36 x\right )\right )\right ) \left (-50 x^4-350 x^3+1800 x+e^6 \left (-x^2-4 x+12\right )+15 e^3 \left (-x^3-6 x^2+4 x+24\right )\right )\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{5} \int \exp \left (\exp \left (\frac {4}{25} \left (25 x^4+300 x^3+900 x^2+e^6 \left (x^2+12 x+36\right )+10 e^3 \left (x^3+12 x^2+36 x\right )\right )\right )\right ) \left (25-8 \exp \left (\frac {4}{25} \left (25 x^4+300 x^3+900 x^2+e^6 \left (x^2+12 x+36\right )+10 e^3 \left (x^3+12 x^2+36 x\right )\right )\right ) \left (-50 x^4-350 x^3+1800 x+e^6 \left (-x^2-4 x+12\right )+15 e^3 \left (-x^3-6 x^2+4 x+24\right )\right )\right )dx\) |
| \(\Big \downarrow \) 2726 |
| \(\displaystyle \frac {5 \left (-50 x^4-350 x^3+e^6 \left (-x^2-4 x+12\right )+15 e^3 \left (-x^3-6 x^2+4 x+24\right )+1800 x\right ) \exp \left (\exp \left (\frac {4}{25} \left (25 x^4+300 x^3+900 x^2+e^6 \left (x^2+12 x+36\right )+10 e^3 \left (x^3+12 x^2+36 x\right )\right )\right )\right )}{50 x^3+450 x^2+15 e^3 \left (x^2+8 x+12\right )+900 x+e^6 (x+6)}\) |
Int[(E^E^((3600*x^2 + 1200*x^3 + 100*x^4 + E^6*(144 + 48*x + 4*x^2) + E^3* (1440*x + 480*x^2 + 40*x^3))/25)*(-25 + E^((3600*x^2 + 1200*x^3 + 100*x^4 + E^6*(144 + 48*x + 4*x^2) + E^3*(1440*x + 480*x^2 + 40*x^3))/25)*(14400*x - 2800*x^3 - 400*x^4 + E^6*(96 - 32*x - 8*x^2) + E^3*(2880 + 480*x - 720* x^2 - 120*x^3))))/5,x]
(5*E^E^((4*(900*x^2 + 300*x^3 + 25*x^4 + E^6*(36 + 12*x + x^2) + 10*E^3*(3 6*x + 12*x^2 + x^3)))/25)*(1800*x - 350*x^3 - 50*x^4 + E^6*(12 - 4*x - x^2 ) + 15*E^3*(24 + 4*x - 6*x^2 - x^3)))/(900*x + 450*x^2 + 50*x^3 + E^6*(6 + x) + 15*E^3*(12 + 8*x + x^2))
3.2.59.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Time = 1.40 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94
| method | result | size |
| risch | \(\frac {\left (50-25 x \right ) {\mathrm e}^{{\mathrm e}^{\frac {4 \left (6+x \right )^{2} \left (10 x \,{\mathrm e}^{3}+25 x^{2}+{\mathrm e}^{6}\right )}{25}}}}{5}\) | \(30\) |
| norman | \(-5 x \,{\mathrm e}^{{\mathrm e}^{\frac {\left (4 x^{2}+48 x +144\right ) {\mathrm e}^{6}}{25}+\frac {\left (40 x^{3}+480 x^{2}+1440 x \right ) {\mathrm e}^{3}}{25}+4 x^{4}+48 x^{3}+144 x^{2}}}+10 \,{\mathrm e}^{{\mathrm e}^{\frac {\left (4 x^{2}+48 x +144\right ) {\mathrm e}^{6}}{25}+\frac {\left (40 x^{3}+480 x^{2}+1440 x \right ) {\mathrm e}^{3}}{25}+4 x^{4}+48 x^{3}+144 x^{2}}}\) | \(111\) |
| parallelrisch | \(-5 x \,{\mathrm e}^{{\mathrm e}^{\frac {\left (4 x^{2}+48 x +144\right ) {\mathrm e}^{6}}{25}+\frac {\left (40 x^{3}+480 x^{2}+1440 x \right ) {\mathrm e}^{3}}{25}+4 x^{4}+48 x^{3}+144 x^{2}}}+10 \,{\mathrm e}^{{\mathrm e}^{\frac {\left (4 x^{2}+48 x +144\right ) {\mathrm e}^{6}}{25}+\frac {\left (40 x^{3}+480 x^{2}+1440 x \right ) {\mathrm e}^{3}}{25}+4 x^{4}+48 x^{3}+144 x^{2}}}\) | \(111\) |
int(1/5*(((-8*x^2-32*x+96)*exp(3)^2+(-120*x^3-720*x^2+480*x+2880)*exp(3)-4 00*x^4-2800*x^3+14400*x)*exp(1/25*(4*x^2+48*x+144)*exp(3)^2+1/25*(40*x^3+4 80*x^2+1440*x)*exp(3)+4*x^4+48*x^3+144*x^2)-25)*exp(exp(1/25*(4*x^2+48*x+1 44)*exp(3)^2+1/25*(40*x^3+480*x^2+1440*x)*exp(3)+4*x^4+48*x^3+144*x^2)),x, method=_RETURNVERBOSE)
Timed out. \[ \int \frac {1}{5} e^{e^{\frac {1}{25} \left (3600 x^2+1200 x^3+100 x^4+e^6 \left (144+48 x+4 x^2\right )+e^3 \left (1440 x+480 x^2+40 x^3\right )\right )}} \left (-25+e^{\frac {1}{25} \left (3600 x^2+1200 x^3+100 x^4+e^6 \left (144+48 x+4 x^2\right )+e^3 \left (1440 x+480 x^2+40 x^3\right )\right )} \left (14400 x-2800 x^3-400 x^4+e^6 \left (96-32 x-8 x^2\right )+e^3 \left (2880+480 x-720 x^2-120 x^3\right )\right )\right ) \, dx=\text {Timed out} \]
integrate(1/5*(((-8*x^2-32*x+96)*exp(3)^2+(-120*x^3-720*x^2+480*x+2880)*ex p(3)-400*x^4-2800*x^3+14400*x)*exp(1/25*(4*x^2+48*x+144)*exp(3)^2+1/25*(40 *x^3+480*x^2+1440*x)*exp(3)+4*x^4+48*x^3+144*x^2)-25)*exp(exp(1/25*(4*x^2+ 48*x+144)*exp(3)^2+1/25*(40*x^3+480*x^2+1440*x)*exp(3)+4*x^4+48*x^3+144*x^ 2)),x, algorithm=\
Timed out. \[ \int \frac {1}{5} e^{e^{\frac {1}{25} \left (3600 x^2+1200 x^3+100 x^4+e^6 \left (144+48 x+4 x^2\right )+e^3 \left (1440 x+480 x^2+40 x^3\right )\right )}} \left (-25+e^{\frac {1}{25} \left (3600 x^2+1200 x^3+100 x^4+e^6 \left (144+48 x+4 x^2\right )+e^3 \left (1440 x+480 x^2+40 x^3\right )\right )} \left (14400 x-2800 x^3-400 x^4+e^6 \left (96-32 x-8 x^2\right )+e^3 \left (2880+480 x-720 x^2-120 x^3\right )\right )\right ) \, dx=\text {Timed out} \]
integrate(1/5*(((-8*x**2-32*x+96)*exp(3)**2+(-120*x**3-720*x**2+480*x+2880 )*exp(3)-400*x**4-2800*x**3+14400*x)*exp(1/25*(4*x**2+48*x+144)*exp(3)**2+ 1/25*(40*x**3+480*x**2+1440*x)*exp(3)+4*x**4+48*x**3+144*x**2)-25)*exp(exp (1/25*(4*x**2+48*x+144)*exp(3)**2+1/25*(40*x**3+480*x**2+1440*x)*exp(3)+4* x**4+48*x**3+144*x**2)),x)
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (24) = 48\).
Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.81 \[ \int \frac {1}{5} e^{e^{\frac {1}{25} \left (3600 x^2+1200 x^3+100 x^4+e^6 \left (144+48 x+4 x^2\right )+e^3 \left (1440 x+480 x^2+40 x^3\right )\right )}} \left (-25+e^{\frac {1}{25} \left (3600 x^2+1200 x^3+100 x^4+e^6 \left (144+48 x+4 x^2\right )+e^3 \left (1440 x+480 x^2+40 x^3\right )\right )} \left (14400 x-2800 x^3-400 x^4+e^6 \left (96-32 x-8 x^2\right )+e^3 \left (2880+480 x-720 x^2-120 x^3\right )\right )\right ) \, dx=-5 \, {\left (x - 2\right )} e^{\left (e^{\left (4 \, x^{4} + \frac {8}{5} \, x^{3} e^{3} + 48 \, x^{3} + \frac {4}{25} \, x^{2} e^{6} + \frac {96}{5} \, x^{2} e^{3} + 144 \, x^{2} + \frac {48}{25} \, x e^{6} + \frac {288}{5} \, x e^{3} + \frac {144}{25} \, e^{6}\right )}\right )} \]
integrate(1/5*(((-8*x^2-32*x+96)*exp(3)^2+(-120*x^3-720*x^2+480*x+2880)*ex p(3)-400*x^4-2800*x^3+14400*x)*exp(1/25*(4*x^2+48*x+144)*exp(3)^2+1/25*(40 *x^3+480*x^2+1440*x)*exp(3)+4*x^4+48*x^3+144*x^2)-25)*exp(exp(1/25*(4*x^2+ 48*x+144)*exp(3)^2+1/25*(40*x^3+480*x^2+1440*x)*exp(3)+4*x^4+48*x^3+144*x^ 2)),x, algorithm=\
-5*(x - 2)*e^(e^(4*x^4 + 8/5*x^3*e^3 + 48*x^3 + 4/25*x^2*e^6 + 96/5*x^2*e^ 3 + 144*x^2 + 48/25*x*e^6 + 288/5*x*e^3 + 144/25*e^6))
\[ \int \frac {1}{5} e^{e^{\frac {1}{25} \left (3600 x^2+1200 x^3+100 x^4+e^6 \left (144+48 x+4 x^2\right )+e^3 \left (1440 x+480 x^2+40 x^3\right )\right )}} \left (-25+e^{\frac {1}{25} \left (3600 x^2+1200 x^3+100 x^4+e^6 \left (144+48 x+4 x^2\right )+e^3 \left (1440 x+480 x^2+40 x^3\right )\right )} \left (14400 x-2800 x^3-400 x^4+e^6 \left (96-32 x-8 x^2\right )+e^3 \left (2880+480 x-720 x^2-120 x^3\right )\right )\right ) \, dx=\int { -\frac {1}{5} \, {\left (8 \, {\left (50 \, x^{4} + 350 \, x^{3} + {\left (x^{2} + 4 \, x - 12\right )} e^{6} + 15 \, {\left (x^{3} + 6 \, x^{2} - 4 \, x - 24\right )} e^{3} - 1800 \, x\right )} e^{\left (4 \, x^{4} + 48 \, x^{3} + 144 \, x^{2} + \frac {4}{25} \, {\left (x^{2} + 12 \, x + 36\right )} e^{6} + \frac {8}{5} \, {\left (x^{3} + 12 \, x^{2} + 36 \, x\right )} e^{3}\right )} + 25\right )} e^{\left (e^{\left (4 \, x^{4} + 48 \, x^{3} + 144 \, x^{2} + \frac {4}{25} \, {\left (x^{2} + 12 \, x + 36\right )} e^{6} + \frac {8}{5} \, {\left (x^{3} + 12 \, x^{2} + 36 \, x\right )} e^{3}\right )}\right )} \,d x } \]
integrate(1/5*(((-8*x^2-32*x+96)*exp(3)^2+(-120*x^3-720*x^2+480*x+2880)*ex p(3)-400*x^4-2800*x^3+14400*x)*exp(1/25*(4*x^2+48*x+144)*exp(3)^2+1/25*(40 *x^3+480*x^2+1440*x)*exp(3)+4*x^4+48*x^3+144*x^2)-25)*exp(exp(1/25*(4*x^2+ 48*x+144)*exp(3)^2+1/25*(40*x^3+480*x^2+1440*x)*exp(3)+4*x^4+48*x^3+144*x^ 2)),x, algorithm=\
integrate(-1/5*(8*(50*x^4 + 350*x^3 + (x^2 + 4*x - 12)*e^6 + 15*(x^3 + 6*x ^2 - 4*x - 24)*e^3 - 1800*x)*e^(4*x^4 + 48*x^3 + 144*x^2 + 4/25*(x^2 + 12* x + 36)*e^6 + 8/5*(x^3 + 12*x^2 + 36*x)*e^3) + 25)*e^(e^(4*x^4 + 48*x^3 + 144*x^2 + 4/25*(x^2 + 12*x + 36)*e^6 + 8/5*(x^3 + 12*x^2 + 36*x)*e^3)), x)
Time = 12.10 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.06 \[ \int \frac {1}{5} e^{e^{\frac {1}{25} \left (3600 x^2+1200 x^3+100 x^4+e^6 \left (144+48 x+4 x^2\right )+e^3 \left (1440 x+480 x^2+40 x^3\right )\right )}} \left (-25+e^{\frac {1}{25} \left (3600 x^2+1200 x^3+100 x^4+e^6 \left (144+48 x+4 x^2\right )+e^3 \left (1440 x+480 x^2+40 x^3\right )\right )} \left (14400 x-2800 x^3-400 x^4+e^6 \left (96-32 x-8 x^2\right )+e^3 \left (2880+480 x-720 x^2-120 x^3\right )\right )\right ) \, dx=-5\,{\mathrm {e}}^{{\mathrm {e}}^{\frac {8\,x^3\,{\mathrm {e}}^3}{5}}\,{\mathrm {e}}^{\frac {4\,x^2\,{\mathrm {e}}^6}{25}}\,{\mathrm {e}}^{\frac {96\,x^2\,{\mathrm {e}}^3}{5}}\,{\mathrm {e}}^{\frac {144\,{\mathrm {e}}^6}{25}}\,{\mathrm {e}}^{4\,x^4}\,{\mathrm {e}}^{48\,x^3}\,{\mathrm {e}}^{144\,x^2}\,{\mathrm {e}}^{\frac {48\,x\,{\mathrm {e}}^6}{25}}\,{\mathrm {e}}^{\frac {288\,x\,{\mathrm {e}}^3}{5}}}\,\left (x-2\right ) \]
int(-(exp(exp((exp(6)*(48*x + 4*x^2 + 144))/25 + (exp(3)*(1440*x + 480*x^2 + 40*x^3))/25 + 144*x^2 + 48*x^3 + 4*x^4))*(exp((exp(6)*(48*x + 4*x^2 + 1 44))/25 + (exp(3)*(1440*x + 480*x^2 + 40*x^3))/25 + 144*x^2 + 48*x^3 + 4*x ^4)*(exp(6)*(32*x + 8*x^2 - 96) - 14400*x - exp(3)*(480*x - 720*x^2 - 120* x^3 + 2880) + 2800*x^3 + 400*x^4) + 25))/5,x)