Integrand size = 87, antiderivative size = 23 \[ \int \frac {1}{81} e^{\frac {1}{81} \left (81 e^{-1+2 x}+e^{-1+x} \left (81 x^2+11250 x^5+390625 x^8\right )\right )} \left (162 e^{-1+2 x}+e^{-1+x} \left (162 x+81 x^2+56250 x^4+11250 x^5+3125000 x^7+390625 x^8\right )\right ) \, dx=e^{e^{-1+x} \left (e^x+\left (x+\frac {625 x^4}{9}\right )^2\right )} \] Output:
exp(((625/9*x^4+x)^2+exp(x))*exp(-1+x))
Time = 3.37 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {1}{81} e^{\frac {1}{81} \left (81 e^{-1+2 x}+e^{-1+x} \left (81 x^2+11250 x^5+390625 x^8\right )\right )} \left (162 e^{-1+2 x}+e^{-1+x} \left (162 x+81 x^2+56250 x^4+11250 x^5+3125000 x^7+390625 x^8\right )\right ) \, dx=e^{e^{-1+2 x}+\frac {1}{81} e^{-1+x} x^2 \left (9+625 x^3\right )^2} \] Input:
Integrate[(E^((81*E^(-1 + 2*x) + E^(-1 + x)*(81*x^2 + 11250*x^5 + 390625*x ^8))/81)*(162*E^(-1 + 2*x) + E^(-1 + x)*(162*x + 81*x^2 + 56250*x^4 + 1125 0*x^5 + 3125000*x^7 + 390625*x^8)))/81,x]
Output:
E^(E^(-1 + 2*x) + (E^(-1 + x)*x^2*(9 + 625*x^3)^2)/81)
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.
\(\displaystyle \int \frac {1}{81} \left (e^{x-1} \left (390625 x^8+3125000 x^7+11250 x^5+56250 x^4+81 x^2+162 x\right )+162 e^{2 x-1}\right ) \exp \left (\frac {1}{81} \left (e^{x-1} \left (390625 x^8+11250 x^5+81 x^2\right )+81 e^{2 x-1}\right )\right ) \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{81} \int \exp \left (\frac {1}{81} \left (e^{x-1} \left (390625 x^8+11250 x^5+81 x^2\right )+81 e^{2 x-1}\right )\right ) \left (e^{x-1} \left (390625 x^8+3125000 x^7+11250 x^5+56250 x^4+81 x^2+162 x\right )+162 e^{2 x-1}\right )dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \frac {1}{81} \int \exp \left (x+\frac {1}{81} e^{x-1} \left (390625 x^8+11250 x^5+81 x^2+81 e^x\right )-1\right ) \left (390625 x^8+3125000 x^7+11250 x^5+56250 x^4+81 x^2+162 x+162 e^x\right )dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{81} \int \left (390625 \exp \left (x+\frac {1}{81} e^{x-1} \left (390625 x^8+11250 x^5+81 x^2+81 e^x\right )-1\right ) x^8+3125000 \exp \left (x+\frac {1}{81} e^{x-1} \left (390625 x^8+11250 x^5+81 x^2+81 e^x\right )-1\right ) x^7+11250 \exp \left (x+\frac {1}{81} e^{x-1} \left (390625 x^8+11250 x^5+81 x^2+81 e^x\right )-1\right ) x^5+56250 \exp \left (x+\frac {1}{81} e^{x-1} \left (390625 x^8+11250 x^5+81 x^2+81 e^x\right )-1\right ) x^4+81 \exp \left (x+\frac {1}{81} e^{x-1} \left (390625 x^8+11250 x^5+81 x^2+81 e^x\right )-1\right ) x^2+162 \exp \left (x+\frac {1}{81} e^{x-1} \left (390625 x^8+11250 x^5+81 x^2+81 e^x\right )-1\right ) x+162 \exp \left (2 x+\frac {1}{81} e^{x-1} \left (390625 x^8+11250 x^5+81 x^2+81 e^x\right )-1\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{81} \left (162 \int \exp \left (2 x+\frac {1}{81} e^{x-1} \left (390625 x^8+11250 x^5+81 x^2+81 e^x\right )-1\right )dx+162 \int \exp \left (x+\frac {1}{81} e^{x-1} \left (390625 x^8+11250 x^5+81 x^2+81 e^x\right )-1\right ) xdx+81 \int \exp \left (x+\frac {1}{81} e^{x-1} \left (390625 x^8+11250 x^5+81 x^2+81 e^x\right )-1\right ) x^2dx+11250 \int \exp \left (x+\frac {1}{81} e^{x-1} \left (390625 x^8+11250 x^5+81 x^2+81 e^x\right )-1\right ) x^5dx+390625 \int \exp \left (x+\frac {1}{81} e^{x-1} \left (390625 x^8+11250 x^5+81 x^2+81 e^x\right )-1\right ) x^8dx+3125000 \int \exp \left (x+\frac {1}{81} e^{x-1} \left (390625 x^8+11250 x^5+81 x^2+81 e^x\right )-1\right ) x^7dx+56250 \int \exp \left (x+\frac {1}{81} e^{x-1} \left (390625 x^8+11250 x^5+81 x^2+81 e^x\right )-1\right ) x^4dx\right )\) |
Input:
Int[(E^((81*E^(-1 + 2*x) + E^(-1 + x)*(81*x^2 + 11250*x^5 + 390625*x^8))/8 1)*(162*E^(-1 + 2*x) + E^(-1 + x)*(162*x + 81*x^2 + 56250*x^4 + 11250*x^5 + 3125000*x^7 + 390625*x^8)))/81,x]
Output:
$Aborted
Time = 0.78 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22
method | result | size |
parallelrisch | \({\mathrm e}^{\frac {{\mathrm e}^{-1+x} \left (390625 x^{8}+11250 x^{5}+81 x^{2}+81 \,{\mathrm e}^{x}\right )}{81}}\) | \(28\) |
risch | \({\mathrm e}^{{\mathrm e}^{-1+2 x}+\frac {390625 \,{\mathrm e}^{-1+x} x^{8}}{81}+\frac {1250 \,{\mathrm e}^{-1+x} x^{5}}{9}+{\mathrm e}^{-1+x} x^{2}}\) | \(35\) |
Input:
int(1/81*(162*exp(-1+x)*exp(x)+(390625*x^8+3125000*x^7+11250*x^5+56250*x^4 +81*x^2+162*x)*exp(-1+x))*exp(exp(-1+x)*exp(x)+1/81*(390625*x^8+11250*x^5+ 81*x^2)*exp(-1+x)),x,method=_RETURNVERBOSE)
Output:
exp(1/81*exp(-1+x)*(390625*x^8+11250*x^5+81*x^2+81*exp(x)))
Time = 0.08 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30 \[ \int \frac {1}{81} e^{\frac {1}{81} \left (81 e^{-1+2 x}+e^{-1+x} \left (81 x^2+11250 x^5+390625 x^8\right )\right )} \left (162 e^{-1+2 x}+e^{-1+x} \left (162 x+81 x^2+56250 x^4+11250 x^5+3125000 x^7+390625 x^8\right )\right ) \, dx=e^{\left (\frac {1}{81} \, {\left (390625 \, x^{8} + 11250 \, x^{5} + 81 \, x^{2}\right )} e^{\left (x - 1\right )} + e^{\left (2 \, x - 1\right )}\right )} \] Input:
integrate(1/81*(162*exp(-1+x)*exp(x)+(390625*x^8+3125000*x^7+11250*x^5+562 50*x^4+81*x^2+162*x)*exp(-1+x))*exp(exp(-1+x)*exp(x)+1/81*(390625*x^8+1125 0*x^5+81*x^2)*exp(-1+x)),x, algorithm="fricas")
Output:
e^(1/81*(390625*x^8 + 11250*x^5 + 81*x^2)*e^(x - 1) + e^(2*x - 1))
Time = 0.20 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39 \[ \int \frac {1}{81} e^{\frac {1}{81} \left (81 e^{-1+2 x}+e^{-1+x} \left (81 x^2+11250 x^5+390625 x^8\right )\right )} \left (162 e^{-1+2 x}+e^{-1+x} \left (162 x+81 x^2+56250 x^4+11250 x^5+3125000 x^7+390625 x^8\right )\right ) \, dx=e^{\frac {\left (\frac {390625 x^{8}}{81} + \frac {1250 x^{5}}{9} + x^{2}\right ) e^{x}}{e} + \frac {e^{2 x}}{e}} \] Input:
integrate(1/81*(162*exp(-1+x)*exp(x)+(390625*x**8+3125000*x**7+11250*x**5+ 56250*x**4+81*x**2+162*x)*exp(-1+x))*exp(exp(-1+x)*exp(x)+1/81*(390625*x** 8+11250*x**5+81*x**2)*exp(-1+x)),x)
Output:
exp((390625*x**8/81 + 1250*x**5/9 + x**2)*exp(-1)*exp(x) + exp(-1)*exp(2*x ))
Time = 0.32 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.48 \[ \int \frac {1}{81} e^{\frac {1}{81} \left (81 e^{-1+2 x}+e^{-1+x} \left (81 x^2+11250 x^5+390625 x^8\right )\right )} \left (162 e^{-1+2 x}+e^{-1+x} \left (162 x+81 x^2+56250 x^4+11250 x^5+3125000 x^7+390625 x^8\right )\right ) \, dx=e^{\left (\frac {390625}{81} \, x^{8} e^{\left (x - 1\right )} + \frac {1250}{9} \, x^{5} e^{\left (x - 1\right )} + x^{2} e^{\left (x - 1\right )} + e^{\left (2 \, x - 1\right )}\right )} \] Input:
integrate(1/81*(162*exp(-1+x)*exp(x)+(390625*x^8+3125000*x^7+11250*x^5+562 50*x^4+81*x^2+162*x)*exp(-1+x))*exp(exp(-1+x)*exp(x)+1/81*(390625*x^8+1125 0*x^5+81*x^2)*exp(-1+x)),x, algorithm="maxima")
Output:
e^(390625/81*x^8*e^(x - 1) + 1250/9*x^5*e^(x - 1) + x^2*e^(x - 1) + e^(2*x - 1))
Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (23) = 46\).
Time = 0.28 (sec) , antiderivative size = 100, normalized size of antiderivative = 4.35 \[ \int \frac {1}{81} e^{\frac {1}{81} \left (81 e^{-1+2 x}+e^{-1+x} \left (81 x^2+11250 x^5+390625 x^8\right )\right )} \left (162 e^{-1+2 x}+e^{-1+x} \left (162 x+81 x^2+56250 x^4+11250 x^5+3125000 x^7+390625 x^8\right )\right ) \, dx=e^{\left (\frac {390625}{81} \, {\left (x - 1\right )}^{8} e^{\left (x - 1\right )} + \frac {3125000}{81} \, {\left (x - 1\right )}^{7} e^{\left (x - 1\right )} + \frac {10937500}{81} \, {\left (x - 1\right )}^{6} e^{\left (x - 1\right )} + \frac {21886250}{81} \, {\left (x - 1\right )}^{5} e^{\left (x - 1\right )} + \frac {27400000}{81} \, {\left (x - 1\right )}^{4} e^{\left (x - 1\right )} + \frac {21987500}{81} \, {\left (x - 1\right )}^{3} e^{\left (x - 1\right )} + \frac {11050081}{81} \, {\left (x - 1\right )}^{2} e^{\left (x - 1\right )} + \frac {3181412}{81} \, {\left (x - 1\right )} e^{\left (x - 1\right )} + e^{\left (2 \, x - 1\right )} + \frac {401956}{81} \, e^{\left (x - 1\right )}\right )} \] Input:
integrate(1/81*(162*exp(-1+x)*exp(x)+(390625*x^8+3125000*x^7+11250*x^5+562 50*x^4+81*x^2+162*x)*exp(-1+x))*exp(exp(-1+x)*exp(x)+1/81*(390625*x^8+1125 0*x^5+81*x^2)*exp(-1+x)),x, algorithm="giac")
Output:
e^(390625/81*(x - 1)^8*e^(x - 1) + 3125000/81*(x - 1)^7*e^(x - 1) + 109375 00/81*(x - 1)^6*e^(x - 1) + 21886250/81*(x - 1)^5*e^(x - 1) + 27400000/81* (x - 1)^4*e^(x - 1) + 21987500/81*(x - 1)^3*e^(x - 1) + 11050081/81*(x - 1 )^2*e^(x - 1) + 3181412/81*(x - 1)*e^(x - 1) + e^(2*x - 1) + 401956/81*e^( x - 1))
Time = 3.00 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.65 \[ \int \frac {1}{81} e^{\frac {1}{81} \left (81 e^{-1+2 x}+e^{-1+x} \left (81 x^2+11250 x^5+390625 x^8\right )\right )} \left (162 e^{-1+2 x}+e^{-1+x} \left (162 x+81 x^2+56250 x^4+11250 x^5+3125000 x^7+390625 x^8\right )\right ) \, dx={\mathrm {e}}^{{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-1}}\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^{-1}\,{\mathrm {e}}^x}\,{\mathrm {e}}^{\frac {1250\,x^5\,{\mathrm {e}}^{-1}\,{\mathrm {e}}^x}{9}}\,{\mathrm {e}}^{\frac {390625\,x^8\,{\mathrm {e}}^{-1}\,{\mathrm {e}}^x}{81}} \] Input:
int((exp(exp(x - 1)*exp(x) + (exp(x - 1)*(81*x^2 + 11250*x^5 + 390625*x^8) )/81)*(162*exp(x - 1)*exp(x) + exp(x - 1)*(162*x + 81*x^2 + 56250*x^4 + 11 250*x^5 + 3125000*x^7 + 390625*x^8)))/81,x)
Output:
exp(exp(2*x)*exp(-1))*exp(x^2*exp(-1)*exp(x))*exp((1250*x^5*exp(-1)*exp(x) )/9)*exp((390625*x^8*exp(-1)*exp(x))/81)
Time = 0.20 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.70 \[ \int \frac {1}{81} e^{\frac {1}{81} \left (81 e^{-1+2 x}+e^{-1+x} \left (81 x^2+11250 x^5+390625 x^8\right )\right )} \left (162 e^{-1+2 x}+e^{-1+x} \left (162 x+81 x^2+56250 x^4+11250 x^5+3125000 x^7+390625 x^8\right )\right ) \, dx=e^{\frac {81 e^{2 x}+390625 e^{x} x^{8}+11250 e^{x} x^{5}+81 e^{x} x^{2}}{81 e}} \] Input:
int(1/81*(162*exp(-1+x)*exp(x)+(390625*x^8+3125000*x^7+11250*x^5+56250*x^4 +81*x^2+162*x)*exp(-1+x))*exp(exp(-1+x)*exp(x)+1/81*(390625*x^8+11250*x^5+ 81*x^2)*exp(-1+x)),x)
Output:
e**((81*e**(2*x) + 390625*e**x*x**8 + 11250*e**x*x**5 + 81*e**x*x**2)/(81* e))