Integrand size = 71, antiderivative size = 23 \[ \int \frac {e^{-e^{1+9 x}+x} \left (-3+27 e^{1+9 x}\right )+24 x}{e^{-2 e^{1+9 x}+2 x}-8 e^{-e^{1+9 x}+x} x^2+16 x^4} \, dx=\frac {3}{e^{-e^{1+9 x}+x}-4 x^2} \] Output:
3/(exp(-exp(1)*exp(9*x)+x)-4*x^2)
Time = 0.70 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {e^{-e^{1+9 x}+x} \left (-3+27 e^{1+9 x}\right )+24 x}{e^{-2 e^{1+9 x}+2 x}-8 e^{-e^{1+9 x}+x} x^2+16 x^4} \, dx=\frac {3 e^{e^{1+9 x}}}{e^x-4 e^{e^{1+9 x}} x^2} \] Input:
Integrate[(E^(-E^(1 + 9*x) + x)*(-3 + 27*E^(1 + 9*x)) + 24*x)/(E^(-2*E^(1 + 9*x) + 2*x) - 8*E^(-E^(1 + 9*x) + x)*x^2 + 16*x^4),x]
Output:
(3*E^E^(1 + 9*x))/(E^x - 4*E^E^(1 + 9*x)*x^2)
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.
\(\displaystyle \int \frac {e^{x-e^{9 x+1}} \left (27 e^{9 x+1}-3\right )+24 x}{16 x^4-8 e^{x-e^{9 x+1}} x^2+e^{2 x-2 e^{9 x+1}}} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{2 e^{9 x+1}} \left (e^{x-e^{9 x+1}} \left (27 e^{9 x+1}-3\right )+24 x\right )}{\left (e^x-4 e^{e^{9 x+1}} x^2\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (15925248 e^{9 e^{9 x+1}+1} x^{16}+3538944 e^{x+8 e^{9 x+1}+1} x^{14}+774144 e^{2 x+7 e^{9 x+1}+1} x^{12}+165888 e^{3 x+6 e^{9 x+1}+1} x^{10}+34560 e^{4 x+5 e^{9 x+1}+1} x^8+6912 e^{5 x+4 e^{9 x+1}+1} x^6+1296 e^{6 x+3 e^{9 x+1}+1} x^4+216 e^{7 x+2 e^{9 x+1}+1} x^2+\frac {12 e^{2 e^{9 x+1}} \left (2359296 e^{9 e^{9 x+1}+1} x^{19}-x+2\right ) x}{\left (4 e^{e^{9 x+1}} x^2-e^x\right )^2}-\frac {3 e^{e^{9 x+1}} \left (23592960 e^{9 e^{9 x+1}+1} x^{18}-1\right )}{4 e^{e^{9 x+1}} x^2-e^x}+27 e^{8 x+e^{9 x+1}+1}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 15925248 \int e^{1+9 e^{9 x+1}} x^{16}dx+3538944 \int e^{x+8 e^{9 x+1}+1} x^{14}dx+774144 \int e^{2 x+7 e^{9 x+1}+1} x^{12}dx+165888 \int e^{3 x+6 e^{9 x+1}+1} x^{10}dx+34560 \int e^{4 x+5 e^{9 x+1}+1} x^8dx+6912 \int e^{5 x+4 e^{9 x+1}+1} x^6dx+1296 \int e^{6 x+3 e^{9 x+1}+1} x^4dx+216 \int e^{7 x+2 e^{9 x+1}+1} x^2dx-3 \int \frac {e^{e^{9 x+1}}}{e^x-4 e^{e^{9 x+1}} x^2}dx+24 \int \frac {e^{2 e^{9 x+1}} x}{\left (4 e^{e^{9 x+1}} x^2-e^x\right )^2}dx-12 \int \frac {e^{2 e^{9 x+1}} x^2}{\left (4 e^{e^{9 x+1}} x^2-e^x\right )^2}dx+28311552 \int \frac {e^{1+11 e^{9 x+1}} x^{20}}{\left (4 e^{e^{9 x+1}} x^2-e^x\right )^2}dx-70778880 \int \frac {e^{1+10 e^{9 x+1}} x^{18}}{4 e^{e^{9 x+1}} x^2-e^x}dx-\frac {3 e^{8 x+\frac {1}{9}} \Gamma \left (\frac {8}{9},-e^{9 x+1}\right )}{\left (-e^{9 x}\right )^{8/9}}\) |
Input:
Int[(E^(-E^(1 + 9*x) + x)*(-3 + 27*E^(1 + 9*x)) + 24*x)/(E^(-2*E^(1 + 9*x) + 2*x) - 8*E^(-E^(1 + 9*x) + x)*x^2 + 16*x^4),x]
Output:
$Aborted
Time = 0.32 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04
method | result | size |
norman | \(-\frac {3}{4 x^{2}-{\mathrm e}^{-{\mathrm e} \,{\mathrm e}^{9 x}+x}}\) | \(24\) |
risch | \(-\frac {3}{4 x^{2}-{\mathrm e}^{-{\mathrm e}^{1+9 x}+x}}\) | \(24\) |
parallelrisch | \(-\frac {3}{4 x^{2}-{\mathrm e}^{-{\mathrm e} \,{\mathrm e}^{9 x}+x}}\) | \(24\) |
Input:
int(((27*exp(1)*exp(9*x)-3)*exp(-exp(1)*exp(9*x)+x)+24*x)/(exp(-exp(1)*exp (9*x)+x)^2-8*x^2*exp(-exp(1)*exp(9*x)+x)+16*x^4),x,method=_RETURNVERBOSE)
Output:
-3/(4*x^2-exp(-exp(1)*exp(9*x)+x))
Time = 0.09 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-e^{1+9 x}+x} \left (-3+27 e^{1+9 x}\right )+24 x}{e^{-2 e^{1+9 x}+2 x}-8 e^{-e^{1+9 x}+x} x^2+16 x^4} \, dx=-\frac {3}{4 \, x^{2} - e^{\left (x - e^{\left (9 \, x + 1\right )}\right )}} \] Input:
integrate(((27*exp(1)*exp(9*x)-3)*exp(-exp(1)*exp(9*x)+x)+24*x)/(exp(-exp( 1)*exp(9*x)+x)^2-8*x^2*exp(-exp(1)*exp(9*x)+x)+16*x^4),x, algorithm="frica s")
Output:
-3/(4*x^2 - e^(x - e^(9*x + 1)))
Time = 0.09 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {e^{-e^{1+9 x}+x} \left (-3+27 e^{1+9 x}\right )+24 x}{e^{-2 e^{1+9 x}+2 x}-8 e^{-e^{1+9 x}+x} x^2+16 x^4} \, dx=\frac {3}{- 4 x^{2} + e^{x - e e^{9 x}}} \] Input:
integrate(((27*exp(1)*exp(9*x)-3)*exp(-exp(1)*exp(9*x)+x)+24*x)/(exp(-exp( 1)*exp(9*x)+x)**2-8*x**2*exp(-exp(1)*exp(9*x)+x)+16*x**4),x)
Output:
3/(-4*x**2 + exp(x - E*exp(9*x)))
Time = 0.09 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22 \[ \int \frac {e^{-e^{1+9 x}+x} \left (-3+27 e^{1+9 x}\right )+24 x}{e^{-2 e^{1+9 x}+2 x}-8 e^{-e^{1+9 x}+x} x^2+16 x^4} \, dx=-\frac {3 \, e^{\left (e^{\left (9 \, x + 1\right )}\right )}}{4 \, x^{2} e^{\left (e^{\left (9 \, x + 1\right )}\right )} - e^{x}} \] Input:
integrate(((27*exp(1)*exp(9*x)-3)*exp(-exp(1)*exp(9*x)+x)+24*x)/(exp(-exp( 1)*exp(9*x)+x)^2-8*x^2*exp(-exp(1)*exp(9*x)+x)+16*x^4),x, algorithm="maxim a")
Output:
-3*e^(e^(9*x + 1))/(4*x^2*e^(e^(9*x + 1)) - e^x)
Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (23) = 46\).
Time = 0.13 (sec) , antiderivative size = 242, normalized size of antiderivative = 10.52 \[ \int \frac {e^{-e^{1+9 x}+x} \left (-3+27 e^{1+9 x}\right )+24 x}{e^{-2 e^{1+9 x}+2 x}-8 e^{-e^{1+9 x}+x} x^2+16 x^4} \, dx=-\frac {3 \, {\left (36 \, x^{3} e^{\left (9 \, x + \frac {1}{2} \, e^{\left (9 \, x + 1\right )} + 1\right )} - 4 \, x^{3} e^{\left (\frac {1}{2} \, e^{\left (9 \, x + 1\right )}\right )} + 8 \, x^{2} e^{\left (\frac {1}{2} \, e^{\left (9 \, x + 1\right )}\right )} - 9 \, x e^{\left (10 \, x - \frac {1}{2} \, e^{\left (9 \, x + 1\right )} + 1\right )} + x e^{\left (x - \frac {1}{2} \, e^{\left (9 \, x + 1\right )}\right )} - 2 \, e^{\left (x - \frac {1}{2} \, e^{\left (9 \, x + 1\right )}\right )}\right )}}{144 \, x^{5} e^{\left (9 \, x + \frac {1}{2} \, e^{\left (9 \, x + 1\right )} + 1\right )} - 16 \, x^{5} e^{\left (\frac {1}{2} \, e^{\left (9 \, x + 1\right )}\right )} + 32 \, x^{4} e^{\left (\frac {1}{2} \, e^{\left (9 \, x + 1\right )}\right )} - 72 \, x^{3} e^{\left (10 \, x - \frac {1}{2} \, e^{\left (9 \, x + 1\right )} + 1\right )} + 8 \, x^{3} e^{\left (x - \frac {1}{2} \, e^{\left (9 \, x + 1\right )}\right )} - 16 \, x^{2} e^{\left (x - \frac {1}{2} \, e^{\left (9 \, x + 1\right )}\right )} + 9 \, x e^{\left (11 \, x - \frac {3}{2} \, e^{\left (9 \, x + 1\right )} + 1\right )} - x e^{\left (2 \, x - \frac {3}{2} \, e^{\left (9 \, x + 1\right )}\right )} + 2 \, e^{\left (2 \, x - \frac {3}{2} \, e^{\left (9 \, x + 1\right )}\right )}} \] Input:
integrate(((27*exp(1)*exp(9*x)-3)*exp(-exp(1)*exp(9*x)+x)+24*x)/(exp(-exp( 1)*exp(9*x)+x)^2-8*x^2*exp(-exp(1)*exp(9*x)+x)+16*x^4),x, algorithm="giac" )
Output:
-3*(36*x^3*e^(9*x + 1/2*e^(9*x + 1) + 1) - 4*x^3*e^(1/2*e^(9*x + 1)) + 8*x ^2*e^(1/2*e^(9*x + 1)) - 9*x*e^(10*x - 1/2*e^(9*x + 1) + 1) + x*e^(x - 1/2 *e^(9*x + 1)) - 2*e^(x - 1/2*e^(9*x + 1)))/(144*x^5*e^(9*x + 1/2*e^(9*x + 1) + 1) - 16*x^5*e^(1/2*e^(9*x + 1)) + 32*x^4*e^(1/2*e^(9*x + 1)) - 72*x^3 *e^(10*x - 1/2*e^(9*x + 1) + 1) + 8*x^3*e^(x - 1/2*e^(9*x + 1)) - 16*x^2*e ^(x - 1/2*e^(9*x + 1)) + 9*x*e^(11*x - 3/2*e^(9*x + 1) + 1) - x*e^(2*x - 3 /2*e^(9*x + 1)) + 2*e^(2*x - 3/2*e^(9*x + 1)))
Time = 0.19 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {e^{-e^{1+9 x}+x} \left (-3+27 e^{1+9 x}\right )+24 x}{e^{-2 e^{1+9 x}+2 x}-8 e^{-e^{1+9 x}+x} x^2+16 x^4} \, dx=\frac {3}{{\mathrm {e}}^{x-{\mathrm {e}}^{9\,x}\,\mathrm {e}}-4\,x^2} \] Input:
int((24*x + exp(x - exp(9*x)*exp(1))*(27*exp(9*x)*exp(1) - 3))/(exp(2*x - 2*exp(9*x)*exp(1)) - 8*x^2*exp(x - exp(9*x)*exp(1)) + 16*x^4),x)
Output:
3/(exp(x - exp(9*x)*exp(1)) - 4*x^2)
Time = 0.16 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43 \[ \int \frac {e^{-e^{1+9 x}+x} \left (-3+27 e^{1+9 x}\right )+24 x}{e^{-2 e^{1+9 x}+2 x}-8 e^{-e^{1+9 x}+x} x^2+16 x^4} \, dx=-\frac {3 e^{e^{9 x} e}}{4 e^{e^{9 x} e} x^{2}-e^{x}} \] Input:
int(((27*exp(1)*exp(9*x)-3)*exp(-exp(1)*exp(9*x)+x)+24*x)/(exp(-exp(1)*exp (9*x)+x)^2-8*x^2*exp(-exp(1)*exp(9*x)+x)+16*x^4),x)
Output:
( - 3*e**(e**(9*x)*e))/(4*e**(e**(9*x)*e)*x**2 - e**x)