Integrand size = 106, antiderivative size = 28 \[ \int \frac {4 e^{10}+e^{e^{4-4 x+x^2}-x} \left (-2 e^5+e^{9-4 x+x^2} (-8+4 x)\right )}{4 e^{2 e^{4-4 x+x^2}-2 x}+e^{5+e^{4-4 x+x^2}-x} (44+16 x)+e^{10} \left (121+88 x+16 x^2\right )} \, dx=\frac {1}{5-2 \left (e^{-5+e^{(2-x)^2}-x}+2 (4+x)\right )} \]
Time = 1.18 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {4 e^{10}+e^{e^{4-4 x+x^2}-x} \left (-2 e^5+e^{9-4 x+x^2} (-8+4 x)\right )}{4 e^{2 e^{4-4 x+x^2}-2 x}+e^{5+e^{4-4 x+x^2}-x} (44+16 x)+e^{10} \left (121+88 x+16 x^2\right )} \, dx=-\frac {e^{5+x}}{2 e^{e^{(-2+x)^2}}+e^{5+x} (11+4 x)} \]
Integrate[(4*E^10 + E^(E^(4 - 4*x + x^2) - x)*(-2*E^5 + E^(9 - 4*x + x^2)* (-8 + 4*x)))/(4*E^(2*E^(4 - 4*x + x^2) - 2*x) + E^(5 + E^(4 - 4*x + x^2) - x)*(44 + 16*x) + E^10*(121 + 88*x + 16*x^2)),x]
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.
\(\displaystyle \int \frac {e^{e^{x^2-4 x+4}-x} \left (e^{x^2-4 x+9} (4 x-8)-2 e^5\right )+4 e^{10}}{e^{e^{x^2-4 x+4}-x+5} (16 x+44)+4 e^{2 e^{x^2-4 x+4}-2 x}+e^{10} \left (16 x^2+88 x+121\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{2 x} \left (e^{e^{x^2-4 x+4}-x} \left (e^{x^2-4 x+9} (4 x-8)-2 e^5\right )+4 e^{10}\right )}{\left (4 e^{x+5} x+2 e^{e^{(x-2)^2}}+11 e^{x+5}\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {4 e^{x^2-3 x+e^{(x-2)^2}+9} (x-2)}{\left (4 e^{x+5} x+2 e^{e^{(x-2)^2}}+11 e^{x+5}\right )^2}+\frac {2 e^{x+5} \left (2 e^{x+5}-e^{e^{(x-2)^2}}\right )}{\left (4 e^{x+5} x+2 e^{e^{(x-2)^2}}+11 e^{x+5}\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -8 \int \frac {e^{x^2-3 x+e^{(x-2)^2}+9}}{\left (4 e^{x+5} x+2 e^{e^{(x-2)^2}}+11 e^{x+5}\right )^2}dx+4 \int \frac {e^{x^2-3 x+e^{(x-2)^2}+9} x}{\left (4 e^{x+5} x+2 e^{e^{(x-2)^2}}+11 e^{x+5}\right )^2}dx-2 \int \frac {e^{x+e^{(x-2)^2}+5}}{\left (4 e^{x+5} x+2 e^{e^{(x-2)^2}}+11 e^{x+5}\right )^2}dx-8 \int \frac {e^{x+e^{(x-2)^2}+5}}{(4 x+11) \left (4 e^{x+5} x+2 e^{e^{(x-2)^2}}+11 e^{x+5}\right )^2}dx+4 \int \frac {e^{x+5}}{(4 x+11) \left (4 e^{x+5} x+2 e^{e^{(x-2)^2}}+11 e^{x+5}\right )}dx\) |
Int[(4*E^10 + E^(E^(4 - 4*x + x^2) - x)*(-2*E^5 + E^(9 - 4*x + x^2)*(-8 + 4*x)))/(4*E^(2*E^(4 - 4*x + x^2) - 2*x) + E^(5 + E^(4 - 4*x + x^2) - x)*(4 4 + 16*x) + E^10*(121 + 88*x + 16*x^2)),x]
3.21.26.3.1 Defintions of rubi rules used
Time = 0.33 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07
method | result | size |
risch | \(-\frac {{\mathrm e}^{5}}{4 x \,{\mathrm e}^{5}+2 \,{\mathrm e}^{{\mathrm e}^{\left (-2+x \right )^{2}}-x}+11 \,{\mathrm e}^{5}}\) | \(30\) |
norman | \(-\frac {{\mathrm e}^{5}}{4 x \,{\mathrm e}^{5}+2 \,{\mathrm e}^{{\mathrm e}^{x^{2}-4 x +4}-x}+11 \,{\mathrm e}^{5}}\) | \(33\) |
parallelrisch | \(-\frac {{\mathrm e}^{5}}{4 x \,{\mathrm e}^{5}+2 \,{\mathrm e}^{{\mathrm e}^{x^{2}-4 x +4}-x}+11 \,{\mathrm e}^{5}}\) | \(33\) |
int((((4*x-8)*exp(5)*exp(x^2-4*x+4)-2*exp(5))*exp(exp(x^2-4*x+4)-x)+4*exp( 5)^2)/(4*exp(exp(x^2-4*x+4)-x)^2+(16*x+44)*exp(5)*exp(exp(x^2-4*x+4)-x)+(1 6*x^2+88*x+121)*exp(5)^2),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43 \[ \int \frac {4 e^{10}+e^{e^{4-4 x+x^2}-x} \left (-2 e^5+e^{9-4 x+x^2} (-8+4 x)\right )}{4 e^{2 e^{4-4 x+x^2}-2 x}+e^{5+e^{4-4 x+x^2}-x} (44+16 x)+e^{10} \left (121+88 x+16 x^2\right )} \, dx=-\frac {e^{10}}{{\left (4 \, x + 11\right )} e^{10} + 2 \, e^{\left (-{\left ({\left (x - 5\right )} e^{5} - e^{\left (x^{2} - 4 \, x + 9\right )}\right )} e^{\left (-5\right )}\right )}} \]
integrate((((4*x-8)*exp(5)*exp(x^2-4*x+4)-2*exp(5))*exp(exp(x^2-4*x+4)-x)+ 4*exp(5)^2)/(4*exp(exp(x^2-4*x+4)-x)^2+(16*x+44)*exp(5)*exp(exp(x^2-4*x+4) -x)+(16*x^2+88*x+121)*exp(5)^2),x, algorithm=\
Time = 0.13 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {4 e^{10}+e^{e^{4-4 x+x^2}-x} \left (-2 e^5+e^{9-4 x+x^2} (-8+4 x)\right )}{4 e^{2 e^{4-4 x+x^2}-2 x}+e^{5+e^{4-4 x+x^2}-x} (44+16 x)+e^{10} \left (121+88 x+16 x^2\right )} \, dx=- \frac {e^{5}}{4 x e^{5} + 2 e^{- x + e^{x^{2} - 4 x + 4}} + 11 e^{5}} \]
integrate((((4*x-8)*exp(5)*exp(x**2-4*x+4)-2*exp(5))*exp(exp(x**2-4*x+4)-x )+4*exp(5)**2)/(4*exp(exp(x**2-4*x+4)-x)**2+(16*x+44)*exp(5)*exp(exp(x**2- 4*x+4)-x)+(16*x**2+88*x+121)*exp(5)**2),x)
Time = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {4 e^{10}+e^{e^{4-4 x+x^2}-x} \left (-2 e^5+e^{9-4 x+x^2} (-8+4 x)\right )}{4 e^{2 e^{4-4 x+x^2}-2 x}+e^{5+e^{4-4 x+x^2}-x} (44+16 x)+e^{10} \left (121+88 x+16 x^2\right )} \, dx=-\frac {e^{\left (x + 5\right )}}{{\left (4 \, x e^{5} + 11 \, e^{5}\right )} e^{x} + 2 \, e^{\left (e^{\left (x^{2} - 4 \, x + 4\right )}\right )}} \]
integrate((((4*x-8)*exp(5)*exp(x^2-4*x+4)-2*exp(5))*exp(exp(x^2-4*x+4)-x)+ 4*exp(5)^2)/(4*exp(exp(x^2-4*x+4)-x)^2+(16*x+44)*exp(5)*exp(exp(x^2-4*x+4) -x)+(16*x^2+88*x+121)*exp(5)^2),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 2593 vs. \(2 (23) = 46\).
Time = 0.39 (sec) , antiderivative size = 2593, normalized size of antiderivative = 92.61 \[ \int \frac {4 e^{10}+e^{e^{4-4 x+x^2}-x} \left (-2 e^5+e^{9-4 x+x^2} (-8+4 x)\right )}{4 e^{2 e^{4-4 x+x^2}-2 x}+e^{5+e^{4-4 x+x^2}-x} (44+16 x)+e^{10} \left (121+88 x+16 x^2\right )} \, dx=\text {Too large to display} \]
integrate((((4*x-8)*exp(5)*exp(x^2-4*x+4)-2*exp(5))*exp(exp(x^2-4*x+4)-x)+ 4*exp(5)^2)/(4*exp(exp(x^2-4*x+4)-x)^2+(16*x+44)*exp(5)*exp(exp(x^2-4*x+4) -x)+(16*x^2+88*x+121)*exp(5)^2),x, algorithm=\
-(1024*x^6*e^(2*x^2 + 3/2*x + e^(x^2 - 4*x + 4) + 23) + 7168*x^5*e^(2*x^2 + 3/2*x + e^(x^2 - 4*x + 4) + 23) + 1024*x^5*e^(2*x^2 + 1/2*x + 2*e^(x^2 - 4*x + 4) + 18) - 1024*x^5*e^(x^2 + 11/2*x + e^(x^2 - 4*x + 4) + 19) + 550 4*x^4*e^(2*x^2 + 3/2*x + e^(x^2 - 4*x + 4) + 23) + 4352*x^4*e^(2*x^2 + 1/2 *x + 2*e^(x^2 - 4*x + 4) + 18) + 256*x^4*e^(2*x^2 - 1/2*x + 3*e^(x^2 - 4*x + 4) + 13) - 10240*x^4*e^(x^2 + 11/2*x + e^(x^2 - 4*x + 4) + 19) - 1024*x ^4*e^(x^2 + 9/2*x + 2*e^(x^2 - 4*x + 4) + 14) + 256*x^4*e^(19/2*x + e^(x^2 - 4*x + 4) + 15) - 55616*x^3*e^(2*x^2 + 3/2*x + e^(x^2 - 4*x + 4) + 23) - 6464*x^3*e^(2*x^2 + 1/2*x + 2*e^(x^2 - 4*x + 4) + 18) + 384*x^3*e^(2*x^2 - 1/2*x + 3*e^(x^2 - 4*x + 4) + 13) - 30336*x^3*e^(x^2 + 11/2*x + e^(x^2 - 4*x + 4) + 19) - 7424*x^3*e^(x^2 + 9/2*x + 2*e^(x^2 - 4*x + 4) + 14) - 25 6*x^3*e^(x^2 + 7/2*x + 3*e^(x^2 - 4*x + 4) + 9) + 3328*x^3*e^(19/2*x + e^( x^2 - 4*x + 4) + 15) + 256*x^3*e^(17/2*x + 2*e^(x^2 - 4*x + 4) + 10) - 963 16*x^2*e^(2*x^2 + 3/2*x + e^(x^2 - 4*x + 4) + 23) - 37840*x^2*e^(2*x^2 + 1 /2*x + 2*e^(x^2 - 4*x + 4) + 18) - 2672*x^2*e^(2*x^2 - 1/2*x + 3*e^(x^2 - 4*x + 4) + 13) + 1408*x^2*e^(x^2 + 11/2*x + e^(x^2 - 4*x + 4) + 19) - 9920 *x^2*e^(x^2 + 9/2*x + 2*e^(x^2 - 4*x + 4) + 14) - 1152*x^2*e^(x^2 + 7/2*x + 3*e^(x^2 - 4*x + 4) + 9) + 16096*x^2*e^(19/2*x + e^(x^2 - 4*x + 4) + 15) + 2624*x^2*e^(17/2*x + 2*e^(x^2 - 4*x + 4) + 10) + 64*x^2*e^(15/2*x + 3*e ^(x^2 - 4*x + 4) + 5) + 106480*x*e^(2*x^2 + 3/2*x + e^(x^2 - 4*x + 4) +...
Timed out. \[ \int \frac {4 e^{10}+e^{e^{4-4 x+x^2}-x} \left (-2 e^5+e^{9-4 x+x^2} (-8+4 x)\right )}{4 e^{2 e^{4-4 x+x^2}-2 x}+e^{5+e^{4-4 x+x^2}-x} (44+16 x)+e^{10} \left (121+88 x+16 x^2\right )} \, dx=\int \frac {4\,{\mathrm {e}}^{10}-{\mathrm {e}}^{{\mathrm {e}}^{x^2-4\,x+4}-x}\,\left (2\,{\mathrm {e}}^5-{\mathrm {e}}^5\,{\mathrm {e}}^{x^2-4\,x+4}\,\left (4\,x-8\right )\right )}{4\,{\mathrm {e}}^{2\,{\mathrm {e}}^{x^2-4\,x+4}-2\,x}+{\mathrm {e}}^{10}\,\left (16\,x^2+88\,x+121\right )+{\mathrm {e}}^{{\mathrm {e}}^{x^2-4\,x+4}-x}\,{\mathrm {e}}^5\,\left (16\,x+44\right )} \,d x \]
int((4*exp(10) - exp(exp(x^2 - 4*x + 4) - x)*(2*exp(5) - exp(5)*exp(x^2 - 4*x + 4)*(4*x - 8)))/(4*exp(2*exp(x^2 - 4*x + 4) - 2*x) + exp(10)*(88*x + 16*x^2 + 121) + exp(exp(x^2 - 4*x + 4) - x)*exp(5)*(16*x + 44)),x)