Integrand size = 123, antiderivative size = 31 \[ \int \frac {4 e^{2+2 x}+e^{-61+16 e^x-e^{2 x}+x} \left (-2-2 x-32 e^x x+4 e^{2 x} x\right )}{9+6 e^{2+2 x}+e^{4+4 x}+e^{-122+32 e^x-2 e^{2 x}+2 x} x^2+e^{-61+16 e^x-e^{2 x}+x} \left (-6 x-2 e^{2+2 x} x\right )} \, dx=\frac {2}{-3-e^{2+2 x}+e^{3-\left (-8+e^x\right )^2+x} x} \]
Time = 0.15 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.58 \[ \int \frac {4 e^{2+2 x}+e^{-61+16 e^x-e^{2 x}+x} \left (-2-2 x-32 e^x x+4 e^{2 x} x\right )}{9+6 e^{2+2 x}+e^{4+4 x}+e^{-122+32 e^x-2 e^{2 x}+2 x} x^2+e^{-61+16 e^x-e^{2 x}+x} \left (-6 x-2 e^{2+2 x} x\right )} \, dx=-\frac {2 e^{61+e^{2 x}}}{3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x} \]
Integrate[(4*E^(2 + 2*x) + E^(-61 + 16*E^x - E^(2*x) + x)*(-2 - 2*x - 32*E ^x*x + 4*E^(2*x)*x))/(9 + 6*E^(2 + 2*x) + E^(4 + 4*x) + E^(-122 + 32*E^x - 2*E^(2*x) + 2*x)*x^2 + E^(-61 + 16*E^x - E^(2*x) + x)*(-6*x - 2*E^(2 + 2* x)*x)),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.
| \(\displaystyle \int \frac {e^{x+16 e^x-e^{2 x}-61} \left (-32 e^x x+4 e^{2 x} x-2 x-2\right )+4 e^{2 x+2}}{e^{2 x+32 e^x-2 e^{2 x}-122} x^2+6 e^{2 x+2}+e^{4 x+4}+e^{x+16 e^x-e^{2 x}-61} \left (-2 e^{2 x+2} x-6 x\right )+9} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{2 \left (e^{2 x}+61\right )} \left (e^{x+16 e^x-e^{2 x}-61} \left (-32 e^x x+4 e^{2 x} x-2 x-2\right )+4 e^{2 x+2}\right )}{\left (-e^{x+16 e^x} x+3 e^{e^{2 x}+61}+e^{2 x+e^{2 x}+63}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 e^{2 \left (e^{2 x}+61\right )-3 e^{2 x}-187} \left (e^{32 e^x} x^2-8 e^{16 e^x+e^{2 x}+63} x+e^{x+16 e^x+e^{2 x}+63} x+e^{2 e^{2 x}+126}\right )}{-e^{x+16 e^x} x+3 e^{e^{2 x}+61}+e^{2 x+e^{2 x}+63}}+\frac {2 e^{2 \left (e^{2 x}+61\right )-3 e^{2 x}-187} \left (2 e^{x+48 e^x} x^3-6 e^{32 e^x+e^{2 x}+61} x^2-16 e^{x+32 e^x+e^{2 x}+63} x^2+48 e^{16 e^x+2 e^{2 x}+124} x-6 \left (1-\frac {e^2}{6}\right ) e^{x+16 e^x+2 e^{2 x}+124} x-6 e^{3 e^{2 x}+187}-e^{x+16 e^x+2 e^{2 x}+126}\right )}{\left (-e^{x+16 e^x} x+3 e^{e^{2 x}+61}+e^{2 x+e^{2 x}+63}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 e^{x+e^{2 x}+61} \left (2 e^{2 \left (x+8 e^x\right )} x-16 e^{x+16 e^x} x+2 e^{x+e^{2 x}+63}-e^{16 e^x} (x+1)\right )}{\left (-e^{x+16 e^x} x+3 e^{e^{2 x}+61}+e^{2 x+e^{2 x}+63}\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int \frac {e^{x+e^{2 x}+61} \left (2 e^{2 \left (x+8 e^x\right )} x-16 e^{x+16 e^x} x+2 e^{x+e^{2 x}+63}-e^{16 e^x} (x+1)\right )}{\left (-e^{x+16 e^x} x+3 e^{61+e^{2 x}}+e^{2 x+e^{2 x}+63}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {2 e^{x+16 e^x-2} x}{-e^{x+16 e^x} x+3 e^{61+e^{2 x}}+e^{2 x+e^{2 x}+63}}+\frac {e^{x-2} \left (2 e^{x+32 e^x} x^2-16 e^{x+16 e^x+e^{2 x}+63} x-6 e^{61+16 e^x+e^{2 x}} \left (1+\frac {e^2}{6}\right ) x-e^{63+16 e^x+e^{2 x}}+2 e^{x+2 e^{2 x}+126}\right )}{\left (-e^{x+16 e^x} x+3 e^{61+e^{2 x}}+e^{2 x+e^{2 x}+63}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (2 \int \frac {e^{2 \left (x+16 e^x-1\right )} x^2}{\left (-e^{x+16 e^x} x+3 e^{61+e^{2 x}}+e^{2 x+e^{2 x}+63}\right )^2}dx+2 \int \frac {e^{2 \left (x+e^{2 x}+62\right )}}{\left (-e^{x+16 e^x} x+3 e^{61+e^{2 x}}+e^{2 x+e^{2 x}+63}\right )^2}dx-\int \frac {e^{x+16 e^x+e^{2 x}+61}}{\left (-e^{x+16 e^x} x+3 e^{61+e^{2 x}}+e^{2 x+e^{2 x}+63}\right )^2}dx-\left (6+e^2\right ) \int \frac {e^{x+16 e^x+e^{2 x}+59} x}{\left (-e^{x+16 e^x} x+3 e^{61+e^{2 x}}+e^{2 x+e^{2 x}+63}\right )^2}dx-16 \int \frac {e^{2 x+16 e^x+e^{2 x}+61} x}{\left (-e^{x+16 e^x} x+3 e^{61+e^{2 x}}+e^{2 x+e^{2 x}+63}\right )^2}dx+2 \int \frac {e^{x+16 e^x-2} x}{-e^{x+16 e^x} x+3 e^{61+e^{2 x}}+e^{2 x+e^{2 x}+63}}dx\right )\) |
Int[(4*E^(2 + 2*x) + E^(-61 + 16*E^x - E^(2*x) + x)*(-2 - 2*x - 32*E^x*x + 4*E^(2*x)*x))/(9 + 6*E^(2 + 2*x) + E^(4 + 4*x) + E^(-122 + 32*E^x - 2*E^( 2*x) + 2*x)*x^2 + E^(-61 + 16*E^x - E^(2*x) + x)*(-6*x - 2*E^(2 + 2*x)*x)) ,x]
3.20.79.3.1 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[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.31 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97
| method | result | size |
| risch | \(-\frac {2}{{\mathrm e}^{2+2 x}-{\mathrm e}^{-{\mathrm e}^{2 x}+16 \,{\mathrm e}^{x}+x -61} x +3}\) | \(30\) |
| parallelrisch | \(\frac {2}{{\mathrm e}^{-{\mathrm e}^{2 x}+16 \,{\mathrm e}^{x}+x -61} x -{\mathrm e}^{2+2 x}-3}\) | \(31\) |
int(((4*x*exp(x)^2-32*exp(x)*x-2*x-2)*exp(-exp(x)^2+16*exp(x)+x-61)+4*exp( 2+2*x))/(x^2*exp(-exp(x)^2+16*exp(x)+x-61)^2+(-2*x*exp(2+2*x)-6*x)*exp(-ex p(x)^2+16*exp(x)+x-61)+exp(2+2*x)^2+6*exp(2+2*x)+9),x,method=_RETURNVERBOS E)
Time = 0.24 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \frac {4 e^{2+2 x}+e^{-61+16 e^x-e^{2 x}+x} \left (-2-2 x-32 e^x x+4 e^{2 x} x\right )}{9+6 e^{2+2 x}+e^{4+4 x}+e^{-122+32 e^x-2 e^{2 x}+2 x} x^2+e^{-61+16 e^x-e^{2 x}+x} \left (-6 x-2 e^{2+2 x} x\right )} \, dx=\frac {2}{x e^{\left (x - e^{\left (2 \, x\right )} + 16 \, e^{x} - 61\right )} - e^{\left (2 \, x + 2\right )} - 3} \]
integrate(((4*x*exp(x)^2-32*exp(x)*x-2*x-2)*exp(-exp(x)^2+16*exp(x)+x-61)+ 4*exp(2+2*x))/(x^2*exp(-exp(x)^2+16*exp(x)+x-61)^2+(-2*x*exp(2+2*x)-6*x)*e xp(-exp(x)^2+16*exp(x)+x-61)+exp(2+2*x)^2+6*exp(2+2*x)+9),x, algorithm=\
Time = 0.16 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {4 e^{2+2 x}+e^{-61+16 e^x-e^{2 x}+x} \left (-2-2 x-32 e^x x+4 e^{2 x} x\right )}{9+6 e^{2+2 x}+e^{4+4 x}+e^{-122+32 e^x-2 e^{2 x}+2 x} x^2+e^{-61+16 e^x-e^{2 x}+x} \left (-6 x-2 e^{2+2 x} x\right )} \, dx=\frac {2}{x e^{x - e^{2 x} + 16 e^{x} - 61} - e^{2} e^{2 x} - 3} \]
integrate(((4*x*exp(x)**2-32*exp(x)*x-2*x-2)*exp(-exp(x)**2+16*exp(x)+x-61 )+4*exp(2+2*x))/(x**2*exp(-exp(x)**2+16*exp(x)+x-61)**2+(-2*x*exp(2+2*x)-6 *x)*exp(-exp(x)**2+16*exp(x)+x-61)+exp(2+2*x)**2+6*exp(2+2*x)+9),x)
Time = 0.34 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.26 \[ \int \frac {4 e^{2+2 x}+e^{-61+16 e^x-e^{2 x}+x} \left (-2-2 x-32 e^x x+4 e^{2 x} x\right )}{9+6 e^{2+2 x}+e^{4+4 x}+e^{-122+32 e^x-2 e^{2 x}+2 x} x^2+e^{-61+16 e^x-e^{2 x}+x} \left (-6 x-2 e^{2+2 x} x\right )} \, dx=\frac {2 \, e^{\left (e^{\left (2 \, x\right )} + 61\right )}}{x e^{\left (x + 16 \, e^{x}\right )} - {\left (3 \, e^{61} + e^{\left (2 \, x + 63\right )}\right )} e^{\left (e^{\left (2 \, x\right )}\right )}} \]
integrate(((4*x*exp(x)^2-32*exp(x)*x-2*x-2)*exp(-exp(x)^2+16*exp(x)+x-61)+ 4*exp(2+2*x))/(x^2*exp(-exp(x)^2+16*exp(x)+x-61)^2+(-2*x*exp(2+2*x)-6*x)*e xp(-exp(x)^2+16*exp(x)+x-61)+exp(2+2*x)^2+6*exp(2+2*x)+9),x, algorithm=\
\[ \int \frac {4 e^{2+2 x}+e^{-61+16 e^x-e^{2 x}+x} \left (-2-2 x-32 e^x x+4 e^{2 x} x\right )}{9+6 e^{2+2 x}+e^{4+4 x}+e^{-122+32 e^x-2 e^{2 x}+2 x} x^2+e^{-61+16 e^x-e^{2 x}+x} \left (-6 x-2 e^{2+2 x} x\right )} \, dx=\int { \frac {2 \, {\left ({\left (2 \, x e^{\left (2 \, x\right )} - 16 \, x e^{x} - x - 1\right )} e^{\left (x - e^{\left (2 \, x\right )} + 16 \, e^{x} - 61\right )} + 2 \, e^{\left (2 \, x + 2\right )}\right )}}{x^{2} e^{\left (2 \, x - 2 \, e^{\left (2 \, x\right )} + 32 \, e^{x} - 122\right )} - 2 \, {\left (x e^{\left (2 \, x + 2\right )} + 3 \, x\right )} e^{\left (x - e^{\left (2 \, x\right )} + 16 \, e^{x} - 61\right )} + e^{\left (4 \, x + 4\right )} + 6 \, e^{\left (2 \, x + 2\right )} + 9} \,d x } \]
integrate(((4*x*exp(x)^2-32*exp(x)*x-2*x-2)*exp(-exp(x)^2+16*exp(x)+x-61)+ 4*exp(2+2*x))/(x^2*exp(-exp(x)^2+16*exp(x)+x-61)^2+(-2*x*exp(2+2*x)-6*x)*e xp(-exp(x)^2+16*exp(x)+x-61)+exp(2+2*x)^2+6*exp(2+2*x)+9),x, algorithm=\
integrate(2*((2*x*e^(2*x) - 16*x*e^x - x - 1)*e^(x - e^(2*x) + 16*e^x - 61 ) + 2*e^(2*x + 2))/(x^2*e^(2*x - 2*e^(2*x) + 32*e^x - 122) - 2*(x*e^(2*x + 2) + 3*x)*e^(x - e^(2*x) + 16*e^x - 61) + e^(4*x + 4) + 6*e^(2*x + 2) + 9 ), x)
Time = 10.56 (sec) , antiderivative size = 166, normalized size of antiderivative = 5.35 \[ \int \frac {4 e^{2+2 x}+e^{-61+16 e^x-e^{2 x}+x} \left (-2-2 x-32 e^x x+4 e^{2 x} x\right )}{9+6 e^{2+2 x}+e^{4+4 x}+e^{-122+32 e^x-2 e^{2 x}+2 x} x^2+e^{-61+16 e^x-e^{2 x}+x} \left (-6 x-2 e^{2+2 x} x\right )} \, dx=\frac {6\,x+96\,x^2\,{\mathrm {e}}^x+{\mathrm {e}}^{2\,x+2}\,\left (2\,x-2\,x^2\right )-12\,x^2\,{\mathrm {e}}^{2\,x}+32\,x^2\,{\mathrm {e}}^{3\,x+2}-4\,x^2\,{\mathrm {e}}^{4\,x+2}+6\,x^2}{\left ({\mathrm {e}}^{x-{\mathrm {e}}^{2\,x}+16\,{\mathrm {e}}^x-61}-\frac {{\mathrm {e}}^{2\,x+2}+3}{x}\right )\,\left (48\,x^3\,{\mathrm {e}}^x-6\,x^3\,{\mathrm {e}}^{2\,x}+x^2\,{\mathrm {e}}^{2\,x+2}-x^3\,{\mathrm {e}}^{2\,x+2}+16\,x^3\,{\mathrm {e}}^{3\,x+2}-2\,x^3\,{\mathrm {e}}^{4\,x+2}+3\,x^2+3\,x^3\right )} \]
int((4*exp(2*x + 2) - exp(x - exp(2*x) + 16*exp(x) - 61)*(2*x - 4*x*exp(2* x) + 32*x*exp(x) + 2))/(6*exp(2*x + 2) + exp(4*x + 4) - exp(x - exp(2*x) + 16*exp(x) - 61)*(6*x + 2*x*exp(2*x + 2)) + x^2*exp(2*x - 2*exp(2*x) + 32* exp(x) - 122) + 9),x)
(6*x + 96*x^2*exp(x) + exp(2*x + 2)*(2*x - 2*x^2) - 12*x^2*exp(2*x) + 32*x ^2*exp(3*x + 2) - 4*x^2*exp(4*x + 2) + 6*x^2)/((exp(x - exp(2*x) + 16*exp( x) - 61) - (exp(2*x + 2) + 3)/x)*(48*x^3*exp(x) - 6*x^3*exp(2*x) + x^2*exp (2*x + 2) - x^3*exp(2*x + 2) + 16*x^3*exp(3*x + 2) - 2*x^3*exp(4*x + 2) + 3*x^2 + 3*x^3))