Integrand size = 121, antiderivative size = 25 \[ \int \frac {e^{\frac {-192 x+58 x^2+2 x^3+e^{-6+x^2} \left (-24 x+8 x^2\right )}{8+e^{-6+x^2}}} \left (-1536+928 x+48 x^2+e^{-12+2 x^2} (-24+16 x)+e^{-6+x^2} \left (-384+244 x+6 x^2+12 x^3-4 x^4\right )\right )}{64+16 e^{-6+x^2}+e^{-12+2 x^2}} \, dx=3+e^{2 (-3+x) x \left (4+\frac {x}{8+e^{-6+x^2}}\right )} \] Output:
exp(2*(x/(exp(x^2-6)+8)+4)*(-3+x)*x)+3
Time = 0.13 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {e^{\frac {-192 x+58 x^2+2 x^3+e^{-6+x^2} \left (-24 x+8 x^2\right )}{8+e^{-6+x^2}}} \left (-1536+928 x+48 x^2+e^{-12+2 x^2} (-24+16 x)+e^{-6+x^2} \left (-384+244 x+6 x^2+12 x^3-4 x^4\right )\right )}{64+16 e^{-6+x^2}+e^{-12+2 x^2}} \, dx=e^{\frac {2 (-3+x) x \left (4 e^{x^2}+e^6 (32+x)\right )}{8 e^6+e^{x^2}}} \] Input:
Integrate[(E^((-192*x + 58*x^2 + 2*x^3 + E^(-6 + x^2)*(-24*x + 8*x^2))/(8 + E^(-6 + x^2)))*(-1536 + 928*x + 48*x^2 + E^(-12 + 2*x^2)*(-24 + 16*x) + E^(-6 + x^2)*(-384 + 244*x + 6*x^2 + 12*x^3 - 4*x^4)))/(64 + 16*E^(-6 + x^ 2) + E^(-12 + 2*x^2)),x]
Output:
E^((2*(-3 + x)*x*(4*E^x^2 + E^6*(32 + x)))/(8*E^6 + E^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.
| \(\displaystyle \int \frac {\left (48 x^2+e^{2 x^2-12} (16 x-24)+e^{x^2-6} \left (-4 x^4+12 x^3+6 x^2+244 x-384\right )+928 x-1536\right ) \exp \left (\frac {2 x^3+58 x^2+e^{x^2-6} \left (8 x^2-24 x\right )-192 x}{e^{x^2-6}+8}\right )}{16 e^{x^2-6}+e^{2 x^2-12}+64} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (48 x^2+e^{2 x^2-12} (16 x-24)+e^{x^2-6} \left (-4 x^4+12 x^3+6 x^2+244 x-384\right )+928 x-1536\right ) \exp \left (\frac {2 (x-3) x \left (4 e^{x^2}+e^6 x+32 e^6\right )}{e^6 \left (e^{x^2-6}+8\right )}+12\right )}{\left (e^{x^2}+8 e^6\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (8 (2 x-3) \exp \left (\frac {2 (x-3) x \left (4 e^{x^2}+e^6 x+32 e^6\right )}{e^6 \left (e^{x^2-6}+8\right )}\right )+\frac {32 (x-3) x^3 \exp \left (\frac {2 (x-3) x \left (4 e^{x^2}+e^6 x+32 e^6\right )}{e^6 \left (e^{x^2-6}+8\right )}+12\right )}{\left (e^{x^2}+8 e^6\right )^2}-\frac {2 \left (2 x^3-6 x^2-3 x+6\right ) x \exp \left (\frac {2 (x-3) x \left (4 e^{x^2}+e^6 x+32 e^6\right )}{e^6 \left (e^{x^2-6}+8\right )}+6\right )}{e^{x^2}+8 e^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -24 \int \exp \left (\frac {2 (x-3) x \left (e^6 x+4 e^{x^2}+32 e^6\right )}{e^6 \left (8+e^{x^2-6}\right )}\right )dx+16 \int \exp \left (\frac {2 (x-3) x \left (e^6 x+4 e^{x^2}+32 e^6\right )}{e^6 \left (8+e^{x^2-6}\right )}\right ) xdx-12 \int \frac {\exp \left (\frac {2 (x-3) x \left (e^6 x+4 e^{x^2}+32 e^6\right )}{e^6 \left (8+e^{x^2-6}\right )}+6\right ) x}{8 e^6+e^{x^2}}dx+6 \int \frac {\exp \left (\frac {2 (x-3) x \left (e^6 x+4 e^{x^2}+32 e^6\right )}{e^6 \left (8+e^{x^2-6}\right )}+6\right ) x^2}{8 e^6+e^{x^2}}dx+32 \int \frac {\exp \left (\frac {2 (x-3) x \left (e^6 x+4 e^{x^2}+32 e^6\right )}{e^6 \left (8+e^{x^2-6}\right )}+12\right ) x^4}{\left (8 e^6+e^{x^2}\right )^2}dx-4 \int \frac {\exp \left (\frac {2 (x-3) x \left (e^6 x+4 e^{x^2}+32 e^6\right )}{e^6 \left (8+e^{x^2-6}\right )}+6\right ) x^4}{8 e^6+e^{x^2}}dx-96 \int \frac {\exp \left (\frac {2 (x-3) x \left (e^6 x+4 e^{x^2}+32 e^6\right )}{e^6 \left (8+e^{x^2-6}\right )}+12\right ) x^3}{\left (8 e^6+e^{x^2}\right )^2}dx+12 \int \frac {\exp \left (\frac {2 (x-3) x \left (e^6 x+4 e^{x^2}+32 e^6\right )}{e^6 \left (8+e^{x^2-6}\right )}+6\right ) x^3}{8 e^6+e^{x^2}}dx\) |
Input:
Int[(E^((-192*x + 58*x^2 + 2*x^3 + E^(-6 + x^2)*(-24*x + 8*x^2))/(8 + E^(- 6 + x^2)))*(-1536 + 928*x + 48*x^2 + E^(-12 + 2*x^2)*(-24 + 16*x) + E^(-6 + x^2)*(-384 + 244*x + 6*x^2 + 12*x^3 - 4*x^4)))/(64 + 16*E^(-6 + x^2) + E ^(-12 + 2*x^2)),x]
Output:
$Aborted
Time = 0.58 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16
| method | result | size |
| risch | \({\mathrm e}^{\frac {2 x \left (-3+x \right ) \left (x +4 \,{\mathrm e}^{x^{2}-6}+32\right )}{{\mathrm e}^{x^{2}-6}+8}}\) | \(29\) |
| parallelrisch | \({\mathrm e}^{\frac {\left (8 x^{2}-24 x \right ) {\mathrm e}^{x^{2}-6}+2 x^{3}+58 x^{2}-192 x}{{\mathrm e}^{x^{2}-6}+8}}\) | \(43\) |
| norman | \(\frac {{\mathrm e}^{x^{2}-6} {\mathrm e}^{\frac {\left (8 x^{2}-24 x \right ) {\mathrm e}^{x^{2}-6}+2 x^{3}+58 x^{2}-192 x}{{\mathrm e}^{x^{2}-6}+8}}+8 \,{\mathrm e}^{\frac {\left (8 x^{2}-24 x \right ) {\mathrm e}^{x^{2}-6}+2 x^{3}+58 x^{2}-192 x}{{\mathrm e}^{x^{2}-6}+8}}}{{\mathrm e}^{x^{2}-6}+8}\) | \(106\) |
Input:
int(((16*x-24)*exp(x^2-6)^2+(-4*x^4+12*x^3+6*x^2+244*x-384)*exp(x^2-6)+48* x^2+928*x-1536)*exp(((8*x^2-24*x)*exp(x^2-6)+2*x^3+58*x^2-192*x)/(exp(x^2- 6)+8))/(exp(x^2-6)^2+16*exp(x^2-6)+64),x,method=_RETURNVERBOSE)
Output:
exp(2*x*(-3+x)*(x+4*exp(x^2-6)+32)/(exp(x^2-6)+8))
Time = 0.09 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.60 \[ \int \frac {e^{\frac {-192 x+58 x^2+2 x^3+e^{-6+x^2} \left (-24 x+8 x^2\right )}{8+e^{-6+x^2}}} \left (-1536+928 x+48 x^2+e^{-12+2 x^2} (-24+16 x)+e^{-6+x^2} \left (-384+244 x+6 x^2+12 x^3-4 x^4\right )\right )}{64+16 e^{-6+x^2}+e^{-12+2 x^2}} \, dx=e^{\left (\frac {2 \, {\left (x^{3} + 29 \, x^{2} + 4 \, {\left (x^{2} - 3 \, x\right )} e^{\left (x^{2} - 6\right )} - 96 \, x\right )}}{e^{\left (x^{2} - 6\right )} + 8}\right )} \] Input:
integrate(((16*x-24)*exp(x^2-6)^2+(-4*x^4+12*x^3+6*x^2+244*x-384)*exp(x^2- 6)+48*x^2+928*x-1536)*exp(((8*x^2-24*x)*exp(x^2-6)+2*x^3+58*x^2-192*x)/(ex p(x^2-6)+8))/(exp(x^2-6)^2+16*exp(x^2-6)+64),x, algorithm="fricas")
Output:
e^(2*(x^3 + 29*x^2 + 4*(x^2 - 3*x)*e^(x^2 - 6) - 96*x)/(e^(x^2 - 6) + 8))
Time = 0.30 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {e^{\frac {-192 x+58 x^2+2 x^3+e^{-6+x^2} \left (-24 x+8 x^2\right )}{8+e^{-6+x^2}}} \left (-1536+928 x+48 x^2+e^{-12+2 x^2} (-24+16 x)+e^{-6+x^2} \left (-384+244 x+6 x^2+12 x^3-4 x^4\right )\right )}{64+16 e^{-6+x^2}+e^{-12+2 x^2}} \, dx=e^{\frac {2 x^{3} + 58 x^{2} - 192 x + \left (8 x^{2} - 24 x\right ) e^{x^{2} - 6}}{e^{x^{2} - 6} + 8}} \] Input:
integrate(((16*x-24)*exp(x**2-6)**2+(-4*x**4+12*x**3+6*x**2+244*x-384)*exp (x**2-6)+48*x**2+928*x-1536)*exp(((8*x**2-24*x)*exp(x**2-6)+2*x**3+58*x**2 -192*x)/(exp(x**2-6)+8))/(exp(x**2-6)**2+16*exp(x**2-6)+64),x)
Output:
exp((2*x**3 + 58*x**2 - 192*x + (8*x**2 - 24*x)*exp(x**2 - 6))/(exp(x**2 - 6) + 8))
Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (23) = 46\).
Time = 0.32 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.68 \[ \int \frac {e^{\frac {-192 x+58 x^2+2 x^3+e^{-6+x^2} \left (-24 x+8 x^2\right )}{8+e^{-6+x^2}}} \left (-1536+928 x+48 x^2+e^{-12+2 x^2} (-24+16 x)+e^{-6+x^2} \left (-384+244 x+6 x^2+12 x^3-4 x^4\right )\right )}{64+16 e^{-6+x^2}+e^{-12+2 x^2}} \, dx=e^{\left (\frac {2 \, x^{3} e^{6}}{8 \, e^{6} + e^{\left (x^{2}\right )}} + \frac {58 \, x^{2} e^{6}}{8 \, e^{6} + e^{\left (x^{2}\right )}} + \frac {8 \, x^{2} e^{\left (x^{2}\right )}}{8 \, e^{6} + e^{\left (x^{2}\right )}} - \frac {192 \, x e^{6}}{8 \, e^{6} + e^{\left (x^{2}\right )}} - \frac {24 \, x e^{\left (x^{2}\right )}}{8 \, e^{6} + e^{\left (x^{2}\right )}}\right )} \] Input:
integrate(((16*x-24)*exp(x^2-6)^2+(-4*x^4+12*x^3+6*x^2+244*x-384)*exp(x^2- 6)+48*x^2+928*x-1536)*exp(((8*x^2-24*x)*exp(x^2-6)+2*x^3+58*x^2-192*x)/(ex p(x^2-6)+8))/(exp(x^2-6)^2+16*exp(x^2-6)+64),x, algorithm="maxima")
Output:
e^(2*x^3*e^6/(8*e^6 + e^(x^2)) + 58*x^2*e^6/(8*e^6 + e^(x^2)) + 8*x^2*e^(x ^2)/(8*e^6 + e^(x^2)) - 192*x*e^6/(8*e^6 + e^(x^2)) - 24*x*e^(x^2)/(8*e^6 + e^(x^2)))
\[ \int \frac {e^{\frac {-192 x+58 x^2+2 x^3+e^{-6+x^2} \left (-24 x+8 x^2\right )}{8+e^{-6+x^2}}} \left (-1536+928 x+48 x^2+e^{-12+2 x^2} (-24+16 x)+e^{-6+x^2} \left (-384+244 x+6 x^2+12 x^3-4 x^4\right )\right )}{64+16 e^{-6+x^2}+e^{-12+2 x^2}} \, dx=\int { \frac {2 \, {\left (24 \, x^{2} + 4 \, {\left (2 \, x - 3\right )} e^{\left (2 \, x^{2} - 12\right )} - {\left (2 \, x^{4} - 6 \, x^{3} - 3 \, x^{2} - 122 \, x + 192\right )} e^{\left (x^{2} - 6\right )} + 464 \, x - 768\right )} e^{\left (\frac {2 \, {\left (x^{3} + 29 \, x^{2} + 4 \, {\left (x^{2} - 3 \, x\right )} e^{\left (x^{2} - 6\right )} - 96 \, x\right )}}{e^{\left (x^{2} - 6\right )} + 8}\right )}}{e^{\left (2 \, x^{2} - 12\right )} + 16 \, e^{\left (x^{2} - 6\right )} + 64} \,d x } \] Input:
integrate(((16*x-24)*exp(x^2-6)^2+(-4*x^4+12*x^3+6*x^2+244*x-384)*exp(x^2- 6)+48*x^2+928*x-1536)*exp(((8*x^2-24*x)*exp(x^2-6)+2*x^3+58*x^2-192*x)/(ex p(x^2-6)+8))/(exp(x^2-6)^2+16*exp(x^2-6)+64),x, algorithm="giac")
Output:
integrate(2*(24*x^2 + 4*(2*x - 3)*e^(2*x^2 - 12) - (2*x^4 - 6*x^3 - 3*x^2 - 122*x + 192)*e^(x^2 - 6) + 464*x - 768)*e^(2*(x^3 + 29*x^2 + 4*(x^2 - 3* x)*e^(x^2 - 6) - 96*x)/(e^(x^2 - 6) + 8))/(e^(2*x^2 - 12) + 16*e^(x^2 - 6) + 64), x)
Time = 0.18 (sec) , antiderivative size = 96, normalized size of antiderivative = 3.84 \[ \int \frac {e^{\frac {-192 x+58 x^2+2 x^3+e^{-6+x^2} \left (-24 x+8 x^2\right )}{8+e^{-6+x^2}}} \left (-1536+928 x+48 x^2+e^{-12+2 x^2} (-24+16 x)+e^{-6+x^2} \left (-384+244 x+6 x^2+12 x^3-4 x^4\right )\right )}{64+16 e^{-6+x^2}+e^{-12+2 x^2}} \, dx={\mathrm {e}}^{-\frac {24\,x\,{\mathrm {e}}^{x^2}}{{\mathrm {e}}^{x^2}+8\,{\mathrm {e}}^6}}\,{\mathrm {e}}^{-\frac {192\,x\,{\mathrm {e}}^6}{{\mathrm {e}}^{x^2}+8\,{\mathrm {e}}^6}}\,{\mathrm {e}}^{\frac {8\,x^2\,{\mathrm {e}}^{x^2}}{{\mathrm {e}}^{x^2}+8\,{\mathrm {e}}^6}}\,{\mathrm {e}}^{\frac {2\,x^3\,{\mathrm {e}}^6}{{\mathrm {e}}^{x^2}+8\,{\mathrm {e}}^6}}\,{\mathrm {e}}^{\frac {58\,x^2\,{\mathrm {e}}^6}{{\mathrm {e}}^{x^2}+8\,{\mathrm {e}}^6}} \] Input:
int((exp(-(192*x + exp(x^2 - 6)*(24*x - 8*x^2) - 58*x^2 - 2*x^3)/(exp(x^2 - 6) + 8))*(928*x + exp(2*x^2 - 12)*(16*x - 24) + exp(x^2 - 6)*(244*x + 6* x^2 + 12*x^3 - 4*x^4 - 384) + 48*x^2 - 1536))/(16*exp(x^2 - 6) + exp(2*x^2 - 12) + 64),x)
Output:
exp(-(24*x*exp(x^2))/(exp(x^2) + 8*exp(6)))*exp(-(192*x*exp(6))/(exp(x^2) + 8*exp(6)))*exp((8*x^2*exp(x^2))/(exp(x^2) + 8*exp(6)))*exp((2*x^3*exp(6) )/(exp(x^2) + 8*exp(6)))*exp((58*x^2*exp(6))/(exp(x^2) + 8*exp(6)))
Time = 67.48 (sec) , antiderivative size = 85, normalized size of antiderivative = 3.40 \[ \int \frac {e^{\frac {-192 x+58 x^2+2 x^3+e^{-6+x^2} \left (-24 x+8 x^2\right )}{8+e^{-6+x^2}}} \left (-1536+928 x+48 x^2+e^{-12+2 x^2} (-24+16 x)+e^{-6+x^2} \left (-384+244 x+6 x^2+12 x^3-4 x^4\right )\right )}{64+16 e^{-6+x^2}+e^{-12+2 x^2}} \, dx=\frac {e^{\frac {8 e^{x^{2}} x^{2}+2 e^{6} x^{3}+64 e^{6} x^{2}}{e^{x^{2}}+8 e^{6}}}}{e^{\frac {24 e^{x^{2}} x +6 e^{6} x^{2}+192 e^{6} x}{e^{x^{2}}+8 e^{6}}}} \] Input:
int(((16*x-24)*exp(x^2-6)^2+(-4*x^4+12*x^3+6*x^2+244*x-384)*exp(x^2-6)+48* x^2+928*x-1536)*exp(((8*x^2-24*x)*exp(x^2-6)+2*x^3+58*x^2-192*x)/(exp(x^2- 6)+8))/(exp(x^2-6)^2+16*exp(x^2-6)+64),x)
Output:
e**((8*e**(x**2)*x**2 + 2*e**6*x**3 + 64*e**6*x**2)/(e**(x**2) + 8*e**6))/ e**((24*e**(x**2)*x + 6*e**6*x**2 + 192*e**6*x)/(e**(x**2) + 8*e**6))