Integrand size = 269, antiderivative size = 36 \[ \int \frac {e^{\frac {144 x^2-e^{4 x^2} x^2-288 x^3-2 e^{2 x^2} x^3+143 x^4+\left (288 x-288 x^2\right ) \log (x)+144 \log ^2(x)}{9 e^{4 x^2}+18 e^{2 x^2} x+9 x^2}} \left (288 x^2-2 e^{6 x^2} x^2-288 x^3-6 e^{4 x^2} x^3-288 x^4+286 x^5+e^{2 x^2} \left (288 x-864 x^3-582 x^4+2304 x^5-1152 x^6\right )+\left (288 x-288 x^2+e^{2 x^2} \left (288+288 x-576 x^2-2304 x^3+2304 x^4\right )\right ) \log (x)+\left (-288 x-1152 e^{2 x^2} x^2\right ) \log ^2(x)\right )}{9 e^{6 x^2} x+27 e^{4 x^2} x^2+27 e^{2 x^2} x^3+9 x^4} \, dx=e^{-\frac {x^2}{9}+\frac {16 \left (-x+x^2-\log (x)\right )^2}{\left (e^{2 x^2}+x\right )^2}} \] Output:
exp(16/(exp(x^2)^2+x)^2*(x^2-x-ln(x))^2-1/9*x^2)
Leaf count is larger than twice the leaf count of optimal. \(77\) vs. \(2(36)=72\).
Time = 0.42 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.14 \[ \int \frac {e^{\frac {144 x^2-e^{4 x^2} x^2-288 x^3-2 e^{2 x^2} x^3+143 x^4+\left (288 x-288 x^2\right ) \log (x)+144 \log ^2(x)}{9 e^{4 x^2}+18 e^{2 x^2} x+9 x^2}} \left (288 x^2-2 e^{6 x^2} x^2-288 x^3-6 e^{4 x^2} x^3-288 x^4+286 x^5+e^{2 x^2} \left (288 x-864 x^3-582 x^4+2304 x^5-1152 x^6\right )+\left (288 x-288 x^2+e^{2 x^2} \left (288+288 x-576 x^2-2304 x^3+2304 x^4\right )\right ) \log (x)+\left (-288 x-1152 e^{2 x^2} x^2\right ) \log ^2(x)\right )}{9 e^{6 x^2} x+27 e^{4 x^2} x^2+27 e^{2 x^2} x^3+9 x^4} \, dx=e^{\frac {x^2 \left (144-e^{4 x^2}-288 x-2 e^{2 x^2} x+143 x^2\right )+144 \log ^2(x)}{9 \left (e^{2 x^2}+x\right )^2}} x^{-\frac {32 (-1+x) x}{\left (e^{2 x^2}+x\right )^2}} \] Input:
Integrate[(E^((144*x^2 - E^(4*x^2)*x^2 - 288*x^3 - 2*E^(2*x^2)*x^3 + 143*x ^4 + (288*x - 288*x^2)*Log[x] + 144*Log[x]^2)/(9*E^(4*x^2) + 18*E^(2*x^2)* x + 9*x^2))*(288*x^2 - 2*E^(6*x^2)*x^2 - 288*x^3 - 6*E^(4*x^2)*x^3 - 288*x ^4 + 286*x^5 + E^(2*x^2)*(288*x - 864*x^3 - 582*x^4 + 2304*x^5 - 1152*x^6) + (288*x - 288*x^2 + E^(2*x^2)*(288 + 288*x - 576*x^2 - 2304*x^3 + 2304*x ^4))*Log[x] + (-288*x - 1152*E^(2*x^2)*x^2)*Log[x]^2))/(9*E^(6*x^2)*x + 27 *E^(4*x^2)*x^2 + 27*E^(2*x^2)*x^3 + 9*x^4),x]
Output:
E^((x^2*(144 - E^(4*x^2) - 288*x - 2*E^(2*x^2)*x + 143*x^2) + 144*Log[x]^2 )/(9*(E^(2*x^2) + x)^2))/x^((32*(-1 + x)*x)/(E^(2*x^2) + 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.
| \(\displaystyle \int \frac {\left (286 x^5-288 x^4-288 x^3-2 e^{6 x^2} x^2+288 x^2+\left (-1152 e^{2 x^2} x^2-288 x\right ) \log ^2(x)-6 e^{4 x^2} x^3+\left (-288 x^2+e^{2 x^2} \left (2304 x^4-2304 x^3-576 x^2+288 x+288\right )+288 x\right ) \log (x)+e^{2 x^2} \left (-1152 x^6+2304 x^5-582 x^4-864 x^3+288 x\right )\right ) \exp \left (\frac {143 x^4-288 x^3-e^{4 x^2} x^2+144 x^2+\left (288 x-288 x^2\right ) \log (x)-2 e^{2 x^2} x^3+144 \log ^2(x)}{9 x^2+18 e^{2 x^2} x+9 e^{4 x^2}}\right )}{9 x^4+27 e^{4 x^2} x^2+9 e^{6 x^2} x+27 e^{2 x^2} x^3} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (286 x^5-288 x^4-288 x^3-2 e^{6 x^2} x^2+288 x^2+\left (-1152 e^{2 x^2} x^2-288 x\right ) \log ^2(x)-6 e^{4 x^2} x^3+\left (-288 x^2+e^{2 x^2} \left (2304 x^4-2304 x^3-576 x^2+288 x+288\right )+288 x\right ) \log (x)+e^{2 x^2} \left (-1152 x^6+2304 x^5-582 x^4-864 x^3+288 x\right )\right ) \exp \left (\frac {143 x^4-288 x^3-e^{4 x^2} x^2+144 x^2+\left (288 x-288 x^2\right ) \log (x)-2 e^{2 x^2} x^3+144 \log ^2(x)}{9 \left (e^{2 x^2}+x\right )^2}\right )}{9 x \left (e^{2 x^2}+x\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int \frac {2 \exp \left (\frac {143 x^4-2 e^{2 x^2} x^3-288 x^3-e^{4 x^2} x^2+144 x^2+144 \log ^2(x)}{9 \left (x+e^{2 x^2}\right )^2}\right ) x^{\frac {288 x-288 x^2}{9 \left (x+e^{2 x^2}\right )^2}-1} \left (143 x^5-144 x^4-3 e^{4 x^2} x^3-144 x^3-e^{6 x^2} x^2+144 x^2-144 \left (4 e^{2 x^2} x^2+x\right ) \log ^2(x)+3 e^{2 x^2} \left (-192 x^6+384 x^5-97 x^4-144 x^3+48 x\right )+144 \left (-x^2+x+e^{2 x^2} \left (8 x^4-8 x^3-2 x^2+x+1\right )\right ) \log (x)\right )}{\left (x+e^{2 x^2}\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{9} \int \frac {\exp \left (\frac {143 x^4-2 e^{2 x^2} x^3-288 x^3-e^{4 x^2} x^2+144 x^2+144 \log ^2(x)}{9 \left (x+e^{2 x^2}\right )^2}\right ) x^{\frac {32 \left (x-x^2\right )}{\left (x+e^{2 x^2}\right )^2}-1} \left (143 x^5-144 x^4-3 e^{4 x^2} x^3-144 x^3-e^{6 x^2} x^2+144 x^2-144 \left (4 e^{2 x^2} x^2+x\right ) \log ^2(x)+3 e^{2 x^2} \left (-192 x^6+384 x^5-97 x^4-144 x^3+48 x\right )+144 \left (-x^2+x+e^{2 x^2} \left (8 x^4-8 x^3-2 x^2+x+1\right )\right ) \log (x)\right )}{\left (x+e^{2 x^2}\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2}{9} \int \left (\frac {144 \exp \left (\frac {143 x^4-2 e^{2 x^2} x^3-288 x^3-e^{4 x^2} x^2+144 x^2+144 \log ^2(x)}{9 \left (x+e^{2 x^2}\right )^2}\right ) \left (4 x^2-1\right ) \left (x^2-x-\log (x)\right )^2 x^{\frac {32 \left (x-x^2\right )}{\left (x+e^{2 x^2}\right )^2}}}{\left (x+e^{2 x^2}\right )^3}-\frac {144 \exp \left (\frac {143 x^4-2 e^{2 x^2} x^3-288 x^3-e^{4 x^2} x^2+144 x^2+144 \log ^2(x)}{9 \left (x+e^{2 x^2}\right )^2}\right ) \left (4 x^6-8 x^5-8 \log (x) x^4+2 x^4+8 \log (x) x^3+3 x^3+4 \log ^2(x) x^2+2 \log (x) x^2-\log (x) x-x-\log (x)\right ) x^{\frac {32 \left (x-x^2\right )}{\left (x+e^{2 x^2}\right )^2}-1}}{\left (x+e^{2 x^2}\right )^2}-\exp \left (\frac {143 x^4-2 e^{2 x^2} x^3-288 x^3-e^{4 x^2} x^2+144 x^2+144 \log ^2(x)}{9 \left (x+e^{2 x^2}\right )^2}\right ) x^{\frac {32 \left (x-x^2\right )}{\left (x+e^{2 x^2}\right )^2}+1}\right )dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {2}{9} \int \left (\frac {144 \exp \left (\frac {143 x^4-2 e^{2 x^2} x^3-288 x^3-e^{4 x^2} x^2+144 x^2+144 \log ^2(x)}{9 \left (x+e^{2 x^2}\right )^2}\right ) \left (4 x^2-1\right ) \left (x^2-x-\log (x)\right )^2 x^{\frac {32 (1-x) x}{\left (x+e^{2 x^2}\right )^2}}}{\left (x+e^{2 x^2}\right )^3}-\frac {144 \exp \left (\frac {143 x^4-2 e^{2 x^2} x^3-288 x^3-e^{4 x^2} x^2+144 x^2+144 \log ^2(x)}{9 \left (x+e^{2 x^2}\right )^2}\right ) \left (4 x^6-8 x^5-8 \log (x) x^4+2 x^4+8 \log (x) x^3+3 x^3+4 \log ^2(x) x^2+2 \log (x) x^2-\log (x) x-x-\log (x)\right ) x^{\frac {32 \left (x-x^2\right )}{\left (x+e^{2 x^2}\right )^2}-1}}{\left (x+e^{2 x^2}\right )^2}-\exp \left (\frac {143 x^4-2 e^{2 x^2} x^3-288 x^3-e^{4 x^2} x^2+144 x^2+144 \log ^2(x)}{9 \left (x+e^{2 x^2}\right )^2}\right ) x^{\frac {32 \left (x-x^2\right )}{\left (x+e^{2 x^2}\right )^2}+1}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2}{9} \int \left (\frac {144 \exp \left (-\frac {-143 x^4+2 e^{2 x^2} x^3+288 x^3+e^{4 x^2} x^2-144 x^2-144 \log ^2(x)}{9 \left (x+e^{2 x^2}\right )^2}\right ) \left (4 x^2-1\right ) \left (-x^2+x+\log (x)\right )^2 x^{-\frac {32 (x-1) x}{\left (x+e^{2 x^2}\right )^2}}}{\left (x+e^{2 x^2}\right )^3}-\exp \left (-\frac {-143 x^4+2 e^{2 x^2} x^3+288 x^3+e^{4 x^2} x^2-144 x^2-144 \log ^2(x)}{9 \left (x+e^{2 x^2}\right )^2}\right ) x^{1-\frac {32 (x-1) x}{\left (x+e^{2 x^2}\right )^2}}+\frac {144 \exp \left (-\frac {-143 x^4+2 e^{2 x^2} x^3+288 x^3+e^{4 x^2} x^2-144 x^2-144 \log ^2(x)}{9 \left (x+e^{2 x^2}\right )^2}\right ) \left (-4 x^6+8 x^5+8 \log (x) x^4-2 x^4-8 \log (x) x^3-3 x^3-4 \log ^2(x) x^2-2 \log (x) x^2+\log (x) x+x+\log (x)\right ) x^{-\frac {32 x^2}{\left (x+e^{2 x^2}\right )^2}+\frac {32 x}{\left (x+e^{2 x^2}\right )^2}-1}}{\left (x+e^{2 x^2}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \frac {2}{9} \int \left (\frac {144 \exp \left (-\frac {-143 x^4+2 e^{2 x^2} x^3+288 x^3+e^{4 x^2} x^2-144 x^2-144 \log ^2(x)}{9 \left (x+e^{2 x^2}\right )^2}\right ) \left (4 x^2-1\right ) \left (-x^2+x+\log (x)\right )^2 x^{-\frac {32 (x-1) x}{\left (x+e^{2 x^2}\right )^2}}}{\left (x+e^{2 x^2}\right )^3}-\exp \left (-\frac {-143 x^4+2 e^{2 x^2} x^3+288 x^3+e^{4 x^2} x^2-144 x^2-144 \log ^2(x)}{9 \left (x+e^{2 x^2}\right )^2}\right ) x^{1-\frac {32 (x-1) x}{\left (x+e^{2 x^2}\right )^2}}+\frac {144 \exp \left (-\frac {-143 x^4+2 e^{2 x^2} x^3+288 x^3+e^{4 x^2} x^2-144 x^2-144 \log ^2(x)}{9 \left (x+e^{2 x^2}\right )^2}\right ) \left (-4 x^6+8 x^5+8 \log (x) x^4-2 x^4-8 \log (x) x^3-3 x^3-4 \log ^2(x) x^2-2 \log (x) x^2+\log (x) x+x+\log (x)\right ) x^{-\frac {32 x^2}{\left (x+e^{2 x^2}\right )^2}+\frac {32 x}{\left (x+e^{2 x^2}\right )^2}-1}}{\left (x+e^{2 x^2}\right )^2}\right )dx\) |
Input:
Int[(E^((144*x^2 - E^(4*x^2)*x^2 - 288*x^3 - 2*E^(2*x^2)*x^3 + 143*x^4 + ( 288*x - 288*x^2)*Log[x] + 144*Log[x]^2)/(9*E^(4*x^2) + 18*E^(2*x^2)*x + 9* x^2))*(288*x^2 - 2*E^(6*x^2)*x^2 - 288*x^3 - 6*E^(4*x^2)*x^3 - 288*x^4 + 2 86*x^5 + E^(2*x^2)*(288*x - 864*x^3 - 582*x^4 + 2304*x^5 - 1152*x^6) + (28 8*x - 288*x^2 + E^(2*x^2)*(288 + 288*x - 576*x^2 - 2304*x^3 + 2304*x^4))*L og[x] + (-288*x - 1152*E^(2*x^2)*x^2)*Log[x]^2))/(9*E^(6*x^2)*x + 27*E^(4* x^2)*x^2 + 27*E^(2*x^2)*x^3 + 9*x^4),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(80\) vs. \(2(32)=64\).
Time = 0.07 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.25
\[{\mathrm e}^{\frac {-2 x^{3} {\mathrm e}^{2 x^{2}}+143 x^{4}-x^{2} {\mathrm e}^{4 x^{2}}-288 x^{2} \ln \left (x \right )-288 x^{3}+144 \ln \left (x \right )^{2}+288 x \ln \left (x \right )+144 x^{2}}{9 \,{\mathrm e}^{4 x^{2}}+18 x \,{\mathrm e}^{2 x^{2}}+9 x^{2}}}\]
Input:
int(((-1152*x^2*exp(x^2)^2-288*x)*ln(x)^2+((2304*x^4-2304*x^3-576*x^2+288* x+288)*exp(x^2)^2-288*x^2+288*x)*ln(x)-2*x^2*exp(x^2)^6-6*x^3*exp(x^2)^4+( -1152*x^6+2304*x^5-582*x^4-864*x^3+288*x)*exp(x^2)^2+286*x^5-288*x^4-288*x ^3+288*x^2)*exp((144*ln(x)^2+(-288*x^2+288*x)*ln(x)-x^2*exp(x^2)^4-2*x^3*e xp(x^2)^2+143*x^4-288*x^3+144*x^2)/(9*exp(x^2)^4+18*x*exp(x^2)^2+9*x^2))/( 9*x*exp(x^2)^6+27*x^2*exp(x^2)^4+27*x^3*exp(x^2)^2+9*x^4),x)
Output:
exp(1/9*(-2*x^3*exp(2*x^2)+143*x^4-x^2*exp(4*x^2)-288*x^2*ln(x)-288*x^3+14 4*ln(x)^2+288*x*ln(x)+144*x^2)/(2*x*exp(2*x^2)+x^2+exp(4*x^2)))
Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (32) = 64\).
Time = 0.10 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.19 \[ \int \frac {e^{\frac {144 x^2-e^{4 x^2} x^2-288 x^3-2 e^{2 x^2} x^3+143 x^4+\left (288 x-288 x^2\right ) \log (x)+144 \log ^2(x)}{9 e^{4 x^2}+18 e^{2 x^2} x+9 x^2}} \left (288 x^2-2 e^{6 x^2} x^2-288 x^3-6 e^{4 x^2} x^3-288 x^4+286 x^5+e^{2 x^2} \left (288 x-864 x^3-582 x^4+2304 x^5-1152 x^6\right )+\left (288 x-288 x^2+e^{2 x^2} \left (288+288 x-576 x^2-2304 x^3+2304 x^4\right )\right ) \log (x)+\left (-288 x-1152 e^{2 x^2} x^2\right ) \log ^2(x)\right )}{9 e^{6 x^2} x+27 e^{4 x^2} x^2+27 e^{2 x^2} x^3+9 x^4} \, dx=e^{\left (\frac {143 \, x^{4} - 2 \, x^{3} e^{\left (2 \, x^{2}\right )} - 288 \, x^{3} - x^{2} e^{\left (4 \, x^{2}\right )} + 144 \, x^{2} - 288 \, {\left (x^{2} - x\right )} \log \left (x\right ) + 144 \, \log \left (x\right )^{2}}{9 \, {\left (x^{2} + 2 \, x e^{\left (2 \, x^{2}\right )} + e^{\left (4 \, x^{2}\right )}\right )}}\right )} \] Input:
integrate(((-1152*x^2*exp(x^2)^2-288*x)*log(x)^2+((2304*x^4-2304*x^3-576*x ^2+288*x+288)*exp(x^2)^2-288*x^2+288*x)*log(x)-2*x^2*exp(x^2)^6-6*x^3*exp( x^2)^4+(-1152*x^6+2304*x^5-582*x^4-864*x^3+288*x)*exp(x^2)^2+286*x^5-288*x ^4-288*x^3+288*x^2)*exp((144*log(x)^2+(-288*x^2+288*x)*log(x)-x^2*exp(x^2) ^4-2*x^3*exp(x^2)^2+143*x^4-288*x^3+144*x^2)/(9*exp(x^2)^4+18*x*exp(x^2)^2 +9*x^2))/(9*x*exp(x^2)^6+27*x^2*exp(x^2)^4+27*x^3*exp(x^2)^2+9*x^4),x, alg orithm="fricas")
Output:
e^(1/9*(143*x^4 - 2*x^3*e^(2*x^2) - 288*x^3 - x^2*e^(4*x^2) + 144*x^2 - 28 8*(x^2 - x)*log(x) + 144*log(x)^2)/(x^2 + 2*x*e^(2*x^2) + e^(4*x^2)))
Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (27) = 54\).
Time = 1.40 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.22 \[ \int \frac {e^{\frac {144 x^2-e^{4 x^2} x^2-288 x^3-2 e^{2 x^2} x^3+143 x^4+\left (288 x-288 x^2\right ) \log (x)+144 \log ^2(x)}{9 e^{4 x^2}+18 e^{2 x^2} x+9 x^2}} \left (288 x^2-2 e^{6 x^2} x^2-288 x^3-6 e^{4 x^2} x^3-288 x^4+286 x^5+e^{2 x^2} \left (288 x-864 x^3-582 x^4+2304 x^5-1152 x^6\right )+\left (288 x-288 x^2+e^{2 x^2} \left (288+288 x-576 x^2-2304 x^3+2304 x^4\right )\right ) \log (x)+\left (-288 x-1152 e^{2 x^2} x^2\right ) \log ^2(x)\right )}{9 e^{6 x^2} x+27 e^{4 x^2} x^2+27 e^{2 x^2} x^3+9 x^4} \, dx=e^{\frac {143 x^{4} - 2 x^{3} e^{2 x^{2}} - 288 x^{3} - x^{2} e^{4 x^{2}} + 144 x^{2} + \left (- 288 x^{2} + 288 x\right ) \log {\left (x \right )} + 144 \log {\left (x \right )}^{2}}{9 x^{2} + 18 x e^{2 x^{2}} + 9 e^{4 x^{2}}}} \] Input:
integrate(((-1152*x**2*exp(x**2)**2-288*x)*ln(x)**2+((2304*x**4-2304*x**3- 576*x**2+288*x+288)*exp(x**2)**2-288*x**2+288*x)*ln(x)-2*x**2*exp(x**2)**6 -6*x**3*exp(x**2)**4+(-1152*x**6+2304*x**5-582*x**4-864*x**3+288*x)*exp(x* *2)**2+286*x**5-288*x**4-288*x**3+288*x**2)*exp((144*ln(x)**2+(-288*x**2+2 88*x)*ln(x)-x**2*exp(x**2)**4-2*x**3*exp(x**2)**2+143*x**4-288*x**3+144*x* *2)/(9*exp(x**2)**4+18*x*exp(x**2)**2+9*x**2))/(9*x*exp(x**2)**6+27*x**2*e xp(x**2)**4+27*x**3*exp(x**2)**2+9*x**4),x)
Output:
exp((143*x**4 - 2*x**3*exp(2*x**2) - 288*x**3 - x**2*exp(4*x**2) + 144*x** 2 + (-288*x**2 + 288*x)*log(x) + 144*log(x)**2)/(9*x**2 + 18*x*exp(2*x**2) + 9*exp(4*x**2)))
\[ \int \frac {e^{\frac {144 x^2-e^{4 x^2} x^2-288 x^3-2 e^{2 x^2} x^3+143 x^4+\left (288 x-288 x^2\right ) \log (x)+144 \log ^2(x)}{9 e^{4 x^2}+18 e^{2 x^2} x+9 x^2}} \left (288 x^2-2 e^{6 x^2} x^2-288 x^3-6 e^{4 x^2} x^3-288 x^4+286 x^5+e^{2 x^2} \left (288 x-864 x^3-582 x^4+2304 x^5-1152 x^6\right )+\left (288 x-288 x^2+e^{2 x^2} \left (288+288 x-576 x^2-2304 x^3+2304 x^4\right )\right ) \log (x)+\left (-288 x-1152 e^{2 x^2} x^2\right ) \log ^2(x)\right )}{9 e^{6 x^2} x+27 e^{4 x^2} x^2+27 e^{2 x^2} x^3+9 x^4} \, dx=\int { \frac {2 \, {\left (143 \, x^{5} - 144 \, x^{4} - 3 \, x^{3} e^{\left (4 \, x^{2}\right )} - 144 \, x^{3} - x^{2} e^{\left (6 \, x^{2}\right )} - 144 \, {\left (4 \, x^{2} e^{\left (2 \, x^{2}\right )} + x\right )} \log \left (x\right )^{2} + 144 \, x^{2} - 3 \, {\left (192 \, x^{6} - 384 \, x^{5} + 97 \, x^{4} + 144 \, x^{3} - 48 \, x\right )} e^{\left (2 \, x^{2}\right )} - 144 \, {\left (x^{2} - {\left (8 \, x^{4} - 8 \, x^{3} - 2 \, x^{2} + x + 1\right )} e^{\left (2 \, x^{2}\right )} - x\right )} \log \left (x\right )\right )} e^{\left (\frac {143 \, x^{4} - 2 \, x^{3} e^{\left (2 \, x^{2}\right )} - 288 \, x^{3} - x^{2} e^{\left (4 \, x^{2}\right )} + 144 \, x^{2} - 288 \, {\left (x^{2} - x\right )} \log \left (x\right ) + 144 \, \log \left (x\right )^{2}}{9 \, {\left (x^{2} + 2 \, x e^{\left (2 \, x^{2}\right )} + e^{\left (4 \, x^{2}\right )}\right )}}\right )}}{9 \, {\left (x^{4} + 3 \, x^{3} e^{\left (2 \, x^{2}\right )} + 3 \, x^{2} e^{\left (4 \, x^{2}\right )} + x e^{\left (6 \, x^{2}\right )}\right )}} \,d x } \] Input:
integrate(((-1152*x^2*exp(x^2)^2-288*x)*log(x)^2+((2304*x^4-2304*x^3-576*x ^2+288*x+288)*exp(x^2)^2-288*x^2+288*x)*log(x)-2*x^2*exp(x^2)^6-6*x^3*exp( x^2)^4+(-1152*x^6+2304*x^5-582*x^4-864*x^3+288*x)*exp(x^2)^2+286*x^5-288*x ^4-288*x^3+288*x^2)*exp((144*log(x)^2+(-288*x^2+288*x)*log(x)-x^2*exp(x^2) ^4-2*x^3*exp(x^2)^2+143*x^4-288*x^3+144*x^2)/(9*exp(x^2)^4+18*x*exp(x^2)^2 +9*x^2))/(9*x*exp(x^2)^6+27*x^2*exp(x^2)^4+27*x^3*exp(x^2)^2+9*x^4),x, alg orithm="maxima")
Output:
2/9*integrate((143*x^5 - 144*x^4 - 3*x^3*e^(4*x^2) - 144*x^3 - x^2*e^(6*x^ 2) - 144*(4*x^2*e^(2*x^2) + x)*log(x)^2 + 144*x^2 - 3*(192*x^6 - 384*x^5 + 97*x^4 + 144*x^3 - 48*x)*e^(2*x^2) - 144*(x^2 - (8*x^4 - 8*x^3 - 2*x^2 + x + 1)*e^(2*x^2) - x)*log(x))*e^(1/9*(143*x^4 - 2*x^3*e^(2*x^2) - 288*x^3 - x^2*e^(4*x^2) + 144*x^2 - 288*(x^2 - x)*log(x) + 144*log(x)^2)/(x^2 + 2* x*e^(2*x^2) + e^(4*x^2)))/(x^4 + 3*x^3*e^(2*x^2) + 3*x^2*e^(4*x^2) + x*e^( 6*x^2)), x)
Exception generated. \[ \int \frac {e^{\frac {144 x^2-e^{4 x^2} x^2-288 x^3-2 e^{2 x^2} x^3+143 x^4+\left (288 x-288 x^2\right ) \log (x)+144 \log ^2(x)}{9 e^{4 x^2}+18 e^{2 x^2} x+9 x^2}} \left (288 x^2-2 e^{6 x^2} x^2-288 x^3-6 e^{4 x^2} x^3-288 x^4+286 x^5+e^{2 x^2} \left (288 x-864 x^3-582 x^4+2304 x^5-1152 x^6\right )+\left (288 x-288 x^2+e^{2 x^2} \left (288+288 x-576 x^2-2304 x^3+2304 x^4\right )\right ) \log (x)+\left (-288 x-1152 e^{2 x^2} x^2\right ) \log ^2(x)\right )}{9 e^{6 x^2} x+27 e^{4 x^2} x^2+27 e^{2 x^2} x^3+9 x^4} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(((-1152*x^2*exp(x^2)^2-288*x)*log(x)^2+((2304*x^4-2304*x^3-576*x ^2+288*x+288)*exp(x^2)^2-288*x^2+288*x)*log(x)-2*x^2*exp(x^2)^6-6*x^3*exp( x^2)^4+(-1152*x^6+2304*x^5-582*x^4-864*x^3+288*x)*exp(x^2)^2+286*x^5-288*x ^4-288*x^3+288*x^2)*exp((144*log(x)^2+(-288*x^2+288*x)*log(x)-x^2*exp(x^2) ^4-2*x^3*exp(x^2)^2+143*x^4-288*x^3+144*x^2)/(9*exp(x^2)^4+18*x*exp(x^2)^2 +9*x^2))/(9*x*exp(x^2)^6+27*x^2*exp(x^2)^4+27*x^3*exp(x^2)^2+9*x^4),x, alg orithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{-339738624,[2,9,18]%%%}+%%%{339738624,[2,9,16]%%%}+%%%{-12 7401984,[
Time = 4.56 (sec) , antiderivative size = 232, normalized size of antiderivative = 6.44 \[ \int \frac {e^{\frac {144 x^2-e^{4 x^2} x^2-288 x^3-2 e^{2 x^2} x^3+143 x^4+\left (288 x-288 x^2\right ) \log (x)+144 \log ^2(x)}{9 e^{4 x^2}+18 e^{2 x^2} x+9 x^2}} \left (288 x^2-2 e^{6 x^2} x^2-288 x^3-6 e^{4 x^2} x^3-288 x^4+286 x^5+e^{2 x^2} \left (288 x-864 x^3-582 x^4+2304 x^5-1152 x^6\right )+\left (288 x-288 x^2+e^{2 x^2} \left (288+288 x-576 x^2-2304 x^3+2304 x^4\right )\right ) \log (x)+\left (-288 x-1152 e^{2 x^2} x^2\right ) \log ^2(x)\right )}{9 e^{6 x^2} x+27 e^{4 x^2} x^2+27 e^{2 x^2} x^3+9 x^4} \, dx=x^{\frac {32\,\left (x-x^2\right )}{{\mathrm {e}}^{4\,x^2}+2\,x\,{\mathrm {e}}^{2\,x^2}+x^2}}\,{\mathrm {e}}^{\frac {144\,x^2}{9\,{\mathrm {e}}^{4\,x^2}+18\,x\,{\mathrm {e}}^{2\,x^2}+9\,x^2}}\,{\mathrm {e}}^{\frac {143\,x^4}{9\,{\mathrm {e}}^{4\,x^2}+18\,x\,{\mathrm {e}}^{2\,x^2}+9\,x^2}}\,{\mathrm {e}}^{-\frac {288\,x^3}{9\,{\mathrm {e}}^{4\,x^2}+18\,x\,{\mathrm {e}}^{2\,x^2}+9\,x^2}}\,{\mathrm {e}}^{\frac {144\,{\ln \left (x\right )}^2}{9\,{\mathrm {e}}^{4\,x^2}+18\,x\,{\mathrm {e}}^{2\,x^2}+9\,x^2}}\,{\mathrm {e}}^{-\frac {x^2\,{\mathrm {e}}^{4\,x^2}}{9\,{\mathrm {e}}^{4\,x^2}+18\,x\,{\mathrm {e}}^{2\,x^2}+9\,x^2}}\,{\mathrm {e}}^{-\frac {2\,x^3\,{\mathrm {e}}^{2\,x^2}}{9\,{\mathrm {e}}^{4\,x^2}+18\,x\,{\mathrm {e}}^{2\,x^2}+9\,x^2}} \] Input:
int(-(exp((144*log(x)^2 + log(x)*(288*x - 288*x^2) - 2*x^3*exp(2*x^2) - x^ 2*exp(4*x^2) + 144*x^2 - 288*x^3 + 143*x^4)/(9*exp(4*x^2) + 18*x*exp(2*x^2 ) + 9*x^2))*(exp(2*x^2)*(864*x^3 - 288*x + 582*x^4 - 2304*x^5 + 1152*x^6) - log(x)*(288*x + exp(2*x^2)*(288*x - 576*x^2 - 2304*x^3 + 2304*x^4 + 288) - 288*x^2) + log(x)^2*(288*x + 1152*x^2*exp(2*x^2)) + 6*x^3*exp(4*x^2) + 2*x^2*exp(6*x^2) - 288*x^2 + 288*x^3 + 288*x^4 - 286*x^5))/(9*x*exp(6*x^2) + 27*x^3*exp(2*x^2) + 27*x^2*exp(4*x^2) + 9*x^4),x)
Output:
x^((32*(x - x^2))/(exp(4*x^2) + 2*x*exp(2*x^2) + x^2))*exp((144*x^2)/(9*ex p(4*x^2) + 18*x*exp(2*x^2) + 9*x^2))*exp((143*x^4)/(9*exp(4*x^2) + 18*x*ex p(2*x^2) + 9*x^2))*exp(-(288*x^3)/(9*exp(4*x^2) + 18*x*exp(2*x^2) + 9*x^2) )*exp((144*log(x)^2)/(9*exp(4*x^2) + 18*x*exp(2*x^2) + 9*x^2))*exp(-(x^2*e xp(4*x^2))/(9*exp(4*x^2) + 18*x*exp(2*x^2) + 9*x^2))*exp(-(2*x^3*exp(2*x^2 ))/(9*exp(4*x^2) + 18*x*exp(2*x^2) + 9*x^2))
\[ \int \frac {e^{\frac {144 x^2-e^{4 x^2} x^2-288 x^3-2 e^{2 x^2} x^3+143 x^4+\left (288 x-288 x^2\right ) \log (x)+144 \log ^2(x)}{9 e^{4 x^2}+18 e^{2 x^2} x+9 x^2}} \left (288 x^2-2 e^{6 x^2} x^2-288 x^3-6 e^{4 x^2} x^3-288 x^4+286 x^5+e^{2 x^2} \left (288 x-864 x^3-582 x^4+2304 x^5-1152 x^6\right )+\left (288 x-288 x^2+e^{2 x^2} \left (288+288 x-576 x^2-2304 x^3+2304 x^4\right )\right ) \log (x)+\left (-288 x-1152 e^{2 x^2} x^2\right ) \log ^2(x)\right )}{9 e^{6 x^2} x+27 e^{4 x^2} x^2+27 e^{2 x^2} x^3+9 x^4} \, dx=\int \frac {\left (\left (-1152 x^{2} \left ({\mathrm e}^{x^{2}}\right )^{2}-288 x \right ) \mathrm {log}\left (x \right )^{2}+\left (\left (2304 x^{4}-2304 x^{3}-576 x^{2}+288 x +288\right ) \left ({\mathrm e}^{x^{2}}\right )^{2}-288 x^{2}+288 x \right ) \mathrm {log}\left (x \right )-2 x^{2} \left ({\mathrm e}^{x^{2}}\right )^{6}-6 x^{3} \left ({\mathrm e}^{x^{2}}\right )^{4}+\left (-1152 x^{6}+2304 x^{5}-582 x^{4}-864 x^{3}+288 x \right ) \left ({\mathrm e}^{x^{2}}\right )^{2}+286 x^{5}-288 x^{4}-288 x^{3}+288 x^{2}\right ) {\mathrm e}^{\frac {144 \mathrm {log}\left (x \right )^{2}+\left (-288 x^{2}+288 x \right ) \mathrm {log}\left (x \right )-x^{2} \left ({\mathrm e}^{x^{2}}\right )^{4}-2 x^{3} \left ({\mathrm e}^{x^{2}}\right )^{2}+143 x^{4}-288 x^{3}+144 x^{2}}{9 \left ({\mathrm e}^{x^{2}}\right )^{4}+18 x \left ({\mathrm e}^{x^{2}}\right )^{2}+9 x^{2}}}}{9 x \left ({\mathrm e}^{x^{2}}\right )^{6}+27 x^{2} \left ({\mathrm e}^{x^{2}}\right )^{4}+27 x^{3} \left ({\mathrm e}^{x^{2}}\right )^{2}+9 x^{4}}d x \] Input:
int(((-1152*x^2*exp(x^2)^2-288*x)*log(x)^2+((2304*x^4-2304*x^3-576*x^2+288 *x+288)*exp(x^2)^2-288*x^2+288*x)*log(x)-2*x^2*exp(x^2)^6-6*x^3*exp(x^2)^4 +(-1152*x^6+2304*x^5-582*x^4-864*x^3+288*x)*exp(x^2)^2+286*x^5-288*x^4-288 *x^3+288*x^2)*exp((144*log(x)^2+(-288*x^2+288*x)*log(x)-x^2*exp(x^2)^4-2*x ^3*exp(x^2)^2+143*x^4-288*x^3+144*x^2)/(9*exp(x^2)^4+18*x*exp(x^2)^2+9*x^2 ))/(9*x*exp(x^2)^6+27*x^2*exp(x^2)^4+27*x^3*exp(x^2)^2+9*x^4),x)
Output:
int(((-1152*x^2*exp(x^2)^2-288*x)*log(x)^2+((2304*x^4-2304*x^3-576*x^2+288 *x+288)*exp(x^2)^2-288*x^2+288*x)*log(x)-2*x^2*exp(x^2)^6-6*x^3*exp(x^2)^4 +(-1152*x^6+2304*x^5-582*x^4-864*x^3+288*x)*exp(x^2)^2+286*x^5-288*x^4-288 *x^3+288*x^2)*exp((144*log(x)^2+(-288*x^2+288*x)*log(x)-x^2*exp(x^2)^4-2*x ^3*exp(x^2)^2+143*x^4-288*x^3+144*x^2)/(9*exp(x^2)^4+18*x*exp(x^2)^2+9*x^2 ))/(9*x*exp(x^2)^6+27*x^2*exp(x^2)^4+27*x^3*exp(x^2)^2+9*x^4),x)