Integrand size = 259, antiderivative size = 22 \[ \int \frac {37968750000+16 e^{20 x^2}-31104 x+7593750000 x^2+607500000 x^4+24300000 x^6+486000 x^8+3888 x^{10}+e^{16 x^2} \left (6000+240 x^2\right )+e^{12 x^2} \left (900000+72000 x^2+1440 x^4\right )+e^{8 x^2} \left (67500000+8100000 x^2+324000 x^4+4320 x^6\right )+e^{4 x^2} \left (2531250000-41472 x+405000000 x^2+24300000 x^4+648000 x^6+6480 x^8\right )}{2373046875+e^{20 x^2}+474609375 x^2+37968750 x^4+1518750 x^6+30375 x^8+243 x^{10}+e^{16 x^2} \left (375+15 x^2\right )+e^{12 x^2} \left (56250+4500 x^2+90 x^4\right )+e^{8 x^2} \left (4218750+506250 x^2+20250 x^4+270 x^6\right )+e^{4 x^2} \left (158203125+25312500 x^2+1518750 x^4+40500 x^6+405 x^8\right )} \, dx=16 \left (x+\frac {1}{\left (25+\frac {e^{4 x^2}}{3}+x^2\right )^4}\right ) \] Output:
16/(x^2+1/3*exp(4*x^2)+25)^4+16*x
Leaf count is larger than twice the leaf count of optimal. \(84\) vs. \(2(22)=44\).
Time = 4.09 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.82 \[ \int \frac {37968750000+16 e^{20 x^2}-31104 x+7593750000 x^2+607500000 x^4+24300000 x^6+486000 x^8+3888 x^{10}+e^{16 x^2} \left (6000+240 x^2\right )+e^{12 x^2} \left (900000+72000 x^2+1440 x^4\right )+e^{8 x^2} \left (67500000+8100000 x^2+324000 x^4+4320 x^6\right )+e^{4 x^2} \left (2531250000-41472 x+405000000 x^2+24300000 x^4+648000 x^6+6480 x^8\right )}{2373046875+e^{20 x^2}+474609375 x^2+37968750 x^4+1518750 x^6+30375 x^8+243 x^{10}+e^{16 x^2} \left (375+15 x^2\right )+e^{12 x^2} \left (56250+4500 x^2+90 x^4\right )+e^{8 x^2} \left (4218750+506250 x^2+20250 x^4+270 x^6\right )+e^{4 x^2} \left (158203125+25312500 x^2+1518750 x^4+40500 x^6+405 x^8\right )} \, dx=\frac {16 \left (81+\left (75+e^{4 x^2}\right )^4 x+12 \left (75+e^{4 x^2}\right )^3 x^3+54 \left (75+e^{4 x^2}\right )^2 x^5+108 \left (75+e^{4 x^2}\right ) x^7+81 x^9\right )}{\left (75+e^{4 x^2}+3 x^2\right )^4} \] Input:
Integrate[(37968750000 + 16*E^(20*x^2) - 31104*x + 7593750000*x^2 + 607500 000*x^4 + 24300000*x^6 + 486000*x^8 + 3888*x^10 + E^(16*x^2)*(6000 + 240*x ^2) + E^(12*x^2)*(900000 + 72000*x^2 + 1440*x^4) + E^(8*x^2)*(67500000 + 8 100000*x^2 + 324000*x^4 + 4320*x^6) + E^(4*x^2)*(2531250000 - 41472*x + 40 5000000*x^2 + 24300000*x^4 + 648000*x^6 + 6480*x^8))/(2373046875 + E^(20*x ^2) + 474609375*x^2 + 37968750*x^4 + 1518750*x^6 + 30375*x^8 + 243*x^10 + E^(16*x^2)*(375 + 15*x^2) + E^(12*x^2)*(56250 + 4500*x^2 + 90*x^4) + E^(8* x^2)*(4218750 + 506250*x^2 + 20250*x^4 + 270*x^6) + E^(4*x^2)*(158203125 + 25312500*x^2 + 1518750*x^4 + 40500*x^6 + 405*x^8)),x]
Output:
(16*(81 + (75 + E^(4*x^2))^4*x + 12*(75 + E^(4*x^2))^3*x^3 + 54*(75 + E^(4 *x^2))^2*x^5 + 108*(75 + E^(4*x^2))*x^7 + 81*x^9))/(75 + E^(4*x^2) + 3*x^2 )^4
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.
| \(\displaystyle \int \frac {3888 x^{10}+486000 x^8+24300000 x^6+607500000 x^4+7593750000 x^2+16 e^{20 x^2}+e^{16 x^2} \left (240 x^2+6000\right )+e^{12 x^2} \left (1440 x^4+72000 x^2+900000\right )+e^{8 x^2} \left (4320 x^6+324000 x^4+8100000 x^2+67500000\right )+e^{4 x^2} \left (6480 x^8+648000 x^6+24300000 x^4+405000000 x^2-41472 x+2531250000\right )-31104 x+37968750000}{243 x^{10}+30375 x^8+1518750 x^6+37968750 x^4+474609375 x^2+e^{20 x^2}+e^{16 x^2} \left (15 x^2+375\right )+e^{12 x^2} \left (90 x^4+4500 x^2+56250\right )+e^{8 x^2} \left (270 x^6+20250 x^4+506250 x^2+4218750\right )+e^{4 x^2} \left (405 x^8+40500 x^6+1518750 x^4+25312500 x^2+158203125\right )+2373046875} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {16 \left (270 e^{8 x^2} \left (x^2+25\right )^3+90 e^{12 x^2} \left (x^2+25\right )^2+15 e^{16 x^2} \left (x^2+25\right )+e^{20 x^2}+81 e^{4 x^2} \left (5 x^8+500 x^6+18750 x^4+312500 x^2-32 x+1953125\right )+243 \left (x^{10}+125 x^8+6250 x^6+156250 x^4+1953125 x^2-8 x+9765625\right )\right )}{\left (3 x^2+e^{4 x^2}+75\right )^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 16 \int \frac {270 e^{8 x^2} \left (x^2+25\right )^3+90 e^{12 x^2} \left (x^2+25\right )^2+15 e^{16 x^2} \left (x^2+25\right )+e^{20 x^2}+81 e^{4 x^2} \left (5 x^8+500 x^6+18750 x^4+312500 x^2-32 x+1953125\right )+243 \left (x^{10}+125 x^8+6250 x^6+156250 x^4+1953125 x^2-8 x+9765625\right )}{\left (3 x^2+e^{4 x^2}+75\right )^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 16 \int \left (\frac {1944 \left (4 x^2+99\right ) x}{\left (3 x^2+e^{4 x^2}+75\right )^5}-\frac {2592 x}{\left (3 x^2+e^{4 x^2}+75\right )^4}+1\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 16 \left (96228 \text {Subst}\left (\int \frac {1}{\left (3 x+e^{4 x}+75\right )^5}dx,x,x^2\right )+3888 \text {Subst}\left (\int \frac {x}{\left (3 x+e^{4 x}+75\right )^5}dx,x,x^2\right )-1296 \text {Subst}\left (\int \frac {1}{\left (3 x+e^{4 x}+75\right )^4}dx,x,x^2\right )+x\right )\) |
Input:
Int[(37968750000 + 16*E^(20*x^2) - 31104*x + 7593750000*x^2 + 607500000*x^ 4 + 24300000*x^6 + 486000*x^8 + 3888*x^10 + E^(16*x^2)*(6000 + 240*x^2) + E^(12*x^2)*(900000 + 72000*x^2 + 1440*x^4) + E^(8*x^2)*(67500000 + 8100000 *x^2 + 324000*x^4 + 4320*x^6) + E^(4*x^2)*(2531250000 - 41472*x + 40500000 0*x^2 + 24300000*x^4 + 648000*x^6 + 6480*x^8))/(2373046875 + E^(20*x^2) + 474609375*x^2 + 37968750*x^4 + 1518750*x^6 + 30375*x^8 + 243*x^10 + E^(16* x^2)*(375 + 15*x^2) + E^(12*x^2)*(56250 + 4500*x^2 + 90*x^4) + E^(8*x^2)*( 4218750 + 506250*x^2 + 20250*x^4 + 270*x^6) + E^(4*x^2)*(158203125 + 25312 500*x^2 + 1518750*x^4 + 40500*x^6 + 405*x^8)),x]
Output:
$Aborted
Time = 2.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00
| method | result | size |
| risch | \(16 x +\frac {1296}{\left (3 x^{2}+{\mathrm e}^{4 x^{2}}+75\right )^{4}}\) | \(22\) |
| parallelrisch | \(\frac {1296+506250000 x +27000000 x \,{\mathrm e}^{4 x^{2}}+129600 x^{7}+1296 x^{9}+4860000 x^{5}+81000000 x^{3}+1728 \,{\mathrm e}^{4 x^{2}} x^{7}+864 \,{\mathrm e}^{8 x^{2}} x^{5}+192 \,{\mathrm e}^{12 x^{2}} x^{3}+129600 \,{\mathrm e}^{4 x^{2}} x^{5}+43200 \,{\mathrm e}^{8 x^{2}} x^{3}+4800 x \,{\mathrm e}^{12 x^{2}}+3240000 \,{\mathrm e}^{4 x^{2}} x^{3}+540000 x \,{\mathrm e}^{8 x^{2}}+16 x \,{\mathrm e}^{16 x^{2}}}{81 x^{8}+108 \,{\mathrm e}^{4 x^{2}} x^{6}+8100 x^{6}+54 \,{\mathrm e}^{8 x^{2}} x^{4}+8100 \,{\mathrm e}^{4 x^{2}} x^{4}+12 \,{\mathrm e}^{12 x^{2}} x^{2}+303750 x^{4}+2700 \,{\mathrm e}^{8 x^{2}} x^{2}+{\mathrm e}^{16 x^{2}}+202500 x^{2} {\mathrm e}^{4 x^{2}}+300 \,{\mathrm e}^{12 x^{2}}+5062500 x^{2}+33750 \,{\mathrm e}^{8 x^{2}}+1687500 \,{\mathrm e}^{4 x^{2}}+31640625}\) | \(273\) |
Input:
int((16*exp(4*x^2)^5+(240*x^2+6000)*exp(4*x^2)^4+(1440*x^4+72000*x^2+90000 0)*exp(4*x^2)^3+(4320*x^6+324000*x^4+8100000*x^2+67500000)*exp(4*x^2)^2+(6 480*x^8+648000*x^6+24300000*x^4+405000000*x^2-41472*x+2531250000)*exp(4*x^ 2)+3888*x^10+486000*x^8+24300000*x^6+607500000*x^4+7593750000*x^2-31104*x+ 37968750000)/(exp(4*x^2)^5+(15*x^2+375)*exp(4*x^2)^4+(90*x^4+4500*x^2+5625 0)*exp(4*x^2)^3+(270*x^6+20250*x^4+506250*x^2+4218750)*exp(4*x^2)^2+(405*x ^8+40500*x^6+1518750*x^4+25312500*x^2+158203125)*exp(4*x^2)+243*x^10+30375 *x^8+1518750*x^6+37968750*x^4+474609375*x^2+2373046875),x,method=_RETURNVE RBOSE)
Output:
16*x+1296/(3*x^2+exp(4*x^2)+75)^4
Leaf count of result is larger than twice the leaf count of optimal. 179 vs. \(2 (21) = 42\).
Time = 0.09 (sec) , antiderivative size = 179, normalized size of antiderivative = 8.14 \[ \int \frac {37968750000+16 e^{20 x^2}-31104 x+7593750000 x^2+607500000 x^4+24300000 x^6+486000 x^8+3888 x^{10}+e^{16 x^2} \left (6000+240 x^2\right )+e^{12 x^2} \left (900000+72000 x^2+1440 x^4\right )+e^{8 x^2} \left (67500000+8100000 x^2+324000 x^4+4320 x^6\right )+e^{4 x^2} \left (2531250000-41472 x+405000000 x^2+24300000 x^4+648000 x^6+6480 x^8\right )}{2373046875+e^{20 x^2}+474609375 x^2+37968750 x^4+1518750 x^6+30375 x^8+243 x^{10}+e^{16 x^2} \left (375+15 x^2\right )+e^{12 x^2} \left (56250+4500 x^2+90 x^4\right )+e^{8 x^2} \left (4218750+506250 x^2+20250 x^4+270 x^6\right )+e^{4 x^2} \left (158203125+25312500 x^2+1518750 x^4+40500 x^6+405 x^8\right )} \, dx=\frac {16 \, {\left (81 \, x^{9} + 8100 \, x^{7} + 303750 \, x^{5} + 5062500 \, x^{3} + x e^{\left (16 \, x^{2}\right )} + 12 \, {\left (x^{3} + 25 \, x\right )} e^{\left (12 \, x^{2}\right )} + 54 \, {\left (x^{5} + 50 \, x^{3} + 625 \, x\right )} e^{\left (8 \, x^{2}\right )} + 108 \, {\left (x^{7} + 75 \, x^{5} + 1875 \, x^{3} + 15625 \, x\right )} e^{\left (4 \, x^{2}\right )} + 31640625 \, x + 81\right )}}{81 \, x^{8} + 8100 \, x^{6} + 303750 \, x^{4} + 5062500 \, x^{2} + 12 \, {\left (x^{2} + 25\right )} e^{\left (12 \, x^{2}\right )} + 54 \, {\left (x^{4} + 50 \, x^{2} + 625\right )} e^{\left (8 \, x^{2}\right )} + 108 \, {\left (x^{6} + 75 \, x^{4} + 1875 \, x^{2} + 15625\right )} e^{\left (4 \, x^{2}\right )} + e^{\left (16 \, x^{2}\right )} + 31640625} \] Input:
integrate((16*exp(4*x^2)^5+(240*x^2+6000)*exp(4*x^2)^4+(1440*x^4+72000*x^2 +900000)*exp(4*x^2)^3+(4320*x^6+324000*x^4+8100000*x^2+67500000)*exp(4*x^2 )^2+(6480*x^8+648000*x^6+24300000*x^4+405000000*x^2-41472*x+2531250000)*ex p(4*x^2)+3888*x^10+486000*x^8+24300000*x^6+607500000*x^4+7593750000*x^2-31 104*x+37968750000)/(exp(4*x^2)^5+(15*x^2+375)*exp(4*x^2)^4+(90*x^4+4500*x^ 2+56250)*exp(4*x^2)^3+(270*x^6+20250*x^4+506250*x^2+4218750)*exp(4*x^2)^2+ (405*x^8+40500*x^6+1518750*x^4+25312500*x^2+158203125)*exp(4*x^2)+243*x^10 +30375*x^8+1518750*x^6+37968750*x^4+474609375*x^2+2373046875),x, algorithm ="fricas")
Output:
16*(81*x^9 + 8100*x^7 + 303750*x^5 + 5062500*x^3 + x*e^(16*x^2) + 12*(x^3 + 25*x)*e^(12*x^2) + 54*(x^5 + 50*x^3 + 625*x)*e^(8*x^2) + 108*(x^7 + 75*x ^5 + 1875*x^3 + 15625*x)*e^(4*x^2) + 31640625*x + 81)/(81*x^8 + 8100*x^6 + 303750*x^4 + 5062500*x^2 + 12*(x^2 + 25)*e^(12*x^2) + 54*(x^4 + 50*x^2 + 625)*e^(8*x^2) + 108*(x^6 + 75*x^4 + 1875*x^2 + 15625)*e^(4*x^2) + e^(16*x ^2) + 31640625)
Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (19) = 38\).
Time = 0.21 (sec) , antiderivative size = 88, normalized size of antiderivative = 4.00 \[ \int \frac {37968750000+16 e^{20 x^2}-31104 x+7593750000 x^2+607500000 x^4+24300000 x^6+486000 x^8+3888 x^{10}+e^{16 x^2} \left (6000+240 x^2\right )+e^{12 x^2} \left (900000+72000 x^2+1440 x^4\right )+e^{8 x^2} \left (67500000+8100000 x^2+324000 x^4+4320 x^6\right )+e^{4 x^2} \left (2531250000-41472 x+405000000 x^2+24300000 x^4+648000 x^6+6480 x^8\right )}{2373046875+e^{20 x^2}+474609375 x^2+37968750 x^4+1518750 x^6+30375 x^8+243 x^{10}+e^{16 x^2} \left (375+15 x^2\right )+e^{12 x^2} \left (56250+4500 x^2+90 x^4\right )+e^{8 x^2} \left (4218750+506250 x^2+20250 x^4+270 x^6\right )+e^{4 x^2} \left (158203125+25312500 x^2+1518750 x^4+40500 x^6+405 x^8\right )} \, dx=16 x + \frac {1296}{81 x^{8} + 8100 x^{6} + 303750 x^{4} + 5062500 x^{2} + \left (12 x^{2} + 300\right ) e^{12 x^{2}} + \left (54 x^{4} + 2700 x^{2} + 33750\right ) e^{8 x^{2}} + \left (108 x^{6} + 8100 x^{4} + 202500 x^{2} + 1687500\right ) e^{4 x^{2}} + e^{16 x^{2}} + 31640625} \] Input:
integrate((16*exp(4*x**2)**5+(240*x**2+6000)*exp(4*x**2)**4+(1440*x**4+720 00*x**2+900000)*exp(4*x**2)**3+(4320*x**6+324000*x**4+8100000*x**2+6750000 0)*exp(4*x**2)**2+(6480*x**8+648000*x**6+24300000*x**4+405000000*x**2-4147 2*x+2531250000)*exp(4*x**2)+3888*x**10+486000*x**8+24300000*x**6+607500000 *x**4+7593750000*x**2-31104*x+37968750000)/(exp(4*x**2)**5+(15*x**2+375)*e xp(4*x**2)**4+(90*x**4+4500*x**2+56250)*exp(4*x**2)**3+(270*x**6+20250*x** 4+506250*x**2+4218750)*exp(4*x**2)**2+(405*x**8+40500*x**6+1518750*x**4+25 312500*x**2+158203125)*exp(4*x**2)+243*x**10+30375*x**8+1518750*x**6+37968 750*x**4+474609375*x**2+2373046875),x)
Output:
16*x + 1296/(81*x**8 + 8100*x**6 + 303750*x**4 + 5062500*x**2 + (12*x**2 + 300)*exp(12*x**2) + (54*x**4 + 2700*x**2 + 33750)*exp(8*x**2) + (108*x**6 + 8100*x**4 + 202500*x**2 + 1687500)*exp(4*x**2) + exp(16*x**2) + 3164062 5)
Leaf count of result is larger than twice the leaf count of optimal. 179 vs. \(2 (21) = 42\).
Time = 0.10 (sec) , antiderivative size = 179, normalized size of antiderivative = 8.14 \[ \int \frac {37968750000+16 e^{20 x^2}-31104 x+7593750000 x^2+607500000 x^4+24300000 x^6+486000 x^8+3888 x^{10}+e^{16 x^2} \left (6000+240 x^2\right )+e^{12 x^2} \left (900000+72000 x^2+1440 x^4\right )+e^{8 x^2} \left (67500000+8100000 x^2+324000 x^4+4320 x^6\right )+e^{4 x^2} \left (2531250000-41472 x+405000000 x^2+24300000 x^4+648000 x^6+6480 x^8\right )}{2373046875+e^{20 x^2}+474609375 x^2+37968750 x^4+1518750 x^6+30375 x^8+243 x^{10}+e^{16 x^2} \left (375+15 x^2\right )+e^{12 x^2} \left (56250+4500 x^2+90 x^4\right )+e^{8 x^2} \left (4218750+506250 x^2+20250 x^4+270 x^6\right )+e^{4 x^2} \left (158203125+25312500 x^2+1518750 x^4+40500 x^6+405 x^8\right )} \, dx=\frac {16 \, {\left (81 \, x^{9} + 8100 \, x^{7} + 303750 \, x^{5} + 5062500 \, x^{3} + x e^{\left (16 \, x^{2}\right )} + 12 \, {\left (x^{3} + 25 \, x\right )} e^{\left (12 \, x^{2}\right )} + 54 \, {\left (x^{5} + 50 \, x^{3} + 625 \, x\right )} e^{\left (8 \, x^{2}\right )} + 108 \, {\left (x^{7} + 75 \, x^{5} + 1875 \, x^{3} + 15625 \, x\right )} e^{\left (4 \, x^{2}\right )} + 31640625 \, x + 81\right )}}{81 \, x^{8} + 8100 \, x^{6} + 303750 \, x^{4} + 5062500 \, x^{2} + 12 \, {\left (x^{2} + 25\right )} e^{\left (12 \, x^{2}\right )} + 54 \, {\left (x^{4} + 50 \, x^{2} + 625\right )} e^{\left (8 \, x^{2}\right )} + 108 \, {\left (x^{6} + 75 \, x^{4} + 1875 \, x^{2} + 15625\right )} e^{\left (4 \, x^{2}\right )} + e^{\left (16 \, x^{2}\right )} + 31640625} \] Input:
integrate((16*exp(4*x^2)^5+(240*x^2+6000)*exp(4*x^2)^4+(1440*x^4+72000*x^2 +900000)*exp(4*x^2)^3+(4320*x^6+324000*x^4+8100000*x^2+67500000)*exp(4*x^2 )^2+(6480*x^8+648000*x^6+24300000*x^4+405000000*x^2-41472*x+2531250000)*ex p(4*x^2)+3888*x^10+486000*x^8+24300000*x^6+607500000*x^4+7593750000*x^2-31 104*x+37968750000)/(exp(4*x^2)^5+(15*x^2+375)*exp(4*x^2)^4+(90*x^4+4500*x^ 2+56250)*exp(4*x^2)^3+(270*x^6+20250*x^4+506250*x^2+4218750)*exp(4*x^2)^2+ (405*x^8+40500*x^6+1518750*x^4+25312500*x^2+158203125)*exp(4*x^2)+243*x^10 +30375*x^8+1518750*x^6+37968750*x^4+474609375*x^2+2373046875),x, algorithm ="maxima")
Output:
16*(81*x^9 + 8100*x^7 + 303750*x^5 + 5062500*x^3 + x*e^(16*x^2) + 12*(x^3 + 25*x)*e^(12*x^2) + 54*(x^5 + 50*x^3 + 625*x)*e^(8*x^2) + 108*(x^7 + 75*x ^5 + 1875*x^3 + 15625*x)*e^(4*x^2) + 31640625*x + 81)/(81*x^8 + 8100*x^6 + 303750*x^4 + 5062500*x^2 + 12*(x^2 + 25)*e^(12*x^2) + 54*(x^4 + 50*x^2 + 625)*e^(8*x^2) + 108*(x^6 + 75*x^4 + 1875*x^2 + 15625)*e^(4*x^2) + e^(16*x ^2) + 31640625)
Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (21) = 42\).
Time = 0.20 (sec) , antiderivative size = 248, normalized size of antiderivative = 11.27 \[ \int \frac {37968750000+16 e^{20 x^2}-31104 x+7593750000 x^2+607500000 x^4+24300000 x^6+486000 x^8+3888 x^{10}+e^{16 x^2} \left (6000+240 x^2\right )+e^{12 x^2} \left (900000+72000 x^2+1440 x^4\right )+e^{8 x^2} \left (67500000+8100000 x^2+324000 x^4+4320 x^6\right )+e^{4 x^2} \left (2531250000-41472 x+405000000 x^2+24300000 x^4+648000 x^6+6480 x^8\right )}{2373046875+e^{20 x^2}+474609375 x^2+37968750 x^4+1518750 x^6+30375 x^8+243 x^{10}+e^{16 x^2} \left (375+15 x^2\right )+e^{12 x^2} \left (56250+4500 x^2+90 x^4\right )+e^{8 x^2} \left (4218750+506250 x^2+20250 x^4+270 x^6\right )+e^{4 x^2} \left (158203125+25312500 x^2+1518750 x^4+40500 x^6+405 x^8\right )} \, dx=\frac {16 \, {\left (81 \, x^{9} + 108 \, x^{7} e^{\left (4 \, x^{2}\right )} + 8100 \, x^{7} + 54 \, x^{5} e^{\left (8 \, x^{2}\right )} + 8100 \, x^{5} e^{\left (4 \, x^{2}\right )} + 303750 \, x^{5} + 12 \, x^{3} e^{\left (12 \, x^{2}\right )} + 2700 \, x^{3} e^{\left (8 \, x^{2}\right )} + 202500 \, x^{3} e^{\left (4 \, x^{2}\right )} + 5062500 \, x^{3} + x e^{\left (16 \, x^{2}\right )} + 300 \, x e^{\left (12 \, x^{2}\right )} + 33750 \, x e^{\left (8 \, x^{2}\right )} + 1687500 \, x e^{\left (4 \, x^{2}\right )} + 31640625 \, x + 81\right )}}{81 \, x^{8} + 108 \, x^{6} e^{\left (4 \, x^{2}\right )} + 8100 \, x^{6} + 54 \, x^{4} e^{\left (8 \, x^{2}\right )} + 8100 \, x^{4} e^{\left (4 \, x^{2}\right )} + 303750 \, x^{4} + 12 \, x^{2} e^{\left (12 \, x^{2}\right )} + 2700 \, x^{2} e^{\left (8 \, x^{2}\right )} + 202500 \, x^{2} e^{\left (4 \, x^{2}\right )} + 5062500 \, x^{2} + e^{\left (16 \, x^{2}\right )} + 300 \, e^{\left (12 \, x^{2}\right )} + 33750 \, e^{\left (8 \, x^{2}\right )} + 1687500 \, e^{\left (4 \, x^{2}\right )} + 31640625} \] Input:
integrate((16*exp(4*x^2)^5+(240*x^2+6000)*exp(4*x^2)^4+(1440*x^4+72000*x^2 +900000)*exp(4*x^2)^3+(4320*x^6+324000*x^4+8100000*x^2+67500000)*exp(4*x^2 )^2+(6480*x^8+648000*x^6+24300000*x^4+405000000*x^2-41472*x+2531250000)*ex p(4*x^2)+3888*x^10+486000*x^8+24300000*x^6+607500000*x^4+7593750000*x^2-31 104*x+37968750000)/(exp(4*x^2)^5+(15*x^2+375)*exp(4*x^2)^4+(90*x^4+4500*x^ 2+56250)*exp(4*x^2)^3+(270*x^6+20250*x^4+506250*x^2+4218750)*exp(4*x^2)^2+ (405*x^8+40500*x^6+1518750*x^4+25312500*x^2+158203125)*exp(4*x^2)+243*x^10 +30375*x^8+1518750*x^6+37968750*x^4+474609375*x^2+2373046875),x, algorithm ="giac")
Output:
16*(81*x^9 + 108*x^7*e^(4*x^2) + 8100*x^7 + 54*x^5*e^(8*x^2) + 8100*x^5*e^ (4*x^2) + 303750*x^5 + 12*x^3*e^(12*x^2) + 2700*x^3*e^(8*x^2) + 202500*x^3 *e^(4*x^2) + 5062500*x^3 + x*e^(16*x^2) + 300*x*e^(12*x^2) + 33750*x*e^(8* x^2) + 1687500*x*e^(4*x^2) + 31640625*x + 81)/(81*x^8 + 108*x^6*e^(4*x^2) + 8100*x^6 + 54*x^4*e^(8*x^2) + 8100*x^4*e^(4*x^2) + 303750*x^4 + 12*x^2*e ^(12*x^2) + 2700*x^2*e^(8*x^2) + 202500*x^2*e^(4*x^2) + 5062500*x^2 + e^(1 6*x^2) + 300*e^(12*x^2) + 33750*e^(8*x^2) + 1687500*e^(4*x^2) + 31640625)
Timed out. \[ \int \frac {37968750000+16 e^{20 x^2}-31104 x+7593750000 x^2+607500000 x^4+24300000 x^6+486000 x^8+3888 x^{10}+e^{16 x^2} \left (6000+240 x^2\right )+e^{12 x^2} \left (900000+72000 x^2+1440 x^4\right )+e^{8 x^2} \left (67500000+8100000 x^2+324000 x^4+4320 x^6\right )+e^{4 x^2} \left (2531250000-41472 x+405000000 x^2+24300000 x^4+648000 x^6+6480 x^8\right )}{2373046875+e^{20 x^2}+474609375 x^2+37968750 x^4+1518750 x^6+30375 x^8+243 x^{10}+e^{16 x^2} \left (375+15 x^2\right )+e^{12 x^2} \left (56250+4500 x^2+90 x^4\right )+e^{8 x^2} \left (4218750+506250 x^2+20250 x^4+270 x^6\right )+e^{4 x^2} \left (158203125+25312500 x^2+1518750 x^4+40500 x^6+405 x^8\right )} \, dx=\int \frac {16\,{\mathrm {e}}^{20\,x^2}-31104\,x+{\mathrm {e}}^{8\,x^2}\,\left (4320\,x^6+324000\,x^4+8100000\,x^2+67500000\right )+{\mathrm {e}}^{16\,x^2}\,\left (240\,x^2+6000\right )+{\mathrm {e}}^{4\,x^2}\,\left (6480\,x^8+648000\,x^6+24300000\,x^4+405000000\,x^2-41472\,x+2531250000\right )+{\mathrm {e}}^{12\,x^2}\,\left (1440\,x^4+72000\,x^2+900000\right )+7593750000\,x^2+607500000\,x^4+24300000\,x^6+486000\,x^8+3888\,x^{10}+37968750000}{{\mathrm {e}}^{20\,x^2}+{\mathrm {e}}^{8\,x^2}\,\left (270\,x^6+20250\,x^4+506250\,x^2+4218750\right )+{\mathrm {e}}^{4\,x^2}\,\left (405\,x^8+40500\,x^6+1518750\,x^4+25312500\,x^2+158203125\right )+{\mathrm {e}}^{16\,x^2}\,\left (15\,x^2+375\right )+{\mathrm {e}}^{12\,x^2}\,\left (90\,x^4+4500\,x^2+56250\right )+474609375\,x^2+37968750\,x^4+1518750\,x^6+30375\,x^8+243\,x^{10}+2373046875} \,d x \] Input:
int((16*exp(20*x^2) - 31104*x + exp(8*x^2)*(8100000*x^2 + 324000*x^4 + 432 0*x^6 + 67500000) + exp(16*x^2)*(240*x^2 + 6000) + exp(4*x^2)*(405000000*x ^2 - 41472*x + 24300000*x^4 + 648000*x^6 + 6480*x^8 + 2531250000) + exp(12 *x^2)*(72000*x^2 + 1440*x^4 + 900000) + 7593750000*x^2 + 607500000*x^4 + 2 4300000*x^6 + 486000*x^8 + 3888*x^10 + 37968750000)/(exp(20*x^2) + exp(8*x ^2)*(506250*x^2 + 20250*x^4 + 270*x^6 + 4218750) + exp(4*x^2)*(25312500*x^ 2 + 1518750*x^4 + 40500*x^6 + 405*x^8 + 158203125) + exp(16*x^2)*(15*x^2 + 375) + exp(12*x^2)*(4500*x^2 + 90*x^4 + 56250) + 474609375*x^2 + 37968750 *x^4 + 1518750*x^6 + 30375*x^8 + 243*x^10 + 2373046875),x)
Output:
int((16*exp(20*x^2) - 31104*x + exp(8*x^2)*(8100000*x^2 + 324000*x^4 + 432 0*x^6 + 67500000) + exp(16*x^2)*(240*x^2 + 6000) + exp(4*x^2)*(405000000*x ^2 - 41472*x + 24300000*x^4 + 648000*x^6 + 6480*x^8 + 2531250000) + exp(12 *x^2)*(72000*x^2 + 1440*x^4 + 900000) + 7593750000*x^2 + 607500000*x^4 + 2 4300000*x^6 + 486000*x^8 + 3888*x^10 + 37968750000)/(exp(20*x^2) + exp(8*x ^2)*(506250*x^2 + 20250*x^4 + 270*x^6 + 4218750) + exp(4*x^2)*(25312500*x^ 2 + 1518750*x^4 + 40500*x^6 + 405*x^8 + 158203125) + exp(16*x^2)*(15*x^2 + 375) + exp(12*x^2)*(4500*x^2 + 90*x^4 + 56250) + 474609375*x^2 + 37968750 *x^4 + 1518750*x^6 + 30375*x^8 + 243*x^10 + 2373046875), x)
Time = 0.22 (sec) , antiderivative size = 268, normalized size of antiderivative = 12.18 \[ \int \frac {37968750000+16 e^{20 x^2}-31104 x+7593750000 x^2+607500000 x^4+24300000 x^6+486000 x^8+3888 x^{10}+e^{16 x^2} \left (6000+240 x^2\right )+e^{12 x^2} \left (900000+72000 x^2+1440 x^4\right )+e^{8 x^2} \left (67500000+8100000 x^2+324000 x^4+4320 x^6\right )+e^{4 x^2} \left (2531250000-41472 x+405000000 x^2+24300000 x^4+648000 x^6+6480 x^8\right )}{2373046875+e^{20 x^2}+474609375 x^2+37968750 x^4+1518750 x^6+30375 x^8+243 x^{10}+e^{16 x^2} \left (375+15 x^2\right )+e^{12 x^2} \left (56250+4500 x^2+90 x^4\right )+e^{8 x^2} \left (4218750+506250 x^2+20250 x^4+270 x^6\right )+e^{4 x^2} \left (158203125+25312500 x^2+1518750 x^4+40500 x^6+405 x^8\right )} \, dx=\frac {16 e^{16 x^{2}} x +192 e^{12 x^{2}} x^{3}+4800 e^{12 x^{2}} x +864 e^{8 x^{2}} x^{5}+43200 e^{8 x^{2}} x^{3}+540000 e^{8 x^{2}} x +1728 e^{4 x^{2}} x^{7}+129600 e^{4 x^{2}} x^{5}+3240000 e^{4 x^{2}} x^{3}+27000000 e^{4 x^{2}} x +1296 x^{9}+129600 x^{7}+4860000 x^{5}+81000000 x^{3}+506250000 x +1296}{e^{16 x^{2}}+12 e^{12 x^{2}} x^{2}+300 e^{12 x^{2}}+54 e^{8 x^{2}} x^{4}+2700 e^{8 x^{2}} x^{2}+33750 e^{8 x^{2}}+108 e^{4 x^{2}} x^{6}+8100 e^{4 x^{2}} x^{4}+202500 e^{4 x^{2}} x^{2}+1687500 e^{4 x^{2}}+81 x^{8}+8100 x^{6}+303750 x^{4}+5062500 x^{2}+31640625} \] Input:
int((16*exp(4*x^2)^5+(240*x^2+6000)*exp(4*x^2)^4+(1440*x^4+72000*x^2+90000 0)*exp(4*x^2)^3+(4320*x^6+324000*x^4+8100000*x^2+67500000)*exp(4*x^2)^2+(6 480*x^8+648000*x^6+24300000*x^4+405000000*x^2-41472*x+2531250000)*exp(4*x^ 2)+3888*x^10+486000*x^8+24300000*x^6+607500000*x^4+7593750000*x^2-31104*x+ 37968750000)/(exp(4*x^2)^5+(15*x^2+375)*exp(4*x^2)^4+(90*x^4+4500*x^2+5625 0)*exp(4*x^2)^3+(270*x^6+20250*x^4+506250*x^2+4218750)*exp(4*x^2)^2+(405*x ^8+40500*x^6+1518750*x^4+25312500*x^2+158203125)*exp(4*x^2)+243*x^10+30375 *x^8+1518750*x^6+37968750*x^4+474609375*x^2+2373046875),x)
Output:
(16*(e**(16*x**2)*x + 12*e**(12*x**2)*x**3 + 300*e**(12*x**2)*x + 54*e**(8 *x**2)*x**5 + 2700*e**(8*x**2)*x**3 + 33750*e**(8*x**2)*x + 108*e**(4*x**2 )*x**7 + 8100*e**(4*x**2)*x**5 + 202500*e**(4*x**2)*x**3 + 1687500*e**(4*x **2)*x + 81*x**9 + 8100*x**7 + 303750*x**5 + 5062500*x**3 + 31640625*x + 8 1))/(e**(16*x**2) + 12*e**(12*x**2)*x**2 + 300*e**(12*x**2) + 54*e**(8*x** 2)*x**4 + 2700*e**(8*x**2)*x**2 + 33750*e**(8*x**2) + 108*e**(4*x**2)*x**6 + 8100*e**(4*x**2)*x**4 + 202500*e**(4*x**2)*x**2 + 1687500*e**(4*x**2) + 81*x**8 + 8100*x**6 + 303750*x**4 + 5062500*x**2 + 31640625)