Integrand size = 344, antiderivative size = 37 \[ \int \frac {-4 e^{10 x}-80 e^{8 x} x-640 e^{6 x} x^2-2560 e^{4 x} x^3-5120 e^{2 x} x^4-4096 x^5+e^{\frac {65536 x^2+25 e^{8 x} x^2+400 e^{6 x} x^3-40960 x^4+6400 x^6+e^{4 x} \left (-2560 x^2+2400 x^4\right )+e^{2 x} \left (-20480 x^3+6400 x^5\right )}{e^{8 x}+16 e^{6 x} x+96 e^{4 x} x^2+256 e^{2 x} x^3+256 x^4}} \left (-1572864 x^4+6144 x^6+153600 x^8+e^{10 x} \left (6 x+150 x^3\right )+e^{8 x} \left (120 x^2+3000 x^4\right )+e^{6 x} \left (-14400 x^3+30720 x^4+24000 x^5\right )+e^{4 x} \left (-119040 x^4+245760 x^5+96000 x^6\right )+e^{2 x} \left (393216 x^3-1572864 x^4-238080 x^5+491520 x^6+192000 x^7\right )\right )}{3 e^{10 x}+60 e^{8 x} x+480 e^{6 x} x^2+1920 e^{4 x} x^3+3840 e^{2 x} x^4+3072 x^5} \, dx=\left (e^{x^2 \left (-5+\frac {16}{\left (\frac {e^{2 x}}{4}+x\right )^2}\right )^2}-\frac {4}{3 x}\right ) x^2 \] Output:
(exp(x^2*(16/(1/4*exp(x)^2+x)^2-5)^2)-4/3/x)*x^2
Time = 0.39 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.43 \[ \int \frac {-4 e^{10 x}-80 e^{8 x} x-640 e^{6 x} x^2-2560 e^{4 x} x^3-5120 e^{2 x} x^4-4096 x^5+e^{\frac {65536 x^2+25 e^{8 x} x^2+400 e^{6 x} x^3-40960 x^4+6400 x^6+e^{4 x} \left (-2560 x^2+2400 x^4\right )+e^{2 x} \left (-20480 x^3+6400 x^5\right )}{e^{8 x}+16 e^{6 x} x+96 e^{4 x} x^2+256 e^{2 x} x^3+256 x^4}} \left (-1572864 x^4+6144 x^6+153600 x^8+e^{10 x} \left (6 x+150 x^3\right )+e^{8 x} \left (120 x^2+3000 x^4\right )+e^{6 x} \left (-14400 x^3+30720 x^4+24000 x^5\right )+e^{4 x} \left (-119040 x^4+245760 x^5+96000 x^6\right )+e^{2 x} \left (393216 x^3-1572864 x^4-238080 x^5+491520 x^6+192000 x^7\right )\right )}{3 e^{10 x}+60 e^{8 x} x+480 e^{6 x} x^2+1920 e^{4 x} x^3+3840 e^{2 x} x^4+3072 x^5} \, dx=\frac {1}{3} \left (-4 x+3 e^{25 x^2+\frac {65536 x^2}{\left (e^{2 x}+4 x\right )^4}-\frac {2560 x^2}{\left (e^{2 x}+4 x\right )^2}} x^2\right ) \] Input:
Integrate[(-4*E^(10*x) - 80*E^(8*x)*x - 640*E^(6*x)*x^2 - 2560*E^(4*x)*x^3 - 5120*E^(2*x)*x^4 - 4096*x^5 + E^((65536*x^2 + 25*E^(8*x)*x^2 + 400*E^(6 *x)*x^3 - 40960*x^4 + 6400*x^6 + E^(4*x)*(-2560*x^2 + 2400*x^4) + E^(2*x)* (-20480*x^3 + 6400*x^5))/(E^(8*x) + 16*E^(6*x)*x + 96*E^(4*x)*x^2 + 256*E^ (2*x)*x^3 + 256*x^4))*(-1572864*x^4 + 6144*x^6 + 153600*x^8 + E^(10*x)*(6* x + 150*x^3) + E^(8*x)*(120*x^2 + 3000*x^4) + E^(6*x)*(-14400*x^3 + 30720* x^4 + 24000*x^5) + E^(4*x)*(-119040*x^4 + 245760*x^5 + 96000*x^6) + E^(2*x )*(393216*x^3 - 1572864*x^4 - 238080*x^5 + 491520*x^6 + 192000*x^7)))/(3*E ^(10*x) + 60*E^(8*x)*x + 480*E^(6*x)*x^2 + 1920*E^(4*x)*x^3 + 3840*E^(2*x) *x^4 + 3072*x^5),x]
Output:
(-4*x + 3*E^(25*x^2 + (65536*x^2)/(E^(2*x) + 4*x)^4 - (2560*x^2)/(E^(2*x) + 4*x)^2)*x^2)/3
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.
| \(\displaystyle \int \frac {\left (153600 x^8+6144 x^6-1572864 x^4+e^{10 x} \left (150 x^3+6 x\right )+e^{8 x} \left (3000 x^4+120 x^2\right )+e^{4 x} \left (96000 x^6+245760 x^5-119040 x^4\right )+e^{6 x} \left (24000 x^5+30720 x^4-14400 x^3\right )+e^{2 x} \left (192000 x^7+491520 x^6-238080 x^5-1572864 x^4+393216 x^3\right )\right ) \exp \left (\frac {6400 x^6-40960 x^4+400 e^{6 x} x^3+25 e^{8 x} x^2+65536 x^2+e^{2 x} \left (6400 x^5-20480 x^3\right )+e^{4 x} \left (2400 x^4-2560 x^2\right )}{256 x^4+256 e^{2 x} x^3+96 e^{4 x} x^2+16 e^{6 x} x+e^{8 x}}\right )-4096 x^5-5120 e^{2 x} x^4-2560 e^{4 x} x^3-640 e^{6 x} x^2-80 e^{8 x} x-4 e^{10 x}}{3072 x^5+3840 e^{2 x} x^4+1920 e^{4 x} x^3+480 e^{6 x} x^2+60 e^{8 x} x+3 e^{10 x}} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (153600 x^8+6144 x^6-1572864 x^4+e^{10 x} \left (150 x^3+6 x\right )+e^{8 x} \left (3000 x^4+120 x^2\right )+e^{4 x} \left (96000 x^6+245760 x^5-119040 x^4\right )+e^{6 x} \left (24000 x^5+30720 x^4-14400 x^3\right )+e^{2 x} \left (192000 x^7+491520 x^6-238080 x^5-1572864 x^4+393216 x^3\right )\right ) \exp \left (\frac {6400 x^6-40960 x^4+400 e^{6 x} x^3+25 e^{8 x} x^2+65536 x^2+e^{2 x} \left (6400 x^5-20480 x^3\right )+e^{4 x} \left (2400 x^4-2560 x^2\right )}{256 x^4+256 e^{2 x} x^3+96 e^{4 x} x^2+16 e^{6 x} x+e^{8 x}}\right )-4096 x^5-5120 e^{2 x} x^4-2560 e^{4 x} x^3-640 e^{6 x} x^2-80 e^{8 x} x-4 e^{10 x}}{3 \left (4 x+e^{2 x}\right )^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \int -\frac {2 \left (2048 x^5+2560 e^{2 x} x^4+1280 e^{4 x} x^3+320 e^{6 x} x^2+40 e^{8 x} x+2 e^{10 x}+3 \exp \left (\frac {6400 x^6-40960 x^4+400 e^{6 x} x^3+25 e^{8 x} x^2+65536 x^2-160 e^{4 x} \left (16 x^2-15 x^4\right )-1280 e^{2 x} \left (16 x^3-5 x^5\right )}{256 x^4+256 e^{2 x} x^3+96 e^{4 x} x^2+16 e^{6 x} x+e^{8 x}}\right ) \left (-25600 x^8-1024 x^6+262144 x^4-e^{10 x} \left (25 x^3+x\right )-20 e^{8 x} \left (25 x^4+x^2\right )+160 e^{6 x} \left (-25 x^5-32 x^4+15 x^3\right )+640 e^{4 x} \left (-25 x^6-64 x^5+31 x^4\right )-256 e^{2 x} \left (125 x^7+320 x^6-155 x^5-1024 x^4+256 x^3\right )\right )\right )}{\left (4 x+e^{2 x}\right )^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{3} \int \frac {2048 x^5+2560 e^{2 x} x^4+1280 e^{4 x} x^3+320 e^{6 x} x^2+40 e^{8 x} x+2 e^{10 x}+3 \exp \left (\frac {6400 x^6-40960 x^4+400 e^{6 x} x^3+25 e^{8 x} x^2+65536 x^2-160 e^{4 x} \left (16 x^2-15 x^4\right )-1280 e^{2 x} \left (16 x^3-5 x^5\right )}{256 x^4+256 e^{2 x} x^3+96 e^{4 x} x^2+16 e^{6 x} x+e^{8 x}}\right ) \left (-25600 x^8-1024 x^6+262144 x^4-e^{10 x} \left (25 x^3+x\right )-20 e^{8 x} \left (25 x^4+x^2\right )+160 e^{6 x} \left (-25 x^5-32 x^4+15 x^3\right )+640 e^{4 x} \left (-25 x^6-64 x^5+31 x^4\right )-256 e^{2 x} \left (125 x^7+320 x^6-155 x^5-1024 x^4+256 x^3\right )\right )}{\left (4 x+e^{2 x}\right )^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2}{3} \int \left (\frac {2048 x^5}{\left (4 x+e^{2 x}\right )^5}+\frac {2560 e^{2 x} x^4}{\left (4 x+e^{2 x}\right )^5}+\frac {1280 e^{4 x} x^3}{\left (4 x+e^{2 x}\right )^5}+\frac {320 e^{6 x} x^2}{\left (4 x+e^{2 x}\right )^5}-\frac {3 \exp \left (\frac {x^2 \left (80 x^2+40 e^{2 x} x+5 e^{4 x}-256\right )^2}{\left (4 x+e^{2 x}\right )^4}\right ) \left (25600 x^7+32000 e^{2 x} x^6+81920 e^{2 x} x^5+16000 e^{4 x} x^5+1024 x^5-39680 e^{2 x} x^4+40960 e^{4 x} x^4+4000 e^{6 x} x^4-262144 e^{2 x} x^3-19840 e^{4 x} x^3+5120 e^{6 x} x^3+500 e^{8 x} x^3-262144 x^3+65536 e^{2 x} x^2-2400 e^{6 x} x^2+25 e^{10 x} x^2+20 e^{8 x} x+e^{10 x}\right ) x}{\left (4 x+e^{2 x}\right )^5}+\frac {40 e^{8 x} x}{\left (4 x+e^{2 x}\right )^5}+\frac {2 e^{10 x}}{\left (4 x+e^{2 x}\right )^5}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle -\frac {2}{3} \int \left (\frac {2048 x^5}{\left (4 x+e^{2 x}\right )^5}+\frac {2560 e^{2 x} x^4}{\left (4 x+e^{2 x}\right )^5}+\frac {1280 e^{4 x} x^3}{\left (4 x+e^{2 x}\right )^5}+\frac {320 e^{6 x} x^2}{\left (4 x+e^{2 x}\right )^5}-\frac {3 \exp \left (\frac {x^2 \left (80 x^2+40 e^{2 x} x+5 e^{4 x}-256\right )^2}{\left (4 x+e^{2 x}\right )^4}\right ) \left (25600 x^7+32000 e^{2 x} x^6+81920 e^{2 x} x^5+16000 e^{4 x} x^5+1024 x^5-39680 e^{2 x} x^4+40960 e^{4 x} x^4+4000 e^{6 x} x^4-262144 e^{2 x} x^3-19840 e^{4 x} x^3+5120 e^{6 x} x^3+500 e^{8 x} x^3-262144 x^3+65536 e^{2 x} x^2-2400 e^{6 x} x^2+25 e^{10 x} x^2+20 e^{8 x} x+e^{10 x}\right ) x}{\left (4 x+e^{2 x}\right )^5}+\frac {40 e^{8 x} x}{\left (4 x+e^{2 x}\right )^5}+\frac {2 e^{10 x}}{\left (4 x+e^{2 x}\right )^5}\right )dx\) |
Input:
Int[(-4*E^(10*x) - 80*E^(8*x)*x - 640*E^(6*x)*x^2 - 2560*E^(4*x)*x^3 - 512 0*E^(2*x)*x^4 - 4096*x^5 + E^((65536*x^2 + 25*E^(8*x)*x^2 + 400*E^(6*x)*x^ 3 - 40960*x^4 + 6400*x^6 + E^(4*x)*(-2560*x^2 + 2400*x^4) + E^(2*x)*(-2048 0*x^3 + 6400*x^5))/(E^(8*x) + 16*E^(6*x)*x + 96*E^(4*x)*x^2 + 256*E^(2*x)* x^3 + 256*x^4))*(-1572864*x^4 + 6144*x^6 + 153600*x^8 + E^(10*x)*(6*x + 15 0*x^3) + E^(8*x)*(120*x^2 + 3000*x^4) + E^(6*x)*(-14400*x^3 + 30720*x^4 + 24000*x^5) + E^(4*x)*(-119040*x^4 + 245760*x^5 + 96000*x^6) + E^(2*x)*(393 216*x^3 - 1572864*x^4 - 238080*x^5 + 491520*x^6 + 192000*x^7)))/(3*E^(10*x ) + 60*E^(8*x)*x + 480*E^(6*x)*x^2 + 1920*E^(4*x)*x^3 + 3840*E^(2*x)*x^4 + 3072*x^5),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(106\) vs. \(2(31)=62\).
Time = 71.50 (sec) , antiderivative size = 107, normalized size of antiderivative = 2.89
| method | result | size |
| risch | \(-\frac {4 x}{3}+x^{2} {\mathrm e}^{\frac {x^{2} \left (6400 \,{\mathrm e}^{2 x} x^{3}+6400 x^{4}+2400 x^{2} {\mathrm e}^{4 x}+400 \,{\mathrm e}^{6 x} x -20480 x \,{\mathrm e}^{2 x}-40960 x^{2}+25 \,{\mathrm e}^{8 x}-2560 \,{\mathrm e}^{4 x}+65536\right )}{{\mathrm e}^{8 x}+16 \,{\mathrm e}^{6 x} x +96 x^{2} {\mathrm e}^{4 x}+256 \,{\mathrm e}^{2 x} x^{3}+256 x^{4}}}\) | \(107\) |
| parallelrisch | \(-\frac {4 x}{3}+x^{2} {\mathrm e}^{\frac {25 x^{2} {\mathrm e}^{8 x}+400 x^{3} {\mathrm e}^{6 x}+\left (2400 x^{4}-2560 x^{2}\right ) {\mathrm e}^{4 x}+\left (6400 x^{5}-20480 x^{3}\right ) {\mathrm e}^{2 x}+6400 x^{6}-40960 x^{4}+65536 x^{2}}{{\mathrm e}^{8 x}+16 \,{\mathrm e}^{6 x} x +96 x^{2} {\mathrm e}^{4 x}+256 \,{\mathrm e}^{2 x} x^{3}+256 x^{4}}}\) | \(114\) |
Input:
int((((150*x^3+6*x)*exp(x)^10+(3000*x^4+120*x^2)*exp(x)^8+(24000*x^5+30720 *x^4-14400*x^3)*exp(x)^6+(96000*x^6+245760*x^5-119040*x^4)*exp(x)^4+(19200 0*x^7+491520*x^6-238080*x^5-1572864*x^4+393216*x^3)*exp(x)^2+153600*x^8+61 44*x^6-1572864*x^4)*exp((25*x^2*exp(x)^8+400*x^3*exp(x)^6+(2400*x^4-2560*x ^2)*exp(x)^4+(6400*x^5-20480*x^3)*exp(x)^2+6400*x^6-40960*x^4+65536*x^2)/( exp(x)^8+16*x*exp(x)^6+96*x^2*exp(x)^4+256*exp(x)^2*x^3+256*x^4))-4*exp(x) ^10-80*x*exp(x)^8-640*x^2*exp(x)^6-2560*x^3*exp(x)^4-5120*exp(x)^2*x^4-409 6*x^5)/(3*exp(x)^10+60*x*exp(x)^8+480*x^2*exp(x)^6+1920*x^3*exp(x)^4+3840* exp(x)^2*x^4+3072*x^5),x,method=_RETURNVERBOSE)
Output:
-4/3*x+x^2*exp(x^2*(6400*exp(2*x)*x^3+6400*x^4+2400*x^2*exp(4*x)+400*exp(6 *x)*x-20480*x*exp(2*x)-40960*x^2+25*exp(8*x)-2560*exp(4*x)+65536)/(exp(8*x )+16*exp(6*x)*x+96*x^2*exp(4*x)+256*exp(2*x)*x^3+256*x^4))
Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (34) = 68\).
Time = 0.10 (sec) , antiderivative size = 115, normalized size of antiderivative = 3.11 \[ \int \frac {-4 e^{10 x}-80 e^{8 x} x-640 e^{6 x} x^2-2560 e^{4 x} x^3-5120 e^{2 x} x^4-4096 x^5+e^{\frac {65536 x^2+25 e^{8 x} x^2+400 e^{6 x} x^3-40960 x^4+6400 x^6+e^{4 x} \left (-2560 x^2+2400 x^4\right )+e^{2 x} \left (-20480 x^3+6400 x^5\right )}{e^{8 x}+16 e^{6 x} x+96 e^{4 x} x^2+256 e^{2 x} x^3+256 x^4}} \left (-1572864 x^4+6144 x^6+153600 x^8+e^{10 x} \left (6 x+150 x^3\right )+e^{8 x} \left (120 x^2+3000 x^4\right )+e^{6 x} \left (-14400 x^3+30720 x^4+24000 x^5\right )+e^{4 x} \left (-119040 x^4+245760 x^5+96000 x^6\right )+e^{2 x} \left (393216 x^3-1572864 x^4-238080 x^5+491520 x^6+192000 x^7\right )\right )}{3 e^{10 x}+60 e^{8 x} x+480 e^{6 x} x^2+1920 e^{4 x} x^3+3840 e^{2 x} x^4+3072 x^5} \, dx=x^{2} e^{\left (\frac {6400 \, x^{6} - 40960 \, x^{4} + 400 \, x^{3} e^{\left (6 \, x\right )} + 25 \, x^{2} e^{\left (8 \, x\right )} + 65536 \, x^{2} + 160 \, {\left (15 \, x^{4} - 16 \, x^{2}\right )} e^{\left (4 \, x\right )} + 1280 \, {\left (5 \, x^{5} - 16 \, x^{3}\right )} e^{\left (2 \, x\right )}}{256 \, x^{4} + 256 \, x^{3} e^{\left (2 \, x\right )} + 96 \, x^{2} e^{\left (4 \, x\right )} + 16 \, x e^{\left (6 \, x\right )} + e^{\left (8 \, x\right )}}\right )} - \frac {4}{3} \, x \] Input:
integrate((((150*x^3+6*x)*exp(x)^10+(3000*x^4+120*x^2)*exp(x)^8+(24000*x^5 +30720*x^4-14400*x^3)*exp(x)^6+(96000*x^6+245760*x^5-119040*x^4)*exp(x)^4+ (192000*x^7+491520*x^6-238080*x^5-1572864*x^4+393216*x^3)*exp(x)^2+153600* x^8+6144*x^6-1572864*x^4)*exp((25*x^2*exp(x)^8+400*x^3*exp(x)^6+(2400*x^4- 2560*x^2)*exp(x)^4+(6400*x^5-20480*x^3)*exp(x)^2+6400*x^6-40960*x^4+65536* x^2)/(exp(x)^8+16*x*exp(x)^6+96*x^2*exp(x)^4+256*exp(x)^2*x^3+256*x^4))-4* exp(x)^10-80*x*exp(x)^8-640*x^2*exp(x)^6-2560*x^3*exp(x)^4-5120*exp(x)^2*x ^4-4096*x^5)/(3*exp(x)^10+60*x*exp(x)^8+480*x^2*exp(x)^6+1920*x^3*exp(x)^4 +3840*exp(x)^2*x^4+3072*x^5),x, algorithm="fricas")
Output:
x^2*e^((6400*x^6 - 40960*x^4 + 400*x^3*e^(6*x) + 25*x^2*e^(8*x) + 65536*x^ 2 + 160*(15*x^4 - 16*x^2)*e^(4*x) + 1280*(5*x^5 - 16*x^3)*e^(2*x))/(256*x^ 4 + 256*x^3*e^(2*x) + 96*x^2*e^(4*x) + 16*x*e^(6*x) + e^(8*x))) - 4/3*x
Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (27) = 54\).
Time = 46.06 (sec) , antiderivative size = 114, normalized size of antiderivative = 3.08 \[ \int \frac {-4 e^{10 x}-80 e^{8 x} x-640 e^{6 x} x^2-2560 e^{4 x} x^3-5120 e^{2 x} x^4-4096 x^5+e^{\frac {65536 x^2+25 e^{8 x} x^2+400 e^{6 x} x^3-40960 x^4+6400 x^6+e^{4 x} \left (-2560 x^2+2400 x^4\right )+e^{2 x} \left (-20480 x^3+6400 x^5\right )}{e^{8 x}+16 e^{6 x} x+96 e^{4 x} x^2+256 e^{2 x} x^3+256 x^4}} \left (-1572864 x^4+6144 x^6+153600 x^8+e^{10 x} \left (6 x+150 x^3\right )+e^{8 x} \left (120 x^2+3000 x^4\right )+e^{6 x} \left (-14400 x^3+30720 x^4+24000 x^5\right )+e^{4 x} \left (-119040 x^4+245760 x^5+96000 x^6\right )+e^{2 x} \left (393216 x^3-1572864 x^4-238080 x^5+491520 x^6+192000 x^7\right )\right )}{3 e^{10 x}+60 e^{8 x} x+480 e^{6 x} x^2+1920 e^{4 x} x^3+3840 e^{2 x} x^4+3072 x^5} \, dx=x^{2} e^{\frac {6400 x^{6} - 40960 x^{4} + 400 x^{3} e^{6 x} + 25 x^{2} e^{8 x} + 65536 x^{2} + \left (2400 x^{4} - 2560 x^{2}\right ) e^{4 x} + \left (6400 x^{5} - 20480 x^{3}\right ) e^{2 x}}{256 x^{4} + 256 x^{3} e^{2 x} + 96 x^{2} e^{4 x} + 16 x e^{6 x} + e^{8 x}}} - \frac {4 x}{3} \] Input:
integrate((((150*x**3+6*x)*exp(x)**10+(3000*x**4+120*x**2)*exp(x)**8+(2400 0*x**5+30720*x**4-14400*x**3)*exp(x)**6+(96000*x**6+245760*x**5-119040*x** 4)*exp(x)**4+(192000*x**7+491520*x**6-238080*x**5-1572864*x**4+393216*x**3 )*exp(x)**2+153600*x**8+6144*x**6-1572864*x**4)*exp((25*x**2*exp(x)**8+400 *x**3*exp(x)**6+(2400*x**4-2560*x**2)*exp(x)**4+(6400*x**5-20480*x**3)*exp (x)**2+6400*x**6-40960*x**4+65536*x**2)/(exp(x)**8+16*x*exp(x)**6+96*x**2* exp(x)**4+256*exp(x)**2*x**3+256*x**4))-4*exp(x)**10-80*x*exp(x)**8-640*x* *2*exp(x)**6-2560*x**3*exp(x)**4-5120*exp(x)**2*x**4-4096*x**5)/(3*exp(x)* *10+60*x*exp(x)**8+480*x**2*exp(x)**6+1920*x**3*exp(x)**4+3840*exp(x)**2*x **4+3072*x**5),x)
Output:
x**2*exp((6400*x**6 - 40960*x**4 + 400*x**3*exp(6*x) + 25*x**2*exp(8*x) + 65536*x**2 + (2400*x**4 - 2560*x**2)*exp(4*x) + (6400*x**5 - 20480*x**3)*e xp(2*x))/(256*x**4 + 256*x**3*exp(2*x) + 96*x**2*exp(4*x) + 16*x*exp(6*x) + exp(8*x))) - 4*x/3
Leaf count of result is larger than twice the leaf count of optimal. 222 vs. \(2 (34) = 68\).
Time = 50.28 (sec) , antiderivative size = 222, normalized size of antiderivative = 6.00 \[ \int \frac {-4 e^{10 x}-80 e^{8 x} x-640 e^{6 x} x^2-2560 e^{4 x} x^3-5120 e^{2 x} x^4-4096 x^5+e^{\frac {65536 x^2+25 e^{8 x} x^2+400 e^{6 x} x^3-40960 x^4+6400 x^6+e^{4 x} \left (-2560 x^2+2400 x^4\right )+e^{2 x} \left (-20480 x^3+6400 x^5\right )}{e^{8 x}+16 e^{6 x} x+96 e^{4 x} x^2+256 e^{2 x} x^3+256 x^4}} \left (-1572864 x^4+6144 x^6+153600 x^8+e^{10 x} \left (6 x+150 x^3\right )+e^{8 x} \left (120 x^2+3000 x^4\right )+e^{6 x} \left (-14400 x^3+30720 x^4+24000 x^5\right )+e^{4 x} \left (-119040 x^4+245760 x^5+96000 x^6\right )+e^{2 x} \left (393216 x^3-1572864 x^4-238080 x^5+491520 x^6+192000 x^7\right )\right )}{3 e^{10 x}+60 e^{8 x} x+480 e^{6 x} x^2+1920 e^{4 x} x^3+3840 e^{2 x} x^4+3072 x^5} \, dx=\frac {1}{3} \, {\left (3 \, x^{2} e^{\left (25 \, x^{2} + \frac {4096 \, e^{\left (4 \, x\right )}}{256 \, x^{4} + 256 \, x^{3} e^{\left (2 \, x\right )} + 96 \, x^{2} e^{\left (4 \, x\right )} + 16 \, x e^{\left (6 \, x\right )} + e^{\left (8 \, x\right )}} + \frac {320 \, e^{\left (2 \, x\right )}}{4 \, x + e^{\left (2 \, x\right )}} + \frac {4096}{16 \, x^{2} + 8 \, x e^{\left (2 \, x\right )} + e^{\left (4 \, x\right )}}\right )} - 4 \, x e^{\left (\frac {160 \, e^{\left (4 \, x\right )}}{16 \, x^{2} + 8 \, x e^{\left (2 \, x\right )} + e^{\left (4 \, x\right )}} + \frac {8192 \, e^{\left (2 \, x\right )}}{64 \, x^{3} + 48 \, x^{2} e^{\left (2 \, x\right )} + 12 \, x e^{\left (4 \, x\right )} + e^{\left (6 \, x\right )}} + 160\right )}\right )} e^{\left (-\frac {160 \, e^{\left (4 \, x\right )}}{16 \, x^{2} + 8 \, x e^{\left (2 \, x\right )} + e^{\left (4 \, x\right )}} - \frac {8192 \, e^{\left (2 \, x\right )}}{64 \, x^{3} + 48 \, x^{2} e^{\left (2 \, x\right )} + 12 \, x e^{\left (4 \, x\right )} + e^{\left (6 \, x\right )}} - 160\right )} \] Input:
integrate((((150*x^3+6*x)*exp(x)^10+(3000*x^4+120*x^2)*exp(x)^8+(24000*x^5 +30720*x^4-14400*x^3)*exp(x)^6+(96000*x^6+245760*x^5-119040*x^4)*exp(x)^4+ (192000*x^7+491520*x^6-238080*x^5-1572864*x^4+393216*x^3)*exp(x)^2+153600* x^8+6144*x^6-1572864*x^4)*exp((25*x^2*exp(x)^8+400*x^3*exp(x)^6+(2400*x^4- 2560*x^2)*exp(x)^4+(6400*x^5-20480*x^3)*exp(x)^2+6400*x^6-40960*x^4+65536* x^2)/(exp(x)^8+16*x*exp(x)^6+96*x^2*exp(x)^4+256*exp(x)^2*x^3+256*x^4))-4* exp(x)^10-80*x*exp(x)^8-640*x^2*exp(x)^6-2560*x^3*exp(x)^4-5120*exp(x)^2*x ^4-4096*x^5)/(3*exp(x)^10+60*x*exp(x)^8+480*x^2*exp(x)^6+1920*x^3*exp(x)^4 +3840*exp(x)^2*x^4+3072*x^5),x, algorithm="maxima")
Output:
1/3*(3*x^2*e^(25*x^2 + 4096*e^(4*x)/(256*x^4 + 256*x^3*e^(2*x) + 96*x^2*e^ (4*x) + 16*x*e^(6*x) + e^(8*x)) + 320*e^(2*x)/(4*x + e^(2*x)) + 4096/(16*x ^2 + 8*x*e^(2*x) + e^(4*x))) - 4*x*e^(160*e^(4*x)/(16*x^2 + 8*x*e^(2*x) + e^(4*x)) + 8192*e^(2*x)/(64*x^3 + 48*x^2*e^(2*x) + 12*x*e^(4*x) + e^(6*x)) + 160))*e^(-160*e^(4*x)/(16*x^2 + 8*x*e^(2*x) + e^(4*x)) - 8192*e^(2*x)/( 64*x^3 + 48*x^2*e^(2*x) + 12*x*e^(4*x) + e^(6*x)) - 160)
Exception generated. \[ \int \frac {-4 e^{10 x}-80 e^{8 x} x-640 e^{6 x} x^2-2560 e^{4 x} x^3-5120 e^{2 x} x^4-4096 x^5+e^{\frac {65536 x^2+25 e^{8 x} x^2+400 e^{6 x} x^3-40960 x^4+6400 x^6+e^{4 x} \left (-2560 x^2+2400 x^4\right )+e^{2 x} \left (-20480 x^3+6400 x^5\right )}{e^{8 x}+16 e^{6 x} x+96 e^{4 x} x^2+256 e^{2 x} x^3+256 x^4}} \left (-1572864 x^4+6144 x^6+153600 x^8+e^{10 x} \left (6 x+150 x^3\right )+e^{8 x} \left (120 x^2+3000 x^4\right )+e^{6 x} \left (-14400 x^3+30720 x^4+24000 x^5\right )+e^{4 x} \left (-119040 x^4+245760 x^5+96000 x^6\right )+e^{2 x} \left (393216 x^3-1572864 x^4-238080 x^5+491520 x^6+192000 x^7\right )\right )}{3 e^{10 x}+60 e^{8 x} x+480 e^{6 x} x^2+1920 e^{4 x} x^3+3840 e^{2 x} x^4+3072 x^5} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((((150*x^3+6*x)*exp(x)^10+(3000*x^4+120*x^2)*exp(x)^8+(24000*x^5 +30720*x^4-14400*x^3)*exp(x)^6+(96000*x^6+245760*x^5-119040*x^4)*exp(x)^4+ (192000*x^7+491520*x^6-238080*x^5-1572864*x^4+393216*x^3)*exp(x)^2+153600* x^8+6144*x^6-1572864*x^4)*exp((25*x^2*exp(x)^8+400*x^3*exp(x)^6+(2400*x^4- 2560*x^2)*exp(x)^4+(6400*x^5-20480*x^3)*exp(x)^2+6400*x^6-40960*x^4+65536* x^2)/(exp(x)^8+16*x*exp(x)^6+96*x^2*exp(x)^4+256*exp(x)^2*x^3+256*x^4))-4* exp(x)^10-80*x*exp(x)^8-640*x^2*exp(x)^6-2560*x^3*exp(x)^4-5120*exp(x)^2*x ^4-4096*x^5)/(3*exp(x)^10+60*x*exp(x)^8+480*x^2*exp(x)^6+1920*x^3*exp(x)^4 +3840*exp(x)^2*x^4+3072*x^5),x, algorithm="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%%%{-289480223093290488558927462521719769633174961664101410098 643960019
Time = 4.30 (sec) , antiderivative size = 419, normalized size of antiderivative = 11.32 \[ \int \frac {-4 e^{10 x}-80 e^{8 x} x-640 e^{6 x} x^2-2560 e^{4 x} x^3-5120 e^{2 x} x^4-4096 x^5+e^{\frac {65536 x^2+25 e^{8 x} x^2+400 e^{6 x} x^3-40960 x^4+6400 x^6+e^{4 x} \left (-2560 x^2+2400 x^4\right )+e^{2 x} \left (-20480 x^3+6400 x^5\right )}{e^{8 x}+16 e^{6 x} x+96 e^{4 x} x^2+256 e^{2 x} x^3+256 x^4}} \left (-1572864 x^4+6144 x^6+153600 x^8+e^{10 x} \left (6 x+150 x^3\right )+e^{8 x} \left (120 x^2+3000 x^4\right )+e^{6 x} \left (-14400 x^3+30720 x^4+24000 x^5\right )+e^{4 x} \left (-119040 x^4+245760 x^5+96000 x^6\right )+e^{2 x} \left (393216 x^3-1572864 x^4-238080 x^5+491520 x^6+192000 x^7\right )\right )}{3 e^{10 x}+60 e^{8 x} x+480 e^{6 x} x^2+1920 e^{4 x} x^3+3840 e^{2 x} x^4+3072 x^5} \, dx=x^2\,{\mathrm {e}}^{\frac {25\,x^2\,{\mathrm {e}}^{8\,x}}{{\mathrm {e}}^{8\,x}+16\,x\,{\mathrm {e}}^{6\,x}+256\,x^3\,{\mathrm {e}}^{2\,x}+96\,x^2\,{\mathrm {e}}^{4\,x}+256\,x^4}}\,{\mathrm {e}}^{\frac {400\,x^3\,{\mathrm {e}}^{6\,x}}{{\mathrm {e}}^{8\,x}+16\,x\,{\mathrm {e}}^{6\,x}+256\,x^3\,{\mathrm {e}}^{2\,x}+96\,x^2\,{\mathrm {e}}^{4\,x}+256\,x^4}}\,{\mathrm {e}}^{\frac {2400\,x^4\,{\mathrm {e}}^{4\,x}}{{\mathrm {e}}^{8\,x}+16\,x\,{\mathrm {e}}^{6\,x}+256\,x^3\,{\mathrm {e}}^{2\,x}+96\,x^2\,{\mathrm {e}}^{4\,x}+256\,x^4}}\,{\mathrm {e}}^{-\frac {2560\,x^2\,{\mathrm {e}}^{4\,x}}{{\mathrm {e}}^{8\,x}+16\,x\,{\mathrm {e}}^{6\,x}+256\,x^3\,{\mathrm {e}}^{2\,x}+96\,x^2\,{\mathrm {e}}^{4\,x}+256\,x^4}}\,{\mathrm {e}}^{\frac {6400\,x^5\,{\mathrm {e}}^{2\,x}}{{\mathrm {e}}^{8\,x}+16\,x\,{\mathrm {e}}^{6\,x}+256\,x^3\,{\mathrm {e}}^{2\,x}+96\,x^2\,{\mathrm {e}}^{4\,x}+256\,x^4}}\,{\mathrm {e}}^{-\frac {20480\,x^3\,{\mathrm {e}}^{2\,x}}{{\mathrm {e}}^{8\,x}+16\,x\,{\mathrm {e}}^{6\,x}+256\,x^3\,{\mathrm {e}}^{2\,x}+96\,x^2\,{\mathrm {e}}^{4\,x}+256\,x^4}}\,{\mathrm {e}}^{\frac {6400\,x^6}{{\mathrm {e}}^{8\,x}+16\,x\,{\mathrm {e}}^{6\,x}+256\,x^3\,{\mathrm {e}}^{2\,x}+96\,x^2\,{\mathrm {e}}^{4\,x}+256\,x^4}}\,{\mathrm {e}}^{-\frac {40960\,x^4}{{\mathrm {e}}^{8\,x}+16\,x\,{\mathrm {e}}^{6\,x}+256\,x^3\,{\mathrm {e}}^{2\,x}+96\,x^2\,{\mathrm {e}}^{4\,x}+256\,x^4}}\,{\mathrm {e}}^{\frac {65536\,x^2}{{\mathrm {e}}^{8\,x}+16\,x\,{\mathrm {e}}^{6\,x}+256\,x^3\,{\mathrm {e}}^{2\,x}+96\,x^2\,{\mathrm {e}}^{4\,x}+256\,x^4}}-\frac {4\,x}{3} \] Input:
int(-(4*exp(10*x) + 80*x*exp(8*x) + 5120*x^4*exp(2*x) + 2560*x^3*exp(4*x) + 640*x^2*exp(6*x) + 4096*x^5 - exp((400*x^3*exp(6*x) - exp(2*x)*(20480*x^ 3 - 6400*x^5) - exp(4*x)*(2560*x^2 - 2400*x^4) + 25*x^2*exp(8*x) + 65536*x ^2 - 40960*x^4 + 6400*x^6)/(exp(8*x) + 16*x*exp(6*x) + 256*x^3*exp(2*x) + 96*x^2*exp(4*x) + 256*x^4))*(exp(10*x)*(6*x + 150*x^3) + exp(2*x)*(393216* x^3 - 1572864*x^4 - 238080*x^5 + 491520*x^6 + 192000*x^7) + exp(8*x)*(120* x^2 + 3000*x^4) + exp(6*x)*(30720*x^4 - 14400*x^3 + 24000*x^5) + exp(4*x)* (245760*x^5 - 119040*x^4 + 96000*x^6) - 1572864*x^4 + 6144*x^6 + 153600*x^ 8))/(3*exp(10*x) + 60*x*exp(8*x) + 3840*x^4*exp(2*x) + 1920*x^3*exp(4*x) + 480*x^2*exp(6*x) + 3072*x^5),x)
Output:
x^2*exp((25*x^2*exp(8*x))/(exp(8*x) + 16*x*exp(6*x) + 256*x^3*exp(2*x) + 9 6*x^2*exp(4*x) + 256*x^4))*exp((400*x^3*exp(6*x))/(exp(8*x) + 16*x*exp(6*x ) + 256*x^3*exp(2*x) + 96*x^2*exp(4*x) + 256*x^4))*exp((2400*x^4*exp(4*x)) /(exp(8*x) + 16*x*exp(6*x) + 256*x^3*exp(2*x) + 96*x^2*exp(4*x) + 256*x^4) )*exp(-(2560*x^2*exp(4*x))/(exp(8*x) + 16*x*exp(6*x) + 256*x^3*exp(2*x) + 96*x^2*exp(4*x) + 256*x^4))*exp((6400*x^5*exp(2*x))/(exp(8*x) + 16*x*exp(6 *x) + 256*x^3*exp(2*x) + 96*x^2*exp(4*x) + 256*x^4))*exp(-(20480*x^3*exp(2 *x))/(exp(8*x) + 16*x*exp(6*x) + 256*x^3*exp(2*x) + 96*x^2*exp(4*x) + 256* x^4))*exp((6400*x^6)/(exp(8*x) + 16*x*exp(6*x) + 256*x^3*exp(2*x) + 96*x^2 *exp(4*x) + 256*x^4))*exp(-(40960*x^4)/(exp(8*x) + 16*x*exp(6*x) + 256*x^3 *exp(2*x) + 96*x^2*exp(4*x) + 256*x^4))*exp((65536*x^2)/(exp(8*x) + 16*x*e xp(6*x) + 256*x^3*exp(2*x) + 96*x^2*exp(4*x) + 256*x^4)) - (4*x)/3
\[ \int \frac {-4 e^{10 x}-80 e^{8 x} x-640 e^{6 x} x^2-2560 e^{4 x} x^3-5120 e^{2 x} x^4-4096 x^5+e^{\frac {65536 x^2+25 e^{8 x} x^2+400 e^{6 x} x^3-40960 x^4+6400 x^6+e^{4 x} \left (-2560 x^2+2400 x^4\right )+e^{2 x} \left (-20480 x^3+6400 x^5\right )}{e^{8 x}+16 e^{6 x} x+96 e^{4 x} x^2+256 e^{2 x} x^3+256 x^4}} \left (-1572864 x^4+6144 x^6+153600 x^8+e^{10 x} \left (6 x+150 x^3\right )+e^{8 x} \left (120 x^2+3000 x^4\right )+e^{6 x} \left (-14400 x^3+30720 x^4+24000 x^5\right )+e^{4 x} \left (-119040 x^4+245760 x^5+96000 x^6\right )+e^{2 x} \left (393216 x^3-1572864 x^4-238080 x^5+491520 x^6+192000 x^7\right )\right )}{3 e^{10 x}+60 e^{8 x} x+480 e^{6 x} x^2+1920 e^{4 x} x^3+3840 e^{2 x} x^4+3072 x^5} \, dx=\int \frac {\left (\left (150 x^{3}+6 x \right ) \left ({\mathrm e}^{x}\right )^{10}+\left (3000 x^{4}+120 x^{2}\right ) \left ({\mathrm e}^{x}\right )^{8}+\left (24000 x^{5}+30720 x^{4}-14400 x^{3}\right ) \left ({\mathrm e}^{x}\right )^{6}+\left (96000 x^{6}+245760 x^{5}-119040 x^{4}\right ) \left ({\mathrm e}^{x}\right )^{4}+\left (192000 x^{7}+491520 x^{6}-238080 x^{5}-1572864 x^{4}+393216 x^{3}\right ) \left ({\mathrm e}^{x}\right )^{2}+153600 x^{8}+6144 x^{6}-1572864 x^{4}\right ) {\mathrm e}^{\frac {25 x^{2} \left ({\mathrm e}^{x}\right )^{8}+400 x^{3} \left ({\mathrm e}^{x}\right )^{6}+\left (2400 x^{4}-2560 x^{2}\right ) \left ({\mathrm e}^{x}\right )^{4}+\left (6400 x^{5}-20480 x^{3}\right ) \left ({\mathrm e}^{x}\right )^{2}+6400 x^{6}-40960 x^{4}+65536 x^{2}}{\left ({\mathrm e}^{x}\right )^{8}+16 x \left ({\mathrm e}^{x}\right )^{6}+96 x^{2} \left ({\mathrm e}^{x}\right )^{4}+256 \left ({\mathrm e}^{x}\right )^{2} x^{3}+256 x^{4}}}-4 \left ({\mathrm e}^{x}\right )^{10}-80 x \left ({\mathrm e}^{x}\right )^{8}-640 x^{2} \left ({\mathrm e}^{x}\right )^{6}-2560 x^{3} \left ({\mathrm e}^{x}\right )^{4}-5120 \left ({\mathrm e}^{x}\right )^{2} x^{4}-4096 x^{5}}{3 \left ({\mathrm e}^{x}\right )^{10}+60 x \left ({\mathrm e}^{x}\right )^{8}+480 x^{2} \left ({\mathrm e}^{x}\right )^{6}+1920 x^{3} \left ({\mathrm e}^{x}\right )^{4}+3840 \left ({\mathrm e}^{x}\right )^{2} x^{4}+3072 x^{5}}d x \] Input:
int((((150*x^3+6*x)*exp(x)^10+(3000*x^4+120*x^2)*exp(x)^8+(24000*x^5+30720 *x^4-14400*x^3)*exp(x)^6+(96000*x^6+245760*x^5-119040*x^4)*exp(x)^4+(19200 0*x^7+491520*x^6-238080*x^5-1572864*x^4+393216*x^3)*exp(x)^2+153600*x^8+61 44*x^6-1572864*x^4)*exp((25*x^2*exp(x)^8+400*x^3*exp(x)^6+(2400*x^4-2560*x ^2)*exp(x)^4+(6400*x^5-20480*x^3)*exp(x)^2+6400*x^6-40960*x^4+65536*x^2)/( exp(x)^8+16*x*exp(x)^6+96*x^2*exp(x)^4+256*exp(x)^2*x^3+256*x^4))-4*exp(x) ^10-80*x*exp(x)^8-640*x^2*exp(x)^6-2560*x^3*exp(x)^4-5120*exp(x)^2*x^4-409 6*x^5)/(3*exp(x)^10+60*x*exp(x)^8+480*x^2*exp(x)^6+1920*x^3*exp(x)^4+3840* exp(x)^2*x^4+3072*x^5),x)
Output:
int((((150*x^3+6*x)*exp(x)^10+(3000*x^4+120*x^2)*exp(x)^8+(24000*x^5+30720 *x^4-14400*x^3)*exp(x)^6+(96000*x^6+245760*x^5-119040*x^4)*exp(x)^4+(19200 0*x^7+491520*x^6-238080*x^5-1572864*x^4+393216*x^3)*exp(x)^2+153600*x^8+61 44*x^6-1572864*x^4)*exp((25*x^2*exp(x)^8+400*x^3*exp(x)^6+(2400*x^4-2560*x ^2)*exp(x)^4+(6400*x^5-20480*x^3)*exp(x)^2+6400*x^6-40960*x^4+65536*x^2)/( exp(x)^8+16*x*exp(x)^6+96*x^2*exp(x)^4+256*exp(x)^2*x^3+256*x^4))-4*exp(x) ^10-80*x*exp(x)^8-640*x^2*exp(x)^6-2560*x^3*exp(x)^4-5120*exp(x)^2*x^4-409 6*x^5)/(3*exp(x)^10+60*x*exp(x)^8+480*x^2*exp(x)^6+1920*x^3*exp(x)^4+3840* exp(x)^2*x^4+3072*x^5),x)