Integrand size = 161, antiderivative size = 26 \[ \int \left (\left (3200 x+1600 x^2+200 x^3+e^x \left (-1280 x^2-1920 x^3-720 x^4-80 x^5\right )+e^{2 x} \left (128 x^3+320 x^4+264 x^5+80 x^6+8 x^7\right )\right ) \log (x)+\left (3200 x+2400 x^2+400 x^3+e^x \left (-1920 x^2-4480 x^3-2760 x^4-600 x^5-40 x^6\right )+e^{2 x} \left (256 x^3+928 x^4+1112 x^5+544 x^6+112 x^7+8 x^8\right )\right ) \log ^2(x)\right ) \, dx=4 x^2 (4+x)^2 \left (-5+e^x x (1+x)\right )^2 \log ^2(x) \]
Time = 0.89 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \left (\left (3200 x+1600 x^2+200 x^3+e^x \left (-1280 x^2-1920 x^3-720 x^4-80 x^5\right )+e^{2 x} \left (128 x^3+320 x^4+264 x^5+80 x^6+8 x^7\right )\right ) \log (x)+\left (3200 x+2400 x^2+400 x^3+e^x \left (-1920 x^2-4480 x^3-2760 x^4-600 x^5-40 x^6\right )+e^{2 x} \left (256 x^3+928 x^4+1112 x^5+544 x^6+112 x^7+8 x^8\right )\right ) \log ^2(x)\right ) \, dx=4 x^2 (4+x)^2 \left (-5+e^x x (1+x)\right )^2 \log ^2(x) \]
Integrate[(3200*x + 1600*x^2 + 200*x^3 + E^x*(-1280*x^2 - 1920*x^3 - 720*x ^4 - 80*x^5) + E^(2*x)*(128*x^3 + 320*x^4 + 264*x^5 + 80*x^6 + 8*x^7))*Log [x] + (3200*x + 2400*x^2 + 400*x^3 + E^x*(-1920*x^2 - 4480*x^3 - 2760*x^4 - 600*x^5 - 40*x^6) + E^(2*x)*(256*x^3 + 928*x^4 + 1112*x^5 + 544*x^6 + 11 2*x^7 + 8*x^8))*Log[x]^2,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.
| \(\displaystyle \int \left (\left (200 x^3+1600 x^2+e^x \left (-80 x^5-720 x^4-1920 x^3-1280 x^2\right )+e^{2 x} \left (8 x^7+80 x^6+264 x^5+320 x^4+128 x^3\right )+3200 x\right ) \log (x)+\left (400 x^3+2400 x^2+e^x \left (-40 x^6-600 x^5-2760 x^4-4480 x^3-1920 x^2\right )+e^{2 x} \left (8 x^8+112 x^7+544 x^6+1112 x^5+928 x^4+256 x^3\right )+3200 x\right ) \log ^2(x)\right ) \, dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 8 \int e^{2 x} x^8 \log ^2(x)dx+112 \int e^{2 x} x^7 \log ^2(x)dx-40 \int e^x x^6 \log ^2(x)dx+544 \int e^{2 x} x^6 \log ^2(x)dx-600 \int e^x x^5 \log ^2(x)dx+1112 \int e^{2 x} x^5 \log ^2(x)dx-2760 \int e^x x^4 \log ^2(x)dx+928 \int e^{2 x} x^4 \log ^2(x)dx-4480 \int e^x x^3 \log ^2(x)dx+256 \int e^{2 x} x^3 \log ^2(x)dx-1920 \int e^x x^2 \log ^2(x)dx-1280 \operatorname {ExpIntegralEi}(x)+\frac {21 \operatorname {ExpIntegralEi}(2 x)}{2}+4 e^{2 x} x^7 \log (x)-2 e^{2 x} x^6+26 e^{2 x} x^6 \log (x)-7 e^{2 x} x^5-80 e^x x^5 \log (x)+54 e^{2 x} x^5 \log (x)+80 e^x x^4-\frac {19}{2} e^{2 x} x^4+100 x^4 \log ^2(x)-320 e^x x^4 \log (x)+25 e^{2 x} x^4 \log (x)+\frac {13}{2} e^{2 x} x^3+800 x^3 \log ^2(x)-640 e^x x^3 \log (x)+14 e^{2 x} x^3 \log (x)+640 e^x x^2-\frac {67}{4} e^{2 x} x^2+1600 x^2 \log ^2(x)+640 e^x x^2 \log (x)-21 e^{2 x} x^2 \log (x)-1920 e^x x+\frac {109}{4} e^{2 x} x+3200 e^x-\frac {193 e^{2 x}}{8}-1280 e^x x \log (x)+21 e^{2 x} x \log (x)+1280 e^x \log (x)-\frac {21}{2} e^{2 x} \log (x)\) |
Int[(3200*x + 1600*x^2 + 200*x^3 + E^x*(-1280*x^2 - 1920*x^3 - 720*x^4 - 8 0*x^5) + E^(2*x)*(128*x^3 + 320*x^4 + 264*x^5 + 80*x^6 + 8*x^7))*Log[x] + (3200*x + 2400*x^2 + 400*x^3 + E^x*(-1920*x^2 - 4480*x^3 - 2760*x^4 - 600* x^5 - 40*x^6) + E^(2*x)*(256*x^3 + 928*x^4 + 1112*x^5 + 544*x^6 + 112*x^7 + 8*x^8))*Log[x]^2,x]
3.2.96.3.1 Defintions of rubi rules used
Leaf count of result is larger than twice the leaf count of optimal. \(137\) vs. \(2(25)=50\).
Time = 0.96 (sec) , antiderivative size = 138, normalized size of antiderivative = 5.31
| method | result | size |
| risch | \(4 \,{\mathrm e}^{2 x} \ln \left (x \right )^{2} x^{8}+40 \,{\mathrm e}^{2 x} \ln \left (x \right )^{2} x^{7}+132 \,{\mathrm e}^{2 x} \ln \left (x \right )^{2} x^{6}+160 \,{\mathrm e}^{2 x} \ln \left (x \right )^{2} x^{5}-40 \ln \left (x \right )^{2} {\mathrm e}^{x} x^{6}+64 \,{\mathrm e}^{2 x} \ln \left (x \right )^{2} x^{4}-360 \ln \left (x \right )^{2} {\mathrm e}^{x} x^{5}-960 x^{4} {\mathrm e}^{x} \ln \left (x \right )^{2}-640 \ln \left (x \right )^{2} {\mathrm e}^{x} x^{3}+100 x^{4} \ln \left (x \right )^{2}+800 x^{3} \ln \left (x \right )^{2}+1600 x^{2} \ln \left (x \right )^{2}\) | \(138\) |
| parallelrisch | \(4 \,{\mathrm e}^{2 x} \ln \left (x \right )^{2} x^{8}+40 \,{\mathrm e}^{2 x} \ln \left (x \right )^{2} x^{7}+132 \,{\mathrm e}^{2 x} \ln \left (x \right )^{2} x^{6}+160 \,{\mathrm e}^{2 x} \ln \left (x \right )^{2} x^{5}-40 \ln \left (x \right )^{2} {\mathrm e}^{x} x^{6}+64 \,{\mathrm e}^{2 x} \ln \left (x \right )^{2} x^{4}-360 \ln \left (x \right )^{2} {\mathrm e}^{x} x^{5}-960 x^{4} {\mathrm e}^{x} \ln \left (x \right )^{2}-640 \ln \left (x \right )^{2} {\mathrm e}^{x} x^{3}+100 x^{4} \ln \left (x \right )^{2}+800 x^{3} \ln \left (x \right )^{2}+1600 x^{2} \ln \left (x \right )^{2}\) | \(138\) |
int(((8*x^8+112*x^7+544*x^6+1112*x^5+928*x^4+256*x^3)*exp(x)^2+(-40*x^6-60 0*x^5-2760*x^4-4480*x^3-1920*x^2)*exp(x)+400*x^3+2400*x^2+3200*x)*ln(x)^2+ ((8*x^7+80*x^6+264*x^5+320*x^4+128*x^3)*exp(x)^2+(-80*x^5-720*x^4-1920*x^3 -1280*x^2)*exp(x)+200*x^3+1600*x^2+3200*x)*ln(x),x,method=_RETURNVERBOSE)
4*ln(x)^2*exp(x)^2*x^8+40*ln(x)^2*exp(x)^2*x^7+132*ln(x)^2*exp(x)^2*x^6+16 0*ln(x)^2*exp(x)^2*x^5-40*ln(x)^2*exp(x)*x^6+64*x^4*exp(x)^2*ln(x)^2-360*l n(x)^2*exp(x)*x^5-960*x^4*exp(x)*ln(x)^2-640*ln(x)^2*exp(x)*x^3+100*x^4*ln (x)^2+800*x^3*ln(x)^2+1600*x^2*ln(x)^2
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (25) = 50\).
Time = 0.26 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.85 \[ \int \left (\left (3200 x+1600 x^2+200 x^3+e^x \left (-1280 x^2-1920 x^3-720 x^4-80 x^5\right )+e^{2 x} \left (128 x^3+320 x^4+264 x^5+80 x^6+8 x^7\right )\right ) \log (x)+\left (3200 x+2400 x^2+400 x^3+e^x \left (-1920 x^2-4480 x^3-2760 x^4-600 x^5-40 x^6\right )+e^{2 x} \left (256 x^3+928 x^4+1112 x^5+544 x^6+112 x^7+8 x^8\right )\right ) \log ^2(x)\right ) \, dx=4 \, {\left (25 \, x^{4} + 200 \, x^{3} + 400 \, x^{2} + {\left (x^{8} + 10 \, x^{7} + 33 \, x^{6} + 40 \, x^{5} + 16 \, x^{4}\right )} e^{\left (2 \, x\right )} - 10 \, {\left (x^{6} + 9 \, x^{5} + 24 \, x^{4} + 16 \, x^{3}\right )} e^{x}\right )} \log \left (x\right )^{2} \]
integrate(((8*x^8+112*x^7+544*x^6+1112*x^5+928*x^4+256*x^3)*exp(x)^2+(-40* x^6-600*x^5-2760*x^4-4480*x^3-1920*x^2)*exp(x)+400*x^3+2400*x^2+3200*x)*lo g(x)^2+((8*x^7+80*x^6+264*x^5+320*x^4+128*x^3)*exp(x)^2+(-80*x^5-720*x^4-1 920*x^3-1280*x^2)*exp(x)+200*x^3+1600*x^2+3200*x)*log(x),x, algorithm=\
4*(25*x^4 + 200*x^3 + 400*x^2 + (x^8 + 10*x^7 + 33*x^6 + 40*x^5 + 16*x^4)* e^(2*x) - 10*(x^6 + 9*x^5 + 24*x^4 + 16*x^3)*e^x)*log(x)^2
Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (26) = 52\).
Time = 0.33 (sec) , antiderivative size = 121, normalized size of antiderivative = 4.65 \[ \int \left (\left (3200 x+1600 x^2+200 x^3+e^x \left (-1280 x^2-1920 x^3-720 x^4-80 x^5\right )+e^{2 x} \left (128 x^3+320 x^4+264 x^5+80 x^6+8 x^7\right )\right ) \log (x)+\left (3200 x+2400 x^2+400 x^3+e^x \left (-1920 x^2-4480 x^3-2760 x^4-600 x^5-40 x^6\right )+e^{2 x} \left (256 x^3+928 x^4+1112 x^5+544 x^6+112 x^7+8 x^8\right )\right ) \log ^2(x)\right ) \, dx=\left (100 x^{4} + 800 x^{3} + 1600 x^{2}\right ) \log {\left (x \right )}^{2} + \left (- 40 x^{6} \log {\left (x \right )}^{2} - 360 x^{5} \log {\left (x \right )}^{2} - 960 x^{4} \log {\left (x \right )}^{2} - 640 x^{3} \log {\left (x \right )}^{2}\right ) e^{x} + \left (4 x^{8} \log {\left (x \right )}^{2} + 40 x^{7} \log {\left (x \right )}^{2} + 132 x^{6} \log {\left (x \right )}^{2} + 160 x^{5} \log {\left (x \right )}^{2} + 64 x^{4} \log {\left (x \right )}^{2}\right ) e^{2 x} \]
integrate(((8*x**8+112*x**7+544*x**6+1112*x**5+928*x**4+256*x**3)*exp(x)** 2+(-40*x**6-600*x**5-2760*x**4-4480*x**3-1920*x**2)*exp(x)+400*x**3+2400*x **2+3200*x)*ln(x)**2+((8*x**7+80*x**6+264*x**5+320*x**4+128*x**3)*exp(x)** 2+(-80*x**5-720*x**4-1920*x**3-1280*x**2)*exp(x)+200*x**3+1600*x**2+3200*x )*ln(x),x)
(100*x**4 + 800*x**3 + 1600*x**2)*log(x)**2 + (-40*x**6*log(x)**2 - 360*x* *5*log(x)**2 - 960*x**4*log(x)**2 - 640*x**3*log(x)**2)*exp(x) + (4*x**8*l og(x)**2 + 40*x**7*log(x)**2 + 132*x**6*log(x)**2 + 160*x**5*log(x)**2 + 6 4*x**4*log(x)**2)*exp(2*x)
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (25) = 50\).
Time = 0.23 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.15 \[ \int \left (\left (3200 x+1600 x^2+200 x^3+e^x \left (-1280 x^2-1920 x^3-720 x^4-80 x^5\right )+e^{2 x} \left (128 x^3+320 x^4+264 x^5+80 x^6+8 x^7\right )\right ) \log (x)+\left (3200 x+2400 x^2+400 x^3+e^x \left (-1920 x^2-4480 x^3-2760 x^4-600 x^5-40 x^6\right )+e^{2 x} \left (256 x^3+928 x^4+1112 x^5+544 x^6+112 x^7+8 x^8\right )\right ) \log ^2(x)\right ) \, dx=4 \, {\left (x^{8} + 10 \, x^{7} + 33 \, x^{6} + 40 \, x^{5} + 16 \, x^{4}\right )} e^{\left (2 \, x\right )} \log \left (x\right )^{2} - 40 \, {\left (x^{6} + 9 \, x^{5} + 24 \, x^{4} + 16 \, x^{3}\right )} e^{x} \log \left (x\right )^{2} + 100 \, {\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} \log \left (x\right )^{2} \]
integrate(((8*x^8+112*x^7+544*x^6+1112*x^5+928*x^4+256*x^3)*exp(x)^2+(-40* x^6-600*x^5-2760*x^4-4480*x^3-1920*x^2)*exp(x)+400*x^3+2400*x^2+3200*x)*lo g(x)^2+((8*x^7+80*x^6+264*x^5+320*x^4+128*x^3)*exp(x)^2+(-80*x^5-720*x^4-1 920*x^3-1280*x^2)*exp(x)+200*x^3+1600*x^2+3200*x)*log(x),x, algorithm=\
4*(x^8 + 10*x^7 + 33*x^6 + 40*x^5 + 16*x^4)*e^(2*x)*log(x)^2 - 40*(x^6 + 9 *x^5 + 24*x^4 + 16*x^3)*e^x*log(x)^2 + 100*(x^4 + 8*x^3 + 16*x^2)*log(x)^2
Leaf count of result is larger than twice the leaf count of optimal. 435 vs. \(2 (25) = 50\).
Time = 0.32 (sec) , antiderivative size = 435, normalized size of antiderivative = 16.73 \[ \int \left (\left (3200 x+1600 x^2+200 x^3+e^x \left (-1280 x^2-1920 x^3-720 x^4-80 x^5\right )+e^{2 x} \left (128 x^3+320 x^4+264 x^5+80 x^6+8 x^7\right )\right ) \log (x)+\left (3200 x+2400 x^2+400 x^3+e^x \left (-1920 x^2-4480 x^3-2760 x^4-600 x^5-40 x^6\right )+e^{2 x} \left (256 x^3+928 x^4+1112 x^5+544 x^6+112 x^7+8 x^8\right )\right ) \log ^2(x)\right ) \, dx=-50 \, x^{4} \log \left (x\right ) - \frac {1600}{3} \, x^{3} \log \left (x\right ) - 1600 \, x^{2} \log \left (x\right ) - \frac {1}{2} \, {\left (8 \, x^{7} - 28 \, x^{6} + 84 \, x^{5} - 210 \, x^{4} + 420 \, x^{3} - 630 \, x^{2} + 630 \, x - 315\right )} e^{\left (2 \, x\right )} \log \left (x\right ) - 10 \, {\left (4 \, x^{6} - 12 \, x^{5} + 30 \, x^{4} - 60 \, x^{3} + 90 \, x^{2} - 90 \, x + 45\right )} e^{\left (2 \, x\right )} \log \left (x\right ) - 33 \, {\left (4 \, x^{5} - 10 \, x^{4} + 20 \, x^{3} - 30 \, x^{2} + 30 \, x - 15\right )} e^{\left (2 \, x\right )} \log \left (x\right ) - 80 \, {\left (2 \, x^{4} - 4 \, x^{3} + 6 \, x^{2} - 6 \, x + 3\right )} e^{\left (2 \, x\right )} \log \left (x\right ) - 16 \, {\left (4 \, x^{3} - 6 \, x^{2} + 6 \, x - 3\right )} e^{\left (2 \, x\right )} \log \left (x\right ) + 80 \, {\left (x^{5} - 5 \, x^{4} + 20 \, x^{3} - 60 \, x^{2} + 120 \, x - 120\right )} e^{x} \log \left (x\right ) + 720 \, {\left (x^{4} - 4 \, x^{3} + 12 \, x^{2} - 24 \, x + 24\right )} e^{x} \log \left (x\right ) + 1920 \, {\left (x^{3} - 3 \, x^{2} + 6 \, x - 6\right )} e^{x} \log \left (x\right ) + 1280 \, {\left (x^{2} - 2 \, x + 2\right )} e^{x} \log \left (x\right ) + 4 \, {\left (25 \, x^{4} + 200 \, x^{3} + 400 \, x^{2} + {\left (x^{8} + 10 \, x^{7} + 33 \, x^{6} + 40 \, x^{5} + 16 \, x^{4}\right )} e^{\left (2 \, x\right )} - 10 \, {\left (x^{6} + 9 \, x^{5} + 24 \, x^{4} + 16 \, x^{3}\right )} e^{x}\right )} \log \left (x\right )^{2} + \frac {1}{6} \, {\left (300 \, x^{4} + 3200 \, x^{3} + 9600 \, x^{2} + 3 \, {\left (8 \, x^{7} + 52 \, x^{6} + 108 \, x^{5} + 50 \, x^{4} + 28 \, x^{3} - 42 \, x^{2} + 42 \, x - 21\right )} e^{\left (2 \, x\right )} - 480 \, {\left (x^{5} + 4 \, x^{4} + 8 \, x^{3} - 8 \, x^{2} + 16 \, x - 16\right )} e^{x}\right )} \log \left (x\right ) \]
integrate(((8*x^8+112*x^7+544*x^6+1112*x^5+928*x^4+256*x^3)*exp(x)^2+(-40* x^6-600*x^5-2760*x^4-4480*x^3-1920*x^2)*exp(x)+400*x^3+2400*x^2+3200*x)*lo g(x)^2+((8*x^7+80*x^6+264*x^5+320*x^4+128*x^3)*exp(x)^2+(-80*x^5-720*x^4-1 920*x^3-1280*x^2)*exp(x)+200*x^3+1600*x^2+3200*x)*log(x),x, algorithm=\
-50*x^4*log(x) - 1600/3*x^3*log(x) - 1600*x^2*log(x) - 1/2*(8*x^7 - 28*x^6 + 84*x^5 - 210*x^4 + 420*x^3 - 630*x^2 + 630*x - 315)*e^(2*x)*log(x) - 10 *(4*x^6 - 12*x^5 + 30*x^4 - 60*x^3 + 90*x^2 - 90*x + 45)*e^(2*x)*log(x) - 33*(4*x^5 - 10*x^4 + 20*x^3 - 30*x^2 + 30*x - 15)*e^(2*x)*log(x) - 80*(2*x ^4 - 4*x^3 + 6*x^2 - 6*x + 3)*e^(2*x)*log(x) - 16*(4*x^3 - 6*x^2 + 6*x - 3 )*e^(2*x)*log(x) + 80*(x^5 - 5*x^4 + 20*x^3 - 60*x^2 + 120*x - 120)*e^x*lo g(x) + 720*(x^4 - 4*x^3 + 12*x^2 - 24*x + 24)*e^x*log(x) + 1920*(x^3 - 3*x ^2 + 6*x - 6)*e^x*log(x) + 1280*(x^2 - 2*x + 2)*e^x*log(x) + 4*(25*x^4 + 2 00*x^3 + 400*x^2 + (x^8 + 10*x^7 + 33*x^6 + 40*x^5 + 16*x^4)*e^(2*x) - 10* (x^6 + 9*x^5 + 24*x^4 + 16*x^3)*e^x)*log(x)^2 + 1/6*(300*x^4 + 3200*x^3 + 9600*x^2 + 3*(8*x^7 + 52*x^6 + 108*x^5 + 50*x^4 + 28*x^3 - 42*x^2 + 42*x - 21)*e^(2*x) - 480*(x^5 + 4*x^4 + 8*x^3 - 8*x^2 + 16*x - 16)*e^x)*log(x)
Time = 12.18 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.96 \[ \int \left (\left (3200 x+1600 x^2+200 x^3+e^x \left (-1280 x^2-1920 x^3-720 x^4-80 x^5\right )+e^{2 x} \left (128 x^3+320 x^4+264 x^5+80 x^6+8 x^7\right )\right ) \log (x)+\left (3200 x+2400 x^2+400 x^3+e^x \left (-1920 x^2-4480 x^3-2760 x^4-600 x^5-40 x^6\right )+e^{2 x} \left (256 x^3+928 x^4+1112 x^5+544 x^6+112 x^7+8 x^8\right )\right ) \log ^2(x)\right ) \, dx={\ln \left (x\right )}^2\,\left ({\mathrm {e}}^{2\,x}\,\left (4\,x^8+40\,x^7+132\,x^6+160\,x^5+64\,x^4\right )-{\mathrm {e}}^x\,\left (40\,x^6+360\,x^5+960\,x^4+640\,x^3\right )+1600\,x^2+800\,x^3+100\,x^4\right ) \]
int(log(x)^2*(3200*x + exp(2*x)*(256*x^3 + 928*x^4 + 1112*x^5 + 544*x^6 + 112*x^7 + 8*x^8) - exp(x)*(1920*x^2 + 4480*x^3 + 2760*x^4 + 600*x^5 + 40*x ^6) + 2400*x^2 + 400*x^3) + log(x)*(3200*x + exp(2*x)*(128*x^3 + 320*x^4 + 264*x^5 + 80*x^6 + 8*x^7) - exp(x)*(1280*x^2 + 1920*x^3 + 720*x^4 + 80*x^ 5) + 1600*x^2 + 200*x^3),x)