Integrand size = 429, antiderivative size = 15 \begin {dmath*} \int \frac {1024 x+3584 x^2+5376 x^3+4480 x^4+2240 x^5+672 x^6+112 x^7+8 x^8+e^2 \left (1024+3584 x+5376 x^2+4480 x^3+2240 x^4+672 x^5+112 x^6+8 x^7\right )+\left (e^4 \left (3584+10752 x+13440 x^2+8960 x^3+3360 x^4+672 x^5+56 x^6\right )+e^2 \left (3584 x+10752 x^2+13440 x^3+8960 x^4+3360 x^5+672 x^6+56 x^7\right )\right ) \log (2 x)+\left (e^6 \left (5376+13440 x+13440 x^2+6720 x^3+1680 x^4+168 x^5\right )+e^4 \left (5376 x+13440 x^2+13440 x^3+6720 x^4+1680 x^5+168 x^6\right )\right ) \log ^2(2 x)+\left (e^8 \left (4480+8960 x+6720 x^2+2240 x^3+280 x^4\right )+e^6 \left (4480 x+8960 x^2+6720 x^3+2240 x^4+280 x^5\right )\right ) \log ^3(2 x)+\left (e^{10} \left (2240+3360 x+1680 x^2+280 x^3\right )+e^8 \left (2240 x+3360 x^2+1680 x^3+280 x^4\right )\right ) \log ^4(2 x)+\left (e^{12} \left (672+672 x+168 x^2\right )+e^{10} \left (672 x+672 x^2+168 x^3\right )\right ) \log ^5(2 x)+\left (e^{14} (112+56 x)+e^{12} \left (112 x+56 x^2\right )\right ) \log ^6(2 x)+\left (8 e^{16}+8 e^{14} x\right ) \log ^7(2 x)}{x} \, dx=-1+\left (2+x+e^2 \log (2 x)\right )^8 \end {dmath*}
Time = 0.07 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \begin {dmath*} \int \frac {1024 x+3584 x^2+5376 x^3+4480 x^4+2240 x^5+672 x^6+112 x^7+8 x^8+e^2 \left (1024+3584 x+5376 x^2+4480 x^3+2240 x^4+672 x^5+112 x^6+8 x^7\right )+\left (e^4 \left (3584+10752 x+13440 x^2+8960 x^3+3360 x^4+672 x^5+56 x^6\right )+e^2 \left (3584 x+10752 x^2+13440 x^3+8960 x^4+3360 x^5+672 x^6+56 x^7\right )\right ) \log (2 x)+\left (e^6 \left (5376+13440 x+13440 x^2+6720 x^3+1680 x^4+168 x^5\right )+e^4 \left (5376 x+13440 x^2+13440 x^3+6720 x^4+1680 x^5+168 x^6\right )\right ) \log ^2(2 x)+\left (e^8 \left (4480+8960 x+6720 x^2+2240 x^3+280 x^4\right )+e^6 \left (4480 x+8960 x^2+6720 x^3+2240 x^4+280 x^5\right )\right ) \log ^3(2 x)+\left (e^{10} \left (2240+3360 x+1680 x^2+280 x^3\right )+e^8 \left (2240 x+3360 x^2+1680 x^3+280 x^4\right )\right ) \log ^4(2 x)+\left (e^{12} \left (672+672 x+168 x^2\right )+e^{10} \left (672 x+672 x^2+168 x^3\right )\right ) \log ^5(2 x)+\left (e^{14} (112+56 x)+e^{12} \left (112 x+56 x^2\right )\right ) \log ^6(2 x)+\left (8 e^{16}+8 e^{14} x\right ) \log ^7(2 x)}{x} \, dx=\left (2+x+e^2 \log (2 x)\right )^8 \end {dmath*}
Integrate[(1024*x + 3584*x^2 + 5376*x^3 + 4480*x^4 + 2240*x^5 + 672*x^6 + 112*x^7 + 8*x^8 + E^2*(1024 + 3584*x + 5376*x^2 + 4480*x^3 + 2240*x^4 + 67 2*x^5 + 112*x^6 + 8*x^7) + (E^4*(3584 + 10752*x + 13440*x^2 + 8960*x^3 + 3 360*x^4 + 672*x^5 + 56*x^6) + E^2*(3584*x + 10752*x^2 + 13440*x^3 + 8960*x ^4 + 3360*x^5 + 672*x^6 + 56*x^7))*Log[2*x] + (E^6*(5376 + 13440*x + 13440 *x^2 + 6720*x^3 + 1680*x^4 + 168*x^5) + E^4*(5376*x + 13440*x^2 + 13440*x^ 3 + 6720*x^4 + 1680*x^5 + 168*x^6))*Log[2*x]^2 + (E^8*(4480 + 8960*x + 672 0*x^2 + 2240*x^3 + 280*x^4) + E^6*(4480*x + 8960*x^2 + 6720*x^3 + 2240*x^4 + 280*x^5))*Log[2*x]^3 + (E^10*(2240 + 3360*x + 1680*x^2 + 280*x^3) + E^8 *(2240*x + 3360*x^2 + 1680*x^3 + 280*x^4))*Log[2*x]^4 + (E^12*(672 + 672*x + 168*x^2) + E^10*(672*x + 672*x^2 + 168*x^3))*Log[2*x]^5 + (E^14*(112 + 56*x) + E^12*(112*x + 56*x^2))*Log[2*x]^6 + (8*E^16 + 8*E^14*x)*Log[2*x]^7 )/x,x]
Leaf count is larger than twice the leaf count of optimal. \(1914\) vs. \(2(15)=30\).
Time = 2.53 (sec) , antiderivative size = 1914, normalized size of antiderivative = 127.60, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.005, Rules used = {2010, 2009}
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.
| \(\displaystyle \int \frac {8 x^8+112 x^7+672 x^6+2240 x^5+4480 x^4+5376 x^3+3584 x^2+\left (e^{12} \left (56 x^2+112 x\right )+e^{14} (56 x+112)\right ) \log ^6(2 x)+\left (e^{12} \left (168 x^2+672 x+672\right )+e^{10} \left (168 x^3+672 x^2+672 x\right )\right ) \log ^5(2 x)+\left (e^{10} \left (280 x^3+1680 x^2+3360 x+2240\right )+e^8 \left (280 x^4+1680 x^3+3360 x^2+2240 x\right )\right ) \log ^4(2 x)+\left (e^8 \left (280 x^4+2240 x^3+6720 x^2+8960 x+4480\right )+e^6 \left (280 x^5+2240 x^4+6720 x^3+8960 x^2+4480 x\right )\right ) \log ^3(2 x)+\left (e^6 \left (168 x^5+1680 x^4+6720 x^3+13440 x^2+13440 x+5376\right )+e^4 \left (168 x^6+1680 x^5+6720 x^4+13440 x^3+13440 x^2+5376 x\right )\right ) \log ^2(2 x)+e^2 \left (8 x^7+112 x^6+672 x^5+2240 x^4+4480 x^3+5376 x^2+3584 x+1024\right )+\left (e^4 \left (56 x^6+672 x^5+3360 x^4+8960 x^3+13440 x^2+10752 x+3584\right )+e^2 \left (56 x^7+672 x^6+3360 x^5+8960 x^4+13440 x^3+10752 x^2+3584 x\right )\right ) \log (2 x)+1024 x+\left (8 e^{14} x+8 e^{16}\right ) \log ^7(2 x)}{x} \, dx\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \int \left (\frac {8 \left (x+e^2\right ) (x+2)^7}{x}+\frac {8 e^{14} \left (x+e^2\right ) \log ^7(2 x)}{x}+\frac {56 e^{12} \left (x+e^2\right ) (x+2) \log ^6(2 x)}{x}+\frac {168 e^{10} \left (x+e^2\right ) (x+2)^2 \log ^5(2 x)}{x}+\frac {280 e^8 \left (x+e^2\right ) (x+2)^3 \log ^4(2 x)}{x}+\frac {280 e^6 \left (x+e^2\right ) (x+2)^4 \log ^3(2 x)}{x}+\frac {168 e^4 \left (x+e^2\right ) (x+2)^5 \log ^2(2 x)}{x}+\frac {56 e^2 \left (x+e^2\right ) (x+2)^6 \log (2 x)}{x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle (x+2)^8+e^{16} \log ^8(2 x)+8 e^{14} x \log ^7(2 x)+16 e^{14} \log ^7(2 x)-\frac {14}{9} e^2 \left (12+e^2\right ) x^6+\frac {14 e^4 x^6}{9}+\frac {56 e^2 x^6}{3}+28 e^{12} x^2 \log ^6(2 x)+56 e^{12} \left (2+e^2\right ) x \log ^6(2 x)-56 e^{14} x \log ^6(2 x)+112 e^{12} \log ^6(2 x)+\frac {336}{125} e^4 \left (10+e^2\right ) x^5-\frac {672}{25} e^2 \left (5+e^2\right ) x^5-\frac {336 e^6 x^5}{125}+\frac {672 e^2 x^5}{5}+56 e^{10} x^3 \log ^5(2 x)+84 e^{10} \left (4+e^2\right ) x^2 \log ^5(2 x)-84 e^{12} x^2 \log ^5(2 x)-336 e^{12} \left (2+e^2\right ) x \log ^5(2 x)+672 e^{10} \left (1+e^2\right ) x \log ^5(2 x)+336 e^{14} x \log ^5(2 x)+448 e^{10} \log ^5(2 x)-70 e^2 \left (8+3 e^2\right ) x^4-\frac {105}{16} e^6 \left (8+e^2\right ) x^4+\frac {105}{2} e^4 \left (4+e^2\right ) x^4+\frac {105 e^8 x^4}{16}+560 e^2 x^4+70 e^8 x^4 \log ^4(2 x)+\frac {280}{3} e^8 \left (6+e^2\right ) x^3 \log ^4(2 x)-\frac {280}{3} e^{10} x^3 \log ^4(2 x)-210 e^{10} \left (4+e^2\right ) x^2 \log ^4(2 x)+840 e^8 \left (2+e^2\right ) x^2 \log ^4(2 x)+210 e^{12} x^2 \log ^4(2 x)+1120 e^8 \left (2+3 e^2\right ) x \log ^4(2 x)+1680 e^{12} \left (2+e^2\right ) x \log ^4(2 x)-3360 e^{10} \left (1+e^2\right ) x \log ^4(2 x)-1680 e^{14} x \log ^4(2 x)+1120 e^8 \log ^4(2 x)-\frac {4480}{9} e^2 \left (3+2 e^2\right ) x^3+\frac {2240}{81} e^8 \left (6+e^2\right ) x^3-\frac {4480}{27} e^6 \left (3+e^2\right ) x^3+\frac {4480}{9} e^4 \left (2+e^2\right ) x^3-\frac {2240 e^{10} x^3}{81}+\frac {4480 e^2 x^3}{3}+56 e^6 x^5 \log ^3(2 x)+70 e^6 \left (8+e^2\right ) x^4 \log ^3(2 x)-70 e^8 x^4 \log ^3(2 x)-\frac {1120}{9} e^8 \left (6+e^2\right ) x^3 \log ^3(2 x)+\frac {2240}{3} e^6 \left (3+e^2\right ) x^3 \log ^3(2 x)+\frac {1120}{9} e^{10} x^3 \log ^3(2 x)+1120 e^6 \left (4+3 e^2\right ) x^2 \log ^3(2 x)+420 e^{10} \left (4+e^2\right ) x^2 \log ^3(2 x)-1680 e^8 \left (2+e^2\right ) x^2 \log ^3(2 x)-420 e^{12} x^2 \log ^3(2 x)-4480 e^8 \left (2+3 e^2\right ) x \log ^3(2 x)+4480 e^6 \left (1+2 e^2\right ) x \log ^3(2 x)-6720 e^{12} \left (2+e^2\right ) x \log ^3(2 x)+13440 e^{10} \left (1+e^2\right ) x \log ^3(2 x)+6720 e^{14} x \log ^3(2 x)+1792 e^6 \log ^3(2 x)-672 e^2 \left (4+5 e^2\right ) x^2-840 e^6 \left (4+3 e^2\right ) x^2-315 e^{10} \left (4+e^2\right ) x^2+1260 e^8 \left (2+e^2\right ) x^2+3360 e^4 \left (1+e^2\right ) x^2+315 e^{12} x^2+2688 e^2 x^2+28 e^4 x^6 \log ^2(2 x)+\frac {168}{5} e^4 \left (10+e^2\right ) x^5 \log ^2(2 x)-\frac {168}{5} e^6 x^5 \log ^2(2 x)-\frac {105}{2} e^6 \left (8+e^2\right ) x^4 \log ^2(2 x)+420 e^4 \left (4+e^2\right ) x^4 \log ^2(2 x)+\frac {105}{2} e^8 x^4 \log ^2(2 x)+\frac {1120}{9} e^8 \left (6+e^2\right ) x^3 \log ^2(2 x)-\frac {2240}{3} e^6 \left (3+e^2\right ) x^3 \log ^2(2 x)+2240 e^4 \left (2+e^2\right ) x^3 \log ^2(2 x)-\frac {1120}{9} e^{10} x^3 \log ^2(2 x)-1680 e^6 \left (4+3 e^2\right ) x^2 \log ^2(2 x)-630 e^{10} \left (4+e^2\right ) x^2 \log ^2(2 x)+2520 e^8 \left (2+e^2\right ) x^2 \log ^2(2 x)+6720 e^4 \left (1+e^2\right ) x^2 \log ^2(2 x)+630 e^{12} x^2 \log ^2(2 x)+2688 e^4 \left (2+5 e^2\right ) x \log ^2(2 x)+13440 e^8 \left (2+3 e^2\right ) x \log ^2(2 x)-13440 e^6 \left (1+2 e^2\right ) x \log ^2(2 x)+20160 e^{12} \left (2+e^2\right ) x \log ^2(2 x)-40320 e^{10} \left (1+e^2\right ) x \log ^2(2 x)-20160 e^{14} x \log ^2(2 x)+1792 e^4 \log ^2(2 x)+5376 e^4 \left (2+5 e^2\right ) x+26880 e^8 \left (2+3 e^2\right ) x-3584 e^2 \left (1+3 e^2\right ) x-26880 e^6 \left (1+2 e^2\right ) x+40320 e^{12} \left (2+e^2\right ) x-80640 e^{10} \left (1+e^2\right ) x-40320 e^{14} x+3584 e^2 x+1024 e^2 \log (x)+8 e^2 x^7 \log (2 x)+\frac {28}{3} e^2 \left (12+e^2\right ) x^6 \log (2 x)-\frac {28}{3} e^4 x^6 \log (2 x)-\frac {336}{25} e^4 \left (10+e^2\right ) x^5 \log (2 x)+\frac {672}{5} e^2 \left (5+e^2\right ) x^5 \log (2 x)+\frac {336}{25} e^6 x^5 \log (2 x)+280 e^2 \left (8+3 e^2\right ) x^4 \log (2 x)+\frac {105}{4} e^6 \left (8+e^2\right ) x^4 \log (2 x)-210 e^4 \left (4+e^2\right ) x^4 \log (2 x)-\frac {105}{4} e^8 x^4 \log (2 x)+\frac {4480}{3} e^2 \left (3+2 e^2\right ) x^3 \log (2 x)-\frac {2240}{27} e^8 \left (6+e^2\right ) x^3 \log (2 x)+\frac {4480}{9} e^6 \left (3+e^2\right ) x^3 \log (2 x)-\frac {4480}{3} e^4 \left (2+e^2\right ) x^3 \log (2 x)+\frac {2240}{27} e^{10} x^3 \log (2 x)+1344 e^2 \left (4+5 e^2\right ) x^2 \log (2 x)+1680 e^6 \left (4+3 e^2\right ) x^2 \log (2 x)+630 e^{10} \left (4+e^2\right ) x^2 \log (2 x)-2520 e^8 \left (2+e^2\right ) x^2 \log (2 x)-6720 e^4 \left (1+e^2\right ) x^2 \log (2 x)-630 e^{12} x^2 \log (2 x)-5376 e^4 \left (2+5 e^2\right ) x \log (2 x)-26880 e^8 \left (2+3 e^2\right ) x \log (2 x)+3584 e^2 \left (1+3 e^2\right ) x \log (2 x)+26880 e^6 \left (1+2 e^2\right ) x \log (2 x)-40320 e^{12} \left (2+e^2\right ) x \log (2 x)+80640 e^{10} \left (1+e^2\right ) x \log (2 x)+40320 e^{14} x \log (2 x)\) |
Int[(1024*x + 3584*x^2 + 5376*x^3 + 4480*x^4 + 2240*x^5 + 672*x^6 + 112*x^ 7 + 8*x^8 + E^2*(1024 + 3584*x + 5376*x^2 + 4480*x^3 + 2240*x^4 + 672*x^5 + 112*x^6 + 8*x^7) + (E^4*(3584 + 10752*x + 13440*x^2 + 8960*x^3 + 3360*x^ 4 + 672*x^5 + 56*x^6) + E^2*(3584*x + 10752*x^2 + 13440*x^3 + 8960*x^4 + 3 360*x^5 + 672*x^6 + 56*x^7))*Log[2*x] + (E^6*(5376 + 13440*x + 13440*x^2 + 6720*x^3 + 1680*x^4 + 168*x^5) + E^4*(5376*x + 13440*x^2 + 13440*x^3 + 67 20*x^4 + 1680*x^5 + 168*x^6))*Log[2*x]^2 + (E^8*(4480 + 8960*x + 6720*x^2 + 2240*x^3 + 280*x^4) + E^6*(4480*x + 8960*x^2 + 6720*x^3 + 2240*x^4 + 280 *x^5))*Log[2*x]^3 + (E^10*(2240 + 3360*x + 1680*x^2 + 280*x^3) + E^8*(2240 *x + 3360*x^2 + 1680*x^3 + 280*x^4))*Log[2*x]^4 + (E^12*(672 + 672*x + 168 *x^2) + E^10*(672*x + 672*x^2 + 168*x^3))*Log[2*x]^5 + (E^14*(112 + 56*x) + E^12*(112*x + 56*x^2))*Log[2*x]^6 + (8*E^16 + 8*E^14*x)*Log[2*x]^7)/x,x]
3584*E^2*x - 40320*E^14*x - 80640*E^10*(1 + E^2)*x + 40320*E^12*(2 + E^2)* x - 26880*E^6*(1 + 2*E^2)*x - 3584*E^2*(1 + 3*E^2)*x + 26880*E^8*(2 + 3*E^ 2)*x + 5376*E^4*(2 + 5*E^2)*x + 2688*E^2*x^2 + 315*E^12*x^2 + 3360*E^4*(1 + E^2)*x^2 + 1260*E^8*(2 + E^2)*x^2 - 315*E^10*(4 + E^2)*x^2 - 840*E^6*(4 + 3*E^2)*x^2 - 672*E^2*(4 + 5*E^2)*x^2 + (4480*E^2*x^3)/3 - (2240*E^10*x^3 )/81 + (4480*E^4*(2 + E^2)*x^3)/9 - (4480*E^6*(3 + E^2)*x^3)/27 + (2240*E^ 8*(6 + E^2)*x^3)/81 - (4480*E^2*(3 + 2*E^2)*x^3)/9 + 560*E^2*x^4 + (105*E^ 8*x^4)/16 + (105*E^4*(4 + E^2)*x^4)/2 - (105*E^6*(8 + E^2)*x^4)/16 - 70*E^ 2*(8 + 3*E^2)*x^4 + (672*E^2*x^5)/5 - (336*E^6*x^5)/125 - (672*E^2*(5 + E^ 2)*x^5)/25 + (336*E^4*(10 + E^2)*x^5)/125 + (56*E^2*x^6)/3 + (14*E^4*x^6)/ 9 - (14*E^2*(12 + E^2)*x^6)/9 + (2 + x)^8 + 1024*E^2*Log[x] + 40320*E^14*x *Log[2*x] + 80640*E^10*(1 + E^2)*x*Log[2*x] - 40320*E^12*(2 + E^2)*x*Log[2 *x] + 26880*E^6*(1 + 2*E^2)*x*Log[2*x] + 3584*E^2*(1 + 3*E^2)*x*Log[2*x] - 26880*E^8*(2 + 3*E^2)*x*Log[2*x] - 5376*E^4*(2 + 5*E^2)*x*Log[2*x] - 630* E^12*x^2*Log[2*x] - 6720*E^4*(1 + E^2)*x^2*Log[2*x] - 2520*E^8*(2 + E^2)*x ^2*Log[2*x] + 630*E^10*(4 + E^2)*x^2*Log[2*x] + 1680*E^6*(4 + 3*E^2)*x^2*L og[2*x] + 1344*E^2*(4 + 5*E^2)*x^2*Log[2*x] + (2240*E^10*x^3*Log[2*x])/27 - (4480*E^4*(2 + E^2)*x^3*Log[2*x])/3 + (4480*E^6*(3 + E^2)*x^3*Log[2*x])/ 9 - (2240*E^8*(6 + E^2)*x^3*Log[2*x])/27 + (4480*E^2*(3 + 2*E^2)*x^3*Log[2 *x])/3 - (105*E^8*x^4*Log[2*x])/4 - 210*E^4*(4 + E^2)*x^4*Log[2*x] + (1...
3.7.98.3.1 Defintions of rubi rules used
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Leaf count of result is larger than twice the leaf count of optimal. \(129\) vs. \(2(14)=28\).
Time = 0.47 (sec) , antiderivative size = 130, normalized size of antiderivative = 8.67
| method | result | size |
| risch | \({\mathrm e}^{16} \ln \left (2 x \right )^{8}+\left (8 x \,{\mathrm e}^{14}+16 \,{\mathrm e}^{14}\right ) \ln \left (2 x \right )^{7}+\left (28 x^{2} {\mathrm e}^{12}+112 x \,{\mathrm e}^{12}+112 \,{\mathrm e}^{12}\right ) \ln \left (2 x \right )^{6}+56 \,{\mathrm e}^{10} \left (2+x \right )^{3} \ln \left (2 x \right )^{5}+70 \,{\mathrm e}^{8} \left (2+x \right )^{4} \ln \left (2 x \right )^{4}+56 \,{\mathrm e}^{6} \left (2+x \right )^{5} \ln \left (2 x \right )^{3}+28 \,{\mathrm e}^{4} \left (2+x \right )^{6} \ln \left (2 x \right )^{2}+8 \,{\mathrm e}^{2} \left (2+x \right )^{7} \ln \left (2 x \right )+\left (2+x \right )^{8}\) | \(130\) |
| parallelrisch | \(28 \,{\mathrm e}^{4} \ln \left (2 x \right )^{2} x^{6}+8 \ln \left (2 x \right )^{7} {\mathrm e}^{14} x +28 \ln \left (2 x \right )^{6} {\mathrm e}^{12} x^{2}+56 \ln \left (2 x \right )^{5} {\mathrm e}^{10} x^{3}+70 \ln \left (2 x \right )^{4} {\mathrm e}^{8} x^{4}+56 \ln \left (2 x \right )^{3} {\mathrm e}^{6} x^{5}+336 \,{\mathrm e}^{4} \ln \left (2 x \right )^{2} x^{5}+112 \ln \left (2 x \right )^{6} {\mathrm e}^{12} x +1024 x +1024 \,{\mathrm e}^{2} \ln \left (2 x \right )+16 x^{7}+x^{8}+112 x^{6}+448 x^{5}+1120 x^{4}+1792 x^{3}+1792 x^{2}+3584 \,{\mathrm e}^{2} \ln \left (2 x \right ) x +5376 \,{\mathrm e}^{4} \ln \left (2 x \right )^{2} x +8 \,{\mathrm e}^{2} \ln \left (2 x \right ) x^{7}+112 \,{\mathrm e}^{2} \ln \left (2 x \right ) x^{6}+672 \,{\mathrm e}^{2} \ln \left (2 x \right ) x^{5}+5376 \,{\mathrm e}^{2} \ln \left (2 x \right ) x^{2}+2240 \,{\mathrm e}^{2} \ln \left (2 x \right ) x^{4}+4480 \,{\mathrm e}^{2} \ln \left (2 x \right ) x^{3}+16 \ln \left (2 x \right )^{7} {\mathrm e}^{14}+112 \ln \left (2 x \right )^{6} {\mathrm e}^{12}+448 \,{\mathrm e}^{10} \ln \left (2 x \right )^{5}+1120 \,{\mathrm e}^{8} \ln \left (2 x \right )^{4}+1792 \,{\mathrm e}^{6} \ln \left (2 x \right )^{3}+1792 \,{\mathrm e}^{4} \ln \left (2 x \right )^{2}+336 \ln \left (2 x \right )^{5} {\mathrm e}^{10} x^{2}+560 \ln \left (2 x \right )^{4} {\mathrm e}^{8} x^{3}+560 \ln \left (2 x \right )^{3} {\mathrm e}^{6} x^{4}+1680 \,{\mathrm e}^{4} \ln \left (2 x \right )^{2} x^{4}+672 \ln \left (2 x \right )^{5} {\mathrm e}^{10} x +1680 \ln \left (2 x \right )^{4} {\mathrm e}^{8} x^{2}+2240 \ln \left (2 x \right )^{3} {\mathrm e}^{6} x^{3}+4480 \,{\mathrm e}^{4} \ln \left (2 x \right )^{2} x^{3}+2240 \ln \left (2 x \right )^{4} {\mathrm e}^{8} x +4480 \ln \left (2 x \right )^{3} {\mathrm e}^{6} x^{2}+6720 \,{\mathrm e}^{4} \ln \left (2 x \right )^{2} x^{2}+4480 \ln \left (2 x \right )^{3} {\mathrm e}^{6} x +{\mathrm e}^{16} \ln \left (2 x \right )^{8}\) | \(507\) |
| parts | \(\text {Expression too large to display}\) | \(1872\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(2064\) |
| default | \(\text {Expression too large to display}\) | \(2064\) |
int(((8*exp(2)^8+8*x*exp(2)^7)*ln(2*x)^7+((56*x+112)*exp(2)^7+(56*x^2+112* x)*exp(2)^6)*ln(2*x)^6+((168*x^2+672*x+672)*exp(2)^6+(168*x^3+672*x^2+672* x)*exp(2)^5)*ln(2*x)^5+((280*x^3+1680*x^2+3360*x+2240)*exp(2)^5+(280*x^4+1 680*x^3+3360*x^2+2240*x)*exp(2)^4)*ln(2*x)^4+((280*x^4+2240*x^3+6720*x^2+8 960*x+4480)*exp(2)^4+(280*x^5+2240*x^4+6720*x^3+8960*x^2+4480*x)*exp(2)^3) *ln(2*x)^3+((168*x^5+1680*x^4+6720*x^3+13440*x^2+13440*x+5376)*exp(2)^3+(1 68*x^6+1680*x^5+6720*x^4+13440*x^3+13440*x^2+5376*x)*exp(2)^2)*ln(2*x)^2+( (56*x^6+672*x^5+3360*x^4+8960*x^3+13440*x^2+10752*x+3584)*exp(2)^2+(56*x^7 +672*x^6+3360*x^5+8960*x^4+13440*x^3+10752*x^2+3584*x)*exp(2))*ln(2*x)+(8* x^7+112*x^6+672*x^5+2240*x^4+4480*x^3+5376*x^2+3584*x+1024)*exp(2)+8*x^8+1 12*x^7+672*x^6+2240*x^5+4480*x^4+5376*x^3+3584*x^2+1024*x)/x,x,method=_RET URNVERBOSE)
exp(16)*ln(2*x)^8+(8*x*exp(14)+16*exp(14))*ln(2*x)^7+(28*x^2*exp(12)+112*x *exp(12)+112*exp(12))*ln(2*x)^6+56*exp(10)*(2+x)^3*ln(2*x)^5+70*exp(8)*(2+ x)^4*ln(2*x)^4+56*exp(6)*(2+x)^5*ln(2*x)^3+28*exp(4)*(2+x)^6*ln(2*x)^2+8*e xp(2)*(2+x)^7*ln(2*x)+(2+x)^8
Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (14) = 28\).
Time = 0.27 (sec) , antiderivative size = 240, normalized size of antiderivative = 16.00 \begin {dmath*} \int \frac {1024 x+3584 x^2+5376 x^3+4480 x^4+2240 x^5+672 x^6+112 x^7+8 x^8+e^2 \left (1024+3584 x+5376 x^2+4480 x^3+2240 x^4+672 x^5+112 x^6+8 x^7\right )+\left (e^4 \left (3584+10752 x+13440 x^2+8960 x^3+3360 x^4+672 x^5+56 x^6\right )+e^2 \left (3584 x+10752 x^2+13440 x^3+8960 x^4+3360 x^5+672 x^6+56 x^7\right )\right ) \log (2 x)+\left (e^6 \left (5376+13440 x+13440 x^2+6720 x^3+1680 x^4+168 x^5\right )+e^4 \left (5376 x+13440 x^2+13440 x^3+6720 x^4+1680 x^5+168 x^6\right )\right ) \log ^2(2 x)+\left (e^8 \left (4480+8960 x+6720 x^2+2240 x^3+280 x^4\right )+e^6 \left (4480 x+8960 x^2+6720 x^3+2240 x^4+280 x^5\right )\right ) \log ^3(2 x)+\left (e^{10} \left (2240+3360 x+1680 x^2+280 x^3\right )+e^8 \left (2240 x+3360 x^2+1680 x^3+280 x^4\right )\right ) \log ^4(2 x)+\left (e^{12} \left (672+672 x+168 x^2\right )+e^{10} \left (672 x+672 x^2+168 x^3\right )\right ) \log ^5(2 x)+\left (e^{14} (112+56 x)+e^{12} \left (112 x+56 x^2\right )\right ) \log ^6(2 x)+\left (8 e^{16}+8 e^{14} x\right ) \log ^7(2 x)}{x} \, dx=8 \, {\left (x + 2\right )} e^{14} \log \left (2 \, x\right )^{7} + e^{16} \log \left (2 \, x\right )^{8} + x^{8} + 28 \, {\left (x^{2} + 4 \, x + 4\right )} e^{12} \log \left (2 \, x\right )^{6} + 16 \, x^{7} + 56 \, {\left (x^{3} + 6 \, x^{2} + 12 \, x + 8\right )} e^{10} \log \left (2 \, x\right )^{5} + 112 \, x^{6} + 70 \, {\left (x^{4} + 8 \, x^{3} + 24 \, x^{2} + 32 \, x + 16\right )} e^{8} \log \left (2 \, x\right )^{4} + 448 \, x^{5} + 56 \, {\left (x^{5} + 10 \, x^{4} + 40 \, x^{3} + 80 \, x^{2} + 80 \, x + 32\right )} e^{6} \log \left (2 \, x\right )^{3} + 1120 \, x^{4} + 28 \, {\left (x^{6} + 12 \, x^{5} + 60 \, x^{4} + 160 \, x^{3} + 240 \, x^{2} + 192 \, x + 64\right )} e^{4} \log \left (2 \, x\right )^{2} + 1792 \, x^{3} + 8 \, {\left (x^{7} + 14 \, x^{6} + 84 \, x^{5} + 280 \, x^{4} + 560 \, x^{3} + 672 \, x^{2} + 448 \, x + 128\right )} e^{2} \log \left (2 \, x\right ) + 1792 \, x^{2} + 1024 \, x \end {dmath*}
integrate(((8*exp(2)^8+8*x*exp(2)^7)*log(2*x)^7+((56*x+112)*exp(2)^7+(56*x ^2+112*x)*exp(2)^6)*log(2*x)^6+((168*x^2+672*x+672)*exp(2)^6+(168*x^3+672* x^2+672*x)*exp(2)^5)*log(2*x)^5+((280*x^3+1680*x^2+3360*x+2240)*exp(2)^5+( 280*x^4+1680*x^3+3360*x^2+2240*x)*exp(2)^4)*log(2*x)^4+((280*x^4+2240*x^3+ 6720*x^2+8960*x+4480)*exp(2)^4+(280*x^5+2240*x^4+6720*x^3+8960*x^2+4480*x) *exp(2)^3)*log(2*x)^3+((168*x^5+1680*x^4+6720*x^3+13440*x^2+13440*x+5376)* exp(2)^3+(168*x^6+1680*x^5+6720*x^4+13440*x^3+13440*x^2+5376*x)*exp(2)^2)* log(2*x)^2+((56*x^6+672*x^5+3360*x^4+8960*x^3+13440*x^2+10752*x+3584)*exp( 2)^2+(56*x^7+672*x^6+3360*x^5+8960*x^4+13440*x^3+10752*x^2+3584*x)*exp(2)) *log(2*x)+(8*x^7+112*x^6+672*x^5+2240*x^4+4480*x^3+5376*x^2+3584*x+1024)*e xp(2)+8*x^8+112*x^7+672*x^6+2240*x^5+4480*x^4+5376*x^3+3584*x^2+1024*x)/x, x, algorithm=\
8*(x + 2)*e^14*log(2*x)^7 + e^16*log(2*x)^8 + x^8 + 28*(x^2 + 4*x + 4)*e^1 2*log(2*x)^6 + 16*x^7 + 56*(x^3 + 6*x^2 + 12*x + 8)*e^10*log(2*x)^5 + 112* x^6 + 70*(x^4 + 8*x^3 + 24*x^2 + 32*x + 16)*e^8*log(2*x)^4 + 448*x^5 + 56* (x^5 + 10*x^4 + 40*x^3 + 80*x^2 + 80*x + 32)*e^6*log(2*x)^3 + 1120*x^4 + 2 8*(x^6 + 12*x^5 + 60*x^4 + 160*x^3 + 240*x^2 + 192*x + 64)*e^4*log(2*x)^2 + 1792*x^3 + 8*(x^7 + 14*x^6 + 84*x^5 + 280*x^4 + 560*x^3 + 672*x^2 + 448* x + 128)*e^2*log(2*x) + 1792*x^2 + 1024*x
Leaf count of result is larger than twice the leaf count of optimal. 357 vs. \(2 (14) = 28\).
Time = 0.59 (sec) , antiderivative size = 357, normalized size of antiderivative = 23.80 \begin {dmath*} \int \frac {1024 x+3584 x^2+5376 x^3+4480 x^4+2240 x^5+672 x^6+112 x^7+8 x^8+e^2 \left (1024+3584 x+5376 x^2+4480 x^3+2240 x^4+672 x^5+112 x^6+8 x^7\right )+\left (e^4 \left (3584+10752 x+13440 x^2+8960 x^3+3360 x^4+672 x^5+56 x^6\right )+e^2 \left (3584 x+10752 x^2+13440 x^3+8960 x^4+3360 x^5+672 x^6+56 x^7\right )\right ) \log (2 x)+\left (e^6 \left (5376+13440 x+13440 x^2+6720 x^3+1680 x^4+168 x^5\right )+e^4 \left (5376 x+13440 x^2+13440 x^3+6720 x^4+1680 x^5+168 x^6\right )\right ) \log ^2(2 x)+\left (e^8 \left (4480+8960 x+6720 x^2+2240 x^3+280 x^4\right )+e^6 \left (4480 x+8960 x^2+6720 x^3+2240 x^4+280 x^5\right )\right ) \log ^3(2 x)+\left (e^{10} \left (2240+3360 x+1680 x^2+280 x^3\right )+e^8 \left (2240 x+3360 x^2+1680 x^3+280 x^4\right )\right ) \log ^4(2 x)+\left (e^{12} \left (672+672 x+168 x^2\right )+e^{10} \left (672 x+672 x^2+168 x^3\right )\right ) \log ^5(2 x)+\left (e^{14} (112+56 x)+e^{12} \left (112 x+56 x^2\right )\right ) \log ^6(2 x)+\left (8 e^{16}+8 e^{14} x\right ) \log ^7(2 x)}{x} \, dx=x^{8} + 16 x^{7} + 112 x^{6} + 448 x^{5} + 1120 x^{4} + 1792 x^{3} + 1792 x^{2} + 1024 x + \left (8 x e^{14} + 16 e^{14}\right ) \log {\left (2 x \right )}^{7} + \left (28 x^{2} e^{12} + 112 x e^{12} + 112 e^{12}\right ) \log {\left (2 x \right )}^{6} + \left (56 x^{3} e^{10} + 336 x^{2} e^{10} + 672 x e^{10} + 448 e^{10}\right ) \log {\left (2 x \right )}^{5} + \left (70 x^{4} e^{8} + 560 x^{3} e^{8} + 1680 x^{2} e^{8} + 2240 x e^{8} + 1120 e^{8}\right ) \log {\left (2 x \right )}^{4} + \left (56 x^{5} e^{6} + 560 x^{4} e^{6} + 2240 x^{3} e^{6} + 4480 x^{2} e^{6} + 4480 x e^{6} + 1792 e^{6}\right ) \log {\left (2 x \right )}^{3} + \left (28 x^{6} e^{4} + 336 x^{5} e^{4} + 1680 x^{4} e^{4} + 4480 x^{3} e^{4} + 6720 x^{2} e^{4} + 5376 x e^{4} + 1792 e^{4}\right ) \log {\left (2 x \right )}^{2} + \left (8 x^{7} e^{2} + 112 x^{6} e^{2} + 672 x^{5} e^{2} + 2240 x^{4} e^{2} + 4480 x^{3} e^{2} + 5376 x^{2} e^{2} + 3584 x e^{2}\right ) \log {\left (2 x \right )} + 1024 e^{2} \log {\left (x \right )} + e^{16} \log {\left (2 x \right )}^{8} \end {dmath*}
integrate(((8*exp(2)**8+8*x*exp(2)**7)*ln(2*x)**7+((56*x+112)*exp(2)**7+(5 6*x**2+112*x)*exp(2)**6)*ln(2*x)**6+((168*x**2+672*x+672)*exp(2)**6+(168*x **3+672*x**2+672*x)*exp(2)**5)*ln(2*x)**5+((280*x**3+1680*x**2+3360*x+2240 )*exp(2)**5+(280*x**4+1680*x**3+3360*x**2+2240*x)*exp(2)**4)*ln(2*x)**4+(( 280*x**4+2240*x**3+6720*x**2+8960*x+4480)*exp(2)**4+(280*x**5+2240*x**4+67 20*x**3+8960*x**2+4480*x)*exp(2)**3)*ln(2*x)**3+((168*x**5+1680*x**4+6720* x**3+13440*x**2+13440*x+5376)*exp(2)**3+(168*x**6+1680*x**5+6720*x**4+1344 0*x**3+13440*x**2+5376*x)*exp(2)**2)*ln(2*x)**2+((56*x**6+672*x**5+3360*x* *4+8960*x**3+13440*x**2+10752*x+3584)*exp(2)**2+(56*x**7+672*x**6+3360*x** 5+8960*x**4+13440*x**3+10752*x**2+3584*x)*exp(2))*ln(2*x)+(8*x**7+112*x**6 +672*x**5+2240*x**4+4480*x**3+5376*x**2+3584*x+1024)*exp(2)+8*x**8+112*x** 7+672*x**6+2240*x**5+4480*x**4+5376*x**3+3584*x**2+1024*x)/x,x)
x**8 + 16*x**7 + 112*x**6 + 448*x**5 + 1120*x**4 + 1792*x**3 + 1792*x**2 + 1024*x + (8*x*exp(14) + 16*exp(14))*log(2*x)**7 + (28*x**2*exp(12) + 112* x*exp(12) + 112*exp(12))*log(2*x)**6 + (56*x**3*exp(10) + 336*x**2*exp(10) + 672*x*exp(10) + 448*exp(10))*log(2*x)**5 + (70*x**4*exp(8) + 560*x**3*e xp(8) + 1680*x**2*exp(8) + 2240*x*exp(8) + 1120*exp(8))*log(2*x)**4 + (56* x**5*exp(6) + 560*x**4*exp(6) + 2240*x**3*exp(6) + 4480*x**2*exp(6) + 4480 *x*exp(6) + 1792*exp(6))*log(2*x)**3 + (28*x**6*exp(4) + 336*x**5*exp(4) + 1680*x**4*exp(4) + 4480*x**3*exp(4) + 6720*x**2*exp(4) + 5376*x*exp(4) + 1792*exp(4))*log(2*x)**2 + (8*x**7*exp(2) + 112*x**6*exp(2) + 672*x**5*exp (2) + 2240*x**4*exp(2) + 4480*x**3*exp(2) + 5376*x**2*exp(2) + 3584*x*exp( 2))*log(2*x) + 1024*exp(2)*log(x) + exp(16)*log(2*x)**8
Leaf count of result is larger than twice the leaf count of optimal. 1620 vs. \(2 (14) = 28\).
Time = 0.24 (sec) , antiderivative size = 1620, normalized size of antiderivative = 108.00 \begin {dmath*} \int \frac {1024 x+3584 x^2+5376 x^3+4480 x^4+2240 x^5+672 x^6+112 x^7+8 x^8+e^2 \left (1024+3584 x+5376 x^2+4480 x^3+2240 x^4+672 x^5+112 x^6+8 x^7\right )+\left (e^4 \left (3584+10752 x+13440 x^2+8960 x^3+3360 x^4+672 x^5+56 x^6\right )+e^2 \left (3584 x+10752 x^2+13440 x^3+8960 x^4+3360 x^5+672 x^6+56 x^7\right )\right ) \log (2 x)+\left (e^6 \left (5376+13440 x+13440 x^2+6720 x^3+1680 x^4+168 x^5\right )+e^4 \left (5376 x+13440 x^2+13440 x^3+6720 x^4+1680 x^5+168 x^6\right )\right ) \log ^2(2 x)+\left (e^8 \left (4480+8960 x+6720 x^2+2240 x^3+280 x^4\right )+e^6 \left (4480 x+8960 x^2+6720 x^3+2240 x^4+280 x^5\right )\right ) \log ^3(2 x)+\left (e^{10} \left (2240+3360 x+1680 x^2+280 x^3\right )+e^8 \left (2240 x+3360 x^2+1680 x^3+280 x^4\right )\right ) \log ^4(2 x)+\left (e^{12} \left (672+672 x+168 x^2\right )+e^{10} \left (672 x+672 x^2+168 x^3\right )\right ) \log ^5(2 x)+\left (e^{14} (112+56 x)+e^{12} \left (112 x+56 x^2\right )\right ) \log ^6(2 x)+\left (8 e^{16}+8 e^{14} x\right ) \log ^7(2 x)}{x} \, dx=\text {Too large to display} \end {dmath*}
integrate(((8*exp(2)^8+8*x*exp(2)^7)*log(2*x)^7+((56*x+112)*exp(2)^7+(56*x ^2+112*x)*exp(2)^6)*log(2*x)^6+((168*x^2+672*x+672)*exp(2)^6+(168*x^3+672* x^2+672*x)*exp(2)^5)*log(2*x)^5+((280*x^3+1680*x^2+3360*x+2240)*exp(2)^5+( 280*x^4+1680*x^3+3360*x^2+2240*x)*exp(2)^4)*log(2*x)^4+((280*x^4+2240*x^3+ 6720*x^2+8960*x+4480)*exp(2)^4+(280*x^5+2240*x^4+6720*x^3+8960*x^2+4480*x) *exp(2)^3)*log(2*x)^3+((168*x^5+1680*x^4+6720*x^3+13440*x^2+13440*x+5376)* exp(2)^3+(168*x^6+1680*x^5+6720*x^4+13440*x^3+13440*x^2+5376*x)*exp(2)^2)* log(2*x)^2+((56*x^6+672*x^5+3360*x^4+8960*x^3+13440*x^2+10752*x+3584)*exp( 2)^2+(56*x^7+672*x^6+3360*x^5+8960*x^4+13440*x^3+10752*x^2+3584*x)*exp(2)) *log(2*x)+(8*x^7+112*x^6+672*x^5+2240*x^4+4480*x^3+5376*x^2+3584*x+1024)*e xp(2)+8*x^8+112*x^7+672*x^6+2240*x^5+4480*x^4+5376*x^3+3584*x^2+1024*x)/x, x, algorithm=\
e^16*log(2*x)^8 + x^8 + 14/9*(18*log(2*x)^2 - 6*log(2*x) + 1)*x^6*e^4 + 8/ 7*x^7*e^2 + 16*e^14*log(2*x)^7 + 16*x^7 + 56/125*(125*log(2*x)^3 - 75*log( 2*x)^2 + 30*log(2*x) - 6)*x^5*e^6 + 168/125*(25*log(2*x)^2 - 10*log(2*x) + 2)*x^5*e^6 + 336/25*(25*log(2*x)^2 - 10*log(2*x) + 2)*x^5*e^4 + 56/3*x^6* e^2 + 112*e^12*log(2*x)^6 + 112*x^6 + 35/16*(32*log(2*x)^4 - 32*log(2*x)^3 + 24*log(2*x)^2 - 12*log(2*x) + 3)*x^4*e^8 + 35/16*(32*log(2*x)^3 - 24*lo g(2*x)^2 + 12*log(2*x) - 3)*x^4*e^8 + 35/2*(32*log(2*x)^3 - 24*log(2*x)^2 + 12*log(2*x) - 3)*x^4*e^6 + 105/2*(8*log(2*x)^2 - 4*log(2*x) + 1)*x^4*e^6 + 210*(8*log(2*x)^2 - 4*log(2*x) + 1)*x^4*e^4 + 672/5*x^5*e^2 + 448*e^10* log(2*x)^5 + 448*x^5 + 56/81*(81*log(2*x)^5 - 135*log(2*x)^4 + 180*log(2*x )^3 - 180*log(2*x)^2 + 120*log(2*x) - 40)*x^3*e^10 + 280/81*(27*log(2*x)^4 - 36*log(2*x)^3 + 36*log(2*x)^2 - 24*log(2*x) + 8)*x^3*e^10 + 560/27*(27* log(2*x)^4 - 36*log(2*x)^3 + 36*log(2*x)^2 - 24*log(2*x) + 8)*x^3*e^8 + 22 40/27*(9*log(2*x)^3 - 9*log(2*x)^2 + 6*log(2*x) - 2)*x^3*e^8 + 2240/9*(9*l og(2*x)^3 - 9*log(2*x)^2 + 6*log(2*x) - 2)*x^3*e^6 + 2240/9*(9*log(2*x)^2 - 6*log(2*x) + 2)*x^3*e^6 + 4480/9*(9*log(2*x)^2 - 6*log(2*x) + 2)*x^3*e^4 + 560*x^4*e^2 + 1120*e^8*log(2*x)^4 + 1120*x^4 + 7*(4*log(2*x)^6 - 12*log (2*x)^5 + 30*log(2*x)^4 - 60*log(2*x)^3 + 90*log(2*x)^2 - 90*log(2*x) + 45 )*x^2*e^12 + 21*(4*log(2*x)^5 - 10*log(2*x)^4 + 20*log(2*x)^3 - 30*log(2*x )^2 + 30*log(2*x) - 15)*x^2*e^12 + 84*(4*log(2*x)^5 - 10*log(2*x)^4 + 2...
Leaf count of result is larger than twice the leaf count of optimal. 448 vs. \(2 (14) = 28\).
Time = 0.35 (sec) , antiderivative size = 448, normalized size of antiderivative = 29.87 \begin {dmath*} \int \frac {1024 x+3584 x^2+5376 x^3+4480 x^4+2240 x^5+672 x^6+112 x^7+8 x^8+e^2 \left (1024+3584 x+5376 x^2+4480 x^3+2240 x^4+672 x^5+112 x^6+8 x^7\right )+\left (e^4 \left (3584+10752 x+13440 x^2+8960 x^3+3360 x^4+672 x^5+56 x^6\right )+e^2 \left (3584 x+10752 x^2+13440 x^3+8960 x^4+3360 x^5+672 x^6+56 x^7\right )\right ) \log (2 x)+\left (e^6 \left (5376+13440 x+13440 x^2+6720 x^3+1680 x^4+168 x^5\right )+e^4 \left (5376 x+13440 x^2+13440 x^3+6720 x^4+1680 x^5+168 x^6\right )\right ) \log ^2(2 x)+\left (e^8 \left (4480+8960 x+6720 x^2+2240 x^3+280 x^4\right )+e^6 \left (4480 x+8960 x^2+6720 x^3+2240 x^4+280 x^5\right )\right ) \log ^3(2 x)+\left (e^{10} \left (2240+3360 x+1680 x^2+280 x^3\right )+e^8 \left (2240 x+3360 x^2+1680 x^3+280 x^4\right )\right ) \log ^4(2 x)+\left (e^{12} \left (672+672 x+168 x^2\right )+e^{10} \left (672 x+672 x^2+168 x^3\right )\right ) \log ^5(2 x)+\left (e^{14} (112+56 x)+e^{12} \left (112 x+56 x^2\right )\right ) \log ^6(2 x)+\left (8 e^{16}+8 e^{14} x\right ) \log ^7(2 x)}{x} \, dx=8 \, x^{7} e^{2} \log \left (2 \, x\right ) + 28 \, x^{6} e^{4} \log \left (2 \, x\right )^{2} + 56 \, x^{5} e^{6} \log \left (2 \, x\right )^{3} + 70 \, x^{4} e^{8} \log \left (2 \, x\right )^{4} + 56 \, x^{3} e^{10} \log \left (2 \, x\right )^{5} + 28 \, x^{2} e^{12} \log \left (2 \, x\right )^{6} + 8 \, x e^{14} \log \left (2 \, x\right )^{7} + e^{16} \log \left (2 \, x\right )^{8} + x^{8} + 112 \, x^{6} e^{2} \log \left (2 \, x\right ) + 336 \, x^{5} e^{4} \log \left (2 \, x\right )^{2} + 560 \, x^{4} e^{6} \log \left (2 \, x\right )^{3} + 560 \, x^{3} e^{8} \log \left (2 \, x\right )^{4} + 336 \, x^{2} e^{10} \log \left (2 \, x\right )^{5} + 112 \, x e^{12} \log \left (2 \, x\right )^{6} + 16 \, e^{14} \log \left (2 \, x\right )^{7} + 16 \, x^{7} + 672 \, x^{5} e^{2} \log \left (2 \, x\right ) + 1680 \, x^{4} e^{4} \log \left (2 \, x\right )^{2} + 2240 \, x^{3} e^{6} \log \left (2 \, x\right )^{3} + 1680 \, x^{2} e^{8} \log \left (2 \, x\right )^{4} + 672 \, x e^{10} \log \left (2 \, x\right )^{5} + 112 \, e^{12} \log \left (2 \, x\right )^{6} + 112 \, x^{6} + 2240 \, x^{4} e^{2} \log \left (2 \, x\right ) + 4480 \, x^{3} e^{4} \log \left (2 \, x\right )^{2} + 4480 \, x^{2} e^{6} \log \left (2 \, x\right )^{3} + 2240 \, x e^{8} \log \left (2 \, x\right )^{4} + 448 \, e^{10} \log \left (2 \, x\right )^{5} + 448 \, x^{5} + 4480 \, x^{3} e^{2} \log \left (2 \, x\right ) + 6720 \, x^{2} e^{4} \log \left (2 \, x\right )^{2} + 4480 \, x e^{6} \log \left (2 \, x\right )^{3} + 1120 \, e^{8} \log \left (2 \, x\right )^{4} + 1120 \, x^{4} + 5376 \, x^{2} e^{2} \log \left (2 \, x\right ) + 5376 \, x e^{4} \log \left (2 \, x\right )^{2} + 1792 \, e^{6} \log \left (2 \, x\right )^{3} + 1792 \, x^{3} + 3584 \, x e^{2} \log \left (2 \, x\right ) + 1792 \, e^{4} \log \left (2 \, x\right )^{2} + 1792 \, x^{2} + 1024 \, e^{2} \log \left (x\right ) + 1024 \, x \end {dmath*}
integrate(((8*exp(2)^8+8*x*exp(2)^7)*log(2*x)^7+((56*x+112)*exp(2)^7+(56*x ^2+112*x)*exp(2)^6)*log(2*x)^6+((168*x^2+672*x+672)*exp(2)^6+(168*x^3+672* x^2+672*x)*exp(2)^5)*log(2*x)^5+((280*x^3+1680*x^2+3360*x+2240)*exp(2)^5+( 280*x^4+1680*x^3+3360*x^2+2240*x)*exp(2)^4)*log(2*x)^4+((280*x^4+2240*x^3+ 6720*x^2+8960*x+4480)*exp(2)^4+(280*x^5+2240*x^4+6720*x^3+8960*x^2+4480*x) *exp(2)^3)*log(2*x)^3+((168*x^5+1680*x^4+6720*x^3+13440*x^2+13440*x+5376)* exp(2)^3+(168*x^6+1680*x^5+6720*x^4+13440*x^3+13440*x^2+5376*x)*exp(2)^2)* log(2*x)^2+((56*x^6+672*x^5+3360*x^4+8960*x^3+13440*x^2+10752*x+3584)*exp( 2)^2+(56*x^7+672*x^6+3360*x^5+8960*x^4+13440*x^3+10752*x^2+3584*x)*exp(2)) *log(2*x)+(8*x^7+112*x^6+672*x^5+2240*x^4+4480*x^3+5376*x^2+3584*x+1024)*e xp(2)+8*x^8+112*x^7+672*x^6+2240*x^5+4480*x^4+5376*x^3+3584*x^2+1024*x)/x, x, algorithm=\
8*x^7*e^2*log(2*x) + 28*x^6*e^4*log(2*x)^2 + 56*x^5*e^6*log(2*x)^3 + 70*x^ 4*e^8*log(2*x)^4 + 56*x^3*e^10*log(2*x)^5 + 28*x^2*e^12*log(2*x)^6 + 8*x*e ^14*log(2*x)^7 + e^16*log(2*x)^8 + x^8 + 112*x^6*e^2*log(2*x) + 336*x^5*e^ 4*log(2*x)^2 + 560*x^4*e^6*log(2*x)^3 + 560*x^3*e^8*log(2*x)^4 + 336*x^2*e ^10*log(2*x)^5 + 112*x*e^12*log(2*x)^6 + 16*e^14*log(2*x)^7 + 16*x^7 + 672 *x^5*e^2*log(2*x) + 1680*x^4*e^4*log(2*x)^2 + 2240*x^3*e^6*log(2*x)^3 + 16 80*x^2*e^8*log(2*x)^4 + 672*x*e^10*log(2*x)^5 + 112*e^12*log(2*x)^6 + 112* x^6 + 2240*x^4*e^2*log(2*x) + 4480*x^3*e^4*log(2*x)^2 + 4480*x^2*e^6*log(2 *x)^3 + 2240*x*e^8*log(2*x)^4 + 448*e^10*log(2*x)^5 + 448*x^5 + 4480*x^3*e ^2*log(2*x) + 6720*x^2*e^4*log(2*x)^2 + 4480*x*e^6*log(2*x)^3 + 1120*e^8*l og(2*x)^4 + 1120*x^4 + 5376*x^2*e^2*log(2*x) + 5376*x*e^4*log(2*x)^2 + 179 2*e^6*log(2*x)^3 + 1792*x^3 + 3584*x*e^2*log(2*x) + 1792*e^4*log(2*x)^2 + 1792*x^2 + 1024*e^2*log(x) + 1024*x
Time = 20.56 (sec) , antiderivative size = 448, normalized size of antiderivative = 29.87 \begin {dmath*} \int \frac {1024 x+3584 x^2+5376 x^3+4480 x^4+2240 x^5+672 x^6+112 x^7+8 x^8+e^2 \left (1024+3584 x+5376 x^2+4480 x^3+2240 x^4+672 x^5+112 x^6+8 x^7\right )+\left (e^4 \left (3584+10752 x+13440 x^2+8960 x^3+3360 x^4+672 x^5+56 x^6\right )+e^2 \left (3584 x+10752 x^2+13440 x^3+8960 x^4+3360 x^5+672 x^6+56 x^7\right )\right ) \log (2 x)+\left (e^6 \left (5376+13440 x+13440 x^2+6720 x^3+1680 x^4+168 x^5\right )+e^4 \left (5376 x+13440 x^2+13440 x^3+6720 x^4+1680 x^5+168 x^6\right )\right ) \log ^2(2 x)+\left (e^8 \left (4480+8960 x+6720 x^2+2240 x^3+280 x^4\right )+e^6 \left (4480 x+8960 x^2+6720 x^3+2240 x^4+280 x^5\right )\right ) \log ^3(2 x)+\left (e^{10} \left (2240+3360 x+1680 x^2+280 x^3\right )+e^8 \left (2240 x+3360 x^2+1680 x^3+280 x^4\right )\right ) \log ^4(2 x)+\left (e^{12} \left (672+672 x+168 x^2\right )+e^{10} \left (672 x+672 x^2+168 x^3\right )\right ) \log ^5(2 x)+\left (e^{14} (112+56 x)+e^{12} \left (112 x+56 x^2\right )\right ) \log ^6(2 x)+\left (8 e^{16}+8 e^{14} x\right ) \log ^7(2 x)}{x} \, dx=1024\,x+1792\,{\ln \left (2\,x\right )}^2\,{\mathrm {e}}^4+1792\,{\ln \left (2\,x\right )}^3\,{\mathrm {e}}^6+1120\,{\ln \left (2\,x\right )}^4\,{\mathrm {e}}^8+448\,{\ln \left (2\,x\right )}^5\,{\mathrm {e}}^{10}+112\,{\ln \left (2\,x\right )}^6\,{\mathrm {e}}^{12}+16\,{\ln \left (2\,x\right )}^7\,{\mathrm {e}}^{14}+{\ln \left (2\,x\right )}^8\,{\mathrm {e}}^{16}+1024\,{\mathrm {e}}^2\,\ln \left (x\right )+1792\,x^2+1792\,x^3+1120\,x^4+448\,x^5+112\,x^6+16\,x^7+x^8+6720\,x^2\,{\ln \left (2\,x\right )}^2\,{\mathrm {e}}^4+4480\,x^3\,{\ln \left (2\,x\right )}^2\,{\mathrm {e}}^4+1680\,x^4\,{\ln \left (2\,x\right )}^2\,{\mathrm {e}}^4+4480\,x^2\,{\ln \left (2\,x\right )}^3\,{\mathrm {e}}^6+336\,x^5\,{\ln \left (2\,x\right )}^2\,{\mathrm {e}}^4+2240\,x^3\,{\ln \left (2\,x\right )}^3\,{\mathrm {e}}^6+28\,x^6\,{\ln \left (2\,x\right )}^2\,{\mathrm {e}}^4+560\,x^4\,{\ln \left (2\,x\right )}^3\,{\mathrm {e}}^6+1680\,x^2\,{\ln \left (2\,x\right )}^4\,{\mathrm {e}}^8+56\,x^5\,{\ln \left (2\,x\right )}^3\,{\mathrm {e}}^6+560\,x^3\,{\ln \left (2\,x\right )}^4\,{\mathrm {e}}^8+70\,x^4\,{\ln \left (2\,x\right )}^4\,{\mathrm {e}}^8+336\,x^2\,{\ln \left (2\,x\right )}^5\,{\mathrm {e}}^{10}+56\,x^3\,{\ln \left (2\,x\right )}^5\,{\mathrm {e}}^{10}+28\,x^2\,{\ln \left (2\,x\right )}^6\,{\mathrm {e}}^{12}+3584\,x\,\ln \left (2\,x\right )\,{\mathrm {e}}^2+5376\,x^2\,\ln \left (2\,x\right )\,{\mathrm {e}}^2+4480\,x^3\,\ln \left (2\,x\right )\,{\mathrm {e}}^2+5376\,x\,{\ln \left (2\,x\right )}^2\,{\mathrm {e}}^4+2240\,x^4\,\ln \left (2\,x\right )\,{\mathrm {e}}^2+672\,x^5\,\ln \left (2\,x\right )\,{\mathrm {e}}^2+112\,x^6\,\ln \left (2\,x\right )\,{\mathrm {e}}^2+4480\,x\,{\ln \left (2\,x\right )}^3\,{\mathrm {e}}^6+8\,x^7\,\ln \left (2\,x\right )\,{\mathrm {e}}^2+2240\,x\,{\ln \left (2\,x\right )}^4\,{\mathrm {e}}^8+672\,x\,{\ln \left (2\,x\right )}^5\,{\mathrm {e}}^{10}+112\,x\,{\ln \left (2\,x\right )}^6\,{\mathrm {e}}^{12}+8\,x\,{\ln \left (2\,x\right )}^7\,{\mathrm {e}}^{14} \end {dmath*}
int((1024*x + exp(2)*(3584*x + 5376*x^2 + 4480*x^3 + 2240*x^4 + 672*x^5 + 112*x^6 + 8*x^7 + 1024) + log(2*x)^2*(exp(4)*(5376*x + 13440*x^2 + 13440*x ^3 + 6720*x^4 + 1680*x^5 + 168*x^6) + exp(6)*(13440*x + 13440*x^2 + 6720*x ^3 + 1680*x^4 + 168*x^5 + 5376)) + log(2*x)*(exp(4)*(10752*x + 13440*x^2 + 8960*x^3 + 3360*x^4 + 672*x^5 + 56*x^6 + 3584) + exp(2)*(3584*x + 10752*x ^2 + 13440*x^3 + 8960*x^4 + 3360*x^5 + 672*x^6 + 56*x^7)) + log(2*x)^7*(8* exp(16) + 8*x*exp(14)) + log(2*x)^4*(exp(10)*(3360*x + 1680*x^2 + 280*x^3 + 2240) + exp(8)*(2240*x + 3360*x^2 + 1680*x^3 + 280*x^4)) + log(2*x)^6*(e xp(12)*(112*x + 56*x^2) + exp(14)*(56*x + 112)) + 3584*x^2 + 5376*x^3 + 44 80*x^4 + 2240*x^5 + 672*x^6 + 112*x^7 + 8*x^8 + log(2*x)^3*(exp(8)*(8960*x + 6720*x^2 + 2240*x^3 + 280*x^4 + 4480) + exp(6)*(4480*x + 8960*x^2 + 672 0*x^3 + 2240*x^4 + 280*x^5)) + log(2*x)^5*(exp(12)*(672*x + 168*x^2 + 672) + exp(10)*(672*x + 672*x^2 + 168*x^3)))/x,x)
1024*x + 1792*log(2*x)^2*exp(4) + 1792*log(2*x)^3*exp(6) + 1120*log(2*x)^4 *exp(8) + 448*log(2*x)^5*exp(10) + 112*log(2*x)^6*exp(12) + 16*log(2*x)^7* exp(14) + log(2*x)^8*exp(16) + 1024*exp(2)*log(x) + 1792*x^2 + 1792*x^3 + 1120*x^4 + 448*x^5 + 112*x^6 + 16*x^7 + x^8 + 6720*x^2*log(2*x)^2*exp(4) + 4480*x^3*log(2*x)^2*exp(4) + 1680*x^4*log(2*x)^2*exp(4) + 4480*x^2*log(2* x)^3*exp(6) + 336*x^5*log(2*x)^2*exp(4) + 2240*x^3*log(2*x)^3*exp(6) + 28* x^6*log(2*x)^2*exp(4) + 560*x^4*log(2*x)^3*exp(6) + 1680*x^2*log(2*x)^4*ex p(8) + 56*x^5*log(2*x)^3*exp(6) + 560*x^3*log(2*x)^4*exp(8) + 70*x^4*log(2 *x)^4*exp(8) + 336*x^2*log(2*x)^5*exp(10) + 56*x^3*log(2*x)^5*exp(10) + 28 *x^2*log(2*x)^6*exp(12) + 3584*x*log(2*x)*exp(2) + 5376*x^2*log(2*x)*exp(2 ) + 4480*x^3*log(2*x)*exp(2) + 5376*x*log(2*x)^2*exp(4) + 2240*x^4*log(2*x )*exp(2) + 672*x^5*log(2*x)*exp(2) + 112*x^6*log(2*x)*exp(2) + 4480*x*log( 2*x)^3*exp(6) + 8*x^7*log(2*x)*exp(2) + 2240*x*log(2*x)^4*exp(8) + 672*x*l og(2*x)^5*exp(10) + 112*x*log(2*x)^6*exp(12) + 8*x*log(2*x)^7*exp(14)