\(\int (1152+1800 x+2032 x^2+1130 x^3+934 x^4+526 x^5+114 x^6+8 x^7+e^8 (8 x-4 x^2+4 x^3)+e^6 (16+112 x-12 x^2+40 x^3+20 x^4)+e^4 (200+616 x+218 x^2+192 x^3+202 x^4+36 x^5)+e^2 (832+1632 x+1312 x^2+664 x^3+716 x^4+272 x^5+28 x^6)+(1024+1792 x+1740 x^2+1914 x^3+1006 x^4+206 x^5+14 x^6+e^8 (4-2 x+6 x^2)+e^6 (64+80 x^2+32 x^3)+e^4 (384+208 x+448 x^2+364 x^3+60 x^4)+e^2 (1024+1152 x+1328 x^2+1416 x^3+488 x^4+48 x^5)) \log (x)+(512 x+2 e^8 x+864 x^2+452 x^3+90 x^4+6 x^5+e^6 (32 x+12 x^2)+e^4 (192 x+150 x^2+24 x^3)+e^2 (512 x+624 x^2+208 x^3+20 x^4)) \log ^2(x)) \, dx\) [401]

Optimal result

Integrand size = 345, antiderivative size = 29 \[ \int \left (1152+1800 x+2032 x^2+1130 x^3+934 x^4+526 x^5+114 x^6+8 x^7+e^8 \left (8 x-4 x^2+4 x^3\right )+e^6 \left (16+112 x-12 x^2+40 x^3+20 x^4\right )+e^4 \left (200+616 x+218 x^2+192 x^3+202 x^4+36 x^5\right )+e^2 \left (832+1632 x+1312 x^2+664 x^3+716 x^4+272 x^5+28 x^6\right )+\left (1024+1792 x+1740 x^2+1914 x^3+1006 x^4+206 x^5+14 x^6+e^8 \left (4-2 x+6 x^2\right )+e^6 \left (64+80 x^2+32 x^3\right )+e^4 \left (384+208 x+448 x^2+364 x^3+60 x^4\right )+e^2 \left (1024+1152 x+1328 x^2+1416 x^3+488 x^4+48 x^5\right )\right ) \log (x)+\left (512 x+2 e^8 x+864 x^2+452 x^3+90 x^4+6 x^5+e^6 \left (32 x+12 x^2\right )+e^4 \left (192 x+150 x^2+24 x^3\right )+e^2 \left (512 x+624 x^2+208 x^3+20 x^4\right )\right ) \log ^2(x)\right ) \, dx=5+\left (x+\left (4+e^2+x\right )^2\right )^2 \left (2-x+x^2+x \log (x)\right )^2 \] Output:

5+(x*ln(x)+x^2-x+2)^2*((exp(2)+x+4)^2+x)^2
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(207\) vs. \(2(29)=58\).

Time = 0.06 (sec) , antiderivative size = 207, normalized size of antiderivative = 7.14 \[ \int \left (1152+1800 x+2032 x^2+1130 x^3+934 x^4+526 x^5+114 x^6+8 x^7+e^8 \left (8 x-4 x^2+4 x^3\right )+e^6 \left (16+112 x-12 x^2+40 x^3+20 x^4\right )+e^4 \left (200+616 x+218 x^2+192 x^3+202 x^4+36 x^5\right )+e^2 \left (832+1632 x+1312 x^2+664 x^3+716 x^4+272 x^5+28 x^6\right )+\left (1024+1792 x+1740 x^2+1914 x^3+1006 x^4+206 x^5+14 x^6+e^8 \left (4-2 x+6 x^2\right )+e^6 \left (64+80 x^2+32 x^3\right )+e^4 \left (384+208 x+448 x^2+364 x^3+60 x^4\right )+e^2 \left (1024+1152 x+1328 x^2+1416 x^3+488 x^4+48 x^5\right )\right ) \log (x)+\left (512 x+2 e^8 x+864 x^2+452 x^3+90 x^4+6 x^5+e^6 \left (32 x+12 x^2\right )+e^4 \left (192 x+150 x^2+24 x^3\right )+e^2 \left (512 x+624 x^2+208 x^3+20 x^4\right )\right ) \log ^2(x)\right ) \, dx=x \left (-4 \left (4+e^2\right )^2 \left (-2+4 e^2+e^4\right )+\left (580+656 e^2+304 e^4+64 e^6+5 e^8\right ) x-2 \left (-274-168 e^2-17 e^4+6 e^6+e^8\right ) x^2+\left (177+84 e^2+26 e^4+8 e^6+e^8\right ) x^3+2 \left (74+62 e^2+19 e^4+2 e^6\right ) x^4+2 \left (41+22 e^2+3 e^4\right ) x^5+4 \left (4+e^2\right ) x^6+x^7+2 \left (2-x+x^2\right ) \left (16+e^4+9 x+x^2+2 e^2 (4+x)\right )^2 \log (x)+x \left (16+e^4+9 x+x^2+2 e^2 (4+x)\right )^2 \log ^2(x)\right ) \] Input:

Integrate[1152 + 1800*x + 2032*x^2 + 1130*x^3 + 934*x^4 + 526*x^5 + 114*x^ 
6 + 8*x^7 + E^8*(8*x - 4*x^2 + 4*x^3) + E^6*(16 + 112*x - 12*x^2 + 40*x^3 
+ 20*x^4) + E^4*(200 + 616*x + 218*x^2 + 192*x^3 + 202*x^4 + 36*x^5) + E^2 
*(832 + 1632*x + 1312*x^2 + 664*x^3 + 716*x^4 + 272*x^5 + 28*x^6) + (1024 
+ 1792*x + 1740*x^2 + 1914*x^3 + 1006*x^4 + 206*x^5 + 14*x^6 + E^8*(4 - 2* 
x + 6*x^2) + E^6*(64 + 80*x^2 + 32*x^3) + E^4*(384 + 208*x + 448*x^2 + 364 
*x^3 + 60*x^4) + E^2*(1024 + 1152*x + 1328*x^2 + 1416*x^3 + 488*x^4 + 48*x 
^5))*Log[x] + (512*x + 2*E^8*x + 864*x^2 + 452*x^3 + 90*x^4 + 6*x^5 + E^6* 
(32*x + 12*x^2) + E^4*(192*x + 150*x^2 + 24*x^3) + E^2*(512*x + 624*x^2 + 
208*x^3 + 20*x^4))*Log[x]^2,x]
 

Output:

x*(-4*(4 + E^2)^2*(-2 + 4*E^2 + E^4) + (580 + 656*E^2 + 304*E^4 + 64*E^6 + 
 5*E^8)*x - 2*(-274 - 168*E^2 - 17*E^4 + 6*E^6 + E^8)*x^2 + (177 + 84*E^2 
+ 26*E^4 + 8*E^6 + E^8)*x^3 + 2*(74 + 62*E^2 + 19*E^4 + 2*E^6)*x^4 + 2*(41 
 + 22*E^2 + 3*E^4)*x^5 + 4*(4 + E^2)*x^6 + x^7 + 2*(2 - x + x^2)*(16 + E^4 
 + 9*x + x^2 + 2*E^2*(4 + x))^2*Log[x] + x*(16 + E^4 + 9*x + x^2 + 2*E^2*( 
4 + x))^2*Log[x]^2)
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(711\) vs. \(2(29)=58\).

Time = 0.98 (sec) , antiderivative size = 711, normalized size of antiderivative = 24.52, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.003, Rules used = {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.

\(\displaystyle \int \left (8 x^7+114 x^6+526 x^5+934 x^4+1130 x^3+2032 x^2+e^8 \left (4 x^3-4 x^2+8 x\right )+e^6 \left (20 x^4+40 x^3-12 x^2+112 x+16\right )+e^4 \left (36 x^5+202 x^4+192 x^3+218 x^2+616 x+200\right )+\left (6 x^5+90 x^4+452 x^3+864 x^2+e^6 \left (12 x^2+32 x\right )+e^4 \left (24 x^3+150 x^2+192 x\right )+e^2 \left (20 x^4+208 x^3+624 x^2+512 x\right )+2 e^8 x+512 x\right ) \log ^2(x)+e^2 \left (28 x^6+272 x^5+716 x^4+664 x^3+1312 x^2+1632 x+832\right )+\left (14 x^6+206 x^5+1006 x^4+1914 x^3+1740 x^2+e^8 \left (6 x^2-2 x+4\right )+e^6 \left (32 x^3+80 x^2+64\right )+e^4 \left (60 x^4+364 x^3+448 x^2+208 x+384\right )+e^2 \left (48 x^5+488 x^4+1416 x^3+1328 x^2+1152 x+1024\right )+1792 x+1024\right ) \log (x)+1800 x+1152\right ) \, dx\)

\(\Big \downarrow \) 2009

\(\displaystyle x^8+4 e^2 x^7+16 x^7+2 x^7 \log (x)-\frac {1}{18} \left (103+24 e^2\right ) x^6+6 e^4 x^6+\frac {136 e^2 x^6}{3}+\frac {1579 x^6}{18}+x^6 \log ^2(x)+\frac {1}{3} \left (103+24 e^2\right ) x^6 \log (x)-\frac {1}{3} x^6 \log (x)-\frac {2}{25} \left (503+244 e^2+30 e^4\right ) x^5+\frac {4}{25} \left (9+2 e^2\right ) x^5+4 e^6 x^5+\frac {202 e^4 x^5}{5}+\frac {716 e^2 x^5}{5}+\frac {934 x^5}{5}+2 \left (9+2 e^2\right ) x^5 \log ^2(x)+\frac {2}{5} \left (503+244 e^2+30 e^4\right ) x^5 \log (x)-\frac {4}{5} \left (9+2 e^2\right ) x^5 \log (x)-\frac {1}{8} \left (957+708 e^2+182 e^4+16 e^6\right ) x^4+\frac {1}{8} \left (113+52 e^2+6 e^4\right ) x^4+e^8 x^4+10 e^6 x^4+48 e^4 x^4+166 e^2 x^4+\frac {565 x^4}{2}+\left (113+52 e^2+6 e^4\right ) x^4 \log ^2(x)+\frac {1}{2} \left (957+708 e^2+182 e^4+16 e^6\right ) x^4 \log (x)-\frac {1}{2} \left (113+52 e^2+6 e^4\right ) x^4 \log (x)-\frac {2}{9} \left (870+664 e^2+224 e^4+40 e^6+3 e^8\right ) x^3+\frac {4}{9} \left (4+e^2\right )^2 \left (9+2 e^2\right ) x^3-\frac {4 e^8 x^3}{3}-4 e^6 x^3+\frac {218 e^4 x^3}{3}+\frac {1312 e^2 x^3}{3}+\frac {2032 x^3}{3}+2 \left (4+e^2\right )^2 \left (9+2 e^2\right ) x^3 \log ^2(x)+\frac {2}{3} \left (870+664 e^2+224 e^4+40 e^6+3 e^8\right ) x^3 \log (x)-\frac {4}{3} \left (4+e^2\right )^2 \left (9+2 e^2\right ) x^3 \log (x)-\frac {1}{2} \left (4+e^2\right )^2 \left (56+8 e^2-e^4\right ) x^2+\frac {1}{2} \left (4+e^2\right )^4 x^2+4 e^8 x^2+56 e^6 x^2+308 e^4 x^2+816 e^2 x^2+900 x^2+\left (4+e^2\right )^4 x^2 \log ^2(x)+\left (4+e^2\right )^2 \left (56+8 e^2-e^4\right ) x^2 \log (x)-\left (4+e^2\right )^4 x^2 \log (x)-4 \left (4+e^2\right )^4 x+16 e^6 x+200 e^4 x+832 e^2 x+1152 x+4 \left (4+e^2\right )^4 x \log (x)\)

Input:

Int[1152 + 1800*x + 2032*x^2 + 1130*x^3 + 934*x^4 + 526*x^5 + 114*x^6 + 8* 
x^7 + E^8*(8*x - 4*x^2 + 4*x^3) + E^6*(16 + 112*x - 12*x^2 + 40*x^3 + 20*x 
^4) + E^4*(200 + 616*x + 218*x^2 + 192*x^3 + 202*x^4 + 36*x^5) + E^2*(832 
+ 1632*x + 1312*x^2 + 664*x^3 + 716*x^4 + 272*x^5 + 28*x^6) + (1024 + 1792 
*x + 1740*x^2 + 1914*x^3 + 1006*x^4 + 206*x^5 + 14*x^6 + E^8*(4 - 2*x + 6* 
x^2) + E^6*(64 + 80*x^2 + 32*x^3) + E^4*(384 + 208*x + 448*x^2 + 364*x^3 + 
 60*x^4) + E^2*(1024 + 1152*x + 1328*x^2 + 1416*x^3 + 488*x^4 + 48*x^5))*L 
og[x] + (512*x + 2*E^8*x + 864*x^2 + 452*x^3 + 90*x^4 + 6*x^5 + E^6*(32*x 
+ 12*x^2) + E^4*(192*x + 150*x^2 + 24*x^3) + E^2*(512*x + 624*x^2 + 208*x^ 
3 + 20*x^4))*Log[x]^2,x]
 

Output:

1152*x + 832*E^2*x + 200*E^4*x + 16*E^6*x - 4*(4 + E^2)^4*x + 900*x^2 + 81 
6*E^2*x^2 + 308*E^4*x^2 + 56*E^6*x^2 + 4*E^8*x^2 + ((4 + E^2)^4*x^2)/2 - ( 
(4 + E^2)^2*(56 + 8*E^2 - E^4)*x^2)/2 + (2032*x^3)/3 + (1312*E^2*x^3)/3 + 
(218*E^4*x^3)/3 - 4*E^6*x^3 - (4*E^8*x^3)/3 + (4*(4 + E^2)^2*(9 + 2*E^2)*x 
^3)/9 - (2*(870 + 664*E^2 + 224*E^4 + 40*E^6 + 3*E^8)*x^3)/9 + (565*x^4)/2 
 + 166*E^2*x^4 + 48*E^4*x^4 + 10*E^6*x^4 + E^8*x^4 + ((113 + 52*E^2 + 6*E^ 
4)*x^4)/8 - ((957 + 708*E^2 + 182*E^4 + 16*E^6)*x^4)/8 + (934*x^5)/5 + (71 
6*E^2*x^5)/5 + (202*E^4*x^5)/5 + 4*E^6*x^5 + (4*(9 + 2*E^2)*x^5)/25 - (2*( 
503 + 244*E^2 + 30*E^4)*x^5)/25 + (1579*x^6)/18 + (136*E^2*x^6)/3 + 6*E^4* 
x^6 - ((103 + 24*E^2)*x^6)/18 + 16*x^7 + 4*E^2*x^7 + x^8 + 4*(4 + E^2)^4*x 
*Log[x] - (4 + E^2)^4*x^2*Log[x] + (4 + E^2)^2*(56 + 8*E^2 - E^4)*x^2*Log[ 
x] - (4*(4 + E^2)^2*(9 + 2*E^2)*x^3*Log[x])/3 + (2*(870 + 664*E^2 + 224*E^ 
4 + 40*E^6 + 3*E^8)*x^3*Log[x])/3 - ((113 + 52*E^2 + 6*E^4)*x^4*Log[x])/2 
+ ((957 + 708*E^2 + 182*E^4 + 16*E^6)*x^4*Log[x])/2 - (4*(9 + 2*E^2)*x^5*L 
og[x])/5 + (2*(503 + 244*E^2 + 30*E^4)*x^5*Log[x])/5 - (x^6*Log[x])/3 + (( 
103 + 24*E^2)*x^6*Log[x])/3 + 2*x^7*Log[x] + (4 + E^2)^4*x^2*Log[x]^2 + 2* 
(4 + E^2)^2*(9 + 2*E^2)*x^3*Log[x]^2 + (113 + 52*E^2 + 6*E^4)*x^4*Log[x]^2 
 + 2*(9 + 2*E^2)*x^5*Log[x]^2 + x^6*Log[x]^2
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(602\) vs. \(2(28)=56\).

Time = 0.12 (sec) , antiderivative size = 603, normalized size of antiderivative = 20.79

\[64 x \,{\mathrm e}^{6} \ln \left (x \right )+1024 x \,{\mathrm e}^{2} \ln \left (x \right )+128 x -48 x \,{\mathrm e}^{6}+x^{6} \ln \left (x \right )^{2}-4 x \,{\mathrm e}^{8}+44 x^{6} {\mathrm e}^{2}+16 \,{\mathrm e}^{6} \ln \left (x \right )^{2} x^{2}+24 \ln \left (x \right ) {\mathrm e}^{6} x^{3}+6 \,{\mathrm e}^{4} x^{6}+113 x^{4} \ln \left (x \right )^{2}+194 x^{5} \ln \left (x \right )+34 x^{6} \ln \left (x \right )+2 x^{7} \ln \left (x \right )+422 x^{4} \ln \left (x \right )+288 x^{3} \ln \left (x \right )^{2}+18 x^{5} \ln \left (x \right )^{2}+26 x^{4} {\mathrm e}^{4}+656 x^{2} {\mathrm e}^{2}+124 \,{\mathrm e}^{2} x^{5}+336 x^{3} {\mathrm e}^{2}+84 x^{4} {\mathrm e}^{2}-192 \,{\mathrm e}^{2} x +256 x^{2} \ln \left (x \right )^{2}+5 x^{2} {\mathrm e}^{8}+x^{4} {\mathrm e}^{8}-2 x^{3} {\mathrm e}^{8}+384 x \,{\mathrm e}^{4} \ln \left (x \right )+34 x^{3} {\mathrm e}^{4}+304 x^{2} {\mathrm e}^{4}+38 \,{\mathrm e}^{4} x^{5}-184 x \,{\mathrm e}^{4}+388 x^{3} \ln \left (x \right )+640 x^{2} \ln \left (x \right )+1024 x \ln \left (x \right )+16 x^{7}+x^{8}+82 x^{6}+580 x^{2}+548 x^{3}+177 x^{4}+148 x^{5}+50 x^{3} {\mathrm e}^{4} \ln \left (x \right )^{2}+8 \ln \left (x \right ) {\mathrm e}^{4} x^{2}+52 \,{\mathrm e}^{2} \ln \left (x \right )^{2} x^{4}+96 \,{\mathrm e}^{2} \ln \left (x \right ) x^{5}+208 \,{\mathrm e}^{2} \ln \left (x \right )^{2} x^{3}+328 \,{\mathrm e}^{2} \ln \left (x \right ) x^{4}+256 \,{\mathrm e}^{2} \ln \left (x \right )^{2} x^{2}+304 \,{\mathrm e}^{2} \ln \left (x \right ) x^{3}+320 \,{\mathrm e}^{2} x^{2} \ln \left (x \right )-12 \,{\mathrm e}^{6} x^{3}+64 x^{2} {\mathrm e}^{6}-2 \ln \left (x \right ) x^{2} {\mathrm e}^{8}-16 \ln \left (x \right ) x^{2} {\mathrm e}^{6}+4 \ln \left (x \right ) x \,{\mathrm e}^{8}+4 \,{\mathrm e}^{2} \ln \left (x \right )^{2} x^{5}+8 \ln \left (x \right ) {\mathrm e}^{2} x^{6}+{\mathrm e}^{8} \ln \left (x \right )^{2} x^{2}+2 \ln \left (x \right ) {\mathrm e}^{8} x^{3}+4 \,{\mathrm e}^{6} \ln \left (x \right )^{2} x^{3}+8 \ln \left (x \right ) {\mathrm e}^{6} x^{4}+6 \,{\mathrm e}^{4} \ln \left (x \right )^{2} x^{4}+12 \ln \left (x \right ) {\mathrm e}^{4} x^{5}+96 \ln \left (x \right )^{2} {\mathrm e}^{4} x^{2}+4 \,{\mathrm e}^{2} x^{7}+4 \,{\mathrm e}^{6} x^{5}+8 \,{\mathrm e}^{6} x^{4}+116 \,{\mathrm e}^{4} \ln \left (x \right ) x^{3}+88 \,{\mathrm e}^{4} \ln \left (x \right ) x^{4}\]

Input:

int((2*x*exp(2)^4+(12*x^2+32*x)*exp(2)^3+(24*x^3+150*x^2+192*x)*exp(2)^2+( 
20*x^4+208*x^3+624*x^2+512*x)*exp(2)+6*x^5+90*x^4+452*x^3+864*x^2+512*x)*l 
n(x)^2+((6*x^2-2*x+4)*exp(2)^4+(32*x^3+80*x^2+64)*exp(2)^3+(60*x^4+364*x^3 
+448*x^2+208*x+384)*exp(2)^2+(48*x^5+488*x^4+1416*x^3+1328*x^2+1152*x+1024 
)*exp(2)+14*x^6+206*x^5+1006*x^4+1914*x^3+1740*x^2+1792*x+1024)*ln(x)+(4*x 
^3-4*x^2+8*x)*exp(2)^4+(20*x^4+40*x^3-12*x^2+112*x+16)*exp(2)^3+(36*x^5+20 
2*x^4+192*x^3+218*x^2+616*x+200)*exp(2)^2+(28*x^6+272*x^5+716*x^4+664*x^3+ 
1312*x^2+1632*x+832)*exp(2)+8*x^7+114*x^6+526*x^5+934*x^4+1130*x^3+2032*x^ 
2+1800*x+1152,x)
 

Output:

8*x^2*exp(2)^2*ln(x)+1024*x*exp(2)*ln(x)+128*x+x^6*ln(x)^2+44*x^6*exp(2)+x 
^4*exp(2)^4+113*x^4*ln(x)^2+194*x^5*ln(x)+34*x^6*ln(x)+2*x^7*ln(x)-4*x*exp 
(2)^4+422*x^4*ln(x)+288*x^3*ln(x)^2+18*x^5*ln(x)^2-48*x*exp(2)^3+64*x^2*ex 
p(2)^3+656*x^2*exp(2)+124*exp(2)*x^5-184*x*exp(2)^2+38*x^5*exp(2)^2+336*x^ 
3*exp(2)+84*x^4*exp(2)-192*exp(2)*x+256*x^2*ln(x)^2+4*exp(2)^3*x^5+8*exp(2 
)^3*x^4+34*x^3*exp(2)^2-12*x^3*exp(2)^3+26*x^4*exp(2)^2+304*x^2*exp(2)^2+6 
*x^6*exp(2)^2+388*x^3*ln(x)+640*x^2*ln(x)+1024*x*ln(x)+16*x^7+x^8+82*x^6+5 
80*x^2+548*x^3+177*x^4+148*x^5+384*x*exp(2)^2*ln(x)+5*exp(2)^4*x^2-2*exp(2 
)^4*ln(x)*x^2+16*exp(2)^3*ln(x)^2*x^2+24*exp(2)^3*ln(x)*x^3+50*exp(2)^2*ln 
(x)^2*x^3+88*exp(2)^2*ln(x)*x^4+52*exp(2)*ln(x)^2*x^4+96*exp(2)*ln(x)*x^5+ 
4*exp(2)^4*ln(x)*x-16*exp(2)^3*ln(x)*x^2+208*exp(2)*ln(x)^2*x^3+328*exp(2) 
*ln(x)*x^4+64*exp(2)^3*ln(x)*x+256*exp(2)*ln(x)^2*x^2+304*exp(2)*ln(x)*x^3 
+96*x^2*exp(2)^2*ln(x)^2+116*exp(2)^2*ln(x)*x^3+320*exp(2)*x^2*ln(x)+exp(2 
)^4*ln(x)^2*x^2+2*ln(x)*exp(2)^4*x^3+4*exp(2)^3*ln(x)^2*x^3+8*ln(x)*exp(2) 
^3*x^4+6*exp(2)^2*ln(x)^2*x^4+12*ln(x)*exp(2)^2*x^5+4*exp(2)*ln(x)^2*x^5+8 
*ln(x)*exp(2)*x^6+4*exp(2)*x^7-2*x^3*exp(2)^4
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 374 vs. \(2 (28) = 56\).

Time = 0.09 (sec) , antiderivative size = 374, normalized size of antiderivative = 12.90 \[ \int \left (1152+1800 x+2032 x^2+1130 x^3+934 x^4+526 x^5+114 x^6+8 x^7+e^8 \left (8 x-4 x^2+4 x^3\right )+e^6 \left (16+112 x-12 x^2+40 x^3+20 x^4\right )+e^4 \left (200+616 x+218 x^2+192 x^3+202 x^4+36 x^5\right )+e^2 \left (832+1632 x+1312 x^2+664 x^3+716 x^4+272 x^5+28 x^6\right )+\left (1024+1792 x+1740 x^2+1914 x^3+1006 x^4+206 x^5+14 x^6+e^8 \left (4-2 x+6 x^2\right )+e^6 \left (64+80 x^2+32 x^3\right )+e^4 \left (384+208 x+448 x^2+364 x^3+60 x^4\right )+e^2 \left (1024+1152 x+1328 x^2+1416 x^3+488 x^4+48 x^5\right )\right ) \log (x)+\left (512 x+2 e^8 x+864 x^2+452 x^3+90 x^4+6 x^5+e^6 \left (32 x+12 x^2\right )+e^4 \left (192 x+150 x^2+24 x^3\right )+e^2 \left (512 x+624 x^2+208 x^3+20 x^4\right )\right ) \log ^2(x)\right ) \, dx=x^{8} + 16 \, x^{7} + 82 \, x^{6} + 148 \, x^{5} + 177 \, x^{4} + 548 \, x^{3} + {\left (x^{6} + 18 \, x^{5} + 113 \, x^{4} + 288 \, x^{3} + x^{2} e^{8} + 256 \, x^{2} + 4 \, {\left (x^{3} + 4 \, x^{2}\right )} e^{6} + 2 \, {\left (3 \, x^{4} + 25 \, x^{3} + 48 \, x^{2}\right )} e^{4} + 4 \, {\left (x^{5} + 13 \, x^{4} + 52 \, x^{3} + 64 \, x^{2}\right )} e^{2}\right )} \log \left (x\right )^{2} + 580 \, x^{2} + {\left (x^{4} - 2 \, x^{3} + 5 \, x^{2} - 4 \, x\right )} e^{8} + 4 \, {\left (x^{5} + 2 \, x^{4} - 3 \, x^{3} + 16 \, x^{2} - 12 \, x\right )} e^{6} + 2 \, {\left (3 \, x^{6} + 19 \, x^{5} + 13 \, x^{4} + 17 \, x^{3} + 152 \, x^{2} - 92 \, x\right )} e^{4} + 4 \, {\left (x^{7} + 11 \, x^{6} + 31 \, x^{5} + 21 \, x^{4} + 84 \, x^{3} + 164 \, x^{2} - 48 \, x\right )} e^{2} + 2 \, {\left (x^{7} + 17 \, x^{6} + 97 \, x^{5} + 211 \, x^{4} + 194 \, x^{3} + 320 \, x^{2} + {\left (x^{3} - x^{2} + 2 \, x\right )} e^{8} + 4 \, {\left (x^{4} + 3 \, x^{3} - 2 \, x^{2} + 8 \, x\right )} e^{6} + 2 \, {\left (3 \, x^{5} + 22 \, x^{4} + 29 \, x^{3} + 2 \, x^{2} + 96 \, x\right )} e^{4} + 4 \, {\left (x^{6} + 12 \, x^{5} + 41 \, x^{4} + 38 \, x^{3} + 40 \, x^{2} + 128 \, x\right )} e^{2} + 512 \, x\right )} \log \left (x\right ) + 128 \, x \] Input:

integrate((2*x*exp(2)^4+(12*x^2+32*x)*exp(2)^3+(24*x^3+150*x^2+192*x)*exp( 
2)^2+(20*x^4+208*x^3+624*x^2+512*x)*exp(2)+6*x^5+90*x^4+452*x^3+864*x^2+51 
2*x)*log(x)^2+((6*x^2-2*x+4)*exp(2)^4+(32*x^3+80*x^2+64)*exp(2)^3+(60*x^4+ 
364*x^3+448*x^2+208*x+384)*exp(2)^2+(48*x^5+488*x^4+1416*x^3+1328*x^2+1152 
*x+1024)*exp(2)+14*x^6+206*x^5+1006*x^4+1914*x^3+1740*x^2+1792*x+1024)*log 
(x)+(4*x^3-4*x^2+8*x)*exp(2)^4+(20*x^4+40*x^3-12*x^2+112*x+16)*exp(2)^3+(3 
6*x^5+202*x^4+192*x^3+218*x^2+616*x+200)*exp(2)^2+(28*x^6+272*x^5+716*x^4+ 
664*x^3+1312*x^2+1632*x+832)*exp(2)+8*x^7+114*x^6+526*x^5+934*x^4+1130*x^3 
+2032*x^2+1800*x+1152,x, algorithm="fricas")
 

Output:

x^8 + 16*x^7 + 82*x^6 + 148*x^5 + 177*x^4 + 548*x^3 + (x^6 + 18*x^5 + 113* 
x^4 + 288*x^3 + x^2*e^8 + 256*x^2 + 4*(x^3 + 4*x^2)*e^6 + 2*(3*x^4 + 25*x^ 
3 + 48*x^2)*e^4 + 4*(x^5 + 13*x^4 + 52*x^3 + 64*x^2)*e^2)*log(x)^2 + 580*x 
^2 + (x^4 - 2*x^3 + 5*x^2 - 4*x)*e^8 + 4*(x^5 + 2*x^4 - 3*x^3 + 16*x^2 - 1 
2*x)*e^6 + 2*(3*x^6 + 19*x^5 + 13*x^4 + 17*x^3 + 152*x^2 - 92*x)*e^4 + 4*( 
x^7 + 11*x^6 + 31*x^5 + 21*x^4 + 84*x^3 + 164*x^2 - 48*x)*e^2 + 2*(x^7 + 1 
7*x^6 + 97*x^5 + 211*x^4 + 194*x^3 + 320*x^2 + (x^3 - x^2 + 2*x)*e^8 + 4*( 
x^4 + 3*x^3 - 2*x^2 + 8*x)*e^6 + 2*(3*x^5 + 22*x^4 + 29*x^3 + 2*x^2 + 96*x 
)*e^4 + 4*(x^6 + 12*x^5 + 41*x^4 + 38*x^3 + 40*x^2 + 128*x)*e^2 + 512*x)*l 
og(x) + 128*x
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (26) = 52\).

Time = 0.43 (sec) , antiderivative size = 442, normalized size of antiderivative = 15.24 \[ \int \left (1152+1800 x+2032 x^2+1130 x^3+934 x^4+526 x^5+114 x^6+8 x^7+e^8 \left (8 x-4 x^2+4 x^3\right )+e^6 \left (16+112 x-12 x^2+40 x^3+20 x^4\right )+e^4 \left (200+616 x+218 x^2+192 x^3+202 x^4+36 x^5\right )+e^2 \left (832+1632 x+1312 x^2+664 x^3+716 x^4+272 x^5+28 x^6\right )+\left (1024+1792 x+1740 x^2+1914 x^3+1006 x^4+206 x^5+14 x^6+e^8 \left (4-2 x+6 x^2\right )+e^6 \left (64+80 x^2+32 x^3\right )+e^4 \left (384+208 x+448 x^2+364 x^3+60 x^4\right )+e^2 \left (1024+1152 x+1328 x^2+1416 x^3+488 x^4+48 x^5\right )\right ) \log (x)+\left (512 x+2 e^8 x+864 x^2+452 x^3+90 x^4+6 x^5+e^6 \left (32 x+12 x^2\right )+e^4 \left (192 x+150 x^2+24 x^3\right )+e^2 \left (512 x+624 x^2+208 x^3+20 x^4\right )\right ) \log ^2(x)\right ) \, dx=x^{8} + x^{7} \cdot \left (16 + 4 e^{2}\right ) + x^{6} \cdot \left (82 + 44 e^{2} + 6 e^{4}\right ) + x^{5} \cdot \left (148 + 124 e^{2} + 4 e^{6} + 38 e^{4}\right ) + x^{4} \cdot \left (177 + 84 e^{2} + 26 e^{4} + e^{8} + 8 e^{6}\right ) + x^{3} \left (- 2 e^{8} - 12 e^{6} + 548 + 34 e^{4} + 336 e^{2}\right ) + x^{2} \cdot \left (580 + 656 e^{2} + 5 e^{8} + 304 e^{4} + 64 e^{6}\right ) + x \left (- 48 e^{6} - 4 e^{8} - 184 e^{4} - 192 e^{2} + 128\right ) + \left (x^{6} + 18 x^{5} + 4 x^{5} e^{2} + 113 x^{4} + 6 x^{4} e^{4} + 52 x^{4} e^{2} + 288 x^{3} + 208 x^{3} e^{2} + 4 x^{3} e^{6} + 50 x^{3} e^{4} + 256 x^{2} + 256 x^{2} e^{2} + x^{2} e^{8} + 96 x^{2} e^{4} + 16 x^{2} e^{6}\right ) \log {\left (x \right )}^{2} + \left (2 x^{7} + 34 x^{6} + 8 x^{6} e^{2} + 194 x^{5} + 12 x^{5} e^{4} + 96 x^{5} e^{2} + 422 x^{4} + 328 x^{4} e^{2} + 8 x^{4} e^{6} + 88 x^{4} e^{4} + 388 x^{3} + 304 x^{3} e^{2} + 2 x^{3} e^{8} + 116 x^{3} e^{4} + 24 x^{3} e^{6} - 16 x^{2} e^{6} - 2 x^{2} e^{8} + 8 x^{2} e^{4} + 640 x^{2} + 320 x^{2} e^{2} + 1024 x + 1024 x e^{2} + 4 x e^{8} + 384 x e^{4} + 64 x e^{6}\right ) \log {\left (x \right )} \] Input:

integrate((2*x*exp(2)**4+(12*x**2+32*x)*exp(2)**3+(24*x**3+150*x**2+192*x) 
*exp(2)**2+(20*x**4+208*x**3+624*x**2+512*x)*exp(2)+6*x**5+90*x**4+452*x** 
3+864*x**2+512*x)*ln(x)**2+((6*x**2-2*x+4)*exp(2)**4+(32*x**3+80*x**2+64)* 
exp(2)**3+(60*x**4+364*x**3+448*x**2+208*x+384)*exp(2)**2+(48*x**5+488*x** 
4+1416*x**3+1328*x**2+1152*x+1024)*exp(2)+14*x**6+206*x**5+1006*x**4+1914* 
x**3+1740*x**2+1792*x+1024)*ln(x)+(4*x**3-4*x**2+8*x)*exp(2)**4+(20*x**4+4 
0*x**3-12*x**2+112*x+16)*exp(2)**3+(36*x**5+202*x**4+192*x**3+218*x**2+616 
*x+200)*exp(2)**2+(28*x**6+272*x**5+716*x**4+664*x**3+1312*x**2+1632*x+832 
)*exp(2)+8*x**7+114*x**6+526*x**5+934*x**4+1130*x**3+2032*x**2+1800*x+1152 
,x)
 

Output:

x**8 + x**7*(16 + 4*exp(2)) + x**6*(82 + 44*exp(2) + 6*exp(4)) + x**5*(148 
 + 124*exp(2) + 4*exp(6) + 38*exp(4)) + x**4*(177 + 84*exp(2) + 26*exp(4) 
+ exp(8) + 8*exp(6)) + x**3*(-2*exp(8) - 12*exp(6) + 548 + 34*exp(4) + 336 
*exp(2)) + x**2*(580 + 656*exp(2) + 5*exp(8) + 304*exp(4) + 64*exp(6)) + x 
*(-48*exp(6) - 4*exp(8) - 184*exp(4) - 192*exp(2) + 128) + (x**6 + 18*x**5 
 + 4*x**5*exp(2) + 113*x**4 + 6*x**4*exp(4) + 52*x**4*exp(2) + 288*x**3 + 
208*x**3*exp(2) + 4*x**3*exp(6) + 50*x**3*exp(4) + 256*x**2 + 256*x**2*exp 
(2) + x**2*exp(8) + 96*x**2*exp(4) + 16*x**2*exp(6))*log(x)**2 + (2*x**7 + 
 34*x**6 + 8*x**6*exp(2) + 194*x**5 + 12*x**5*exp(4) + 96*x**5*exp(2) + 42 
2*x**4 + 328*x**4*exp(2) + 8*x**4*exp(6) + 88*x**4*exp(4) + 388*x**3 + 304 
*x**3*exp(2) + 2*x**3*exp(8) + 116*x**3*exp(4) + 24*x**3*exp(6) - 16*x**2* 
exp(6) - 2*x**2*exp(8) + 8*x**2*exp(4) + 640*x**2 + 320*x**2*exp(2) + 1024 
*x + 1024*x*exp(2) + 4*x*exp(8) + 384*x*exp(4) + 64*x*exp(6))*log(x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 610 vs. \(2 (28) = 56\).

Time = 0.05 (sec) , antiderivative size = 610, normalized size of antiderivative = 21.03 \[ \int \left (1152+1800 x+2032 x^2+1130 x^3+934 x^4+526 x^5+114 x^6+8 x^7+e^8 \left (8 x-4 x^2+4 x^3\right )+e^6 \left (16+112 x-12 x^2+40 x^3+20 x^4\right )+e^4 \left (200+616 x+218 x^2+192 x^3+202 x^4+36 x^5\right )+e^2 \left (832+1632 x+1312 x^2+664 x^3+716 x^4+272 x^5+28 x^6\right )+\left (1024+1792 x+1740 x^2+1914 x^3+1006 x^4+206 x^5+14 x^6+e^8 \left (4-2 x+6 x^2\right )+e^6 \left (64+80 x^2+32 x^3\right )+e^4 \left (384+208 x+448 x^2+364 x^3+60 x^4\right )+e^2 \left (1024+1152 x+1328 x^2+1416 x^3+488 x^4+48 x^5\right )\right ) \log (x)+\left (512 x+2 e^8 x+864 x^2+452 x^3+90 x^4+6 x^5+e^6 \left (32 x+12 x^2\right )+e^4 \left (192 x+150 x^2+24 x^3\right )+e^2 \left (512 x+624 x^2+208 x^3+20 x^4\right )\right ) \log ^2(x)\right ) \, dx =\text {Too large to display} \] Input:

integrate((2*x*exp(2)^4+(12*x^2+32*x)*exp(2)^3+(24*x^3+150*x^2+192*x)*exp( 
2)^2+(20*x^4+208*x^3+624*x^2+512*x)*exp(2)+6*x^5+90*x^4+452*x^3+864*x^2+51 
2*x)*log(x)^2+((6*x^2-2*x+4)*exp(2)^4+(32*x^3+80*x^2+64)*exp(2)^3+(60*x^4+ 
364*x^3+448*x^2+208*x+384)*exp(2)^2+(48*x^5+488*x^4+1416*x^3+1328*x^2+1152 
*x+1024)*exp(2)+14*x^6+206*x^5+1006*x^4+1914*x^3+1740*x^2+1792*x+1024)*log 
(x)+(4*x^3-4*x^2+8*x)*exp(2)^4+(20*x^4+40*x^3-12*x^2+112*x+16)*exp(2)^3+(3 
6*x^5+202*x^4+192*x^3+218*x^2+616*x+200)*exp(2)^2+(28*x^6+272*x^5+716*x^4+ 
664*x^3+1312*x^2+1632*x+832)*exp(2)+8*x^7+114*x^6+526*x^5+934*x^4+1130*x^3 
+2032*x^2+1800*x+1152,x, algorithm="maxima")
 

Output:

x^8 + 1/18*(18*log(x)^2 - 6*log(x) + 1)*x^6 + 16*x^7 - 1/18*x^6*(24*e^2 + 
103) + 2/25*(25*(2*e^2 + 9)*log(x)^2 - 10*(2*e^2 + 9)*log(x) + 4*e^2 + 18) 
*x^5 + 263/3*x^6 - 2/25*x^5*(30*e^4 + 244*e^2 + 503) + 1/8*(8*(6*e^4 + 52* 
e^2 + 113)*log(x)^2 - 4*(6*e^4 + 52*e^2 + 113)*log(x) + 6*e^4 + 52*e^2 + 1 
13)*x^4 + 934/5*x^5 - 1/8*x^4*(16*e^6 + 182*e^4 + 708*e^2 + 957) + 2/9*(9* 
(2*e^6 + 25*e^4 + 104*e^2 + 144)*log(x)^2 - 6*(2*e^6 + 25*e^4 + 104*e^2 + 
144)*log(x) + 4*e^6 + 50*e^4 + 208*e^2 + 288)*x^3 + 565/2*x^4 - 2/9*x^3*(3 
*e^8 + 40*e^6 + 224*e^4 + 664*e^2 + 870) + 1/2*(2*(e^8 + 16*e^6 + 96*e^4 + 
 256*e^2 + 256)*log(x)^2 - 2*(e^8 + 16*e^6 + 96*e^4 + 256*e^2 + 256)*log(x 
) + e^8 + 16*e^6 + 96*e^4 + 256*e^2 + 256)*x^2 + 2032/3*x^3 + 1/2*x^2*(e^8 
 - 104*e^4 - 576*e^2 - 896) + 900*x^2 - 4*x*(e^8 + 16*e^6 + 96*e^4 + 256*e 
^2 + 256) + 1/3*(3*x^4 - 4*x^3 + 12*x^2)*e^8 + 2*(2*x^5 + 5*x^4 - 2*x^3 + 
28*x^2 + 8*x)*e^6 + 2/15*(45*x^6 + 303*x^5 + 360*x^4 + 545*x^3 + 2310*x^2 
+ 1500*x)*e^4 + 2/15*(30*x^7 + 340*x^6 + 1074*x^5 + 1245*x^4 + 3280*x^3 + 
6120*x^2 + 6240*x)*e^2 + 1/30*(60*x^7 + 1030*x^6 + 6036*x^5 + 14355*x^4 + 
17400*x^3 + 26880*x^2 + 30*(2*x^3 - x^2 + 4*x)*e^8 + 80*(3*x^4 + 10*x^3 + 
24*x)*e^6 + 10*(36*x^5 + 273*x^4 + 448*x^3 + 312*x^2 + 1152*x)*e^4 + 4*(60 
*x^6 + 732*x^5 + 2655*x^4 + 3320*x^3 + 4320*x^2 + 7680*x)*e^2 + 30720*x)*l 
og(x) + 1152*x
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 627 vs. \(2 (28) = 56\).

Time = 0.13 (sec) , antiderivative size = 627, normalized size of antiderivative = 21.62 \[ \int \left (1152+1800 x+2032 x^2+1130 x^3+934 x^4+526 x^5+114 x^6+8 x^7+e^8 \left (8 x-4 x^2+4 x^3\right )+e^6 \left (16+112 x-12 x^2+40 x^3+20 x^4\right )+e^4 \left (200+616 x+218 x^2+192 x^3+202 x^4+36 x^5\right )+e^2 \left (832+1632 x+1312 x^2+664 x^3+716 x^4+272 x^5+28 x^6\right )+\left (1024+1792 x+1740 x^2+1914 x^3+1006 x^4+206 x^5+14 x^6+e^8 \left (4-2 x+6 x^2\right )+e^6 \left (64+80 x^2+32 x^3\right )+e^4 \left (384+208 x+448 x^2+364 x^3+60 x^4\right )+e^2 \left (1024+1152 x+1328 x^2+1416 x^3+488 x^4+48 x^5\right )\right ) \log (x)+\left (512 x+2 e^8 x+864 x^2+452 x^3+90 x^4+6 x^5+e^6 \left (32 x+12 x^2\right )+e^4 \left (192 x+150 x^2+24 x^3\right )+e^2 \left (512 x+624 x^2+208 x^3+20 x^4\right )\right ) \log ^2(x)\right ) \, dx =\text {Too large to display} \] Input:

integrate((2*x*exp(2)^4+(12*x^2+32*x)*exp(2)^3+(24*x^3+150*x^2+192*x)*exp( 
2)^2+(20*x^4+208*x^3+624*x^2+512*x)*exp(2)+6*x^5+90*x^4+452*x^3+864*x^2+51 
2*x)*log(x)^2+((6*x^2-2*x+4)*exp(2)^4+(32*x^3+80*x^2+64)*exp(2)^3+(60*x^4+ 
364*x^3+448*x^2+208*x+384)*exp(2)^2+(48*x^5+488*x^4+1416*x^3+1328*x^2+1152 
*x+1024)*exp(2)+14*x^6+206*x^5+1006*x^4+1914*x^3+1740*x^2+1792*x+1024)*log 
(x)+(4*x^3-4*x^2+8*x)*exp(2)^4+(20*x^4+40*x^3-12*x^2+112*x+16)*exp(2)^3+(3 
6*x^5+202*x^4+192*x^3+218*x^2+616*x+200)*exp(2)^2+(28*x^6+272*x^5+716*x^4+ 
664*x^3+1312*x^2+1632*x+832)*exp(2)+8*x^7+114*x^6+526*x^5+934*x^4+1130*x^3 
+2032*x^2+1800*x+1152,x, algorithm="giac")
 

Output:

x^8 + 2*x^7*log(x) + 8*x^6*e^2*log(x) + x^6*log(x)^2 + 4*x^5*e^2*log(x)^2 
+ 16*x^7 - 4/3*x^6*e^2 + 34*x^6*log(x) + 12*x^5*e^4*log(x) + 96*x^5*e^2*lo 
g(x) + 18*x^5*log(x)^2 + 6*x^4*e^4*log(x)^2 + 52*x^4*e^2*log(x)^2 + 82*x^6 
 - 12/5*x^5*e^4 - 96/5*x^5*e^2 + 194*x^5*log(x) + 8*x^4*e^6*log(x) + 88*x^ 
4*e^4*log(x) + 328*x^4*e^2*log(x) + 113*x^4*log(x)^2 + 4*x^3*e^6*log(x)^2 
+ 50*x^3*e^4*log(x)^2 + 208*x^3*e^2*log(x)^2 + 148*x^5 - 2*x^4*e^6 - 22*x^ 
4*e^4 - 82*x^4*e^2 + 422*x^4*log(x) + 2*x^3*e^8*log(x) + 24*x^3*e^6*log(x) 
 + 116*x^3*e^4*log(x) + 304*x^3*e^2*log(x) + 288*x^3*log(x)^2 + x^2*e^8*lo 
g(x)^2 + 16*x^2*e^6*log(x)^2 + 96*x^2*e^4*log(x)^2 + 256*x^2*e^2*log(x)^2 
+ 177*x^4 - 2/3*x^3*e^8 - 8*x^3*e^6 - 116/3*x^3*e^4 - 304/3*x^3*e^2 + 388* 
x^3*log(x) - 2*x^2*e^8*log(x) - 16*x^2*e^6*log(x) + 8*x^2*e^4*log(x) + 320 
*x^2*e^2*log(x) + 256*x^2*log(x)^2 + 548*x^3 + x^2*e^8 + 8*x^2*e^6 - 4*x^2 
*e^4 - 160*x^2*e^2 + 640*x^2*log(x) + 4*x*e^8*log(x) + 64*x*e^6*log(x) + 3 
84*x*e^4*log(x) + 1024*x*e^2*log(x) + 580*x^2 + 1/3*(3*x^4 - 4*x^3 + 12*x^ 
2)*e^8 - 4*x*e^8 + 2*(2*x^5 + 5*x^4 - 2*x^3 + 28*x^2 + 8*x)*e^6 - 64*x*e^6 
 + 2/15*(45*x^6 + 303*x^5 + 360*x^4 + 545*x^3 + 2310*x^2 + 1500*x)*e^4 - 3 
84*x*e^4 + 2/15*(30*x^7 + 340*x^6 + 1074*x^5 + 1245*x^4 + 3280*x^3 + 6120* 
x^2 + 6240*x)*e^2 - 1024*x*e^2 + 1024*x*log(x) + 128*x
 

Mupad [B] (verification not implemented)

Time = 3.04 (sec) , antiderivative size = 309, normalized size of antiderivative = 10.66 \[ \int \left (1152+1800 x+2032 x^2+1130 x^3+934 x^4+526 x^5+114 x^6+8 x^7+e^8 \left (8 x-4 x^2+4 x^3\right )+e^6 \left (16+112 x-12 x^2+40 x^3+20 x^4\right )+e^4 \left (200+616 x+218 x^2+192 x^3+202 x^4+36 x^5\right )+e^2 \left (832+1632 x+1312 x^2+664 x^3+716 x^4+272 x^5+28 x^6\right )+\left (1024+1792 x+1740 x^2+1914 x^3+1006 x^4+206 x^5+14 x^6+e^8 \left (4-2 x+6 x^2\right )+e^6 \left (64+80 x^2+32 x^3\right )+e^4 \left (384+208 x+448 x^2+364 x^3+60 x^4\right )+e^2 \left (1024+1152 x+1328 x^2+1416 x^3+488 x^4+48 x^5\right )\right ) \log (x)+\left (512 x+2 e^8 x+864 x^2+452 x^3+90 x^4+6 x^5+e^6 \left (32 x+12 x^2\right )+e^4 \left (192 x+150 x^2+24 x^3\right )+e^2 \left (512 x+624 x^2+208 x^3+20 x^4\right )\right ) \log ^2(x)\right ) \, dx=x^8+2\,x^7\,\ln \left (x\right )+\left (4\,{\mathrm {e}}^2+16\right )\,x^7+x^6\,{\ln \left (x\right )}^2+\left (8\,{\mathrm {e}}^2+34\right )\,x^6\,\ln \left (x\right )+\left (44\,{\mathrm {e}}^2+6\,{\mathrm {e}}^4+82\right )\,x^6+\left (4\,{\mathrm {e}}^2+18\right )\,x^5\,{\ln \left (x\right )}^2+\left (96\,{\mathrm {e}}^2+12\,{\mathrm {e}}^4+194\right )\,x^5\,\ln \left (x\right )+\left (124\,{\mathrm {e}}^2+38\,{\mathrm {e}}^4+4\,{\mathrm {e}}^6+148\right )\,x^5+\left (52\,{\mathrm {e}}^2+6\,{\mathrm {e}}^4+113\right )\,x^4\,{\ln \left (x\right )}^2+\left (328\,{\mathrm {e}}^2+88\,{\mathrm {e}}^4+8\,{\mathrm {e}}^6+422\right )\,x^4\,\ln \left (x\right )+\left (84\,{\mathrm {e}}^2+26\,{\mathrm {e}}^4+8\,{\mathrm {e}}^6+{\mathrm {e}}^8+177\right )\,x^4+2\,\left (2\,{\mathrm {e}}^2+9\right )\,{\left ({\mathrm {e}}^2+4\right )}^2\,x^3\,{\ln \left (x\right )}^2+\left (304\,{\mathrm {e}}^2+116\,{\mathrm {e}}^4+24\,{\mathrm {e}}^6+2\,{\mathrm {e}}^8+388\right )\,x^3\,\ln \left (x\right )+\left (336\,{\mathrm {e}}^2+34\,{\mathrm {e}}^4-12\,{\mathrm {e}}^6-2\,{\mathrm {e}}^8+548\right )\,x^3+{\left ({\mathrm {e}}^2+4\right )}^4\,x^2\,{\ln \left (x\right )}^2-2\,{\left ({\mathrm {e}}^2+4\right )}^2\,\left ({\mathrm {e}}^4-20\right )\,x^2\,\ln \left (x\right )+\left (656\,{\mathrm {e}}^2+304\,{\mathrm {e}}^4+64\,{\mathrm {e}}^6+5\,{\mathrm {e}}^8+580\right )\,x^2+4\,{\left ({\mathrm {e}}^2+4\right )}^4\,x\,\ln \left (x\right )-4\,{\left ({\mathrm {e}}^2+4\right )}^2\,\left (4\,{\mathrm {e}}^2+{\mathrm {e}}^4-2\right )\,x \] Input:

int(1800*x + exp(2)*(1632*x + 1312*x^2 + 664*x^3 + 716*x^4 + 272*x^5 + 28* 
x^6 + 832) + log(x)^2*(512*x + exp(6)*(32*x + 12*x^2) + 2*x*exp(8) + exp(4 
)*(192*x + 150*x^2 + 24*x^3) + exp(2)*(512*x + 624*x^2 + 208*x^3 + 20*x^4) 
 + 864*x^2 + 452*x^3 + 90*x^4 + 6*x^5) + exp(8)*(8*x - 4*x^2 + 4*x^3) + ex 
p(6)*(112*x - 12*x^2 + 40*x^3 + 20*x^4 + 16) + log(x)*(1792*x + exp(8)*(6* 
x^2 - 2*x + 4) + exp(6)*(80*x^2 + 32*x^3 + 64) + exp(4)*(208*x + 448*x^2 + 
 364*x^3 + 60*x^4 + 384) + exp(2)*(1152*x + 1328*x^2 + 1416*x^3 + 488*x^4 
+ 48*x^5 + 1024) + 1740*x^2 + 1914*x^3 + 1006*x^4 + 206*x^5 + 14*x^6 + 102 
4) + exp(4)*(616*x + 218*x^2 + 192*x^3 + 202*x^4 + 36*x^5 + 200) + 2032*x^ 
2 + 1130*x^3 + 934*x^4 + 526*x^5 + 114*x^6 + 8*x^7 + 1152,x)
 

Output:

2*x^7*log(x) + x^6*(44*exp(2) + 6*exp(4) + 82) + x^4*(84*exp(2) + 26*exp(4 
) + 8*exp(6) + exp(8) + 177) + x^7*(4*exp(2) + 16) + x^6*log(x)^2 + x^3*(3 
36*exp(2) + 34*exp(4) - 12*exp(6) - 2*exp(8) + 548) + x^2*(656*exp(2) + 30 
4*exp(4) + 64*exp(6) + 5*exp(8) + 580) + x^8 + x^5*(124*exp(2) + 38*exp(4) 
 + 4*exp(6) + 148) + x^5*log(x)*(96*exp(2) + 12*exp(4) + 194) + x^6*log(x) 
*(8*exp(2) + 34) + x^4*log(x)^2*(52*exp(2) + 6*exp(4) + 113) - 4*x*(exp(2) 
 + 4)^2*(4*exp(2) + exp(4) - 2) + x^3*log(x)*(304*exp(2) + 116*exp(4) + 24 
*exp(6) + 2*exp(8) + 388) + x^5*log(x)^2*(4*exp(2) + 18) + x^2*log(x)^2*(e 
xp(2) + 4)^4 + x^4*log(x)*(328*exp(2) + 88*exp(4) + 8*exp(6) + 422) + 4*x* 
log(x)*(exp(2) + 4)^4 - 2*x^2*log(x)*(exp(2) + 4)^2*(exp(4) - 20) + 2*x^3* 
log(x)^2*(2*exp(2) + 9)*(exp(2) + 4)^2
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 547, normalized size of antiderivative = 18.86 \[ \int \left (1152+1800 x+2032 x^2+1130 x^3+934 x^4+526 x^5+114 x^6+8 x^7+e^8 \left (8 x-4 x^2+4 x^3\right )+e^6 \left (16+112 x-12 x^2+40 x^3+20 x^4\right )+e^4 \left (200+616 x+218 x^2+192 x^3+202 x^4+36 x^5\right )+e^2 \left (832+1632 x+1312 x^2+664 x^3+716 x^4+272 x^5+28 x^6\right )+\left (1024+1792 x+1740 x^2+1914 x^3+1006 x^4+206 x^5+14 x^6+e^8 \left (4-2 x+6 x^2\right )+e^6 \left (64+80 x^2+32 x^3\right )+e^4 \left (384+208 x+448 x^2+364 x^3+60 x^4\right )+e^2 \left (1024+1152 x+1328 x^2+1416 x^3+488 x^4+48 x^5\right )\right ) \log (x)+\left (512 x+2 e^8 x+864 x^2+452 x^3+90 x^4+6 x^5+e^6 \left (32 x+12 x^2\right )+e^4 \left (192 x+150 x^2+24 x^3\right )+e^2 \left (512 x+624 x^2+208 x^3+20 x^4\right )\right ) \log ^2(x)\right ) \, dx =\text {Too large to display} \] Input:

int((2*x*exp(2)^4+(12*x^2+32*x)*exp(2)^3+(24*x^3+150*x^2+192*x)*exp(2)^2+( 
20*x^4+208*x^3+624*x^2+512*x)*exp(2)+6*x^5+90*x^4+452*x^3+864*x^2+512*x)*l 
og(x)^2+((6*x^2-2*x+4)*exp(2)^4+(32*x^3+80*x^2+64)*exp(2)^3+(60*x^4+364*x^ 
3+448*x^2+208*x+384)*exp(2)^2+(48*x^5+488*x^4+1416*x^3+1328*x^2+1152*x+102 
4)*exp(2)+14*x^6+206*x^5+1006*x^4+1914*x^3+1740*x^2+1792*x+1024)*log(x)+(4 
*x^3-4*x^2+8*x)*exp(2)^4+(20*x^4+40*x^3-12*x^2+112*x+16)*exp(2)^3+(36*x^5+ 
202*x^4+192*x^3+218*x^2+616*x+200)*exp(2)^2+(28*x^6+272*x^5+716*x^4+664*x^ 
3+1312*x^2+1632*x+832)*exp(2)+8*x^7+114*x^6+526*x^5+934*x^4+1130*x^3+2032* 
x^2+1800*x+1152,x)
 

Output:

x*(log(x)**2*e**8*x + 4*log(x)**2*e**6*x**2 + 16*log(x)**2*e**6*x + 6*log( 
x)**2*e**4*x**3 + 50*log(x)**2*e**4*x**2 + 96*log(x)**2*e**4*x + 4*log(x)* 
*2*e**2*x**4 + 52*log(x)**2*e**2*x**3 + 208*log(x)**2*e**2*x**2 + 256*log( 
x)**2*e**2*x + log(x)**2*x**5 + 18*log(x)**2*x**4 + 113*log(x)**2*x**3 + 2 
88*log(x)**2*x**2 + 256*log(x)**2*x + 2*log(x)*e**8*x**2 - 2*log(x)*e**8*x 
 + 4*log(x)*e**8 + 8*log(x)*e**6*x**3 + 24*log(x)*e**6*x**2 - 16*log(x)*e* 
*6*x + 64*log(x)*e**6 + 12*log(x)*e**4*x**4 + 88*log(x)*e**4*x**3 + 116*lo 
g(x)*e**4*x**2 + 8*log(x)*e**4*x + 384*log(x)*e**4 + 8*log(x)*e**2*x**5 + 
96*log(x)*e**2*x**4 + 328*log(x)*e**2*x**3 + 304*log(x)*e**2*x**2 + 320*lo 
g(x)*e**2*x + 1024*log(x)*e**2 + 2*log(x)*x**6 + 34*log(x)*x**5 + 194*log( 
x)*x**4 + 422*log(x)*x**3 + 388*log(x)*x**2 + 640*log(x)*x + 1024*log(x) + 
 e**8*x**3 - 2*e**8*x**2 + 5*e**8*x - 4*e**8 + 4*e**6*x**4 + 8*e**6*x**3 - 
 12*e**6*x**2 + 64*e**6*x - 48*e**6 + 6*e**4*x**5 + 38*e**4*x**4 + 26*e**4 
*x**3 + 34*e**4*x**2 + 304*e**4*x - 184*e**4 + 4*e**2*x**6 + 44*e**2*x**5 
+ 124*e**2*x**4 + 84*e**2*x**3 + 336*e**2*x**2 + 656*e**2*x - 192*e**2 + x 
**7 + 16*x**6 + 82*x**5 + 148*x**4 + 177*x**3 + 548*x**2 + 580*x + 128)