Integrand size = 180, antiderivative size = 31 \[ \int \frac {1}{36} \left (324 x^3+225 x^4+360 x^6+240 x^7+84 x^9+55 x^{10}+e^8 \left (52 x^5+35 x^6\right )+e^6 \left (-240 x^6-160 x^7\right )+e^4 \left (264 x^4+180 x^5+408 x^7+270 x^8\right )+e^2 \left (-624 x^5-420 x^6-304 x^8-200 x^9\right )+\left (144 x^3+24 e^8 x^5+168 x^6-112 e^6 x^6+40 x^9+e^4 \left (120 x^4+192 x^7\right )+e^2 \left (-288 x^5-144 x^8\right )\right ) \log (x)\right ) \, dx=\frac {1}{9} x^4 \left (3+\left (e^2-x\right )^2 x\right )^2 \left (2+\frac {5 x}{4}+\log (x)\right ) \] Output:
1/9*(ln(x)+5/4*x+2)*x^4*((exp(2)-x)^2*x+3)^2
Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {1}{36} \left (324 x^3+225 x^4+360 x^6+240 x^7+84 x^9+55 x^{10}+e^8 \left (52 x^5+35 x^6\right )+e^6 \left (-240 x^6-160 x^7\right )+e^4 \left (264 x^4+180 x^5+408 x^7+270 x^8\right )+e^2 \left (-624 x^5-420 x^6-304 x^8-200 x^9\right )+\left (144 x^3+24 e^8 x^5+168 x^6-112 e^6 x^6+40 x^9+e^4 \left (120 x^4+192 x^7\right )+e^2 \left (-288 x^5-144 x^8\right )\right ) \log (x)\right ) \, dx=\frac {1}{36} x^4 \left (3+e^4 x-2 e^2 x^2+x^3\right )^2 (8+5 x+4 \log (x)) \] Input:
Integrate[(324*x^3 + 225*x^4 + 360*x^6 + 240*x^7 + 84*x^9 + 55*x^10 + E^8* (52*x^5 + 35*x^6) + E^6*(-240*x^6 - 160*x^7) + E^4*(264*x^4 + 180*x^5 + 40 8*x^7 + 270*x^8) + E^2*(-624*x^5 - 420*x^6 - 304*x^8 - 200*x^9) + (144*x^3 + 24*E^8*x^5 + 168*x^6 - 112*E^6*x^6 + 40*x^9 + E^4*(120*x^4 + 192*x^7) + E^2*(-288*x^5 - 144*x^8))*Log[x])/36,x]
Output:
(x^4*(3 + E^4*x - 2*E^2*x^2 + x^3)^2*(8 + 5*x + 4*Log[x]))/36
Leaf count is larger than twice the leaf count of optimal. \(243\) vs. \(2(31)=62\).
Time = 0.75 (sec) , antiderivative size = 243, normalized size of antiderivative = 7.84, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.011, Rules used = {27, 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.
| \(\displaystyle \int \frac {1}{36} \left (55 x^{10}+84 x^9+240 x^7+360 x^6+225 x^4+324 x^3+e^6 \left (-160 x^7-240 x^6\right )+e^8 \left (35 x^6+52 x^5\right )+e^2 \left (-200 x^9-304 x^8-420 x^6-624 x^5\right )+e^4 \left (270 x^8+408 x^7+180 x^5+264 x^4\right )+\left (40 x^9-112 e^6 x^6+168 x^6+24 e^8 x^5+144 x^3+e^2 \left (-144 x^8-288 x^5\right )+e^4 \left (192 x^7+120 x^4\right )\right ) \log (x)\right ) \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{36} \int \left (55 x^{10}+84 x^9+240 x^7+360 x^6+225 x^4+324 x^3+e^8 \left (35 x^6+52 x^5\right )-80 e^6 \left (2 x^7+3 x^6\right )+6 e^4 \left (45 x^8+68 x^7+30 x^5+44 x^4\right )-4 e^2 \left (50 x^9+76 x^8+105 x^6+156 x^5\right )+8 \left (5 x^9-14 e^6 x^6+21 x^6+3 e^8 x^5+18 x^3+3 e^4 \left (8 x^7+5 x^4\right )-18 e^2 \left (x^8+2 x^5\right )\right ) \log (x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{36} \left (5 x^{11}-20 e^2 x^{10}+8 x^{10}+4 x^{10} \log (x)+30 e^4 x^9-32 e^2 x^9-16 e^2 x^9 \log (x)-20 e^6 x^8+48 e^4 x^8+30 x^8+24 e^4 x^8 \log (x)-\frac {8}{7} \left (3-2 e^6\right ) x^7+5 e^8 x^7-\frac {240 e^6 x^7}{7}-60 e^2 x^7+\frac {360 x^7}{7}+8 \left (3-2 e^6\right ) x^7 \log (x)+\frac {2}{3} e^2 \left (12-e^6\right ) x^6+\frac {26 e^8 x^6}{3}+30 e^4 x^6-104 e^2 x^6-4 e^2 \left (12-e^6\right ) x^6 \log (x)+48 e^4 x^5+45 x^5+24 e^4 x^5 \log (x)+72 x^4+36 x^4 \log (x)\right )\) |
Input:
Int[(324*x^3 + 225*x^4 + 360*x^6 + 240*x^7 + 84*x^9 + 55*x^10 + E^8*(52*x^ 5 + 35*x^6) + E^6*(-240*x^6 - 160*x^7) + E^4*(264*x^4 + 180*x^5 + 408*x^7 + 270*x^8) + E^2*(-624*x^5 - 420*x^6 - 304*x^8 - 200*x^9) + (144*x^3 + 24* E^8*x^5 + 168*x^6 - 112*E^6*x^6 + 40*x^9 + E^4*(120*x^4 + 192*x^7) + E^2*( -288*x^5 - 144*x^8))*Log[x])/36,x]
Output:
(72*x^4 + 45*x^5 + 48*E^4*x^5 - 104*E^2*x^6 + 30*E^4*x^6 + (26*E^8*x^6)/3 + (2*E^2*(12 - E^6)*x^6)/3 + (360*x^7)/7 - 60*E^2*x^7 - (240*E^6*x^7)/7 + 5*E^8*x^7 - (8*(3 - 2*E^6)*x^7)/7 + 30*x^8 + 48*E^4*x^8 - 20*E^6*x^8 - 32* E^2*x^9 + 30*E^4*x^9 + 8*x^10 - 20*E^2*x^10 + 5*x^11 + 36*x^4*Log[x] + 24* E^4*x^5*Log[x] - 4*E^2*(12 - E^6)*x^6*Log[x] + 8*(3 - 2*E^6)*x^7*Log[x] + 24*E^4*x^8*Log[x] - 16*E^2*x^9*Log[x] + 4*x^10*Log[x])/36
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(177\) vs. \(2(26)=52\).
Time = 240.44 (sec) , antiderivative size = 178, normalized size of antiderivative = 5.74
| method | result | size |
| risch | \(\frac {\left (4 \,{\mathrm e}^{8} x^{6}-16 \,{\mathrm e}^{6} x^{7}+24 \,{\mathrm e}^{4} x^{8}-16 \,{\mathrm e}^{2} x^{9}+4 x^{10}+24 x^{5} {\mathrm e}^{4}-48 x^{6} {\mathrm e}^{2}+24 x^{7}+36 x^{4}\right ) \ln \left (x \right )}{36}+\frac {2 x^{10}}{9}-\frac {8 \,{\mathrm e}^{2} x^{9}}{9}+\frac {4 \,{\mathrm e}^{4} x^{8}}{3}-\frac {8 \,{\mathrm e}^{6} x^{7}}{9}+\frac {4 x^{7}}{3}+\frac {2 \,{\mathrm e}^{8} x^{6}}{9}-\frac {8 x^{6} {\mathrm e}^{2}}{3}+\frac {4 x^{5} {\mathrm e}^{4}}{3}+2 x^{4}+\frac {5 \,{\mathrm e}^{8} x^{7}}{36}-\frac {5 \,{\mathrm e}^{6} x^{8}}{9}+\frac {5 \,{\mathrm e}^{4} x^{9}}{6}+\frac {5 \,{\mathrm e}^{4} x^{6}}{6}-\frac {5 \,{\mathrm e}^{2} x^{10}}{9}-\frac {5 \,{\mathrm e}^{2} x^{7}}{3}+\frac {5 x^{11}}{36}+\frac {5 x^{8}}{6}+\frac {5 x^{5}}{4}\) | \(178\) |
| norman | \(x^{4} \ln \left (x \right )+\left (\frac {2}{9}-\frac {5 \,{\mathrm e}^{2}}{9}\right ) x^{10}+\left (\frac {5}{4}+\frac {4 \,{\mathrm e}^{4}}{3}\right ) x^{5}+\left (-\frac {8 \,{\mathrm e}^{2}}{9}+\frac {5 \,{\mathrm e}^{4}}{6}\right ) x^{9}+\left (\frac {5}{6}+\frac {4 \,{\mathrm e}^{4}}{3}-\frac {5 \,{\mathrm e}^{6}}{9}\right ) x^{8}+\left (\frac {2 \,{\mathrm e}^{8}}{9}-\frac {8 \,{\mathrm e}^{2}}{3}+\frac {5 \,{\mathrm e}^{4}}{6}\right ) x^{6}+\left (\frac {4}{3}-\frac {8 \,{\mathrm e}^{6}}{9}+\frac {5 \,{\mathrm e}^{8}}{36}-\frac {5 \,{\mathrm e}^{2}}{3}\right ) x^{7}+\left (-\frac {4 \,{\mathrm e}^{6}}{9}+\frac {2}{3}\right ) x^{7} \ln \left (x \right )+\left (\frac {{\mathrm e}^{8}}{9}-\frac {4 \,{\mathrm e}^{2}}{3}\right ) x^{6} \ln \left (x \right )+2 x^{4}+\frac {5 x^{11}}{36}+\frac {x^{10} \ln \left (x \right )}{9}-\frac {4 \,{\mathrm e}^{2} x^{9} \ln \left (x \right )}{9}+\frac {2 \,{\mathrm e}^{4} \ln \left (x \right ) x^{5}}{3}+\frac {2 \,{\mathrm e}^{4} \ln \left (x \right ) x^{8}}{3}\) | \(185\) |
| parallelrisch | \(\frac {2 \,{\mathrm e}^{4} \ln \left (x \right ) x^{8}}{3}-\frac {4 \,{\mathrm e}^{6} \ln \left (x \right ) x^{7}}{9}+\frac {2 \,{\mathrm e}^{4} \ln \left (x \right ) x^{5}}{3}+x^{4} \ln \left (x \right )-\frac {8 x^{6} {\mathrm e}^{2}}{3}+\frac {4 x^{5} {\mathrm e}^{4}}{3}+\frac {x^{10} \ln \left (x \right )}{9}+\frac {2 x^{7} \ln \left (x \right )}{3}-\frac {8 \,{\mathrm e}^{2} x^{9}}{9}+\frac {5 x^{11}}{36}+\frac {5 x^{8}}{6}+\frac {4 x^{7}}{3}+\frac {2 x^{10}}{9}+\frac {5 x^{5}}{4}+2 x^{4}-\frac {8 \,{\mathrm e}^{6} x^{7}}{9}-\frac {5 \,{\mathrm e}^{2} x^{10}}{9}+\frac {5 \,{\mathrm e}^{4} x^{9}}{6}+\frac {2 \,{\mathrm e}^{8} x^{6}}{9}-\frac {4 \,{\mathrm e}^{2} x^{6} \ln \left (x \right )}{3}-\frac {4 \,{\mathrm e}^{2} x^{9} \ln \left (x \right )}{9}+\frac {5 \,{\mathrm e}^{8} x^{7}}{36}-\frac {5 \,{\mathrm e}^{6} x^{8}}{9}+\frac {4 \,{\mathrm e}^{4} x^{8}}{3}+\frac {5 \,{\mathrm e}^{4} x^{6}}{6}+\frac {\ln \left (x \right ) {\mathrm e}^{8} x^{6}}{9}-\frac {5 \,{\mathrm e}^{2} x^{7}}{3}\) | \(214\) |
| default | \(\frac {2 \,{\mathrm e}^{4} \ln \left (x \right ) x^{8}}{3}-\frac {4 \,{\mathrm e}^{6} \ln \left (x \right ) x^{7}}{9}+\frac {2 \,{\mathrm e}^{4} \ln \left (x \right ) x^{5}}{3}+x^{4} \ln \left (x \right )+\frac {2 x^{6} {\mathrm e}^{2}}{9}-\frac {2 x^{5} {\mathrm e}^{4}}{15}+\frac {x^{10} \ln \left (x \right )}{9}+\frac {2 x^{7} \ln \left (x \right )}{3}+\frac {4 \,{\mathrm e}^{2} x^{9}}{81}+\frac {5 x^{11}}{36}+\frac {5 x^{8}}{6}+\frac {4 x^{7}}{3}+\frac {2 x^{10}}{9}+\frac {5 x^{5}}{4}+2 x^{4}+\frac {4 \,{\mathrm e}^{6} x^{7}}{63}+\frac {{\mathrm e}^{2} \left (-5 x^{10}-\frac {76}{9} x^{9}-15 x^{7}-26 x^{6}\right )}{9}-\frac {{\mathrm e}^{8} x^{6}}{54}-\frac {4 \,{\mathrm e}^{2} x^{6} \ln \left (x \right )}{3}-\frac {4 \,{\mathrm e}^{2} x^{9} \ln \left (x \right )}{9}+\frac {{\mathrm e}^{4} \left (5 x^{9}+\frac {17}{2} x^{8}+5 x^{6}+\frac {44}{5} x^{5}\right )}{6}+\frac {20 \,{\mathrm e}^{6} \left (-\frac {1}{4} x^{8}-\frac {3}{7} x^{7}\right )}{9}+\frac {{\mathrm e}^{8} \left (5 x^{7}+\frac {26}{3} x^{6}\right )}{36}-\frac {{\mathrm e}^{4} x^{8}}{12}+\frac {\ln \left (x \right ) {\mathrm e}^{8} x^{6}}{9}\) | \(250\) |
| parts | \(\frac {2 \,{\mathrm e}^{8} \left (\frac {x^{6} \ln \left (x \right )}{6}-\frac {x^{6}}{36}\right )}{3}-\frac {28 \,{\mathrm e}^{6} \left (\frac {x^{7} \ln \left (x \right )}{7}-\frac {x^{7}}{49}\right )}{9}+\frac {16 \,{\mathrm e}^{4} \left (\frac {x^{8} \ln \left (x \right )}{8}-\frac {x^{8}}{64}\right )}{3}+\frac {10 \,{\mathrm e}^{4} \left (\frac {x^{5} \ln \left (x \right )}{5}-\frac {x^{5}}{25}\right )}{3}+x^{4} \ln \left (x \right )-\frac {26 x^{6} {\mathrm e}^{2}}{9}+\frac {22 x^{5} {\mathrm e}^{4}}{15}+\frac {x^{10} \ln \left (x \right )}{9}+\frac {2 x^{7} \ln \left (x \right )}{3}-\frac {76 \,{\mathrm e}^{2} x^{9}}{81}+\frac {5 x^{11}}{36}+\frac {5 x^{8}}{6}+\frac {4 x^{7}}{3}+\frac {2 x^{10}}{9}+\frac {5 x^{5}}{4}+2 x^{4}-4 \,{\mathrm e}^{2} \left (\frac {x^{9} \ln \left (x \right )}{9}-\frac {x^{9}}{81}\right )-8 \,{\mathrm e}^{2} \left (\frac {x^{6} \ln \left (x \right )}{6}-\frac {x^{6}}{36}\right )-\frac {20 \,{\mathrm e}^{6} x^{7}}{21}-\frac {5 \,{\mathrm e}^{2} x^{10}}{9}+\frac {5 \,{\mathrm e}^{4} x^{9}}{6}+\frac {13 \,{\mathrm e}^{8} x^{6}}{54}+\frac {5 \,{\mathrm e}^{8} x^{7}}{36}-\frac {5 \,{\mathrm e}^{6} x^{8}}{9}+\frac {17 \,{\mathrm e}^{4} x^{8}}{12}+\frac {5 \,{\mathrm e}^{4} x^{6}}{6}-\frac {5 \,{\mathrm e}^{2} x^{7}}{3}\) | \(262\) |
Input:
int(1/36*(24*x^5*exp(2)^4-112*x^6*exp(2)^3+(192*x^7+120*x^4)*exp(2)^2+(-14 4*x^8-288*x^5)*exp(2)+40*x^9+168*x^6+144*x^3)*ln(x)+1/36*(35*x^6+52*x^5)*e xp(2)^4+1/36*(-160*x^7-240*x^6)*exp(2)^3+1/36*(270*x^8+408*x^7+180*x^5+264 *x^4)*exp(2)^2+1/36*(-200*x^9-304*x^8-420*x^6-624*x^5)*exp(2)+55/36*x^10+7 /3*x^9+20/3*x^7+10*x^6+25/4*x^4+9*x^3,x,method=_RETURNVERBOSE)
Output:
1/36*(4*exp(8)*x^6-16*exp(6)*x^7+24*exp(4)*x^8-16*exp(2)*x^9+4*x^10+24*x^5 *exp(4)-48*x^6*exp(2)+24*x^7+36*x^4)*ln(x)+2/9*x^10-8/9*exp(2)*x^9+4/3*exp (4)*x^8-8/9*exp(6)*x^7+4/3*x^7+2/9*exp(8)*x^6-8/3*x^6*exp(2)+4/3*x^5*exp(4 )+2*x^4+5/36*exp(8)*x^7-5/9*exp(6)*x^8+5/6*exp(4)*x^9+5/6*exp(4)*x^6-5/9*e xp(2)*x^10-5/3*exp(2)*x^7+5/36*x^11+5/6*x^8+5/4*x^5
Leaf count of result is larger than twice the leaf count of optimal. 166 vs. \(2 (28) = 56\).
Time = 0.09 (sec) , antiderivative size = 166, normalized size of antiderivative = 5.35 \[ \int \frac {1}{36} \left (324 x^3+225 x^4+360 x^6+240 x^7+84 x^9+55 x^{10}+e^8 \left (52 x^5+35 x^6\right )+e^6 \left (-240 x^6-160 x^7\right )+e^4 \left (264 x^4+180 x^5+408 x^7+270 x^8\right )+e^2 \left (-624 x^5-420 x^6-304 x^8-200 x^9\right )+\left (144 x^3+24 e^8 x^5+168 x^6-112 e^6 x^6+40 x^9+e^4 \left (120 x^4+192 x^7\right )+e^2 \left (-288 x^5-144 x^8\right )\right ) \log (x)\right ) \, dx=\frac {5}{36} \, x^{11} + \frac {2}{9} \, x^{10} + \frac {5}{6} \, x^{8} + \frac {4}{3} \, x^{7} + \frac {5}{4} \, x^{5} + 2 \, x^{4} + \frac {1}{36} \, {\left (5 \, x^{7} + 8 \, x^{6}\right )} e^{8} - \frac {1}{9} \, {\left (5 \, x^{8} + 8 \, x^{7}\right )} e^{6} + \frac {1}{6} \, {\left (5 \, x^{9} + 8 \, x^{8} + 5 \, x^{6} + 8 \, x^{5}\right )} e^{4} - \frac {1}{9} \, {\left (5 \, x^{10} + 8 \, x^{9} + 15 \, x^{7} + 24 \, x^{6}\right )} e^{2} + \frac {1}{9} \, {\left (x^{10} - 4 \, x^{7} e^{6} + 6 \, x^{7} + x^{6} e^{8} + 9 \, x^{4} + 6 \, {\left (x^{8} + x^{5}\right )} e^{4} - 4 \, {\left (x^{9} + 3 \, x^{6}\right )} e^{2}\right )} \log \left (x\right ) \] Input:
integrate(1/36*(24*x^5*exp(2)^4-112*x^6*exp(2)^3+(192*x^7+120*x^4)*exp(2)^ 2+(-144*x^8-288*x^5)*exp(2)+40*x^9+168*x^6+144*x^3)*log(x)+1/36*(35*x^6+52 *x^5)*exp(2)^4+1/36*(-160*x^7-240*x^6)*exp(2)^3+1/36*(270*x^8+408*x^7+180* x^5+264*x^4)*exp(2)^2+1/36*(-200*x^9-304*x^8-420*x^6-624*x^5)*exp(2)+55/36 *x^10+7/3*x^9+20/3*x^7+10*x^6+25/4*x^4+9*x^3,x, algorithm="fricas")
Output:
5/36*x^11 + 2/9*x^10 + 5/6*x^8 + 4/3*x^7 + 5/4*x^5 + 2*x^4 + 1/36*(5*x^7 + 8*x^6)*e^8 - 1/9*(5*x^8 + 8*x^7)*e^6 + 1/6*(5*x^9 + 8*x^8 + 5*x^6 + 8*x^5 )*e^4 - 1/9*(5*x^10 + 8*x^9 + 15*x^7 + 24*x^6)*e^2 + 1/9*(x^10 - 4*x^7*e^6 + 6*x^7 + x^6*e^8 + 9*x^4 + 6*(x^8 + x^5)*e^4 - 4*(x^9 + 3*x^6)*e^2)*log( x)
Leaf count of result is larger than twice the leaf count of optimal. 204 vs. \(2 (27) = 54\).
Time = 0.16 (sec) , antiderivative size = 204, normalized size of antiderivative = 6.58 \[ \int \frac {1}{36} \left (324 x^3+225 x^4+360 x^6+240 x^7+84 x^9+55 x^{10}+e^8 \left (52 x^5+35 x^6\right )+e^6 \left (-240 x^6-160 x^7\right )+e^4 \left (264 x^4+180 x^5+408 x^7+270 x^8\right )+e^2 \left (-624 x^5-420 x^6-304 x^8-200 x^9\right )+\left (144 x^3+24 e^8 x^5+168 x^6-112 e^6 x^6+40 x^9+e^4 \left (120 x^4+192 x^7\right )+e^2 \left (-288 x^5-144 x^8\right )\right ) \log (x)\right ) \, dx=\frac {5 x^{11}}{36} + x^{10} \cdot \left (\frac {2}{9} - \frac {5 e^{2}}{9}\right ) + x^{9} \left (- \frac {8 e^{2}}{9} + \frac {5 e^{4}}{6}\right ) + x^{8} \left (- \frac {5 e^{6}}{9} + \frac {5}{6} + \frac {4 e^{4}}{3}\right ) + x^{7} \left (- \frac {8 e^{6}}{9} - \frac {5 e^{2}}{3} + \frac {4}{3} + \frac {5 e^{8}}{36}\right ) + x^{6} \left (- \frac {8 e^{2}}{3} + \frac {5 e^{4}}{6} + \frac {2 e^{8}}{9}\right ) + x^{5} \cdot \left (\frac {5}{4} + \frac {4 e^{4}}{3}\right ) + 2 x^{4} + \left (\frac {x^{10}}{9} - \frac {4 x^{9} e^{2}}{9} + \frac {2 x^{8} e^{4}}{3} - \frac {4 x^{7} e^{6}}{9} + \frac {2 x^{7}}{3} - \frac {4 x^{6} e^{2}}{3} + \frac {x^{6} e^{8}}{9} + \frac {2 x^{5} e^{4}}{3} + x^{4}\right ) \log {\left (x \right )} \] Input:
integrate(1/36*(24*x**5*exp(2)**4-112*x**6*exp(2)**3+(192*x**7+120*x**4)*e xp(2)**2+(-144*x**8-288*x**5)*exp(2)+40*x**9+168*x**6+144*x**3)*ln(x)+1/36 *(35*x**6+52*x**5)*exp(2)**4+1/36*(-160*x**7-240*x**6)*exp(2)**3+1/36*(270 *x**8+408*x**7+180*x**5+264*x**4)*exp(2)**2+1/36*(-200*x**9-304*x**8-420*x **6-624*x**5)*exp(2)+55/36*x**10+7/3*x**9+20/3*x**7+10*x**6+25/4*x**4+9*x* *3,x)
Output:
5*x**11/36 + x**10*(2/9 - 5*exp(2)/9) + x**9*(-8*exp(2)/9 + 5*exp(4)/6) + x**8*(-5*exp(6)/9 + 5/6 + 4*exp(4)/3) + x**7*(-8*exp(6)/9 - 5*exp(2)/3 + 4 /3 + 5*exp(8)/36) + x**6*(-8*exp(2)/3 + 5*exp(4)/6 + 2*exp(8)/9) + x**5*(5 /4 + 4*exp(4)/3) + 2*x**4 + (x**10/9 - 4*x**9*exp(2)/9 + 2*x**8*exp(4)/3 - 4*x**7*exp(6)/9 + 2*x**7/3 - 4*x**6*exp(2)/3 + x**6*exp(8)/9 + 2*x**5*exp (4)/3 + x**4)*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 210 vs. \(2 (28) = 56\).
Time = 0.03 (sec) , antiderivative size = 210, normalized size of antiderivative = 6.77 \[ \int \frac {1}{36} \left (324 x^3+225 x^4+360 x^6+240 x^7+84 x^9+55 x^{10}+e^8 \left (52 x^5+35 x^6\right )+e^6 \left (-240 x^6-160 x^7\right )+e^4 \left (264 x^4+180 x^5+408 x^7+270 x^8\right )+e^2 \left (-624 x^5-420 x^6-304 x^8-200 x^9\right )+\left (144 x^3+24 e^8 x^5+168 x^6-112 e^6 x^6+40 x^9+e^4 \left (120 x^4+192 x^7\right )+e^2 \left (-288 x^5-144 x^8\right )\right ) \log (x)\right ) \, dx=\frac {5}{36} \, x^{11} + \frac {2}{9} \, x^{10} + \frac {4}{81} \, x^{9} e^{2} - \frac {1}{12} \, x^{8} e^{4} + \frac {5}{6} \, x^{8} + \frac {2}{63} \, x^{7} {\left (2 \, e^{6} - 3\right )} + \frac {10}{7} \, x^{7} - \frac {1}{54} \, x^{6} {\left (e^{8} - 12 \, e^{2}\right )} - \frac {2}{15} \, x^{5} e^{4} + \frac {5}{4} \, x^{5} + 2 \, x^{4} + \frac {1}{108} \, {\left (15 \, x^{7} + 26 \, x^{6}\right )} e^{8} - \frac {5}{63} \, {\left (7 \, x^{8} + 12 \, x^{7}\right )} e^{6} + \frac {1}{60} \, {\left (50 \, x^{9} + 85 \, x^{8} + 50 \, x^{6} + 88 \, x^{5}\right )} e^{4} - \frac {1}{81} \, {\left (45 \, x^{10} + 76 \, x^{9} + 135 \, x^{7} + 234 \, x^{6}\right )} e^{2} + \frac {1}{9} \, {\left (x^{10} - 4 \, x^{7} e^{6} + 6 \, x^{7} + x^{6} e^{8} + 9 \, x^{4} + 6 \, {\left (x^{8} + x^{5}\right )} e^{4} - 4 \, {\left (x^{9} + 3 \, x^{6}\right )} e^{2}\right )} \log \left (x\right ) \] Input:
integrate(1/36*(24*x^5*exp(2)^4-112*x^6*exp(2)^3+(192*x^7+120*x^4)*exp(2)^ 2+(-144*x^8-288*x^5)*exp(2)+40*x^9+168*x^6+144*x^3)*log(x)+1/36*(35*x^6+52 *x^5)*exp(2)^4+1/36*(-160*x^7-240*x^6)*exp(2)^3+1/36*(270*x^8+408*x^7+180* x^5+264*x^4)*exp(2)^2+1/36*(-200*x^9-304*x^8-420*x^6-624*x^5)*exp(2)+55/36 *x^10+7/3*x^9+20/3*x^7+10*x^6+25/4*x^4+9*x^3,x, algorithm="maxima")
Output:
5/36*x^11 + 2/9*x^10 + 4/81*x^9*e^2 - 1/12*x^8*e^4 + 5/6*x^8 + 2/63*x^7*(2 *e^6 - 3) + 10/7*x^7 - 1/54*x^6*(e^8 - 12*e^2) - 2/15*x^5*e^4 + 5/4*x^5 + 2*x^4 + 1/108*(15*x^7 + 26*x^6)*e^8 - 5/63*(7*x^8 + 12*x^7)*e^6 + 1/60*(50 *x^9 + 85*x^8 + 50*x^6 + 88*x^5)*e^4 - 1/81*(45*x^10 + 76*x^9 + 135*x^7 + 234*x^6)*e^2 + 1/9*(x^10 - 4*x^7*e^6 + 6*x^7 + x^6*e^8 + 9*x^4 + 6*(x^8 + x^5)*e^4 - 4*(x^9 + 3*x^6)*e^2)*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (28) = 56\).
Time = 0.12 (sec) , antiderivative size = 227, normalized size of antiderivative = 7.32 \[ \int \frac {1}{36} \left (324 x^3+225 x^4+360 x^6+240 x^7+84 x^9+55 x^{10}+e^8 \left (52 x^5+35 x^6\right )+e^6 \left (-240 x^6-160 x^7\right )+e^4 \left (264 x^4+180 x^5+408 x^7+270 x^8\right )+e^2 \left (-624 x^5-420 x^6-304 x^8-200 x^9\right )+\left (144 x^3+24 e^8 x^5+168 x^6-112 e^6 x^6+40 x^9+e^4 \left (120 x^4+192 x^7\right )+e^2 \left (-288 x^5-144 x^8\right )\right ) \log (x)\right ) \, dx=\frac {5}{36} \, x^{11} + \frac {1}{9} \, x^{10} \log \left (x\right ) - \frac {4}{9} \, x^{9} e^{2} \log \left (x\right ) + \frac {2}{9} \, x^{10} + \frac {4}{81} \, x^{9} e^{2} + \frac {2}{3} \, x^{8} e^{4} \log \left (x\right ) - \frac {1}{12} \, x^{8} e^{4} - \frac {4}{9} \, x^{7} e^{6} \log \left (x\right ) + \frac {5}{6} \, x^{8} + \frac {4}{63} \, x^{7} e^{6} + \frac {2}{3} \, x^{7} \log \left (x\right ) + \frac {1}{9} \, x^{6} e^{8} \log \left (x\right ) - \frac {4}{3} \, x^{6} e^{2} \log \left (x\right ) + \frac {4}{3} \, x^{7} - \frac {1}{54} \, x^{6} e^{8} + \frac {2}{9} \, x^{6} e^{2} + \frac {2}{3} \, x^{5} e^{4} \log \left (x\right ) - \frac {2}{15} \, x^{5} e^{4} + \frac {5}{4} \, x^{5} + x^{4} \log \left (x\right ) + 2 \, x^{4} + \frac {1}{108} \, {\left (15 \, x^{7} + 26 \, x^{6}\right )} e^{8} - \frac {5}{63} \, {\left (7 \, x^{8} + 12 \, x^{7}\right )} e^{6} + \frac {1}{60} \, {\left (50 \, x^{9} + 85 \, x^{8} + 50 \, x^{6} + 88 \, x^{5}\right )} e^{4} - \frac {1}{81} \, {\left (45 \, x^{10} + 76 \, x^{9} + 135 \, x^{7} + 234 \, x^{6}\right )} e^{2} \] Input:
integrate(1/36*(24*x^5*exp(2)^4-112*x^6*exp(2)^3+(192*x^7+120*x^4)*exp(2)^ 2+(-144*x^8-288*x^5)*exp(2)+40*x^9+168*x^6+144*x^3)*log(x)+1/36*(35*x^6+52 *x^5)*exp(2)^4+1/36*(-160*x^7-240*x^6)*exp(2)^3+1/36*(270*x^8+408*x^7+180* x^5+264*x^4)*exp(2)^2+1/36*(-200*x^9-304*x^8-420*x^6-624*x^5)*exp(2)+55/36 *x^10+7/3*x^9+20/3*x^7+10*x^6+25/4*x^4+9*x^3,x, algorithm="giac")
Output:
5/36*x^11 + 1/9*x^10*log(x) - 4/9*x^9*e^2*log(x) + 2/9*x^10 + 4/81*x^9*e^2 + 2/3*x^8*e^4*log(x) - 1/12*x^8*e^4 - 4/9*x^7*e^6*log(x) + 5/6*x^8 + 4/63 *x^7*e^6 + 2/3*x^7*log(x) + 1/9*x^6*e^8*log(x) - 4/3*x^6*e^2*log(x) + 4/3* x^7 - 1/54*x^6*e^8 + 2/9*x^6*e^2 + 2/3*x^5*e^4*log(x) - 2/15*x^5*e^4 + 5/4 *x^5 + x^4*log(x) + 2*x^4 + 1/108*(15*x^7 + 26*x^6)*e^8 - 5/63*(7*x^8 + 12 *x^7)*e^6 + 1/60*(50*x^9 + 85*x^8 + 50*x^6 + 88*x^5)*e^4 - 1/81*(45*x^10 + 76*x^9 + 135*x^7 + 234*x^6)*e^2
Time = 3.68 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {1}{36} \left (324 x^3+225 x^4+360 x^6+240 x^7+84 x^9+55 x^{10}+e^8 \left (52 x^5+35 x^6\right )+e^6 \left (-240 x^6-160 x^7\right )+e^4 \left (264 x^4+180 x^5+408 x^7+270 x^8\right )+e^2 \left (-624 x^5-420 x^6-304 x^8-200 x^9\right )+\left (144 x^3+24 e^8 x^5+168 x^6-112 e^6 x^6+40 x^9+e^4 \left (120 x^4+192 x^7\right )+e^2 \left (-288 x^5-144 x^8\right )\right ) \log (x)\right ) \, dx=\frac {x^4\,\left (5\,x+4\,\ln \left (x\right )+8\right )\,{\left (x^3-2\,{\mathrm {e}}^2\,x^2+{\mathrm {e}}^4\,x+3\right )}^2}{36} \] Input:
int((exp(8)*(52*x^5 + 35*x^6))/36 - (exp(6)*(240*x^6 + 160*x^7))/36 + (log (x)*(exp(4)*(120*x^4 + 192*x^7) - exp(2)*(288*x^5 + 144*x^8) - 112*x^6*exp (6) + 24*x^5*exp(8) + 144*x^3 + 168*x^6 + 40*x^9))/36 + 9*x^3 + (25*x^4)/4 + 10*x^6 + (20*x^7)/3 + (7*x^9)/3 + (55*x^10)/36 + (exp(4)*(264*x^4 + 180 *x^5 + 408*x^7 + 270*x^8))/36 - (exp(2)*(624*x^5 + 420*x^6 + 304*x^8 + 200 *x^9))/36,x)
Output:
(x^4*(5*x + 4*log(x) + 8)*(x*exp(4) - 2*x^2*exp(2) + x^3 + 3)^2)/36
Time = 0.17 (sec) , antiderivative size = 200, normalized size of antiderivative = 6.45 \[ \int \frac {1}{36} \left (324 x^3+225 x^4+360 x^6+240 x^7+84 x^9+55 x^{10}+e^8 \left (52 x^5+35 x^6\right )+e^6 \left (-240 x^6-160 x^7\right )+e^4 \left (264 x^4+180 x^5+408 x^7+270 x^8\right )+e^2 \left (-624 x^5-420 x^6-304 x^8-200 x^9\right )+\left (144 x^3+24 e^8 x^5+168 x^6-112 e^6 x^6+40 x^9+e^4 \left (120 x^4+192 x^7\right )+e^2 \left (-288 x^5-144 x^8\right )\right ) \log (x)\right ) \, dx=\frac {x^{4} \left (72+45 x -96 e^{2} x^{2}+30 e^{4} x^{2}+5 x^{7}+48 e^{4} x +48 x^{3}+5 e^{8} x^{3}+8 x^{6}+36 \,\mathrm {log}\left (x \right )-20 e^{6} x^{4}+30 e^{4} x^{5}-20 e^{2} x^{6}-32 e^{6} x^{3}+48 e^{4} x^{4}+4 \,\mathrm {log}\left (x \right ) e^{8} x^{2}-16 \,\mathrm {log}\left (x \right ) e^{6} x^{3}+24 \,\mathrm {log}\left (x \right ) e^{4} x^{4}+24 \,\mathrm {log}\left (x \right ) e^{4} x -16 \,\mathrm {log}\left (x \right ) e^{2} x^{5}-48 \,\mathrm {log}\left (x \right ) e^{2} x^{2}+24 \,\mathrm {log}\left (x \right ) x^{3}+8 e^{8} x^{2}+30 x^{4}-60 e^{2} x^{3}+4 \,\mathrm {log}\left (x \right ) x^{6}-32 e^{2} x^{5}\right )}{36} \] Input:
int(1/36*(24*x^5*exp(2)^4-112*x^6*exp(2)^3+(192*x^7+120*x^4)*exp(2)^2+(-14 4*x^8-288*x^5)*exp(2)+40*x^9+168*x^6+144*x^3)*log(x)+1/36*(35*x^6+52*x^5)* exp(2)^4+1/36*(-160*x^7-240*x^6)*exp(2)^3+1/36*(270*x^8+408*x^7+180*x^5+26 4*x^4)*exp(2)^2+1/36*(-200*x^9-304*x^8-420*x^6-624*x^5)*exp(2)+55/36*x^10+ 7/3*x^9+20/3*x^7+10*x^6+25/4*x^4+9*x^3,x)
Output:
(x**4*(4*log(x)*e**8*x**2 - 16*log(x)*e**6*x**3 + 24*log(x)*e**4*x**4 + 24 *log(x)*e**4*x - 16*log(x)*e**2*x**5 - 48*log(x)*e**2*x**2 + 4*log(x)*x**6 + 24*log(x)*x**3 + 36*log(x) + 5*e**8*x**3 + 8*e**8*x**2 - 20*e**6*x**4 - 32*e**6*x**3 + 30*e**4*x**5 + 48*e**4*x**4 + 30*e**4*x**2 + 48*e**4*x - 2 0*e**2*x**6 - 32*e**2*x**5 - 60*e**2*x**3 - 96*e**2*x**2 + 5*x**7 + 8*x**6 + 30*x**4 + 48*x**3 + 45*x + 72))/36