Integrand size = 445, antiderivative size = 39 \[ \int \frac {e^x \left (-90+180 x-1090 x^2+2165 x^3-1835 x^4+580 x^5-10 x^6-10 x^7+5 x^8\right )+e^x \left (45-60 x+165 x^2-300 x^3+150 x^4+75 x^5-105 x^6+30 x^7\right ) \log \left (x^2-2 x^3+9 x^4-20 x^5+36 x^6-58 x^7+70 x^8-56 x^9+28 x^{10}-8 x^{11}+x^{12}\right )+e^x \left (45-90 x+225 x^2-450 x^3+450 x^4-225 x^5+45 x^6\right ) \log ^2\left (x^2-2 x^3+9 x^4-20 x^5+36 x^6-58 x^7+70 x^8-56 x^9+28 x^{10}-8 x^{11}+x^{12}\right )}{-9 x^2+3 x^3-31 x^4+31 x^5-4 x^6-9 x^7+2 x^8+x^9+\left (-18 x^2+12 x^3-66 x^4+84 x^5-36 x^6-6 x^7+6 x^8\right ) \log \left (x^2-2 x^3+9 x^4-20 x^5+36 x^6-58 x^7+70 x^8-56 x^9+28 x^{10}-8 x^{11}+x^{12}\right )+\left (-9 x^2+9 x^3-36 x^4+54 x^5-36 x^6+9 x^7\right ) \log ^2\left (x^2-2 x^3+9 x^4-20 x^5+36 x^6-58 x^7+70 x^8-56 x^9+28 x^{10}-8 x^{11}+x^{12}\right )} \, dx=-4+\frac {5 e^x}{x+\frac {x}{\frac {x}{3}+\log \left (\left (-x+(1-x)^4 x^2\right )^2\right )}} \] Output:
5*exp(x)/(x/(1/3*x+ln(((1-x)^4*x^2-x)^2))+x)-4
Time = 0.13 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.21 \[ \int \frac {e^x \left (-90+180 x-1090 x^2+2165 x^3-1835 x^4+580 x^5-10 x^6-10 x^7+5 x^8\right )+e^x \left (45-60 x+165 x^2-300 x^3+150 x^4+75 x^5-105 x^6+30 x^7\right ) \log \left (x^2-2 x^3+9 x^4-20 x^5+36 x^6-58 x^7+70 x^8-56 x^9+28 x^{10}-8 x^{11}+x^{12}\right )+e^x \left (45-90 x+225 x^2-450 x^3+450 x^4-225 x^5+45 x^6\right ) \log ^2\left (x^2-2 x^3+9 x^4-20 x^5+36 x^6-58 x^7+70 x^8-56 x^9+28 x^{10}-8 x^{11}+x^{12}\right )}{-9 x^2+3 x^3-31 x^4+31 x^5-4 x^6-9 x^7+2 x^8+x^9+\left (-18 x^2+12 x^3-66 x^4+84 x^5-36 x^6-6 x^7+6 x^8\right ) \log \left (x^2-2 x^3+9 x^4-20 x^5+36 x^6-58 x^7+70 x^8-56 x^9+28 x^{10}-8 x^{11}+x^{12}\right )+\left (-9 x^2+9 x^3-36 x^4+54 x^5-36 x^6+9 x^7\right ) \log ^2\left (x^2-2 x^3+9 x^4-20 x^5+36 x^6-58 x^7+70 x^8-56 x^9+28 x^{10}-8 x^{11}+x^{12}\right )} \, dx=\frac {5 e^x \left (1-\frac {3}{3+x+3 \log \left (x^2 \left (-1+x-4 x^2+6 x^3-4 x^4+x^5\right )^2\right )}\right )}{x} \] Input:
Integrate[(E^x*(-90 + 180*x - 1090*x^2 + 2165*x^3 - 1835*x^4 + 580*x^5 - 1 0*x^6 - 10*x^7 + 5*x^8) + E^x*(45 - 60*x + 165*x^2 - 300*x^3 + 150*x^4 + 7 5*x^5 - 105*x^6 + 30*x^7)*Log[x^2 - 2*x^3 + 9*x^4 - 20*x^5 + 36*x^6 - 58*x ^7 + 70*x^8 - 56*x^9 + 28*x^10 - 8*x^11 + x^12] + E^x*(45 - 90*x + 225*x^2 - 450*x^3 + 450*x^4 - 225*x^5 + 45*x^6)*Log[x^2 - 2*x^3 + 9*x^4 - 20*x^5 + 36*x^6 - 58*x^7 + 70*x^8 - 56*x^9 + 28*x^10 - 8*x^11 + x^12]^2)/(-9*x^2 + 3*x^3 - 31*x^4 + 31*x^5 - 4*x^6 - 9*x^7 + 2*x^8 + x^9 + (-18*x^2 + 12*x^ 3 - 66*x^4 + 84*x^5 - 36*x^6 - 6*x^7 + 6*x^8)*Log[x^2 - 2*x^3 + 9*x^4 - 20 *x^5 + 36*x^6 - 58*x^7 + 70*x^8 - 56*x^9 + 28*x^10 - 8*x^11 + x^12] + (-9* x^2 + 9*x^3 - 36*x^4 + 54*x^5 - 36*x^6 + 9*x^7)*Log[x^2 - 2*x^3 + 9*x^4 - 20*x^5 + 36*x^6 - 58*x^7 + 70*x^8 - 56*x^9 + 28*x^10 - 8*x^11 + x^12]^2),x ]
Output:
(5*E^x*(1 - 3/(3 + x + 3*Log[x^2*(-1 + x - 4*x^2 + 6*x^3 - 4*x^4 + x^5)^2] )))/x
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {e^x \left (5 x^8-10 x^7-10 x^6+580 x^5-1835 x^4+2165 x^3-1090 x^2+180 x-90\right )+e^x \left (45 x^6-225 x^5+450 x^4-450 x^3+225 x^2-90 x+45\right ) \log ^2\left (x^{12}-8 x^{11}+28 x^{10}-56 x^9+70 x^8-58 x^7+36 x^6-20 x^5+9 x^4-2 x^3+x^2\right )+e^x \left (30 x^7-105 x^6+75 x^5+150 x^4-300 x^3+165 x^2-60 x+45\right ) \log \left (x^{12}-8 x^{11}+28 x^{10}-56 x^9+70 x^8-58 x^7+36 x^6-20 x^5+9 x^4-2 x^3+x^2\right )}{x^9+2 x^8-9 x^7-4 x^6+31 x^5-31 x^4+3 x^3-9 x^2+\left (9 x^7-36 x^6+54 x^5-36 x^4+9 x^3-9 x^2\right ) \log ^2\left (x^{12}-8 x^{11}+28 x^{10}-56 x^9+70 x^8-58 x^7+36 x^6-20 x^5+9 x^4-2 x^3+x^2\right )+\left (6 x^8-6 x^7-36 x^6+84 x^5-66 x^4+12 x^3-18 x^2\right ) \log \left (x^{12}-8 x^{11}+28 x^{10}-56 x^9+70 x^8-58 x^7+36 x^6-20 x^5+9 x^4-2 x^3+x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {5 e^x \left (-x^8+2 x^7+2 x^6-116 x^5+367 x^4-433 x^3+218 x^2-9 \left (x^6-5 x^5+10 x^4-10 x^3+5 x^2-2 x+1\right ) \log ^2\left (x^2 \left (x^5-4 x^4+6 x^3-4 x^2+x-1\right )^2\right )-3 \left (2 x^7-7 x^6+5 x^5+10 x^4-20 x^3+11 x^2-4 x+3\right ) \log \left (x^2 \left (x^5-4 x^4+6 x^3-4 x^2+x-1\right )^2\right )-36 x+18\right )}{x^2 \left (-x^5+4 x^4-6 x^3+4 x^2-x+1\right ) \left (3 \log \left (x^2 \left (x^5-4 x^4+6 x^3-4 x^2+x-1\right )^2\right )+x+3\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 5 \int \frac {e^x \left (-x^8+2 x^7+2 x^6-116 x^5+367 x^4-433 x^3+218 x^2-36 x-9 \left (x^6-5 x^5+10 x^4-10 x^3+5 x^2-2 x+1\right ) \log ^2\left (x^2 \left (-x^5+4 x^4-6 x^3+4 x^2-x+1\right )^2\right )-3 \left (2 x^7-7 x^6+5 x^5+10 x^4-20 x^3+11 x^2-4 x+3\right ) \log \left (x^2 \left (-x^5+4 x^4-6 x^3+4 x^2-x+1\right )^2\right )+18\right )}{x^2 \left (-x^5+4 x^4-6 x^3+4 x^2-x+1\right ) \left (x+3 \log \left (x^2 \left (-x^5+4 x^4-6 x^3+4 x^2-x+1\right )^2\right )+3\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 5 \int \left (-\frac {3 e^x (x-1)}{x^2 \left (x+3 \log \left (x^2 \left (x^5-4 x^4+6 x^3-4 x^2+x-1\right )^2\right )+3\right )}+\frac {e^x (x-1)}{x^2}+\frac {3 e^x \left (x^6+32 x^5-114 x^4+140 x^3-71 x^2+11 x-6\right )}{x^2 \left (x^5-4 x^4+6 x^3-4 x^2+x-1\right ) \left (x+3 \log \left (x^2 \left (x^5-4 x^4+6 x^3-4 x^2+x-1\right )^2\right )+3\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 5 \left (18 \int \frac {e^x}{x^2 \left (x+3 \log \left (x^2 \left (x^5-4 x^4+6 x^3-4 x^2+x-1\right )^2\right )+3\right )^2}dx-15 \int \frac {e^x}{x \left (x+3 \log \left (x^2 \left (x^5-4 x^4+6 x^3-4 x^2+x-1\right )^2\right )+3\right )^2}dx-126 \int \frac {e^x}{\left (x^5-4 x^4+6 x^3-4 x^2+x-1\right ) \left (x+3 \log \left (x^2 \left (x^5-4 x^4+6 x^3-4 x^2+x-1\right )^2\right )+3\right )^2}dx+252 \int \frac {e^x x}{\left (x^5-4 x^4+6 x^3-4 x^2+x-1\right ) \left (x+3 \log \left (x^2 \left (x^5-4 x^4+6 x^3-4 x^2+x-1\right )^2\right )+3\right )^2}dx-180 \int \frac {e^x x^2}{\left (x^5-4 x^4+6 x^3-4 x^2+x-1\right ) \left (x+3 \log \left (x^2 \left (x^5-4 x^4+6 x^3-4 x^2+x-1\right )^2\right )+3\right )^2}dx+18 \int \frac {e^x x^3}{\left (x^5-4 x^4+6 x^3-4 x^2+x-1\right ) \left (x+3 \log \left (x^2 \left (x^5-4 x^4+6 x^3-4 x^2+x-1\right )^2\right )+3\right )^2}dx+18 \int \frac {e^x x^4}{\left (x^5-4 x^4+6 x^3-4 x^2+x-1\right ) \left (x+3 \log \left (x^2 \left (x^5-4 x^4+6 x^3-4 x^2+x-1\right )^2\right )+3\right )^2}dx+3 \int \frac {e^x}{x^2 \left (x+3 \log \left (x^2 \left (x^5-4 x^4+6 x^3-4 x^2+x-1\right )^2\right )+3\right )}dx-3 \int \frac {e^x}{x \left (x+3 \log \left (x^2 \left (x^5-4 x^4+6 x^3-4 x^2+x-1\right )^2\right )+3\right )}dx+\frac {e^x}{x}\right )\) |
Input:
Int[(E^x*(-90 + 180*x - 1090*x^2 + 2165*x^3 - 1835*x^4 + 580*x^5 - 10*x^6 - 10*x^7 + 5*x^8) + E^x*(45 - 60*x + 165*x^2 - 300*x^3 + 150*x^4 + 75*x^5 - 105*x^6 + 30*x^7)*Log[x^2 - 2*x^3 + 9*x^4 - 20*x^5 + 36*x^6 - 58*x^7 + 7 0*x^8 - 56*x^9 + 28*x^10 - 8*x^11 + x^12] + E^x*(45 - 90*x + 225*x^2 - 450 *x^3 + 450*x^4 - 225*x^5 + 45*x^6)*Log[x^2 - 2*x^3 + 9*x^4 - 20*x^5 + 36*x ^6 - 58*x^7 + 70*x^8 - 56*x^9 + 28*x^10 - 8*x^11 + x^12]^2)/(-9*x^2 + 3*x^ 3 - 31*x^4 + 31*x^5 - 4*x^6 - 9*x^7 + 2*x^8 + x^9 + (-18*x^2 + 12*x^3 - 66 *x^4 + 84*x^5 - 36*x^6 - 6*x^7 + 6*x^8)*Log[x^2 - 2*x^3 + 9*x^4 - 20*x^5 + 36*x^6 - 58*x^7 + 70*x^8 - 56*x^9 + 28*x^10 - 8*x^11 + x^12] + (-9*x^2 + 9*x^3 - 36*x^4 + 54*x^5 - 36*x^6 + 9*x^7)*Log[x^2 - 2*x^3 + 9*x^4 - 20*x^5 + 36*x^6 - 58*x^7 + 70*x^8 - 56*x^9 + 28*x^10 - 8*x^11 + x^12]^2),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(128\) vs. \(2(36)=72\).
Time = 14.57 (sec) , antiderivative size = 129, normalized size of antiderivative = 3.31
| method | result | size |
| parallelrisch | \(-\frac {-1260 \,{\mathrm e}^{x} x -3780 \,{\mathrm e}^{x} \ln \left (x^{2} \left (x^{10}-8 x^{9}+28 x^{8}-56 x^{7}+70 x^{6}-58 x^{5}+36 x^{4}-20 x^{3}+9 x^{2}-2 x +1\right )\right )}{252 x \left (x +3 \ln \left (x^{2} \left (x^{10}-8 x^{9}+28 x^{8}-56 x^{7}+70 x^{6}-58 x^{5}+36 x^{4}-20 x^{3}+9 x^{2}-2 x +1\right )\right )+3\right )}\) | \(129\) |
| risch | \(\frac {5 \,{\mathrm e}^{x}}{x}-\frac {30 i {\mathrm e}^{x}}{x \left (3 \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-6 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+3 \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+3 \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i \left (x^{5}-4 x^{4}+6 x^{3}-4 x^{2}+x -1\right )^{2}\right ) \operatorname {csgn}\left (i x^{2} \left (x^{5}-4 x^{4}+6 x^{3}-4 x^{2}+x -1\right )^{2}\right )-3 \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{2} \left (x^{5}-4 x^{4}+6 x^{3}-4 x^{2}+x -1\right )^{2}\right )^{2}+3 \pi {\operatorname {csgn}\left (i \left (x^{5}-4 x^{4}+6 x^{3}-4 x^{2}+x -1\right )\right )}^{2} \operatorname {csgn}\left (i \left (x^{5}-4 x^{4}+6 x^{3}-4 x^{2}+x -1\right )^{2}\right )-6 \pi \,\operatorname {csgn}\left (i \left (x^{5}-4 x^{4}+6 x^{3}-4 x^{2}+x -1\right )\right ) {\operatorname {csgn}\left (i \left (x^{5}-4 x^{4}+6 x^{3}-4 x^{2}+x -1\right )^{2}\right )}^{2}+3 \pi {\operatorname {csgn}\left (i \left (x^{5}-4 x^{4}+6 x^{3}-4 x^{2}+x -1\right )^{2}\right )}^{3}-3 \pi \,\operatorname {csgn}\left (i \left (x^{5}-4 x^{4}+6 x^{3}-4 x^{2}+x -1\right )^{2}\right ) \operatorname {csgn}\left (i x^{2} \left (x^{5}-4 x^{4}+6 x^{3}-4 x^{2}+x -1\right )^{2}\right )^{2}+3 \pi \operatorname {csgn}\left (i x^{2} \left (x^{5}-4 x^{4}+6 x^{3}-4 x^{2}+x -1\right )^{2}\right )^{3}+2 i x +12 i \ln \left (x \right )+12 i \ln \left (x^{5}-4 x^{4}+6 x^{3}-4 x^{2}+x -1\right )+6 i\right )}\) | \(454\) |
Input:
int(((45*x^6-225*x^5+450*x^4-450*x^3+225*x^2-90*x+45)*exp(x)*ln(x^12-8*x^1 1+28*x^10-56*x^9+70*x^8-58*x^7+36*x^6-20*x^5+9*x^4-2*x^3+x^2)^2+(30*x^7-10 5*x^6+75*x^5+150*x^4-300*x^3+165*x^2-60*x+45)*exp(x)*ln(x^12-8*x^11+28*x^1 0-56*x^9+70*x^8-58*x^7+36*x^6-20*x^5+9*x^4-2*x^3+x^2)+(5*x^8-10*x^7-10*x^6 +580*x^5-1835*x^4+2165*x^3-1090*x^2+180*x-90)*exp(x))/((9*x^7-36*x^6+54*x^ 5-36*x^4+9*x^3-9*x^2)*ln(x^12-8*x^11+28*x^10-56*x^9+70*x^8-58*x^7+36*x^6-2 0*x^5+9*x^4-2*x^3+x^2)^2+(6*x^8-6*x^7-36*x^6+84*x^5-66*x^4+12*x^3-18*x^2)* ln(x^12-8*x^11+28*x^10-56*x^9+70*x^8-58*x^7+36*x^6-20*x^5+9*x^4-2*x^3+x^2) +x^9+2*x^8-9*x^7-4*x^6+31*x^5-31*x^4+3*x^3-9*x^2),x,method=_RETURNVERBOSE)
Output:
-1/252*(-1260*exp(x)*x-3780*exp(x)*ln(x^2*(x^10-8*x^9+28*x^8-56*x^7+70*x^6 -58*x^5+36*x^4-20*x^3+9*x^2-2*x+1)))/x/(x+3*ln(x^2*(x^10-8*x^9+28*x^8-56*x ^7+70*x^6-58*x^5+36*x^4-20*x^3+9*x^2-2*x+1))+3)
Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (35) = 70\).
Time = 0.10 (sec) , antiderivative size = 129, normalized size of antiderivative = 3.31 \[ \int \frac {e^x \left (-90+180 x-1090 x^2+2165 x^3-1835 x^4+580 x^5-10 x^6-10 x^7+5 x^8\right )+e^x \left (45-60 x+165 x^2-300 x^3+150 x^4+75 x^5-105 x^6+30 x^7\right ) \log \left (x^2-2 x^3+9 x^4-20 x^5+36 x^6-58 x^7+70 x^8-56 x^9+28 x^{10}-8 x^{11}+x^{12}\right )+e^x \left (45-90 x+225 x^2-450 x^3+450 x^4-225 x^5+45 x^6\right ) \log ^2\left (x^2-2 x^3+9 x^4-20 x^5+36 x^6-58 x^7+70 x^8-56 x^9+28 x^{10}-8 x^{11}+x^{12}\right )}{-9 x^2+3 x^3-31 x^4+31 x^5-4 x^6-9 x^7+2 x^8+x^9+\left (-18 x^2+12 x^3-66 x^4+84 x^5-36 x^6-6 x^7+6 x^8\right ) \log \left (x^2-2 x^3+9 x^4-20 x^5+36 x^6-58 x^7+70 x^8-56 x^9+28 x^{10}-8 x^{11}+x^{12}\right )+\left (-9 x^2+9 x^3-36 x^4+54 x^5-36 x^6+9 x^7\right ) \log ^2\left (x^2-2 x^3+9 x^4-20 x^5+36 x^6-58 x^7+70 x^8-56 x^9+28 x^{10}-8 x^{11}+x^{12}\right )} \, dx=\frac {5 \, {\left (x e^{x} + 3 \, e^{x} \log \left (x^{12} - 8 \, x^{11} + 28 \, x^{10} - 56 \, x^{9} + 70 \, x^{8} - 58 \, x^{7} + 36 \, x^{6} - 20 \, x^{5} + 9 \, x^{4} - 2 \, x^{3} + x^{2}\right )\right )}}{x^{2} + 3 \, x \log \left (x^{12} - 8 \, x^{11} + 28 \, x^{10} - 56 \, x^{9} + 70 \, x^{8} - 58 \, x^{7} + 36 \, x^{6} - 20 \, x^{5} + 9 \, x^{4} - 2 \, x^{3} + x^{2}\right ) + 3 \, x} \] Input:
integrate(((45*x^6-225*x^5+450*x^4-450*x^3+225*x^2-90*x+45)*exp(x)*log(x^1 2-8*x^11+28*x^10-56*x^9+70*x^8-58*x^7+36*x^6-20*x^5+9*x^4-2*x^3+x^2)^2+(30 *x^7-105*x^6+75*x^5+150*x^4-300*x^3+165*x^2-60*x+45)*exp(x)*log(x^12-8*x^1 1+28*x^10-56*x^9+70*x^8-58*x^7+36*x^6-20*x^5+9*x^4-2*x^3+x^2)+(5*x^8-10*x^ 7-10*x^6+580*x^5-1835*x^4+2165*x^3-1090*x^2+180*x-90)*exp(x))/((9*x^7-36*x ^6+54*x^5-36*x^4+9*x^3-9*x^2)*log(x^12-8*x^11+28*x^10-56*x^9+70*x^8-58*x^7 +36*x^6-20*x^5+9*x^4-2*x^3+x^2)^2+(6*x^8-6*x^7-36*x^6+84*x^5-66*x^4+12*x^3 -18*x^2)*log(x^12-8*x^11+28*x^10-56*x^9+70*x^8-58*x^7+36*x^6-20*x^5+9*x^4- 2*x^3+x^2)+x^9+2*x^8-9*x^7-4*x^6+31*x^5-31*x^4+3*x^3-9*x^2),x, algorithm=" fricas")
Output:
5*(x*e^x + 3*e^x*log(x^12 - 8*x^11 + 28*x^10 - 56*x^9 + 70*x^8 - 58*x^7 + 36*x^6 - 20*x^5 + 9*x^4 - 2*x^3 + x^2))/(x^2 + 3*x*log(x^12 - 8*x^11 + 28* x^10 - 56*x^9 + 70*x^8 - 58*x^7 + 36*x^6 - 20*x^5 + 9*x^4 - 2*x^3 + x^2) + 3*x)
Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (26) = 52\).
Time = 0.36 (sec) , antiderivative size = 126, normalized size of antiderivative = 3.23 \[ \int \frac {e^x \left (-90+180 x-1090 x^2+2165 x^3-1835 x^4+580 x^5-10 x^6-10 x^7+5 x^8\right )+e^x \left (45-60 x+165 x^2-300 x^3+150 x^4+75 x^5-105 x^6+30 x^7\right ) \log \left (x^2-2 x^3+9 x^4-20 x^5+36 x^6-58 x^7+70 x^8-56 x^9+28 x^{10}-8 x^{11}+x^{12}\right )+e^x \left (45-90 x+225 x^2-450 x^3+450 x^4-225 x^5+45 x^6\right ) \log ^2\left (x^2-2 x^3+9 x^4-20 x^5+36 x^6-58 x^7+70 x^8-56 x^9+28 x^{10}-8 x^{11}+x^{12}\right )}{-9 x^2+3 x^3-31 x^4+31 x^5-4 x^6-9 x^7+2 x^8+x^9+\left (-18 x^2+12 x^3-66 x^4+84 x^5-36 x^6-6 x^7+6 x^8\right ) \log \left (x^2-2 x^3+9 x^4-20 x^5+36 x^6-58 x^7+70 x^8-56 x^9+28 x^{10}-8 x^{11}+x^{12}\right )+\left (-9 x^2+9 x^3-36 x^4+54 x^5-36 x^6+9 x^7\right ) \log ^2\left (x^2-2 x^3+9 x^4-20 x^5+36 x^6-58 x^7+70 x^8-56 x^9+28 x^{10}-8 x^{11}+x^{12}\right )} \, dx=\frac {\left (5 x + 15 \log {\left (x^{12} - 8 x^{11} + 28 x^{10} - 56 x^{9} + 70 x^{8} - 58 x^{7} + 36 x^{6} - 20 x^{5} + 9 x^{4} - 2 x^{3} + x^{2} \right )}\right ) e^{x}}{x^{2} + 3 x \log {\left (x^{12} - 8 x^{11} + 28 x^{10} - 56 x^{9} + 70 x^{8} - 58 x^{7} + 36 x^{6} - 20 x^{5} + 9 x^{4} - 2 x^{3} + x^{2} \right )} + 3 x} \] Input:
integrate(((45*x**6-225*x**5+450*x**4-450*x**3+225*x**2-90*x+45)*exp(x)*ln (x**12-8*x**11+28*x**10-56*x**9+70*x**8-58*x**7+36*x**6-20*x**5+9*x**4-2*x **3+x**2)**2+(30*x**7-105*x**6+75*x**5+150*x**4-300*x**3+165*x**2-60*x+45) *exp(x)*ln(x**12-8*x**11+28*x**10-56*x**9+70*x**8-58*x**7+36*x**6-20*x**5+ 9*x**4-2*x**3+x**2)+(5*x**8-10*x**7-10*x**6+580*x**5-1835*x**4+2165*x**3-1 090*x**2+180*x-90)*exp(x))/((9*x**7-36*x**6+54*x**5-36*x**4+9*x**3-9*x**2) *ln(x**12-8*x**11+28*x**10-56*x**9+70*x**8-58*x**7+36*x**6-20*x**5+9*x**4- 2*x**3+x**2)**2+(6*x**8-6*x**7-36*x**6+84*x**5-66*x**4+12*x**3-18*x**2)*ln (x**12-8*x**11+28*x**10-56*x**9+70*x**8-58*x**7+36*x**6-20*x**5+9*x**4-2*x **3+x**2)+x**9+2*x**8-9*x**7-4*x**6+31*x**5-31*x**4+3*x**3-9*x**2),x)
Output:
(5*x + 15*log(x**12 - 8*x**11 + 28*x**10 - 56*x**9 + 70*x**8 - 58*x**7 + 3 6*x**6 - 20*x**5 + 9*x**4 - 2*x**3 + x**2))*exp(x)/(x**2 + 3*x*log(x**12 - 8*x**11 + 28*x**10 - 56*x**9 + 70*x**8 - 58*x**7 + 36*x**6 - 20*x**5 + 9* x**4 - 2*x**3 + x**2) + 3*x)
Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (35) = 70\).
Time = 0.13 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.97 \[ \int \frac {e^x \left (-90+180 x-1090 x^2+2165 x^3-1835 x^4+580 x^5-10 x^6-10 x^7+5 x^8\right )+e^x \left (45-60 x+165 x^2-300 x^3+150 x^4+75 x^5-105 x^6+30 x^7\right ) \log \left (x^2-2 x^3+9 x^4-20 x^5+36 x^6-58 x^7+70 x^8-56 x^9+28 x^{10}-8 x^{11}+x^{12}\right )+e^x \left (45-90 x+225 x^2-450 x^3+450 x^4-225 x^5+45 x^6\right ) \log ^2\left (x^2-2 x^3+9 x^4-20 x^5+36 x^6-58 x^7+70 x^8-56 x^9+28 x^{10}-8 x^{11}+x^{12}\right )}{-9 x^2+3 x^3-31 x^4+31 x^5-4 x^6-9 x^7+2 x^8+x^9+\left (-18 x^2+12 x^3-66 x^4+84 x^5-36 x^6-6 x^7+6 x^8\right ) \log \left (x^2-2 x^3+9 x^4-20 x^5+36 x^6-58 x^7+70 x^8-56 x^9+28 x^{10}-8 x^{11}+x^{12}\right )+\left (-9 x^2+9 x^3-36 x^4+54 x^5-36 x^6+9 x^7\right ) \log ^2\left (x^2-2 x^3+9 x^4-20 x^5+36 x^6-58 x^7+70 x^8-56 x^9+28 x^{10}-8 x^{11}+x^{12}\right )} \, dx=\frac {5 \, {\left ({\left (x + 6 \, \log \left (x\right )\right )} e^{x} + 6 \, e^{x} \log \left (x^{5} - 4 \, x^{4} + 6 \, x^{3} - 4 \, x^{2} + x - 1\right )\right )}}{x^{2} + 6 \, x \log \left (x^{5} - 4 \, x^{4} + 6 \, x^{3} - 4 \, x^{2} + x - 1\right ) + 6 \, x \log \left (x\right ) + 3 \, x} \] Input:
integrate(((45*x^6-225*x^5+450*x^4-450*x^3+225*x^2-90*x+45)*exp(x)*log(x^1 2-8*x^11+28*x^10-56*x^9+70*x^8-58*x^7+36*x^6-20*x^5+9*x^4-2*x^3+x^2)^2+(30 *x^7-105*x^6+75*x^5+150*x^4-300*x^3+165*x^2-60*x+45)*exp(x)*log(x^12-8*x^1 1+28*x^10-56*x^9+70*x^8-58*x^7+36*x^6-20*x^5+9*x^4-2*x^3+x^2)+(5*x^8-10*x^ 7-10*x^6+580*x^5-1835*x^4+2165*x^3-1090*x^2+180*x-90)*exp(x))/((9*x^7-36*x ^6+54*x^5-36*x^4+9*x^3-9*x^2)*log(x^12-8*x^11+28*x^10-56*x^9+70*x^8-58*x^7 +36*x^6-20*x^5+9*x^4-2*x^3+x^2)^2+(6*x^8-6*x^7-36*x^6+84*x^5-66*x^4+12*x^3 -18*x^2)*log(x^12-8*x^11+28*x^10-56*x^9+70*x^8-58*x^7+36*x^6-20*x^5+9*x^4- 2*x^3+x^2)+x^9+2*x^8-9*x^7-4*x^6+31*x^5-31*x^4+3*x^3-9*x^2),x, algorithm=" maxima")
Output:
5*((x + 6*log(x))*e^x + 6*e^x*log(x^5 - 4*x^4 + 6*x^3 - 4*x^2 + x - 1))/(x ^2 + 6*x*log(x^5 - 4*x^4 + 6*x^3 - 4*x^2 + x - 1) + 6*x*log(x) + 3*x)
Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (35) = 70\).
Time = 4.35 (sec) , antiderivative size = 129, normalized size of antiderivative = 3.31 \[ \int \frac {e^x \left (-90+180 x-1090 x^2+2165 x^3-1835 x^4+580 x^5-10 x^6-10 x^7+5 x^8\right )+e^x \left (45-60 x+165 x^2-300 x^3+150 x^4+75 x^5-105 x^6+30 x^7\right ) \log \left (x^2-2 x^3+9 x^4-20 x^5+36 x^6-58 x^7+70 x^8-56 x^9+28 x^{10}-8 x^{11}+x^{12}\right )+e^x \left (45-90 x+225 x^2-450 x^3+450 x^4-225 x^5+45 x^6\right ) \log ^2\left (x^2-2 x^3+9 x^4-20 x^5+36 x^6-58 x^7+70 x^8-56 x^9+28 x^{10}-8 x^{11}+x^{12}\right )}{-9 x^2+3 x^3-31 x^4+31 x^5-4 x^6-9 x^7+2 x^8+x^9+\left (-18 x^2+12 x^3-66 x^4+84 x^5-36 x^6-6 x^7+6 x^8\right ) \log \left (x^2-2 x^3+9 x^4-20 x^5+36 x^6-58 x^7+70 x^8-56 x^9+28 x^{10}-8 x^{11}+x^{12}\right )+\left (-9 x^2+9 x^3-36 x^4+54 x^5-36 x^6+9 x^7\right ) \log ^2\left (x^2-2 x^3+9 x^4-20 x^5+36 x^6-58 x^7+70 x^8-56 x^9+28 x^{10}-8 x^{11}+x^{12}\right )} \, dx=\frac {5 \, {\left (x e^{x} + 3 \, e^{x} \log \left (x^{12} - 8 \, x^{11} + 28 \, x^{10} - 56 \, x^{9} + 70 \, x^{8} - 58 \, x^{7} + 36 \, x^{6} - 20 \, x^{5} + 9 \, x^{4} - 2 \, x^{3} + x^{2}\right )\right )}}{x^{2} + 3 \, x \log \left (x^{12} - 8 \, x^{11} + 28 \, x^{10} - 56 \, x^{9} + 70 \, x^{8} - 58 \, x^{7} + 36 \, x^{6} - 20 \, x^{5} + 9 \, x^{4} - 2 \, x^{3} + x^{2}\right ) + 3 \, x} \] Input:
integrate(((45*x^6-225*x^5+450*x^4-450*x^3+225*x^2-90*x+45)*exp(x)*log(x^1 2-8*x^11+28*x^10-56*x^9+70*x^8-58*x^7+36*x^6-20*x^5+9*x^4-2*x^3+x^2)^2+(30 *x^7-105*x^6+75*x^5+150*x^4-300*x^3+165*x^2-60*x+45)*exp(x)*log(x^12-8*x^1 1+28*x^10-56*x^9+70*x^8-58*x^7+36*x^6-20*x^5+9*x^4-2*x^3+x^2)+(5*x^8-10*x^ 7-10*x^6+580*x^5-1835*x^4+2165*x^3-1090*x^2+180*x-90)*exp(x))/((9*x^7-36*x ^6+54*x^5-36*x^4+9*x^3-9*x^2)*log(x^12-8*x^11+28*x^10-56*x^9+70*x^8-58*x^7 +36*x^6-20*x^5+9*x^4-2*x^3+x^2)^2+(6*x^8-6*x^7-36*x^6+84*x^5-66*x^4+12*x^3 -18*x^2)*log(x^12-8*x^11+28*x^10-56*x^9+70*x^8-58*x^7+36*x^6-20*x^5+9*x^4- 2*x^3+x^2)+x^9+2*x^8-9*x^7-4*x^6+31*x^5-31*x^4+3*x^3-9*x^2),x, algorithm=" giac")
Output:
5*(x*e^x + 3*e^x*log(x^12 - 8*x^11 + 28*x^10 - 56*x^9 + 70*x^8 - 58*x^7 + 36*x^6 - 20*x^5 + 9*x^4 - 2*x^3 + x^2))/(x^2 + 3*x*log(x^12 - 8*x^11 + 28* x^10 - 56*x^9 + 70*x^8 - 58*x^7 + 36*x^6 - 20*x^5 + 9*x^4 - 2*x^3 + x^2) + 3*x)
Time = 4.62 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.97 \[ \int \frac {e^x \left (-90+180 x-1090 x^2+2165 x^3-1835 x^4+580 x^5-10 x^6-10 x^7+5 x^8\right )+e^x \left (45-60 x+165 x^2-300 x^3+150 x^4+75 x^5-105 x^6+30 x^7\right ) \log \left (x^2-2 x^3+9 x^4-20 x^5+36 x^6-58 x^7+70 x^8-56 x^9+28 x^{10}-8 x^{11}+x^{12}\right )+e^x \left (45-90 x+225 x^2-450 x^3+450 x^4-225 x^5+45 x^6\right ) \log ^2\left (x^2-2 x^3+9 x^4-20 x^5+36 x^6-58 x^7+70 x^8-56 x^9+28 x^{10}-8 x^{11}+x^{12}\right )}{-9 x^2+3 x^3-31 x^4+31 x^5-4 x^6-9 x^7+2 x^8+x^9+\left (-18 x^2+12 x^3-66 x^4+84 x^5-36 x^6-6 x^7+6 x^8\right ) \log \left (x^2-2 x^3+9 x^4-20 x^5+36 x^6-58 x^7+70 x^8-56 x^9+28 x^{10}-8 x^{11}+x^{12}\right )+\left (-9 x^2+9 x^3-36 x^4+54 x^5-36 x^6+9 x^7\right ) \log ^2\left (x^2-2 x^3+9 x^4-20 x^5+36 x^6-58 x^7+70 x^8-56 x^9+28 x^{10}-8 x^{11}+x^{12}\right )} \, dx=\frac {5\,{\mathrm {e}}^x}{x}-\frac {15\,{\mathrm {e}}^x}{3\,x+3\,x\,\ln \left (x^{12}-8\,x^{11}+28\,x^{10}-56\,x^9+70\,x^8-58\,x^7+36\,x^6-20\,x^5+9\,x^4-2\,x^3+x^2\right )+x^2} \] Input:
int(-(exp(x)*log(x^2 - 2*x^3 + 9*x^4 - 20*x^5 + 36*x^6 - 58*x^7 + 70*x^8 - 56*x^9 + 28*x^10 - 8*x^11 + x^12)*(165*x^2 - 60*x - 300*x^3 + 150*x^4 + 7 5*x^5 - 105*x^6 + 30*x^7 + 45) - exp(x)*(1090*x^2 - 180*x - 2165*x^3 + 183 5*x^4 - 580*x^5 + 10*x^6 + 10*x^7 - 5*x^8 + 90) + exp(x)*log(x^2 - 2*x^3 + 9*x^4 - 20*x^5 + 36*x^6 - 58*x^7 + 70*x^8 - 56*x^9 + 28*x^10 - 8*x^11 + x ^12)^2*(225*x^2 - 90*x - 450*x^3 + 450*x^4 - 225*x^5 + 45*x^6 + 45))/(log( x^2 - 2*x^3 + 9*x^4 - 20*x^5 + 36*x^6 - 58*x^7 + 70*x^8 - 56*x^9 + 28*x^10 - 8*x^11 + x^12)*(18*x^2 - 12*x^3 + 66*x^4 - 84*x^5 + 36*x^6 + 6*x^7 - 6* x^8) + log(x^2 - 2*x^3 + 9*x^4 - 20*x^5 + 36*x^6 - 58*x^7 + 70*x^8 - 56*x^ 9 + 28*x^10 - 8*x^11 + x^12)^2*(9*x^2 - 9*x^3 + 36*x^4 - 54*x^5 + 36*x^6 - 9*x^7) + 9*x^2 - 3*x^3 + 31*x^4 - 31*x^5 + 4*x^6 + 9*x^7 - 2*x^8 - x^9),x )
Output:
(5*exp(x))/x - (15*exp(x))/(3*x + 3*x*log(x^2 - 2*x^3 + 9*x^4 - 20*x^5 + 3 6*x^6 - 58*x^7 + 70*x^8 - 56*x^9 + 28*x^10 - 8*x^11 + x^12) + x^2)
Time = 0.17 (sec) , antiderivative size = 125, normalized size of antiderivative = 3.21 \[ \int \frac {e^x \left (-90+180 x-1090 x^2+2165 x^3-1835 x^4+580 x^5-10 x^6-10 x^7+5 x^8\right )+e^x \left (45-60 x+165 x^2-300 x^3+150 x^4+75 x^5-105 x^6+30 x^7\right ) \log \left (x^2-2 x^3+9 x^4-20 x^5+36 x^6-58 x^7+70 x^8-56 x^9+28 x^{10}-8 x^{11}+x^{12}\right )+e^x \left (45-90 x+225 x^2-450 x^3+450 x^4-225 x^5+45 x^6\right ) \log ^2\left (x^2-2 x^3+9 x^4-20 x^5+36 x^6-58 x^7+70 x^8-56 x^9+28 x^{10}-8 x^{11}+x^{12}\right )}{-9 x^2+3 x^3-31 x^4+31 x^5-4 x^6-9 x^7+2 x^8+x^9+\left (-18 x^2+12 x^3-66 x^4+84 x^5-36 x^6-6 x^7+6 x^8\right ) \log \left (x^2-2 x^3+9 x^4-20 x^5+36 x^6-58 x^7+70 x^8-56 x^9+28 x^{10}-8 x^{11}+x^{12}\right )+\left (-9 x^2+9 x^3-36 x^4+54 x^5-36 x^6+9 x^7\right ) \log ^2\left (x^2-2 x^3+9 x^4-20 x^5+36 x^6-58 x^7+70 x^8-56 x^9+28 x^{10}-8 x^{11}+x^{12}\right )} \, dx=\frac {5 e^{x} \left (3 \,\mathrm {log}\left (x^{12}-8 x^{11}+28 x^{10}-56 x^{9}+70 x^{8}-58 x^{7}+36 x^{6}-20 x^{5}+9 x^{4}-2 x^{3}+x^{2}\right )+x \right )}{x \left (3 \,\mathrm {log}\left (x^{12}-8 x^{11}+28 x^{10}-56 x^{9}+70 x^{8}-58 x^{7}+36 x^{6}-20 x^{5}+9 x^{4}-2 x^{3}+x^{2}\right )+x +3\right )} \] Input:
int(((45*x^6-225*x^5+450*x^4-450*x^3+225*x^2-90*x+45)*exp(x)*log(x^12-8*x^ 11+28*x^10-56*x^9+70*x^8-58*x^7+36*x^6-20*x^5+9*x^4-2*x^3+x^2)^2+(30*x^7-1 05*x^6+75*x^5+150*x^4-300*x^3+165*x^2-60*x+45)*exp(x)*log(x^12-8*x^11+28*x ^10-56*x^9+70*x^8-58*x^7+36*x^6-20*x^5+9*x^4-2*x^3+x^2)+(5*x^8-10*x^7-10*x ^6+580*x^5-1835*x^4+2165*x^3-1090*x^2+180*x-90)*exp(x))/((9*x^7-36*x^6+54* x^5-36*x^4+9*x^3-9*x^2)*log(x^12-8*x^11+28*x^10-56*x^9+70*x^8-58*x^7+36*x^ 6-20*x^5+9*x^4-2*x^3+x^2)^2+(6*x^8-6*x^7-36*x^6+84*x^5-66*x^4+12*x^3-18*x^ 2)*log(x^12-8*x^11+28*x^10-56*x^9+70*x^8-58*x^7+36*x^6-20*x^5+9*x^4-2*x^3+ x^2)+x^9+2*x^8-9*x^7-4*x^6+31*x^5-31*x^4+3*x^3-9*x^2),x)
Output:
(5*e**x*(3*log(x**12 - 8*x**11 + 28*x**10 - 56*x**9 + 70*x**8 - 58*x**7 + 36*x**6 - 20*x**5 + 9*x**4 - 2*x**3 + x**2) + x))/(x*(3*log(x**12 - 8*x**1 1 + 28*x**10 - 56*x**9 + 70*x**8 - 58*x**7 + 36*x**6 - 20*x**5 + 9*x**4 - 2*x**3 + x**2) + x + 3))