Integrand size = 344, antiderivative size = 23 \[ \int \frac {-4 x^4+16 x^5+16 e^4 x^5+16 x^6+\left (4 x^3+4 e^4 x^3+4 x^4\right ) \log \left (1+e^4+x\right )}{1024 x^5-4096 x^6+5120 x^7-5120 x^9+4096 x^{10}-1024 x^{11}+e^4 \left (1024 x^5-5120 x^6+10240 x^7-10240 x^8+5120 x^9-1024 x^{10}\right )+\left (1280 x^4-3840 x^5+2560 x^6+2560 x^7-3840 x^8+1280 x^9+e^4 \left (1280 x^4-5120 x^5+7680 x^6-5120 x^7+1280 x^8\right )\right ) \log \left (1+e^4+x\right )+\left (640 x^3-1280 x^4+1280 x^6-640 x^7+e^4 \left (640 x^3-1920 x^4+1920 x^5-640 x^6\right )\right ) \log ^2\left (1+e^4+x\right )+\left (160 x^2-160 x^3-160 x^4+160 x^5+e^4 \left (160 x^2-320 x^3+160 x^4\right )\right ) \log ^3\left (1+e^4+x\right )+\left (20 x-20 x^3+e^4 \left (20 x-20 x^2\right )\right ) \log ^4\left (1+e^4+x\right )+\left (1+e^4+x\right ) \log ^5\left (1+e^4+x\right )} \, dx=\frac {x^4}{\left (4 \left (x-x^2\right )+\log \left (1+e^4+x\right )\right )^4} \] Output:
x^4/(-4*x^2+4*x+ln(exp(4)+x+1))^4
Time = 3.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {-4 x^4+16 x^5+16 e^4 x^5+16 x^6+\left (4 x^3+4 e^4 x^3+4 x^4\right ) \log \left (1+e^4+x\right )}{1024 x^5-4096 x^6+5120 x^7-5120 x^9+4096 x^{10}-1024 x^{11}+e^4 \left (1024 x^5-5120 x^6+10240 x^7-10240 x^8+5120 x^9-1024 x^{10}\right )+\left (1280 x^4-3840 x^5+2560 x^6+2560 x^7-3840 x^8+1280 x^9+e^4 \left (1280 x^4-5120 x^5+7680 x^6-5120 x^7+1280 x^8\right )\right ) \log \left (1+e^4+x\right )+\left (640 x^3-1280 x^4+1280 x^6-640 x^7+e^4 \left (640 x^3-1920 x^4+1920 x^5-640 x^6\right )\right ) \log ^2\left (1+e^4+x\right )+\left (160 x^2-160 x^3-160 x^4+160 x^5+e^4 \left (160 x^2-320 x^3+160 x^4\right )\right ) \log ^3\left (1+e^4+x\right )+\left (20 x-20 x^3+e^4 \left (20 x-20 x^2\right )\right ) \log ^4\left (1+e^4+x\right )+\left (1+e^4+x\right ) \log ^5\left (1+e^4+x\right )} \, dx=\frac {x^4}{\left (-4 (-1+x) x+\log \left (1+e^4+x\right )\right )^4} \] Input:
Integrate[(-4*x^4 + 16*x^5 + 16*E^4*x^5 + 16*x^6 + (4*x^3 + 4*E^4*x^3 + 4* x^4)*Log[1 + E^4 + x])/(1024*x^5 - 4096*x^6 + 5120*x^7 - 5120*x^9 + 4096*x ^10 - 1024*x^11 + E^4*(1024*x^5 - 5120*x^6 + 10240*x^7 - 10240*x^8 + 5120* x^9 - 1024*x^10) + (1280*x^4 - 3840*x^5 + 2560*x^6 + 2560*x^7 - 3840*x^8 + 1280*x^9 + E^4*(1280*x^4 - 5120*x^5 + 7680*x^6 - 5120*x^7 + 1280*x^8))*Lo g[1 + E^4 + x] + (640*x^3 - 1280*x^4 + 1280*x^6 - 640*x^7 + E^4*(640*x^3 - 1920*x^4 + 1920*x^5 - 640*x^6))*Log[1 + E^4 + x]^2 + (160*x^2 - 160*x^3 - 160*x^4 + 160*x^5 + E^4*(160*x^2 - 320*x^3 + 160*x^4))*Log[1 + E^4 + x]^3 + (20*x - 20*x^3 + E^4*(20*x - 20*x^2))*Log[1 + E^4 + x]^4 + (1 + E^4 + x )*Log[1 + E^4 + x]^5),x]
Output:
x^4/(-4*(-1 + x)*x + Log[1 + E^4 + x])^4
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.
| \(\displaystyle \int \frac {16 x^6+16 e^4 x^5+16 x^5-4 x^4+\left (4 x^4+4 e^4 x^3+4 x^3\right ) \log \left (x+e^4+1\right )}{-1024 x^{11}+4096 x^{10}-5120 x^9+5120 x^7-4096 x^6+1024 x^5+\left (-20 x^3+e^4 \left (20 x-20 x^2\right )+20 x\right ) \log ^4\left (x+e^4+1\right )+\left (160 x^5-160 x^4-160 x^3+160 x^2+e^4 \left (160 x^4-320 x^3+160 x^2\right )\right ) \log ^3\left (x+e^4+1\right )+\left (-640 x^7+1280 x^6-1280 x^4+640 x^3+e^4 \left (-640 x^6+1920 x^5-1920 x^4+640 x^3\right )\right ) \log ^2\left (x+e^4+1\right )+e^4 \left (-1024 x^{10}+5120 x^9-10240 x^8+10240 x^7-5120 x^6+1024 x^5\right )+\left (1280 x^9-3840 x^8+2560 x^7+2560 x^6-3840 x^5+1280 x^4+e^4 \left (1280 x^8-5120 x^7+7680 x^6-5120 x^5+1280 x^4\right )\right ) \log \left (x+e^4+1\right )+\left (x+e^4+1\right ) \log ^5\left (x+e^4+1\right )} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {16 x^6+\left (16+16 e^4\right ) x^5-4 x^4+\left (4 x^4+4 e^4 x^3+4 x^3\right ) \log \left (x+e^4+1\right )}{-1024 x^{11}+4096 x^{10}-5120 x^9+5120 x^7-4096 x^6+1024 x^5+\left (-20 x^3+e^4 \left (20 x-20 x^2\right )+20 x\right ) \log ^4\left (x+e^4+1\right )+\left (160 x^5-160 x^4-160 x^3+160 x^2+e^4 \left (160 x^4-320 x^3+160 x^2\right )\right ) \log ^3\left (x+e^4+1\right )+\left (-640 x^7+1280 x^6-1280 x^4+640 x^3+e^4 \left (-640 x^6+1920 x^5-1920 x^4+640 x^3\right )\right ) \log ^2\left (x+e^4+1\right )+e^4 \left (-1024 x^{10}+5120 x^9-10240 x^8+10240 x^7-5120 x^6+1024 x^5\right )+\left (1280 x^9-3840 x^8+2560 x^7+2560 x^6-3840 x^5+1280 x^4+e^4 \left (1280 x^8-5120 x^7+7680 x^6-5120 x^5+1280 x^4\right )\right ) \log \left (x+e^4+1\right )+\left (x+e^4+1\right ) \log ^5\left (x+e^4+1\right )}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {4 x^3 \left (-x \left (4 x^2+4 \left (1+e^4\right ) x-1\right )-\left (x+e^4+1\right ) \log \left (x+e^4+1\right )\right )}{\left (x+e^4+1\right ) \left (4 (x-1) x-\log \left (x+e^4+1\right )\right )^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \int -\frac {x^3 \left (x \left (-4 x^2-4 \left (1+e^4\right ) x+1\right )-\left (x+e^4+1\right ) \log \left (x+e^4+1\right )\right )}{\left (x+e^4+1\right ) \left (4 (1-x) x+\log \left (x+e^4+1\right )\right )^5}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -4 \int \frac {x^3 \left (x \left (-4 x^2-4 \left (1+e^4\right ) x+1\right )-\left (x+e^4+1\right ) \log \left (x+e^4+1\right )\right )}{\left (x+e^4+1\right ) \left (4 (1-x) x+\log \left (x+e^4+1\right )\right )^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (\frac {x^4 \left (-8 x^2-4 \left (1+2 e^4\right ) x+4 e^4+5\right )}{\left (x+e^4+1\right ) \left (-4 x^2+4 x+\log \left (x+e^4+1\right )\right )^5}-\frac {x^3}{\left (4 x^2-4 x-\log \left (x+e^4+1\right )\right )^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -4 \left (\left (1+e^4\right )^3 \int \frac {1}{\left (4 x^2-4 x-\log \left (x+e^4+1\right )\right )^5}dx-\left (1+e^4\right )^2 \int \frac {x}{\left (4 x^2-4 x-\log \left (x+e^4+1\right )\right )^5}dx+\left (1+e^4\right ) \int \frac {x^2}{\left (4 x^2-4 x-\log \left (x+e^4+1\right )\right )^5}dx-\left (1+e^4\right )^4 \int \frac {1}{\left (x+e^4+1\right ) \left (4 x^2-4 x-\log \left (x+e^4+1\right )\right )^5}dx+8 \int \frac {x^5}{\left (4 x^2-4 x-\log \left (x+e^4+1\right )\right )^5}dx-4 \int \frac {x^4}{\left (4 x^2-4 x-\log \left (x+e^4+1\right )\right )^5}dx-\int \frac {x^3}{\left (4 x^2-4 x-\log \left (x+e^4+1\right )\right )^5}dx-\int \frac {x^3}{\left (4 x^2-4 x-\log \left (x+e^4+1\right )\right )^4}dx\right )\) |
Input:
Int[(-4*x^4 + 16*x^5 + 16*E^4*x^5 + 16*x^6 + (4*x^3 + 4*E^4*x^3 + 4*x^4)*L og[1 + E^4 + x])/(1024*x^5 - 4096*x^6 + 5120*x^7 - 5120*x^9 + 4096*x^10 - 1024*x^11 + E^4*(1024*x^5 - 5120*x^6 + 10240*x^7 - 10240*x^8 + 5120*x^9 - 1024*x^10) + (1280*x^4 - 3840*x^5 + 2560*x^6 + 2560*x^7 - 3840*x^8 + 1280* x^9 + E^4*(1280*x^4 - 5120*x^5 + 7680*x^6 - 5120*x^7 + 1280*x^8))*Log[1 + E^4 + x] + (640*x^3 - 1280*x^4 + 1280*x^6 - 640*x^7 + E^4*(640*x^3 - 1920* x^4 + 1920*x^5 - 640*x^6))*Log[1 + E^4 + x]^2 + (160*x^2 - 160*x^3 - 160*x ^4 + 160*x^5 + E^4*(160*x^2 - 320*x^3 + 160*x^4))*Log[1 + E^4 + x]^3 + (20 *x - 20*x^3 + E^4*(20*x - 20*x^2))*Log[1 + E^4 + x]^4 + (1 + E^4 + x)*Log[ 1 + E^4 + x]^5),x]
Output:
$Aborted
Time = 2.14 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04
| method | result | size |
| risch | \(\frac {x^{4}}{\left (4 x^{2}-4 x -\ln \left ({\mathrm e}^{4}+x +1\right )\right )^{4}}\) | \(24\) |
| parallelrisch | \(\frac {x^{4}}{256 x^{8}-1024 x^{7}-256 \ln \left ({\mathrm e}^{4}+x +1\right ) x^{6}+1536 x^{6}+768 \ln \left ({\mathrm e}^{4}+x +1\right ) x^{5}+96 \ln \left ({\mathrm e}^{4}+x +1\right )^{2} x^{4}-1024 x^{5}-768 \ln \left ({\mathrm e}^{4}+x +1\right ) x^{4}-192 \ln \left ({\mathrm e}^{4}+x +1\right )^{2} x^{3}-16 \ln \left ({\mathrm e}^{4}+x +1\right )^{3} x^{2}+256 x^{4}+256 x^{3} \ln \left ({\mathrm e}^{4}+x +1\right )+96 x^{2} \ln \left ({\mathrm e}^{4}+x +1\right )^{2}+16 x \ln \left ({\mathrm e}^{4}+x +1\right )^{3}+\ln \left ({\mathrm e}^{4}+x +1\right )^{4}}\) | \(148\) |
| default | \(\frac {{\mathrm e}^{16}-4 \,{\mathrm e}^{12+\ln \left ({\mathrm e}^{4}+x +1\right )}+6 \,{\mathrm e}^{8+2 \ln \left ({\mathrm e}^{4}+x +1\right )}-4 \,{\mathrm e}^{4+3 \ln \left ({\mathrm e}^{4}+x +1\right )}+\left ({\mathrm e}^{4}+x +1\right )^{4}+4 \,{\mathrm e}^{12}-12 \,{\mathrm e}^{8+\ln \left ({\mathrm e}^{4}+x +1\right )}+12 \,{\mathrm e}^{4+2 \ln \left ({\mathrm e}^{4}+x +1\right )}-4 \left ({\mathrm e}^{4}+x +1\right )^{3}+6 \,{\mathrm e}^{8}-12 \,{\mathrm e}^{\ln \left ({\mathrm e}^{4}+x +1\right )+4}+6 \left ({\mathrm e}^{4}+x +1\right )^{2}-4 x -3}{\left (4 \,{\mathrm e}^{8}-8 \,{\mathrm e}^{\ln \left ({\mathrm e}^{4}+x +1\right )+4}+4 \left ({\mathrm e}^{4}+x +1\right )^{2}-12 x -4-\ln \left ({\mathrm e}^{4}+x +1\right )\right )^{4}}\) | \(153\) |
Input:
int(((4*x^3*exp(4)+4*x^4+4*x^3)*ln(exp(4)+x+1)+16*x^5*exp(4)+16*x^6+16*x^5 -4*x^4)/((exp(4)+x+1)*ln(exp(4)+x+1)^5+((-20*x^2+20*x)*exp(4)-20*x^3+20*x) *ln(exp(4)+x+1)^4+((160*x^4-320*x^3+160*x^2)*exp(4)+160*x^5-160*x^4-160*x^ 3+160*x^2)*ln(exp(4)+x+1)^3+((-640*x^6+1920*x^5-1920*x^4+640*x^3)*exp(4)-6 40*x^7+1280*x^6-1280*x^4+640*x^3)*ln(exp(4)+x+1)^2+((1280*x^8-5120*x^7+768 0*x^6-5120*x^5+1280*x^4)*exp(4)+1280*x^9-3840*x^8+2560*x^7+2560*x^6-3840*x ^5+1280*x^4)*ln(exp(4)+x+1)+(-1024*x^10+5120*x^9-10240*x^8+10240*x^7-5120* x^6+1024*x^5)*exp(4)-1024*x^11+4096*x^10-5120*x^9+5120*x^7-4096*x^6+1024*x ^5),x,method=_RETURNVERBOSE)
Output:
x^4/(4*x^2-4*x-ln(exp(4)+x+1))^4
Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (23) = 46\).
Time = 0.10 (sec) , antiderivative size = 106, normalized size of antiderivative = 4.61 \[ \int \frac {-4 x^4+16 x^5+16 e^4 x^5+16 x^6+\left (4 x^3+4 e^4 x^3+4 x^4\right ) \log \left (1+e^4+x\right )}{1024 x^5-4096 x^6+5120 x^7-5120 x^9+4096 x^{10}-1024 x^{11}+e^4 \left (1024 x^5-5120 x^6+10240 x^7-10240 x^8+5120 x^9-1024 x^{10}\right )+\left (1280 x^4-3840 x^5+2560 x^6+2560 x^7-3840 x^8+1280 x^9+e^4 \left (1280 x^4-5120 x^5+7680 x^6-5120 x^7+1280 x^8\right )\right ) \log \left (1+e^4+x\right )+\left (640 x^3-1280 x^4+1280 x^6-640 x^7+e^4 \left (640 x^3-1920 x^4+1920 x^5-640 x^6\right )\right ) \log ^2\left (1+e^4+x\right )+\left (160 x^2-160 x^3-160 x^4+160 x^5+e^4 \left (160 x^2-320 x^3+160 x^4\right )\right ) \log ^3\left (1+e^4+x\right )+\left (20 x-20 x^3+e^4 \left (20 x-20 x^2\right )\right ) \log ^4\left (1+e^4+x\right )+\left (1+e^4+x\right ) \log ^5\left (1+e^4+x\right )} \, dx=\frac {x^{4}}{256 \, x^{8} - 1024 \, x^{7} + 1536 \, x^{6} - 1024 \, x^{5} + 256 \, x^{4} - 16 \, {\left (x^{2} - x\right )} \log \left (x + e^{4} + 1\right )^{3} + \log \left (x + e^{4} + 1\right )^{4} + 96 \, {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} \log \left (x + e^{4} + 1\right )^{2} - 256 \, {\left (x^{6} - 3 \, x^{5} + 3 \, x^{4} - x^{3}\right )} \log \left (x + e^{4} + 1\right )} \] Input:
integrate(((4*x^3*exp(4)+4*x^4+4*x^3)*log(exp(4)+x+1)+16*x^5*exp(4)+16*x^6 +16*x^5-4*x^4)/((exp(4)+x+1)*log(exp(4)+x+1)^5+((-20*x^2+20*x)*exp(4)-20*x ^3+20*x)*log(exp(4)+x+1)^4+((160*x^4-320*x^3+160*x^2)*exp(4)+160*x^5-160*x ^4-160*x^3+160*x^2)*log(exp(4)+x+1)^3+((-640*x^6+1920*x^5-1920*x^4+640*x^3 )*exp(4)-640*x^7+1280*x^6-1280*x^4+640*x^3)*log(exp(4)+x+1)^2+((1280*x^8-5 120*x^7+7680*x^6-5120*x^5+1280*x^4)*exp(4)+1280*x^9-3840*x^8+2560*x^7+2560 *x^6-3840*x^5+1280*x^4)*log(exp(4)+x+1)+(-1024*x^10+5120*x^9-10240*x^8+102 40*x^7-5120*x^6+1024*x^5)*exp(4)-1024*x^11+4096*x^10-5120*x^9+5120*x^7-409 6*x^6+1024*x^5),x, algorithm="fricas")
Output:
x^4/(256*x^8 - 1024*x^7 + 1536*x^6 - 1024*x^5 + 256*x^4 - 16*(x^2 - x)*log (x + e^4 + 1)^3 + log(x + e^4 + 1)^4 + 96*(x^4 - 2*x^3 + x^2)*log(x + e^4 + 1)^2 - 256*(x^6 - 3*x^5 + 3*x^4 - x^3)*log(x + e^4 + 1))
Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (20) = 40\).
Time = 0.21 (sec) , antiderivative size = 110, normalized size of antiderivative = 4.78 \[ \int \frac {-4 x^4+16 x^5+16 e^4 x^5+16 x^6+\left (4 x^3+4 e^4 x^3+4 x^4\right ) \log \left (1+e^4+x\right )}{1024 x^5-4096 x^6+5120 x^7-5120 x^9+4096 x^{10}-1024 x^{11}+e^4 \left (1024 x^5-5120 x^6+10240 x^7-10240 x^8+5120 x^9-1024 x^{10}\right )+\left (1280 x^4-3840 x^5+2560 x^6+2560 x^7-3840 x^8+1280 x^9+e^4 \left (1280 x^4-5120 x^5+7680 x^6-5120 x^7+1280 x^8\right )\right ) \log \left (1+e^4+x\right )+\left (640 x^3-1280 x^4+1280 x^6-640 x^7+e^4 \left (640 x^3-1920 x^4+1920 x^5-640 x^6\right )\right ) \log ^2\left (1+e^4+x\right )+\left (160 x^2-160 x^3-160 x^4+160 x^5+e^4 \left (160 x^2-320 x^3+160 x^4\right )\right ) \log ^3\left (1+e^4+x\right )+\left (20 x-20 x^3+e^4 \left (20 x-20 x^2\right )\right ) \log ^4\left (1+e^4+x\right )+\left (1+e^4+x\right ) \log ^5\left (1+e^4+x\right )} \, dx=\frac {x^{4}}{256 x^{8} - 1024 x^{7} + 1536 x^{6} - 1024 x^{5} + 256 x^{4} + \left (- 16 x^{2} + 16 x\right ) \log {\left (x + 1 + e^{4} \right )}^{3} + \left (96 x^{4} - 192 x^{3} + 96 x^{2}\right ) \log {\left (x + 1 + e^{4} \right )}^{2} + \left (- 256 x^{6} + 768 x^{5} - 768 x^{4} + 256 x^{3}\right ) \log {\left (x + 1 + e^{4} \right )} + \log {\left (x + 1 + e^{4} \right )}^{4}} \] Input:
integrate(((4*x**3*exp(4)+4*x**4+4*x**3)*ln(exp(4)+x+1)+16*x**5*exp(4)+16* x**6+16*x**5-4*x**4)/((exp(4)+x+1)*ln(exp(4)+x+1)**5+((-20*x**2+20*x)*exp( 4)-20*x**3+20*x)*ln(exp(4)+x+1)**4+((160*x**4-320*x**3+160*x**2)*exp(4)+16 0*x**5-160*x**4-160*x**3+160*x**2)*ln(exp(4)+x+1)**3+((-640*x**6+1920*x**5 -1920*x**4+640*x**3)*exp(4)-640*x**7+1280*x**6-1280*x**4+640*x**3)*ln(exp( 4)+x+1)**2+((1280*x**8-5120*x**7+7680*x**6-5120*x**5+1280*x**4)*exp(4)+128 0*x**9-3840*x**8+2560*x**7+2560*x**6-3840*x**5+1280*x**4)*ln(exp(4)+x+1)+( -1024*x**10+5120*x**9-10240*x**8+10240*x**7-5120*x**6+1024*x**5)*exp(4)-10 24*x**11+4096*x**10-5120*x**9+5120*x**7-4096*x**6+1024*x**5),x)
Output:
x**4/(256*x**8 - 1024*x**7 + 1536*x**6 - 1024*x**5 + 256*x**4 + (-16*x**2 + 16*x)*log(x + 1 + exp(4))**3 + (96*x**4 - 192*x**3 + 96*x**2)*log(x + 1 + exp(4))**2 + (-256*x**6 + 768*x**5 - 768*x**4 + 256*x**3)*log(x + 1 + ex p(4)) + log(x + 1 + exp(4))**4)
Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (23) = 46\).
Time = 0.33 (sec) , antiderivative size = 106, normalized size of antiderivative = 4.61 \[ \int \frac {-4 x^4+16 x^5+16 e^4 x^5+16 x^6+\left (4 x^3+4 e^4 x^3+4 x^4\right ) \log \left (1+e^4+x\right )}{1024 x^5-4096 x^6+5120 x^7-5120 x^9+4096 x^{10}-1024 x^{11}+e^4 \left (1024 x^5-5120 x^6+10240 x^7-10240 x^8+5120 x^9-1024 x^{10}\right )+\left (1280 x^4-3840 x^5+2560 x^6+2560 x^7-3840 x^8+1280 x^9+e^4 \left (1280 x^4-5120 x^5+7680 x^6-5120 x^7+1280 x^8\right )\right ) \log \left (1+e^4+x\right )+\left (640 x^3-1280 x^4+1280 x^6-640 x^7+e^4 \left (640 x^3-1920 x^4+1920 x^5-640 x^6\right )\right ) \log ^2\left (1+e^4+x\right )+\left (160 x^2-160 x^3-160 x^4+160 x^5+e^4 \left (160 x^2-320 x^3+160 x^4\right )\right ) \log ^3\left (1+e^4+x\right )+\left (20 x-20 x^3+e^4 \left (20 x-20 x^2\right )\right ) \log ^4\left (1+e^4+x\right )+\left (1+e^4+x\right ) \log ^5\left (1+e^4+x\right )} \, dx=\frac {x^{4}}{256 \, x^{8} - 1024 \, x^{7} + 1536 \, x^{6} - 1024 \, x^{5} + 256 \, x^{4} - 16 \, {\left (x^{2} - x\right )} \log \left (x + e^{4} + 1\right )^{3} + \log \left (x + e^{4} + 1\right )^{4} + 96 \, {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} \log \left (x + e^{4} + 1\right )^{2} - 256 \, {\left (x^{6} - 3 \, x^{5} + 3 \, x^{4} - x^{3}\right )} \log \left (x + e^{4} + 1\right )} \] Input:
integrate(((4*x^3*exp(4)+4*x^4+4*x^3)*log(exp(4)+x+1)+16*x^5*exp(4)+16*x^6 +16*x^5-4*x^4)/((exp(4)+x+1)*log(exp(4)+x+1)^5+((-20*x^2+20*x)*exp(4)-20*x ^3+20*x)*log(exp(4)+x+1)^4+((160*x^4-320*x^3+160*x^2)*exp(4)+160*x^5-160*x ^4-160*x^3+160*x^2)*log(exp(4)+x+1)^3+((-640*x^6+1920*x^5-1920*x^4+640*x^3 )*exp(4)-640*x^7+1280*x^6-1280*x^4+640*x^3)*log(exp(4)+x+1)^2+((1280*x^8-5 120*x^7+7680*x^6-5120*x^5+1280*x^4)*exp(4)+1280*x^9-3840*x^8+2560*x^7+2560 *x^6-3840*x^5+1280*x^4)*log(exp(4)+x+1)+(-1024*x^10+5120*x^9-10240*x^8+102 40*x^7-5120*x^6+1024*x^5)*exp(4)-1024*x^11+4096*x^10-5120*x^9+5120*x^7-409 6*x^6+1024*x^5),x, algorithm="maxima")
Output:
x^4/(256*x^8 - 1024*x^7 + 1536*x^6 - 1024*x^5 + 256*x^4 - 16*(x^2 - x)*log (x + e^4 + 1)^3 + log(x + e^4 + 1)^4 + 96*(x^4 - 2*x^3 + x^2)*log(x + e^4 + 1)^2 - 256*(x^6 - 3*x^5 + 3*x^4 - x^3)*log(x + e^4 + 1))
Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (23) = 46\).
Time = 1.10 (sec) , antiderivative size = 147, normalized size of antiderivative = 6.39 \[ \int \frac {-4 x^4+16 x^5+16 e^4 x^5+16 x^6+\left (4 x^3+4 e^4 x^3+4 x^4\right ) \log \left (1+e^4+x\right )}{1024 x^5-4096 x^6+5120 x^7-5120 x^9+4096 x^{10}-1024 x^{11}+e^4 \left (1024 x^5-5120 x^6+10240 x^7-10240 x^8+5120 x^9-1024 x^{10}\right )+\left (1280 x^4-3840 x^5+2560 x^6+2560 x^7-3840 x^8+1280 x^9+e^4 \left (1280 x^4-5120 x^5+7680 x^6-5120 x^7+1280 x^8\right )\right ) \log \left (1+e^4+x\right )+\left (640 x^3-1280 x^4+1280 x^6-640 x^7+e^4 \left (640 x^3-1920 x^4+1920 x^5-640 x^6\right )\right ) \log ^2\left (1+e^4+x\right )+\left (160 x^2-160 x^3-160 x^4+160 x^5+e^4 \left (160 x^2-320 x^3+160 x^4\right )\right ) \log ^3\left (1+e^4+x\right )+\left (20 x-20 x^3+e^4 \left (20 x-20 x^2\right )\right ) \log ^4\left (1+e^4+x\right )+\left (1+e^4+x\right ) \log ^5\left (1+e^4+x\right )} \, dx=\frac {x^{4}}{256 \, x^{8} - 1024 \, x^{7} - 256 \, x^{6} \log \left (x + e^{4} + 1\right ) + 1536 \, x^{6} + 768 \, x^{5} \log \left (x + e^{4} + 1\right ) + 96 \, x^{4} \log \left (x + e^{4} + 1\right )^{2} - 1024 \, x^{5} - 768 \, x^{4} \log \left (x + e^{4} + 1\right ) - 192 \, x^{3} \log \left (x + e^{4} + 1\right )^{2} - 16 \, x^{2} \log \left (x + e^{4} + 1\right )^{3} + 256 \, x^{4} + 256 \, x^{3} \log \left (x + e^{4} + 1\right ) + 96 \, x^{2} \log \left (x + e^{4} + 1\right )^{2} + 16 \, x \log \left (x + e^{4} + 1\right )^{3} + \log \left (x + e^{4} + 1\right )^{4}} \] Input:
integrate(((4*x^3*exp(4)+4*x^4+4*x^3)*log(exp(4)+x+1)+16*x^5*exp(4)+16*x^6 +16*x^5-4*x^4)/((exp(4)+x+1)*log(exp(4)+x+1)^5+((-20*x^2+20*x)*exp(4)-20*x ^3+20*x)*log(exp(4)+x+1)^4+((160*x^4-320*x^3+160*x^2)*exp(4)+160*x^5-160*x ^4-160*x^3+160*x^2)*log(exp(4)+x+1)^3+((-640*x^6+1920*x^5-1920*x^4+640*x^3 )*exp(4)-640*x^7+1280*x^6-1280*x^4+640*x^3)*log(exp(4)+x+1)^2+((1280*x^8-5 120*x^7+7680*x^6-5120*x^5+1280*x^4)*exp(4)+1280*x^9-3840*x^8+2560*x^7+2560 *x^6-3840*x^5+1280*x^4)*log(exp(4)+x+1)+(-1024*x^10+5120*x^9-10240*x^8+102 40*x^7-5120*x^6+1024*x^5)*exp(4)-1024*x^11+4096*x^10-5120*x^9+5120*x^7-409 6*x^6+1024*x^5),x, algorithm="giac")
Output:
x^4/(256*x^8 - 1024*x^7 - 256*x^6*log(x + e^4 + 1) + 1536*x^6 + 768*x^5*lo g(x + e^4 + 1) + 96*x^4*log(x + e^4 + 1)^2 - 1024*x^5 - 768*x^4*log(x + e^ 4 + 1) - 192*x^3*log(x + e^4 + 1)^2 - 16*x^2*log(x + e^4 + 1)^3 + 256*x^4 + 256*x^3*log(x + e^4 + 1) + 96*x^2*log(x + e^4 + 1)^2 + 16*x*log(x + e^4 + 1)^3 + log(x + e^4 + 1)^4)
Timed out. \[ \int \frac {-4 x^4+16 x^5+16 e^4 x^5+16 x^6+\left (4 x^3+4 e^4 x^3+4 x^4\right ) \log \left (1+e^4+x\right )}{1024 x^5-4096 x^6+5120 x^7-5120 x^9+4096 x^{10}-1024 x^{11}+e^4 \left (1024 x^5-5120 x^6+10240 x^7-10240 x^8+5120 x^9-1024 x^{10}\right )+\left (1280 x^4-3840 x^5+2560 x^6+2560 x^7-3840 x^8+1280 x^9+e^4 \left (1280 x^4-5120 x^5+7680 x^6-5120 x^7+1280 x^8\right )\right ) \log \left (1+e^4+x\right )+\left (640 x^3-1280 x^4+1280 x^6-640 x^7+e^4 \left (640 x^3-1920 x^4+1920 x^5-640 x^6\right )\right ) \log ^2\left (1+e^4+x\right )+\left (160 x^2-160 x^3-160 x^4+160 x^5+e^4 \left (160 x^2-320 x^3+160 x^4\right )\right ) \log ^3\left (1+e^4+x\right )+\left (20 x-20 x^3+e^4 \left (20 x-20 x^2\right )\right ) \log ^4\left (1+e^4+x\right )+\left (1+e^4+x\right ) \log ^5\left (1+e^4+x\right )} \, dx=\int \frac {\ln \left (x+{\mathrm {e}}^4+1\right )\,\left (4\,x^3\,{\mathrm {e}}^4+4\,x^3+4\,x^4\right )+16\,x^5\,{\mathrm {e}}^4-4\,x^4+16\,x^5+16\,x^6}{{\ln \left (x+{\mathrm {e}}^4+1\right )}^4\,\left (20\,x+{\mathrm {e}}^4\,\left (20\,x-20\,x^2\right )-20\,x^3\right )+{\ln \left (x+{\mathrm {e}}^4+1\right )}^3\,\left ({\mathrm {e}}^4\,\left (160\,x^4-320\,x^3+160\,x^2\right )+160\,x^2-160\,x^3-160\,x^4+160\,x^5\right )+{\mathrm {e}}^4\,\left (-1024\,x^{10}+5120\,x^9-10240\,x^8+10240\,x^7-5120\,x^6+1024\,x^5\right )+{\ln \left (x+{\mathrm {e}}^4+1\right )}^2\,\left (640\,x^3-1280\,x^4+1280\,x^6-640\,x^7+{\mathrm {e}}^4\,\left (-640\,x^6+1920\,x^5-1920\,x^4+640\,x^3\right )\right )+{\ln \left (x+{\mathrm {e}}^4+1\right )}^5\,\left (x+{\mathrm {e}}^4+1\right )+1024\,x^5-4096\,x^6+5120\,x^7-5120\,x^9+4096\,x^{10}-1024\,x^{11}+\ln \left (x+{\mathrm {e}}^4+1\right )\,\left ({\mathrm {e}}^4\,\left (1280\,x^8-5120\,x^7+7680\,x^6-5120\,x^5+1280\,x^4\right )+1280\,x^4-3840\,x^5+2560\,x^6+2560\,x^7-3840\,x^8+1280\,x^9\right )} \,d x \] Input:
int((log(x + exp(4) + 1)*(4*x^3*exp(4) + 4*x^3 + 4*x^4) + 16*x^5*exp(4) - 4*x^4 + 16*x^5 + 16*x^6)/(log(x + exp(4) + 1)^4*(20*x + exp(4)*(20*x - 20* x^2) - 20*x^3) + log(x + exp(4) + 1)^3*(exp(4)*(160*x^2 - 320*x^3 + 160*x^ 4) + 160*x^2 - 160*x^3 - 160*x^4 + 160*x^5) + exp(4)*(1024*x^5 - 5120*x^6 + 10240*x^7 - 10240*x^8 + 5120*x^9 - 1024*x^10) + log(x + exp(4) + 1)^2*(6 40*x^3 - 1280*x^4 + 1280*x^6 - 640*x^7 + exp(4)*(640*x^3 - 1920*x^4 + 1920 *x^5 - 640*x^6)) + log(x + exp(4) + 1)^5*(x + exp(4) + 1) + 1024*x^5 - 409 6*x^6 + 5120*x^7 - 5120*x^9 + 4096*x^10 - 1024*x^11 + log(x + exp(4) + 1)* (exp(4)*(1280*x^4 - 5120*x^5 + 7680*x^6 - 5120*x^7 + 1280*x^8) + 1280*x^4 - 3840*x^5 + 2560*x^6 + 2560*x^7 - 3840*x^8 + 1280*x^9)),x)
Output:
int((log(x + exp(4) + 1)*(4*x^3*exp(4) + 4*x^3 + 4*x^4) + 16*x^5*exp(4) - 4*x^4 + 16*x^5 + 16*x^6)/(log(x + exp(4) + 1)^4*(20*x + exp(4)*(20*x - 20* x^2) - 20*x^3) + log(x + exp(4) + 1)^3*(exp(4)*(160*x^2 - 320*x^3 + 160*x^ 4) + 160*x^2 - 160*x^3 - 160*x^4 + 160*x^5) + exp(4)*(1024*x^5 - 5120*x^6 + 10240*x^7 - 10240*x^8 + 5120*x^9 - 1024*x^10) + log(x + exp(4) + 1)^2*(6 40*x^3 - 1280*x^4 + 1280*x^6 - 640*x^7 + exp(4)*(640*x^3 - 1920*x^4 + 1920 *x^5 - 640*x^6)) + log(x + exp(4) + 1)^5*(x + exp(4) + 1) + 1024*x^5 - 409 6*x^6 + 5120*x^7 - 5120*x^9 + 4096*x^10 - 1024*x^11 + log(x + exp(4) + 1)* (exp(4)*(1280*x^4 - 5120*x^5 + 7680*x^6 - 5120*x^7 + 1280*x^8) + 1280*x^4 - 3840*x^5 + 2560*x^6 + 2560*x^7 - 3840*x^8 + 1280*x^9)), x)
Time = 0.17 (sec) , antiderivative size = 304, normalized size of antiderivative = 13.22 \[ \int \frac {-4 x^4+16 x^5+16 e^4 x^5+16 x^6+\left (4 x^3+4 e^4 x^3+4 x^4\right ) \log \left (1+e^4+x\right )}{1024 x^5-4096 x^6+5120 x^7-5120 x^9+4096 x^{10}-1024 x^{11}+e^4 \left (1024 x^5-5120 x^6+10240 x^7-10240 x^8+5120 x^9-1024 x^{10}\right )+\left (1280 x^4-3840 x^5+2560 x^6+2560 x^7-3840 x^8+1280 x^9+e^4 \left (1280 x^4-5120 x^5+7680 x^6-5120 x^7+1280 x^8\right )\right ) \log \left (1+e^4+x\right )+\left (640 x^3-1280 x^4+1280 x^6-640 x^7+e^4 \left (640 x^3-1920 x^4+1920 x^5-640 x^6\right )\right ) \log ^2\left (1+e^4+x\right )+\left (160 x^2-160 x^3-160 x^4+160 x^5+e^4 \left (160 x^2-320 x^3+160 x^4\right )\right ) \log ^3\left (1+e^4+x\right )+\left (20 x-20 x^3+e^4 \left (20 x-20 x^2\right )\right ) \log ^4\left (1+e^4+x\right )+\left (1+e^4+x\right ) \log ^5\left (1+e^4+x\right )} \, dx=\frac {-\mathrm {log}\left (e^{4}+x +1\right )^{4}+16 \mathrm {log}\left (e^{4}+x +1\right )^{3} x^{2}-16 \mathrm {log}\left (e^{4}+x +1\right )^{3} x -96 \mathrm {log}\left (e^{4}+x +1\right )^{2} x^{4}+192 \mathrm {log}\left (e^{4}+x +1\right )^{2} x^{3}-96 \mathrm {log}\left (e^{4}+x +1\right )^{2} x^{2}+256 \,\mathrm {log}\left (e^{4}+x +1\right ) x^{6}-768 \,\mathrm {log}\left (e^{4}+x +1\right ) x^{5}+768 \,\mathrm {log}\left (e^{4}+x +1\right ) x^{4}-256 \,\mathrm {log}\left (e^{4}+x +1\right ) x^{3}-256 x^{8}+1024 x^{7}-1536 x^{6}+1024 x^{5}}{256 \mathrm {log}\left (e^{4}+x +1\right )^{4}-4096 \mathrm {log}\left (e^{4}+x +1\right )^{3} x^{2}+4096 \mathrm {log}\left (e^{4}+x +1\right )^{3} x +24576 \mathrm {log}\left (e^{4}+x +1\right )^{2} x^{4}-49152 \mathrm {log}\left (e^{4}+x +1\right )^{2} x^{3}+24576 \mathrm {log}\left (e^{4}+x +1\right )^{2} x^{2}-65536 \,\mathrm {log}\left (e^{4}+x +1\right ) x^{6}+196608 \,\mathrm {log}\left (e^{4}+x +1\right ) x^{5}-196608 \,\mathrm {log}\left (e^{4}+x +1\right ) x^{4}+65536 \,\mathrm {log}\left (e^{4}+x +1\right ) x^{3}+65536 x^{8}-262144 x^{7}+393216 x^{6}-262144 x^{5}+65536 x^{4}} \] Input:
int(((4*x^3*exp(4)+4*x^4+4*x^3)*log(exp(4)+x+1)+16*x^5*exp(4)+16*x^6+16*x^ 5-4*x^4)/((exp(4)+x+1)*log(exp(4)+x+1)^5+((-20*x^2+20*x)*exp(4)-20*x^3+20* x)*log(exp(4)+x+1)^4+((160*x^4-320*x^3+160*x^2)*exp(4)+160*x^5-160*x^4-160 *x^3+160*x^2)*log(exp(4)+x+1)^3+((-640*x^6+1920*x^5-1920*x^4+640*x^3)*exp( 4)-640*x^7+1280*x^6-1280*x^4+640*x^3)*log(exp(4)+x+1)^2+((1280*x^8-5120*x^ 7+7680*x^6-5120*x^5+1280*x^4)*exp(4)+1280*x^9-3840*x^8+2560*x^7+2560*x^6-3 840*x^5+1280*x^4)*log(exp(4)+x+1)+(-1024*x^10+5120*x^9-10240*x^8+10240*x^7 -5120*x^6+1024*x^5)*exp(4)-1024*x^11+4096*x^10-5120*x^9+5120*x^7-4096*x^6+ 1024*x^5),x)
Output:
( - log(e**4 + x + 1)**4 + 16*log(e**4 + x + 1)**3*x**2 - 16*log(e**4 + x + 1)**3*x - 96*log(e**4 + x + 1)**2*x**4 + 192*log(e**4 + x + 1)**2*x**3 - 96*log(e**4 + x + 1)**2*x**2 + 256*log(e**4 + x + 1)*x**6 - 768*log(e**4 + x + 1)*x**5 + 768*log(e**4 + x + 1)*x**4 - 256*log(e**4 + x + 1)*x**3 - 256*x**8 + 1024*x**7 - 1536*x**6 + 1024*x**5)/(256*(log(e**4 + x + 1)**4 - 16*log(e**4 + x + 1)**3*x**2 + 16*log(e**4 + x + 1)**3*x + 96*log(e**4 + x + 1)**2*x**4 - 192*log(e**4 + x + 1)**2*x**3 + 96*log(e**4 + x + 1)**2*x **2 - 256*log(e**4 + x + 1)*x**6 + 768*log(e**4 + x + 1)*x**5 - 768*log(e* *4 + x + 1)*x**4 + 256*log(e**4 + x + 1)*x**3 + 256*x**8 - 1024*x**7 + 153 6*x**6 - 1024*x**5 + 256*x**4))