Integrand size = 284, antiderivative size = 33 \[ \int \frac {\left (-800 x^5+2000 x^6+\left (800 x^4-3200 x^5\right ) \log (x)+1200 x^4 \log ^2(x)\right ) \log ^2(\log (16))+\left (-12800 x^3+38400 x^4-25600 x^3 \log (x)\right ) \log ^3(\log (16))+\left (e^x (256-256 x)+102400 x^2\right ) \log ^4(\log (16))}{625 x^{12}-2500 x^{11} \log (x)+3750 x^{10} \log ^2(x)-2500 x^9 \log ^3(x)+625 x^8 \log ^4(x)+\left (40000 x^{10}-120000 x^9 \log (x)+120000 x^8 \log ^2(x)-40000 x^7 \log ^3(x)\right ) \log (\log (16))+\left (-800 e^x x^6+960000 x^8+\left (1600 e^x x^5-1920000 x^7\right ) \log (x)+\left (-800 e^x x^4+960000 x^6\right ) \log ^2(x)\right ) \log ^2(\log (16))+\left (-25600 e^x x^4+10240000 x^6+\left (25600 e^x x^3-10240000 x^5\right ) \log (x)\right ) \log ^3(\log (16))+\left (256 e^{2 x}-204800 e^x x^2+40960000 x^4\right ) \log ^4(\log (16))} \, dx=\frac {x}{e^x-25 x^2 \left (-4+\frac {x (-x+\log (x))}{4 \log (\log (16))}\right )^2} \] Output:
x/(exp(x)-25*x^2*(1/4*(ln(x)-x)/ln(4*ln(2))*x-4)^2)
Leaf count is larger than twice the leaf count of optimal. \(70\) vs. \(2(33)=66\).
Time = 0.10 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.12 \[ \int \frac {\left (-800 x^5+2000 x^6+\left (800 x^4-3200 x^5\right ) \log (x)+1200 x^4 \log ^2(x)\right ) \log ^2(\log (16))+\left (-12800 x^3+38400 x^4-25600 x^3 \log (x)\right ) \log ^3(\log (16))+\left (e^x (256-256 x)+102400 x^2\right ) \log ^4(\log (16))}{625 x^{12}-2500 x^{11} \log (x)+3750 x^{10} \log ^2(x)-2500 x^9 \log ^3(x)+625 x^8 \log ^4(x)+\left (40000 x^{10}-120000 x^9 \log (x)+120000 x^8 \log ^2(x)-40000 x^7 \log ^3(x)\right ) \log (\log (16))+\left (-800 e^x x^6+960000 x^8+\left (1600 e^x x^5-1920000 x^7\right ) \log (x)+\left (-800 e^x x^4+960000 x^6\right ) \log ^2(x)\right ) \log ^2(\log (16))+\left (-25600 e^x x^4+10240000 x^6+\left (25600 e^x x^3-10240000 x^5\right ) \log (x)\right ) \log ^3(\log (16))+\left (256 e^{2 x}-204800 e^x x^2+40960000 x^4\right ) \log ^4(\log (16))} \, dx=-\frac {16 x \log ^2(\log (16))}{25 x^6-50 x^5 \log (x)+25 x^4 \log ^2(x)+800 x^4 \log (\log (16))-800 x^3 \log (x) \log (\log (16))-16 e^x \log ^2(\log (16))+6400 x^2 \log ^2(\log (16))} \] Input:
Integrate[((-800*x^5 + 2000*x^6 + (800*x^4 - 3200*x^5)*Log[x] + 1200*x^4*L og[x]^2)*Log[Log[16]]^2 + (-12800*x^3 + 38400*x^4 - 25600*x^3*Log[x])*Log[ Log[16]]^3 + (E^x*(256 - 256*x) + 102400*x^2)*Log[Log[16]]^4)/(625*x^12 - 2500*x^11*Log[x] + 3750*x^10*Log[x]^2 - 2500*x^9*Log[x]^3 + 625*x^8*Log[x] ^4 + (40000*x^10 - 120000*x^9*Log[x] + 120000*x^8*Log[x]^2 - 40000*x^7*Log [x]^3)*Log[Log[16]] + (-800*E^x*x^6 + 960000*x^8 + (1600*E^x*x^5 - 1920000 *x^7)*Log[x] + (-800*E^x*x^4 + 960000*x^6)*Log[x]^2)*Log[Log[16]]^2 + (-25 600*E^x*x^4 + 10240000*x^6 + (25600*E^x*x^3 - 10240000*x^5)*Log[x])*Log[Lo g[16]]^3 + (256*E^(2*x) - 204800*E^x*x^2 + 40960000*x^4)*Log[Log[16]]^4),x ]
Output:
(-16*x*Log[Log[16]]^2)/(25*x^6 - 50*x^5*Log[x] + 25*x^4*Log[x]^2 + 800*x^4 *Log[Log[16]] - 800*x^3*Log[x]*Log[Log[16]] - 16*E^x*Log[Log[16]]^2 + 6400 *x^2*Log[Log[16]]^2)
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.
| \(\displaystyle \int \frac {\left (102400 x^2+e^x (256-256 x)\right ) \log ^4(\log (16))+\log ^3(\log (16)) \left (38400 x^4-12800 x^3-25600 x^3 \log (x)\right )+\log ^2(\log (16)) \left (2000 x^6-800 x^5+1200 x^4 \log ^2(x)+\left (800 x^4-3200 x^5\right ) \log (x)\right )}{625 x^{12}-2500 x^{11} \log (x)+3750 x^{10} \log ^2(x)-2500 x^9 \log ^3(x)+625 x^8 \log ^4(x)+\left (40960000 x^4-204800 e^x x^2+256 e^{2 x}\right ) \log ^4(\log (16))+\log (\log (16)) \left (40000 x^{10}-120000 x^9 \log (x)+120000 x^8 \log ^2(x)-40000 x^7 \log ^3(x)\right )+\log ^3(\log (16)) \left (10240000 x^6-25600 e^x x^4+\left (25600 e^x x^3-10240000 x^5\right ) \log (x)\right )+\log ^2(\log (16)) \left (960000 x^8-800 e^x x^6+\left (1600 e^x x^5-1920000 x^7\right ) \log (x)+\left (960000 x^6-800 e^x x^4\right ) \log ^2(x)\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {16 \log ^2(\log (16)) \left (125 x^6-50 x^5+75 x^4 \log ^2(x)+2400 x^4 \log (\log (16))-800 x^3 \log (\log (16))+6400 x^2 \log ^2(\log (16))-50 x^3 \log (x) \left (4 x^2-x+32 \log (\log (16))\right )-16 e^x x \log ^2(\log (16))+16 e^x \log ^2(\log (16))\right )}{\left (25 x^6+25 x^4 \log ^2(x)+800 x^4 \log (\log (16))+6400 x^2 \log ^2(\log (16))-50 \log (x) \left (x^5+16 x^3 \log (\log (16))\right )-16 e^x \log ^2(\log (16))\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 16 \log ^2(\log (16)) \int -\frac {-125 x^6+50 x^5-75 \log ^2(x) x^4-2400 \log (\log (16)) x^4-50 \log (x) \left (-4 x^2+x-32 \log (\log (16))\right ) x^3+800 \log (\log (16)) x^3-6400 \log ^2(\log (16)) x^2+16 e^x \log ^2(\log (16)) x-16 e^x \log ^2(\log (16))}{\left (25 x^6+25 \log ^2(x) x^4+800 \log (\log (16)) x^4+6400 \log ^2(\log (16)) x^2-50 \log (x) \left (x^5+16 \log (\log (16)) x^3\right )-16 e^x \log ^2(\log (16))\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -16 \log ^2(\log (16)) \int \frac {-125 x^6+50 x^5-75 \log ^2(x) x^4-2400 \log (\log (16)) x^4-50 \log (x) \left (-4 x^2+x-32 \log (\log (16))\right ) x^3+800 \log (\log (16)) x^3-6400 \log ^2(\log (16)) x^2+16 e^x \log ^2(\log (16)) x-16 e^x \log ^2(\log (16))}{\left (25 x^6+25 \log ^2(x) x^4+800 \log (\log (16)) x^4+6400 \log ^2(\log (16)) x^2-50 \log (x) \left (x^5+16 \log (\log (16)) x^3\right )-16 e^x \log ^2(\log (16))\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -16 \log ^2(\log (16)) \int \left (\frac {25 x^2 \left (x^5-2 \log (x) x^4-6 x^4+\log ^2(x) x^3+10 \log (x) x^3+2 (1+16 \log (\log (16))) x^3-4 \log ^2(x) x^2-2 \log (x) (1+16 \log (\log (16))) x^2-128 \log (\log (16)) x^2+32 \log (\log (16)) (1+8 \log (\log (16))) x+96 \log (x) \log (\log (16)) x-512 \log ^2(\log (16))\right )}{\left (25 x^6-50 \log (x) x^5+25 \log ^2(x) x^4+800 \log (\log (16)) x^4-800 \log (x) \log (\log (16)) x^3+6400 \log ^2(\log (16)) x^2-16 e^x \log ^2(\log (16))\right )^2}-\frac {x-1}{25 x^6-50 \log (x) x^5+25 \log ^2(x) x^4+800 \log (\log (16)) x^4-800 \log (x) \log (\log (16)) x^3+6400 \log ^2(\log (16)) x^2-16 e^x \log ^2(\log (16))}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle -16 \log ^2(\log (16)) \int \left (\frac {25 x^2 \left (x^5-2 \log (x) x^4-6 x^4+\log ^2(x) x^3+10 \log (x) x^3+2 (1+16 \log (\log (16))) x^3-4 \log ^2(x) x^2-2 \log (x) (1+16 \log (\log (16))) x^2-128 \log (\log (16)) x^2+32 \log (\log (16)) (1+8 \log (\log (16))) x+96 \log (x) \log (\log (16)) x-512 \log ^2(\log (16))\right )}{\left (25 x^6-50 \log (x) x^5+25 \log ^2(x) x^4+800 \log (\log (16)) x^4-800 \log (x) \log (\log (16)) x^3+6400 \log ^2(\log (16)) x^2-16 e^x \log ^2(\log (16))\right )^2}-\frac {x-1}{25 x^6-50 \log (x) x^5+25 \log ^2(x) x^4+800 \log (\log (16)) x^4-800 \log (x) \log (\log (16)) x^3+6400 \log ^2(\log (16)) x^2-16 e^x \log ^2(\log (16))}\right )dx\) |
Input:
Int[((-800*x^5 + 2000*x^6 + (800*x^4 - 3200*x^5)*Log[x] + 1200*x^4*Log[x]^ 2)*Log[Log[16]]^2 + (-12800*x^3 + 38400*x^4 - 25600*x^3*Log[x])*Log[Log[16 ]]^3 + (E^x*(256 - 256*x) + 102400*x^2)*Log[Log[16]]^4)/(625*x^12 - 2500*x ^11*Log[x] + 3750*x^10*Log[x]^2 - 2500*x^9*Log[x]^3 + 625*x^8*Log[x]^4 + ( 40000*x^10 - 120000*x^9*Log[x] + 120000*x^8*Log[x]^2 - 40000*x^7*Log[x]^3) *Log[Log[16]] + (-800*E^x*x^6 + 960000*x^8 + (1600*E^x*x^5 - 1920000*x^7)* Log[x] + (-800*E^x*x^4 + 960000*x^6)*Log[x]^2)*Log[Log[16]]^2 + (-25600*E^ x*x^4 + 10240000*x^6 + (25600*E^x*x^3 - 10240000*x^5)*Log[x])*Log[Log[16]] ^3 + (256*E^(2*x) - 204800*E^x*x^2 + 40960000*x^4)*Log[Log[16]]^4),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(79\) vs. \(2(32)=64\).
Time = 35.49 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.42
| method | result | size |
| parallelrisch | \(-\frac {16 \ln \left (4 \ln \left (2\right )\right )^{2} x}{25 x^{6}-50 x^{5} \ln \left (x \right )+25 x^{4} \ln \left (x \right )^{2}+800 x^{4} \ln \left (4 \ln \left (2\right )\right )-800 x^{3} \ln \left (x \right ) \ln \left (4 \ln \left (2\right )\right )+6400 x^{2} \ln \left (4 \ln \left (2\right )\right )^{2}-16 \ln \left (4 \ln \left (2\right )\right )^{2} {\mathrm e}^{x}}\) | \(80\) |
| risch | \(-\frac {16 x \left (4 \ln \left (2\right )^{2}+4 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right )+\ln \left (\ln \left (2\right )\right )^{2}\right )}{25 x^{6}-50 x^{5} \ln \left (x \right )+25 x^{4} \ln \left (x \right )^{2}+1600 x^{4} \ln \left (2\right )-1600 \ln \left (x \right ) \ln \left (2\right ) x^{3}+800 \ln \left (\ln \left (2\right )\right ) x^{4}-800 \ln \left (\ln \left (2\right )\right ) \ln \left (x \right ) x^{3}+25600 x^{2} \ln \left (2\right )^{2}+25600 \ln \left (\ln \left (2\right )\right ) \ln \left (2\right ) x^{2}+6400 x^{2} \ln \left (\ln \left (2\right )\right )^{2}-64 \ln \left (2\right )^{2} {\mathrm e}^{x}-64 \ln \left (2\right ) {\mathrm e}^{x} \ln \left (\ln \left (2\right )\right )-16 \,{\mathrm e}^{x} \ln \left (\ln \left (2\right )\right )^{2}}\) | \(136\) |
Input:
int((((-256*x+256)*exp(x)+102400*x^2)*ln(4*ln(2))^4+(-25600*x^3*ln(x)+3840 0*x^4-12800*x^3)*ln(4*ln(2))^3+(1200*x^4*ln(x)^2+(-3200*x^5+800*x^4)*ln(x) +2000*x^6-800*x^5)*ln(4*ln(2))^2)/((256*exp(x)^2-204800*exp(x)*x^2+4096000 0*x^4)*ln(4*ln(2))^4+((25600*exp(x)*x^3-10240000*x^5)*ln(x)-25600*exp(x)*x ^4+10240000*x^6)*ln(4*ln(2))^3+((-800*exp(x)*x^4+960000*x^6)*ln(x)^2+(1600 *x^5*exp(x)-1920000*x^7)*ln(x)-800*x^6*exp(x)+960000*x^8)*ln(4*ln(2))^2+(- 40000*x^7*ln(x)^3+120000*x^8*ln(x)^2-120000*x^9*ln(x)+40000*x^10)*ln(4*ln( 2))+625*x^8*ln(x)^4-2500*x^9*ln(x)^3+3750*x^10*ln(x)^2-2500*x^11*ln(x)+625 *x^12),x,method=_RETURNVERBOSE)
Output:
-16*ln(4*ln(2))^2*x/(25*x^6-50*x^5*ln(x)+25*x^4*ln(x)^2+800*x^4*ln(4*ln(2) )-800*x^3*ln(x)*ln(4*ln(2))+6400*x^2*ln(4*ln(2))^2-16*ln(4*ln(2))^2*exp(x) )
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (34) = 68\).
Time = 0.09 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.15 \[ \int \frac {\left (-800 x^5+2000 x^6+\left (800 x^4-3200 x^5\right ) \log (x)+1200 x^4 \log ^2(x)\right ) \log ^2(\log (16))+\left (-12800 x^3+38400 x^4-25600 x^3 \log (x)\right ) \log ^3(\log (16))+\left (e^x (256-256 x)+102400 x^2\right ) \log ^4(\log (16))}{625 x^{12}-2500 x^{11} \log (x)+3750 x^{10} \log ^2(x)-2500 x^9 \log ^3(x)+625 x^8 \log ^4(x)+\left (40000 x^{10}-120000 x^9 \log (x)+120000 x^8 \log ^2(x)-40000 x^7 \log ^3(x)\right ) \log (\log (16))+\left (-800 e^x x^6+960000 x^8+\left (1600 e^x x^5-1920000 x^7\right ) \log (x)+\left (-800 e^x x^4+960000 x^6\right ) \log ^2(x)\right ) \log ^2(\log (16))+\left (-25600 e^x x^4+10240000 x^6+\left (25600 e^x x^3-10240000 x^5\right ) \log (x)\right ) \log ^3(\log (16))+\left (256 e^{2 x}-204800 e^x x^2+40960000 x^4\right ) \log ^4(\log (16))} \, dx=-\frac {16 \, x \log \left (4 \, \log \left (2\right )\right )^{2}}{25 \, x^{6} - 50 \, x^{5} \log \left (x\right ) + 25 \, x^{4} \log \left (x\right )^{2} + 16 \, {\left (400 \, x^{2} - e^{x}\right )} \log \left (4 \, \log \left (2\right )\right )^{2} + 800 \, {\left (x^{4} - x^{3} \log \left (x\right )\right )} \log \left (4 \, \log \left (2\right )\right )} \] Input:
integrate((((-256*x+256)*exp(x)+102400*x^2)*log(4*log(2))^4+(-25600*x^3*lo g(x)+38400*x^4-12800*x^3)*log(4*log(2))^3+(1200*x^4*log(x)^2+(-3200*x^5+80 0*x^4)*log(x)+2000*x^6-800*x^5)*log(4*log(2))^2)/((256*exp(x)^2-204800*exp (x)*x^2+40960000*x^4)*log(4*log(2))^4+((25600*exp(x)*x^3-10240000*x^5)*log (x)-25600*exp(x)*x^4+10240000*x^6)*log(4*log(2))^3+((-800*exp(x)*x^4+96000 0*x^6)*log(x)^2+(1600*x^5*exp(x)-1920000*x^7)*log(x)-800*x^6*exp(x)+960000 *x^8)*log(4*log(2))^2+(-40000*x^7*log(x)^3+120000*x^8*log(x)^2-120000*x^9* log(x)+40000*x^10)*log(4*log(2))+625*x^8*log(x)^4-2500*x^9*log(x)^3+3750*x ^10*log(x)^2-2500*x^11*log(x)+625*x^12),x, algorithm="fricas")
Output:
-16*x*log(4*log(2))^2/(25*x^6 - 50*x^5*log(x) + 25*x^4*log(x)^2 + 16*(400* x^2 - e^x)*log(4*log(2))^2 + 800*(x^4 - x^3*log(x))*log(4*log(2)))
Leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (29) = 58\).
Time = 1.13 (sec) , antiderivative size = 162, normalized size of antiderivative = 4.91 \[ \int \frac {\left (-800 x^5+2000 x^6+\left (800 x^4-3200 x^5\right ) \log (x)+1200 x^4 \log ^2(x)\right ) \log ^2(\log (16))+\left (-12800 x^3+38400 x^4-25600 x^3 \log (x)\right ) \log ^3(\log (16))+\left (e^x (256-256 x)+102400 x^2\right ) \log ^4(\log (16))}{625 x^{12}-2500 x^{11} \log (x)+3750 x^{10} \log ^2(x)-2500 x^9 \log ^3(x)+625 x^8 \log ^4(x)+\left (40000 x^{10}-120000 x^9 \log (x)+120000 x^8 \log ^2(x)-40000 x^7 \log ^3(x)\right ) \log (\log (16))+\left (-800 e^x x^6+960000 x^8+\left (1600 e^x x^5-1920000 x^7\right ) \log (x)+\left (-800 e^x x^4+960000 x^6\right ) \log ^2(x)\right ) \log ^2(\log (16))+\left (-25600 e^x x^4+10240000 x^6+\left (25600 e^x x^3-10240000 x^5\right ) \log (x)\right ) \log ^3(\log (16))+\left (256 e^{2 x}-204800 e^x x^2+40960000 x^4\right ) \log ^4(\log (16))} \, dx=\frac {64 x \log {\left (2 \right )} \log {\left (\log {\left (2 \right )} \right )} + 16 x \log {\left (\log {\left (2 \right )} \right )}^{2} + 64 x \log {\left (2 \right )}^{2}}{- 25 x^{6} + 50 x^{5} \log {\left (x \right )} - 25 x^{4} \log {\left (x \right )}^{2} - 1600 x^{4} \log {\left (2 \right )} - 800 x^{4} \log {\left (\log {\left (2 \right )} \right )} + 800 x^{3} \log {\left (x \right )} \log {\left (\log {\left (2 \right )} \right )} + 1600 x^{3} \log {\left (2 \right )} \log {\left (x \right )} - 25600 x^{2} \log {\left (2 \right )}^{2} - 6400 x^{2} \log {\left (\log {\left (2 \right )} \right )}^{2} - 25600 x^{2} \log {\left (2 \right )} \log {\left (\log {\left (2 \right )} \right )} + \left (64 \log {\left (2 \right )} \log {\left (\log {\left (2 \right )} \right )} + 16 \log {\left (\log {\left (2 \right )} \right )}^{2} + 64 \log {\left (2 \right )}^{2}\right ) e^{x}} \] Input:
integrate((((-256*x+256)*exp(x)+102400*x**2)*ln(4*ln(2))**4+(-25600*x**3*l n(x)+38400*x**4-12800*x**3)*ln(4*ln(2))**3+(1200*x**4*ln(x)**2+(-3200*x**5 +800*x**4)*ln(x)+2000*x**6-800*x**5)*ln(4*ln(2))**2)/((256*exp(x)**2-20480 0*exp(x)*x**2+40960000*x**4)*ln(4*ln(2))**4+((25600*exp(x)*x**3-10240000*x **5)*ln(x)-25600*exp(x)*x**4+10240000*x**6)*ln(4*ln(2))**3+((-800*exp(x)*x **4+960000*x**6)*ln(x)**2+(1600*x**5*exp(x)-1920000*x**7)*ln(x)-800*x**6*e xp(x)+960000*x**8)*ln(4*ln(2))**2+(-40000*x**7*ln(x)**3+120000*x**8*ln(x)* *2-120000*x**9*ln(x)+40000*x**10)*ln(4*ln(2))+625*x**8*ln(x)**4-2500*x**9* ln(x)**3+3750*x**10*ln(x)**2-2500*x**11*ln(x)+625*x**12),x)
Output:
(64*x*log(2)*log(log(2)) + 16*x*log(log(2))**2 + 64*x*log(2)**2)/(-25*x**6 + 50*x**5*log(x) - 25*x**4*log(x)**2 - 1600*x**4*log(2) - 800*x**4*log(lo g(2)) + 800*x**3*log(x)*log(log(2)) + 1600*x**3*log(2)*log(x) - 25600*x**2 *log(2)**2 - 6400*x**2*log(log(2))**2 - 25600*x**2*log(2)*log(log(2)) + (6 4*log(2)*log(log(2)) + 16*log(log(2))**2 + 64*log(2)**2)*exp(x))
Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (34) = 68\).
Time = 0.26 (sec) , antiderivative size = 120, normalized size of antiderivative = 3.64 \[ \int \frac {\left (-800 x^5+2000 x^6+\left (800 x^4-3200 x^5\right ) \log (x)+1200 x^4 \log ^2(x)\right ) \log ^2(\log (16))+\left (-12800 x^3+38400 x^4-25600 x^3 \log (x)\right ) \log ^3(\log (16))+\left (e^x (256-256 x)+102400 x^2\right ) \log ^4(\log (16))}{625 x^{12}-2500 x^{11} \log (x)+3750 x^{10} \log ^2(x)-2500 x^9 \log ^3(x)+625 x^8 \log ^4(x)+\left (40000 x^{10}-120000 x^9 \log (x)+120000 x^8 \log ^2(x)-40000 x^7 \log ^3(x)\right ) \log (\log (16))+\left (-800 e^x x^6+960000 x^8+\left (1600 e^x x^5-1920000 x^7\right ) \log (x)+\left (-800 e^x x^4+960000 x^6\right ) \log ^2(x)\right ) \log ^2(\log (16))+\left (-25600 e^x x^4+10240000 x^6+\left (25600 e^x x^3-10240000 x^5\right ) \log (x)\right ) \log ^3(\log (16))+\left (256 e^{2 x}-204800 e^x x^2+40960000 x^4\right ) \log ^4(\log (16))} \, dx=-\frac {16 \, {\left (4 \, \log \left (2\right )^{2} + 4 \, \log \left (2\right ) \log \left (\log \left (2\right )\right ) + \log \left (\log \left (2\right )\right )^{2}\right )} x}{25 \, x^{6} + 25 \, x^{4} \log \left (x\right )^{2} + 800 \, x^{4} {\left (2 \, \log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )} + 6400 \, {\left (4 \, \log \left (2\right )^{2} + 4 \, \log \left (2\right ) \log \left (\log \left (2\right )\right ) + \log \left (\log \left (2\right )\right )^{2}\right )} x^{2} - 16 \, {\left (4 \, \log \left (2\right )^{2} + 4 \, \log \left (2\right ) \log \left (\log \left (2\right )\right ) + \log \left (\log \left (2\right )\right )^{2}\right )} e^{x} - 50 \, {\left (x^{5} + 16 \, x^{3} {\left (2 \, \log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )}\right )} \log \left (x\right )} \] Input:
integrate((((-256*x+256)*exp(x)+102400*x^2)*log(4*log(2))^4+(-25600*x^3*lo g(x)+38400*x^4-12800*x^3)*log(4*log(2))^3+(1200*x^4*log(x)^2+(-3200*x^5+80 0*x^4)*log(x)+2000*x^6-800*x^5)*log(4*log(2))^2)/((256*exp(x)^2-204800*exp (x)*x^2+40960000*x^4)*log(4*log(2))^4+((25600*exp(x)*x^3-10240000*x^5)*log (x)-25600*exp(x)*x^4+10240000*x^6)*log(4*log(2))^3+((-800*exp(x)*x^4+96000 0*x^6)*log(x)^2+(1600*x^5*exp(x)-1920000*x^7)*log(x)-800*x^6*exp(x)+960000 *x^8)*log(4*log(2))^2+(-40000*x^7*log(x)^3+120000*x^8*log(x)^2-120000*x^9* log(x)+40000*x^10)*log(4*log(2))+625*x^8*log(x)^4-2500*x^9*log(x)^3+3750*x ^10*log(x)^2-2500*x^11*log(x)+625*x^12),x, algorithm="maxima")
Output:
-16*(4*log(2)^2 + 4*log(2)*log(log(2)) + log(log(2))^2)*x/(25*x^6 + 25*x^4 *log(x)^2 + 800*x^4*(2*log(2) + log(log(2))) + 6400*(4*log(2)^2 + 4*log(2) *log(log(2)) + log(log(2))^2)*x^2 - 16*(4*log(2)^2 + 4*log(2)*log(log(2)) + log(log(2))^2)*e^x - 50*(x^5 + 16*x^3*(2*log(2) + log(log(2))))*log(x))
Timed out. \[ \int \frac {\left (-800 x^5+2000 x^6+\left (800 x^4-3200 x^5\right ) \log (x)+1200 x^4 \log ^2(x)\right ) \log ^2(\log (16))+\left (-12800 x^3+38400 x^4-25600 x^3 \log (x)\right ) \log ^3(\log (16))+\left (e^x (256-256 x)+102400 x^2\right ) \log ^4(\log (16))}{625 x^{12}-2500 x^{11} \log (x)+3750 x^{10} \log ^2(x)-2500 x^9 \log ^3(x)+625 x^8 \log ^4(x)+\left (40000 x^{10}-120000 x^9 \log (x)+120000 x^8 \log ^2(x)-40000 x^7 \log ^3(x)\right ) \log (\log (16))+\left (-800 e^x x^6+960000 x^8+\left (1600 e^x x^5-1920000 x^7\right ) \log (x)+\left (-800 e^x x^4+960000 x^6\right ) \log ^2(x)\right ) \log ^2(\log (16))+\left (-25600 e^x x^4+10240000 x^6+\left (25600 e^x x^3-10240000 x^5\right ) \log (x)\right ) \log ^3(\log (16))+\left (256 e^{2 x}-204800 e^x x^2+40960000 x^4\right ) \log ^4(\log (16))} \, dx=\text {Timed out} \] Input:
integrate((((-256*x+256)*exp(x)+102400*x^2)*log(4*log(2))^4+(-25600*x^3*lo g(x)+38400*x^4-12800*x^3)*log(4*log(2))^3+(1200*x^4*log(x)^2+(-3200*x^5+80 0*x^4)*log(x)+2000*x^6-800*x^5)*log(4*log(2))^2)/((256*exp(x)^2-204800*exp (x)*x^2+40960000*x^4)*log(4*log(2))^4+((25600*exp(x)*x^3-10240000*x^5)*log (x)-25600*exp(x)*x^4+10240000*x^6)*log(4*log(2))^3+((-800*exp(x)*x^4+96000 0*x^6)*log(x)^2+(1600*x^5*exp(x)-1920000*x^7)*log(x)-800*x^6*exp(x)+960000 *x^8)*log(4*log(2))^2+(-40000*x^7*log(x)^3+120000*x^8*log(x)^2-120000*x^9* log(x)+40000*x^10)*log(4*log(2))+625*x^8*log(x)^4-2500*x^9*log(x)^3+3750*x ^10*log(x)^2-2500*x^11*log(x)+625*x^12),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {\left (-800 x^5+2000 x^6+\left (800 x^4-3200 x^5\right ) \log (x)+1200 x^4 \log ^2(x)\right ) \log ^2(\log (16))+\left (-12800 x^3+38400 x^4-25600 x^3 \log (x)\right ) \log ^3(\log (16))+\left (e^x (256-256 x)+102400 x^2\right ) \log ^4(\log (16))}{625 x^{12}-2500 x^{11} \log (x)+3750 x^{10} \log ^2(x)-2500 x^9 \log ^3(x)+625 x^8 \log ^4(x)+\left (40000 x^{10}-120000 x^9 \log (x)+120000 x^8 \log ^2(x)-40000 x^7 \log ^3(x)\right ) \log (\log (16))+\left (-800 e^x x^6+960000 x^8+\left (1600 e^x x^5-1920000 x^7\right ) \log (x)+\left (-800 e^x x^4+960000 x^6\right ) \log ^2(x)\right ) \log ^2(\log (16))+\left (-25600 e^x x^4+10240000 x^6+\left (25600 e^x x^3-10240000 x^5\right ) \log (x)\right ) \log ^3(\log (16))+\left (256 e^{2 x}-204800 e^x x^2+40960000 x^4\right ) \log ^4(\log (16))} \, dx=\int -\frac {{\ln \left (4\,\ln \left (2\right )\right )}^4\,\left ({\mathrm {e}}^x\,\left (256\,x-256\right )-102400\,x^2\right )-{\ln \left (4\,\ln \left (2\right )\right )}^2\,\left (\ln \left (x\right )\,\left (800\,x^4-3200\,x^5\right )+1200\,x^4\,{\ln \left (x\right )}^2-800\,x^5+2000\,x^6\right )+{\ln \left (4\,\ln \left (2\right )\right )}^3\,\left (25600\,x^3\,\ln \left (x\right )+12800\,x^3-38400\,x^4\right )}{{\ln \left (4\,\ln \left (2\right )\right )}^4\,\left (256\,{\mathrm {e}}^{2\,x}-204800\,x^2\,{\mathrm {e}}^x+40960000\,x^4\right )-\ln \left (4\,\ln \left (2\right )\right )\,\left (-40000\,x^{10}+120000\,x^9\,\ln \left (x\right )-120000\,x^8\,{\ln \left (x\right )}^2+40000\,x^7\,{\ln \left (x\right )}^3\right )-2500\,x^{11}\,\ln \left (x\right )+625\,x^8\,{\ln \left (x\right )}^4-2500\,x^9\,{\ln \left (x\right )}^3+3750\,x^{10}\,{\ln \left (x\right )}^2+{\ln \left (4\,\ln \left (2\right )\right )}^3\,\left (10240000\,x^6-25600\,x^4\,{\mathrm {e}}^x+\ln \left (x\right )\,\left (25600\,x^3\,{\mathrm {e}}^x-10240000\,x^5\right )\right )+625\,x^{12}-{\ln \left (4\,\ln \left (2\right )\right )}^2\,\left (800\,x^6\,{\mathrm {e}}^x+{\ln \left (x\right )}^2\,\left (800\,x^4\,{\mathrm {e}}^x-960000\,x^6\right )-960000\,x^8-\ln \left (x\right )\,\left (1600\,x^5\,{\mathrm {e}}^x-1920000\,x^7\right )\right )} \,d x \] Input:
int(-(log(4*log(2))^4*(exp(x)*(256*x - 256) - 102400*x^2) - log(4*log(2))^ 2*(log(x)*(800*x^4 - 3200*x^5) + 1200*x^4*log(x)^2 - 800*x^5 + 2000*x^6) + log(4*log(2))^3*(25600*x^3*log(x) + 12800*x^3 - 38400*x^4))/(log(4*log(2) )^4*(256*exp(2*x) - 204800*x^2*exp(x) + 40960000*x^4) - log(4*log(2))*(120 000*x^9*log(x) + 40000*x^7*log(x)^3 - 120000*x^8*log(x)^2 - 40000*x^10) - 2500*x^11*log(x) + 625*x^8*log(x)^4 - 2500*x^9*log(x)^3 + 3750*x^10*log(x) ^2 + log(4*log(2))^3*(10240000*x^6 - 25600*x^4*exp(x) + log(x)*(25600*x^3* exp(x) - 10240000*x^5)) + 625*x^12 - log(4*log(2))^2*(800*x^6*exp(x) + log (x)^2*(800*x^4*exp(x) - 960000*x^6) - 960000*x^8 - log(x)*(1600*x^5*exp(x) - 1920000*x^7))),x)
Output:
int(-(log(4*log(2))^4*(exp(x)*(256*x - 256) - 102400*x^2) - log(4*log(2))^ 2*(log(x)*(800*x^4 - 3200*x^5) + 1200*x^4*log(x)^2 - 800*x^5 + 2000*x^6) + log(4*log(2))^3*(25600*x^3*log(x) + 12800*x^3 - 38400*x^4))/(log(4*log(2) )^4*(256*exp(2*x) - 204800*x^2*exp(x) + 40960000*x^4) - log(4*log(2))*(120 000*x^9*log(x) + 40000*x^7*log(x)^3 - 120000*x^8*log(x)^2 - 40000*x^10) - 2500*x^11*log(x) + 625*x^8*log(x)^4 - 2500*x^9*log(x)^3 + 3750*x^10*log(x) ^2 + log(4*log(2))^3*(10240000*x^6 - 25600*x^4*exp(x) + log(x)*(25600*x^3* exp(x) - 10240000*x^5)) + 625*x^12 - log(4*log(2))^2*(800*x^6*exp(x) + log (x)^2*(800*x^4*exp(x) - 960000*x^6) - 960000*x^8 - log(x)*(1600*x^5*exp(x) - 1920000*x^7))), x)
Time = 0.23 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.42 \[ \int \frac {\left (-800 x^5+2000 x^6+\left (800 x^4-3200 x^5\right ) \log (x)+1200 x^4 \log ^2(x)\right ) \log ^2(\log (16))+\left (-12800 x^3+38400 x^4-25600 x^3 \log (x)\right ) \log ^3(\log (16))+\left (e^x (256-256 x)+102400 x^2\right ) \log ^4(\log (16))}{625 x^{12}-2500 x^{11} \log (x)+3750 x^{10} \log ^2(x)-2500 x^9 \log ^3(x)+625 x^8 \log ^4(x)+\left (40000 x^{10}-120000 x^9 \log (x)+120000 x^8 \log ^2(x)-40000 x^7 \log ^3(x)\right ) \log (\log (16))+\left (-800 e^x x^6+960000 x^8+\left (1600 e^x x^5-1920000 x^7\right ) \log (x)+\left (-800 e^x x^4+960000 x^6\right ) \log ^2(x)\right ) \log ^2(\log (16))+\left (-25600 e^x x^4+10240000 x^6+\left (25600 e^x x^3-10240000 x^5\right ) \log (x)\right ) \log ^3(\log (16))+\left (256 e^{2 x}-204800 e^x x^2+40960000 x^4\right ) \log ^4(\log (16))} \, dx=\frac {16 \mathrm {log}\left (4 \,\mathrm {log}\left (2\right )\right )^{2} x}{16 e^{x} \mathrm {log}\left (4 \,\mathrm {log}\left (2\right )\right )^{2}-6400 \mathrm {log}\left (4 \,\mathrm {log}\left (2\right )\right )^{2} x^{2}+800 \,\mathrm {log}\left (4 \,\mathrm {log}\left (2\right )\right ) \mathrm {log}\left (x \right ) x^{3}-800 \,\mathrm {log}\left (4 \,\mathrm {log}\left (2\right )\right ) x^{4}-25 \mathrm {log}\left (x \right )^{2} x^{4}+50 \,\mathrm {log}\left (x \right ) x^{5}-25 x^{6}} \] Input:
int((((-256*x+256)*exp(x)+102400*x^2)*log(4*log(2))^4+(-25600*x^3*log(x)+3 8400*x^4-12800*x^3)*log(4*log(2))^3+(1200*x^4*log(x)^2+(-3200*x^5+800*x^4) *log(x)+2000*x^6-800*x^5)*log(4*log(2))^2)/((256*exp(x)^2-204800*exp(x)*x^ 2+40960000*x^4)*log(4*log(2))^4+((25600*exp(x)*x^3-10240000*x^5)*log(x)-25 600*exp(x)*x^4+10240000*x^6)*log(4*log(2))^3+((-800*exp(x)*x^4+960000*x^6) *log(x)^2+(1600*x^5*exp(x)-1920000*x^7)*log(x)-800*x^6*exp(x)+960000*x^8)* log(4*log(2))^2+(-40000*x^7*log(x)^3+120000*x^8*log(x)^2-120000*x^9*log(x) +40000*x^10)*log(4*log(2))+625*x^8*log(x)^4-2500*x^9*log(x)^3+3750*x^10*lo g(x)^2-2500*x^11*log(x)+625*x^12),x)
Output:
(16*log(4*log(2))**2*x)/(16*e**x*log(4*log(2))**2 - 6400*log(4*log(2))**2* x**2 + 800*log(4*log(2))*log(x)*x**3 - 800*log(4*log(2))*x**4 - 25*log(x)* *2*x**4 + 50*log(x)*x**5 - 25*x**6)