Integrand size = 319, antiderivative size = 28 \[ \int \frac {1536 x+640 x^2-96 x^3-72 x^4-8 x^5+e^x \left (-384-64 x+40 x^2+8 x^3\right )+\left (4 e^{3 x}+e^{2 x} \left (96-112 x-16 x^2+8 x^3\right )+e^x \left (-384 x+288 x^2+144 x^3-20 x^4-8 x^5\right )\right ) \log (x)+\left (-1536-640 x+96 x^2+72 x^3+8 x^4+\left (-1536 x-384 x^2+288 x^3+120 x^4+12 x^5+e^{2 x} (48+12 x)+e^x \left (768-416 x-280 x^2+8 x^4\right )\right ) \log (x)\right ) \log (\log (x))+\left (1536+128 x-480 x^2-168 x^3-16 x^4+e^x \left (192+96 x+12 x^2\right )\right ) \log (x) \log ^2(\log (x))+\left (256+192 x+48 x^2+4 x^3\right ) \log (x) \log ^3(\log (x))}{e^{3 x} \log (x)+e^{2 x} (12+3 x) \log (x) \log (\log (x))+e^x \left (48+24 x+3 x^2\right ) \log (x) \log ^2(\log (x))+\left (64+48 x+12 x^2+x^3\right ) \log (x) \log ^3(\log (x))} \, dx=4+(-3+x) \left (2-\frac {2 x}{\frac {e^x}{4+x}+\log (\log (x))}\right )^2 \] Output:
4+(2-2*x/(ln(ln(x))+exp(x)/(4+x)))^2*(-3+x)
Time = 0.14 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75 \[ \int \frac {1536 x+640 x^2-96 x^3-72 x^4-8 x^5+e^x \left (-384-64 x+40 x^2+8 x^3\right )+\left (4 e^{3 x}+e^{2 x} \left (96-112 x-16 x^2+8 x^3\right )+e^x \left (-384 x+288 x^2+144 x^3-20 x^4-8 x^5\right )\right ) \log (x)+\left (-1536-640 x+96 x^2+72 x^3+8 x^4+\left (-1536 x-384 x^2+288 x^3+120 x^4+12 x^5+e^{2 x} (48+12 x)+e^x \left (768-416 x-280 x^2+8 x^4\right )\right ) \log (x)\right ) \log (\log (x))+\left (1536+128 x-480 x^2-168 x^3-16 x^4+e^x \left (192+96 x+12 x^2\right )\right ) \log (x) \log ^2(\log (x))+\left (256+192 x+48 x^2+4 x^3\right ) \log (x) \log ^3(\log (x))}{e^{3 x} \log (x)+e^{2 x} (12+3 x) \log (x) \log (\log (x))+e^x \left (48+24 x+3 x^2\right ) \log (x) \log ^2(\log (x))+\left (64+48 x+12 x^2+x^3\right ) \log (x) \log ^3(\log (x))} \, dx=4 x \left (1+\frac {(-3+x) x (4+x)^2}{\left (e^x+(4+x) \log (\log (x))\right )^2}-\frac {2 (-3+x) (4+x)}{e^x+(4+x) \log (\log (x))}\right ) \] Input:
Integrate[(1536*x + 640*x^2 - 96*x^3 - 72*x^4 - 8*x^5 + E^x*(-384 - 64*x + 40*x^2 + 8*x^3) + (4*E^(3*x) + E^(2*x)*(96 - 112*x - 16*x^2 + 8*x^3) + E^ x*(-384*x + 288*x^2 + 144*x^3 - 20*x^4 - 8*x^5))*Log[x] + (-1536 - 640*x + 96*x^2 + 72*x^3 + 8*x^4 + (-1536*x - 384*x^2 + 288*x^3 + 120*x^4 + 12*x^5 + E^(2*x)*(48 + 12*x) + E^x*(768 - 416*x - 280*x^2 + 8*x^4))*Log[x])*Log[ Log[x]] + (1536 + 128*x - 480*x^2 - 168*x^3 - 16*x^4 + E^x*(192 + 96*x + 1 2*x^2))*Log[x]*Log[Log[x]]^2 + (256 + 192*x + 48*x^2 + 4*x^3)*Log[x]*Log[L og[x]]^3)/(E^(3*x)*Log[x] + E^(2*x)*(12 + 3*x)*Log[x]*Log[Log[x]] + E^x*(4 8 + 24*x + 3*x^2)*Log[x]*Log[Log[x]]^2 + (64 + 48*x + 12*x^2 + x^3)*Log[x] *Log[Log[x]]^3),x]
Output:
4*x*(1 + ((-3 + x)*x*(4 + x)^2)/(E^x + (4 + x)*Log[Log[x]])^2 - (2*(-3 + x )*(4 + x))/(E^x + (4 + x)*Log[Log[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.
| \(\displaystyle \int \frac {-8 x^5-72 x^4-96 x^3+640 x^2+e^x \left (8 x^3+40 x^2-64 x-384\right )+\left (4 x^3+48 x^2+192 x+256\right ) \log (x) \log ^3(\log (x))+\left (-16 x^4-168 x^3-480 x^2+e^x \left (12 x^2+96 x+192\right )+128 x+1536\right ) \log (x) \log ^2(\log (x))+\left (e^{2 x} \left (8 x^3-16 x^2-112 x+96\right )+e^x \left (-8 x^5-20 x^4+144 x^3+288 x^2-384 x\right )+4 e^{3 x}\right ) \log (x)+\left (8 x^4+72 x^3+96 x^2+\left (12 x^5+120 x^4+288 x^3-384 x^2+e^x \left (8 x^4-280 x^2-416 x+768\right )-1536 x+e^{2 x} (12 x+48)\right ) \log (x)-640 x-1536\right ) \log (\log (x))+1536 x}{e^x \left (3 x^2+24 x+48\right ) \log (x) \log ^2(\log (x))+\left (x^3+12 x^2+48 x+64\right ) \log (x) \log ^3(\log (x))+e^{2 x} (3 x+12) \log (x) \log (\log (x))+e^{3 x} \log (x)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {4 \left (-x (x+4)+e^x+(x+4) \log (\log (x))\right ) \left (\log (x) \left ((x+4) \left (-3 x^2-6 x+2 e^x+24\right ) \log (\log (x))+e^x \left (2 x^3-3 x^2-24 x+e^x+24\right )+(x+4)^2 \log ^2(\log (x))\right )+2 (x-3) (x+4)^2\right )}{\log (x) \left (e^x+(x+4) \log (\log (x))\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \int -\frac {\left (-x (x+4)+\log (\log (x)) (x+4)+e^x\right ) \left (2 (3-x) (x+4)^2-\log (x) \left ((x+4)^2 \log ^2(\log (x))+(x+4) \left (-3 x^2-6 x+2 e^x+24\right ) \log (\log (x))+e^x \left (2 x^3-3 x^2-24 x+e^x+24\right )\right )\right )}{\log (x) \left ((x+4) \log (\log (x))+e^x\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -4 \int \frac {\left (-x (x+4)+\log (\log (x)) (x+4)+e^x\right ) \left (2 (3-x) (x+4)^2-\log (x) \left ((x+4)^2 \log ^2(\log (x))+(x+4) \left (-3 x^2-6 x+2 e^x+24\right ) \log (\log (x))+e^x \left (2 x^3-3 x^2-24 x+e^x+24\right )\right )\right )}{\log (x) \left ((x+4) \log (\log (x))+e^x\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (-\frac {2 (x-3) x \left (\log (x) \log (\log (x)) x^2+3 \log (x) \log (\log (x)) x-x-4\right ) (x+4)^2}{\log (x) \left (x \log (\log (x))+4 \log (\log (x))+e^x\right )^3}+\frac {\left (2 \log (x) x^4-3 \log (x) x^3+2 \log (x) \log (\log (x)) x^3-24 \log (x) x^2-2 x^2+24 \log (x) x-18 \log (x) \log (\log (x)) x-2 x+24\right ) (x+4)}{\log (x) \left (x \log (\log (x))+4 \log (\log (x))+e^x\right )^2}-\frac {2 \left (x^3-2 x^2-14 x+12\right )}{x \log (\log (x))+4 \log (\log (x))+e^x}-1\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle -4 \int \left (-\frac {2 (x-3) x \left (\log (x) \log (\log (x)) x^2+3 \log (x) \log (\log (x)) x-x-4\right ) (x+4)^2}{\log (x) \left (x \log (\log (x))+4 \log (\log (x))+e^x\right )^3}+\frac {\left (2 \log (x) x^4-3 \log (x) x^3+2 \log (x) \log (\log (x)) x^3-24 \log (x) x^2-2 x^2+24 \log (x) x-18 \log (x) \log (\log (x)) x-2 x+24\right ) (x+4)}{\log (x) \left (x \log (\log (x))+4 \log (\log (x))+e^x\right )^2}-\frac {2 \left (x^3-2 x^2-14 x+12\right )}{x \log (\log (x))+4 \log (\log (x))+e^x}-1\right )dx\) |
Input:
Int[(1536*x + 640*x^2 - 96*x^3 - 72*x^4 - 8*x^5 + E^x*(-384 - 64*x + 40*x^ 2 + 8*x^3) + (4*E^(3*x) + E^(2*x)*(96 - 112*x - 16*x^2 + 8*x^3) + E^x*(-38 4*x + 288*x^2 + 144*x^3 - 20*x^4 - 8*x^5))*Log[x] + (-1536 - 640*x + 96*x^ 2 + 72*x^3 + 8*x^4 + (-1536*x - 384*x^2 + 288*x^3 + 120*x^4 + 12*x^5 + E^( 2*x)*(48 + 12*x) + E^x*(768 - 416*x - 280*x^2 + 8*x^4))*Log[x])*Log[Log[x] ] + (1536 + 128*x - 480*x^2 - 168*x^3 - 16*x^4 + E^x*(192 + 96*x + 12*x^2) )*Log[x]*Log[Log[x]]^2 + (256 + 192*x + 48*x^2 + 4*x^3)*Log[x]*Log[Log[x]] ^3)/(E^(3*x)*Log[x] + E^(2*x)*(12 + 3*x)*Log[x]*Log[Log[x]] + E^x*(48 + 24 *x + 3*x^2)*Log[x]*Log[Log[x]]^2 + (64 + 48*x + 12*x^2 + x^3)*Log[x]*Log[L og[x]]^3),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(63\) vs. \(2(27)=54\).
Time = 13.37 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.29
| method | result | size |
| risch | \(4 x +\frac {4 \left (x^{3}-2 x^{2} \ln \left (\ln \left (x \right )\right )+x^{2}-2 \,{\mathrm e}^{x} x -2 x \ln \left (\ln \left (x \right )\right )-12 x +6 \,{\mathrm e}^{x}+24 \ln \left (\ln \left (x \right )\right )\right ) \left (4+x \right ) x}{\left (x \ln \left (\ln \left (x \right )\right )+{\mathrm e}^{x}+4 \ln \left (\ln \left (x \right )\right )\right )^{2}}\) | \(64\) |
| parallelrisch | \(-\frac {16 \,{\mathrm e}^{x} x^{2}-192 \,{\mathrm e}^{x} x +80 \ln \left (\ln \left (x \right )\right ) x^{3}-128 x^{2} \ln \left (\ln \left (x \right )\right )+16 \,{\mathrm e}^{x} x^{3}+512 \ln \left (\ln \left (x \right )\right )^{2}-768 x \ln \left (\ln \left (x \right )\right )-8 x \,{\mathrm e}^{2 x}-8 x^{5}+384 x^{2}+64 x^{3}-40 x^{4}+32 \,{\mathrm e}^{2 x}-16 \,{\mathrm e}^{x} \ln \left (\ln \left (x \right )\right ) x^{2}-32 x^{2} \ln \left (\ln \left (x \right )\right )^{2}-8 \ln \left (\ln \left (x \right )\right )^{2} x^{3}+16 \ln \left (\ln \left (x \right )\right ) x^{4}+256 \,{\mathrm e}^{x} \ln \left (\ln \left (x \right )\right )+128 x \ln \left (\ln \left (x \right )\right )^{2}}{2 \left (x^{2} \ln \left (\ln \left (x \right )\right )^{2}+2 \,{\mathrm e}^{x} \ln \left (\ln \left (x \right )\right ) x +8 x \ln \left (\ln \left (x \right )\right )^{2}+{\mathrm e}^{2 x}+8 \,{\mathrm e}^{x} \ln \left (\ln \left (x \right )\right )+16 \ln \left (\ln \left (x \right )\right )^{2}\right )}\) | \(184\) |
Input:
int(((4*x^3+48*x^2+192*x+256)*ln(x)*ln(ln(x))^3+((12*x^2+96*x+192)*exp(x)- 16*x^4-168*x^3-480*x^2+128*x+1536)*ln(x)*ln(ln(x))^2+(((12*x+48)*exp(x)^2+ (8*x^4-280*x^2-416*x+768)*exp(x)+12*x^5+120*x^4+288*x^3-384*x^2-1536*x)*ln (x)+8*x^4+72*x^3+96*x^2-640*x-1536)*ln(ln(x))+(4*exp(x)^3+(8*x^3-16*x^2-11 2*x+96)*exp(x)^2+(-8*x^5-20*x^4+144*x^3+288*x^2-384*x)*exp(x))*ln(x)+(8*x^ 3+40*x^2-64*x-384)*exp(x)-8*x^5-72*x^4-96*x^3+640*x^2+1536*x)/((x^3+12*x^2 +48*x+64)*ln(x)*ln(ln(x))^3+(3*x^2+24*x+48)*exp(x)*ln(x)*ln(ln(x))^2+(3*x+ 12)*exp(x)^2*ln(x)*ln(ln(x))+exp(x)^3*ln(x)),x,method=_RETURNVERBOSE)
Output:
4*x+4*(x^3-2*x^2*ln(ln(x))+x^2-2*exp(x)*x-2*x*ln(ln(x))-12*x+6*exp(x)+24*l n(ln(x)))*(4+x)*x/(x*ln(ln(x))+exp(x)+4*ln(ln(x)))^2
Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (27) = 54\).
Time = 0.10 (sec) , antiderivative size = 123, normalized size of antiderivative = 4.39 \[ \int \frac {1536 x+640 x^2-96 x^3-72 x^4-8 x^5+e^x \left (-384-64 x+40 x^2+8 x^3\right )+\left (4 e^{3 x}+e^{2 x} \left (96-112 x-16 x^2+8 x^3\right )+e^x \left (-384 x+288 x^2+144 x^3-20 x^4-8 x^5\right )\right ) \log (x)+\left (-1536-640 x+96 x^2+72 x^3+8 x^4+\left (-1536 x-384 x^2+288 x^3+120 x^4+12 x^5+e^{2 x} (48+12 x)+e^x \left (768-416 x-280 x^2+8 x^4\right )\right ) \log (x)\right ) \log (\log (x))+\left (1536+128 x-480 x^2-168 x^3-16 x^4+e^x \left (192+96 x+12 x^2\right )\right ) \log (x) \log ^2(\log (x))+\left (256+192 x+48 x^2+4 x^3\right ) \log (x) \log ^3(\log (x))}{e^{3 x} \log (x)+e^{2 x} (12+3 x) \log (x) \log (\log (x))+e^x \left (48+24 x+3 x^2\right ) \log (x) \log ^2(\log (x))+\left (64+48 x+12 x^2+x^3\right ) \log (x) \log ^3(\log (x))} \, dx=\frac {4 \, {\left (x^{5} + 5 \, x^{4} - 8 \, x^{3} + {\left (x^{3} + 8 \, x^{2} + 16 \, x\right )} \log \left (\log \left (x\right )\right )^{2} - 48 \, x^{2} + x e^{\left (2 \, x\right )} - 2 \, {\left (x^{3} + x^{2} - 12 \, x\right )} e^{x} - 2 \, {\left (x^{4} + 5 \, x^{3} - 8 \, x^{2} - {\left (x^{2} + 4 \, x\right )} e^{x} - 48 \, x\right )} \log \left (\log \left (x\right )\right )\right )}}{2 \, {\left (x + 4\right )} e^{x} \log \left (\log \left (x\right )\right ) + {\left (x^{2} + 8 \, x + 16\right )} \log \left (\log \left (x\right )\right )^{2} + e^{\left (2 \, x\right )}} \] Input:
integrate(((4*x^3+48*x^2+192*x+256)*log(x)*log(log(x))^3+((12*x^2+96*x+192 )*exp(x)-16*x^4-168*x^3-480*x^2+128*x+1536)*log(x)*log(log(x))^2+(((12*x+4 8)*exp(x)^2+(8*x^4-280*x^2-416*x+768)*exp(x)+12*x^5+120*x^4+288*x^3-384*x^ 2-1536*x)*log(x)+8*x^4+72*x^3+96*x^2-640*x-1536)*log(log(x))+(4*exp(x)^3+( 8*x^3-16*x^2-112*x+96)*exp(x)^2+(-8*x^5-20*x^4+144*x^3+288*x^2-384*x)*exp( x))*log(x)+(8*x^3+40*x^2-64*x-384)*exp(x)-8*x^5-72*x^4-96*x^3+640*x^2+1536 *x)/((x^3+12*x^2+48*x+64)*log(x)*log(log(x))^3+(3*x^2+24*x+48)*exp(x)*log( x)*log(log(x))^2+(3*x+12)*exp(x)^2*log(x)*log(log(x))+exp(x)^3*log(x)),x, algorithm="fricas")
Output:
4*(x^5 + 5*x^4 - 8*x^3 + (x^3 + 8*x^2 + 16*x)*log(log(x))^2 - 48*x^2 + x*e ^(2*x) - 2*(x^3 + x^2 - 12*x)*e^x - 2*(x^4 + 5*x^3 - 8*x^2 - (x^2 + 4*x)*e ^x - 48*x)*log(log(x)))/(2*(x + 4)*e^x*log(log(x)) + (x^2 + 8*x + 16)*log( log(x))^2 + e^(2*x))
Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (22) = 44\).
Time = 0.24 (sec) , antiderivative size = 131, normalized size of antiderivative = 4.68 \[ \int \frac {1536 x+640 x^2-96 x^3-72 x^4-8 x^5+e^x \left (-384-64 x+40 x^2+8 x^3\right )+\left (4 e^{3 x}+e^{2 x} \left (96-112 x-16 x^2+8 x^3\right )+e^x \left (-384 x+288 x^2+144 x^3-20 x^4-8 x^5\right )\right ) \log (x)+\left (-1536-640 x+96 x^2+72 x^3+8 x^4+\left (-1536 x-384 x^2+288 x^3+120 x^4+12 x^5+e^{2 x} (48+12 x)+e^x \left (768-416 x-280 x^2+8 x^4\right )\right ) \log (x)\right ) \log (\log (x))+\left (1536+128 x-480 x^2-168 x^3-16 x^4+e^x \left (192+96 x+12 x^2\right )\right ) \log (x) \log ^2(\log (x))+\left (256+192 x+48 x^2+4 x^3\right ) \log (x) \log ^3(\log (x))}{e^{3 x} \log (x)+e^{2 x} (12+3 x) \log (x) \log (\log (x))+e^x \left (48+24 x+3 x^2\right ) \log (x) \log ^2(\log (x))+\left (64+48 x+12 x^2+x^3\right ) \log (x) \log ^3(\log (x))} \, dx=4 x + \frac {4 x^{5} - 8 x^{4} \log {\left (\log {\left (x \right )} \right )} + 20 x^{4} - 40 x^{3} \log {\left (\log {\left (x \right )} \right )} - 32 x^{3} + 64 x^{2} \log {\left (\log {\left (x \right )} \right )} - 192 x^{2} + 384 x \log {\left (\log {\left (x \right )} \right )} + \left (- 8 x^{3} - 8 x^{2} + 96 x\right ) e^{x}}{x^{2} \log {\left (\log {\left (x \right )} \right )}^{2} + 8 x \log {\left (\log {\left (x \right )} \right )}^{2} + \left (2 x \log {\left (\log {\left (x \right )} \right )} + 8 \log {\left (\log {\left (x \right )} \right )}\right ) e^{x} + e^{2 x} + 16 \log {\left (\log {\left (x \right )} \right )}^{2}} \] Input:
integrate(((4*x**3+48*x**2+192*x+256)*ln(x)*ln(ln(x))**3+((12*x**2+96*x+19 2)*exp(x)-16*x**4-168*x**3-480*x**2+128*x+1536)*ln(x)*ln(ln(x))**2+(((12*x +48)*exp(x)**2+(8*x**4-280*x**2-416*x+768)*exp(x)+12*x**5+120*x**4+288*x** 3-384*x**2-1536*x)*ln(x)+8*x**4+72*x**3+96*x**2-640*x-1536)*ln(ln(x))+(4*e xp(x)**3+(8*x**3-16*x**2-112*x+96)*exp(x)**2+(-8*x**5-20*x**4+144*x**3+288 *x**2-384*x)*exp(x))*ln(x)+(8*x**3+40*x**2-64*x-384)*exp(x)-8*x**5-72*x**4 -96*x**3+640*x**2+1536*x)/((x**3+12*x**2+48*x+64)*ln(x)*ln(ln(x))**3+(3*x* *2+24*x+48)*exp(x)*ln(x)*ln(ln(x))**2+(3*x+12)*exp(x)**2*ln(x)*ln(ln(x))+e xp(x)**3*ln(x)),x)
Output:
4*x + (4*x**5 - 8*x**4*log(log(x)) + 20*x**4 - 40*x**3*log(log(x)) - 32*x* *3 + 64*x**2*log(log(x)) - 192*x**2 + 384*x*log(log(x)) + (-8*x**3 - 8*x** 2 + 96*x)*exp(x))/(x**2*log(log(x))**2 + 8*x*log(log(x))**2 + (2*x*log(log (x)) + 8*log(log(x)))*exp(x) + exp(2*x) + 16*log(log(x))**2)
Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (27) = 54\).
Time = 0.22 (sec) , antiderivative size = 123, normalized size of antiderivative = 4.39 \[ \int \frac {1536 x+640 x^2-96 x^3-72 x^4-8 x^5+e^x \left (-384-64 x+40 x^2+8 x^3\right )+\left (4 e^{3 x}+e^{2 x} \left (96-112 x-16 x^2+8 x^3\right )+e^x \left (-384 x+288 x^2+144 x^3-20 x^4-8 x^5\right )\right ) \log (x)+\left (-1536-640 x+96 x^2+72 x^3+8 x^4+\left (-1536 x-384 x^2+288 x^3+120 x^4+12 x^5+e^{2 x} (48+12 x)+e^x \left (768-416 x-280 x^2+8 x^4\right )\right ) \log (x)\right ) \log (\log (x))+\left (1536+128 x-480 x^2-168 x^3-16 x^4+e^x \left (192+96 x+12 x^2\right )\right ) \log (x) \log ^2(\log (x))+\left (256+192 x+48 x^2+4 x^3\right ) \log (x) \log ^3(\log (x))}{e^{3 x} \log (x)+e^{2 x} (12+3 x) \log (x) \log (\log (x))+e^x \left (48+24 x+3 x^2\right ) \log (x) \log ^2(\log (x))+\left (64+48 x+12 x^2+x^3\right ) \log (x) \log ^3(\log (x))} \, dx=\frac {4 \, {\left (x^{5} + 5 \, x^{4} - 8 \, x^{3} + {\left (x^{3} + 8 \, x^{2} + 16 \, x\right )} \log \left (\log \left (x\right )\right )^{2} - 48 \, x^{2} + x e^{\left (2 \, x\right )} - 2 \, {\left (x^{3} + x^{2} - 12 \, x\right )} e^{x} - 2 \, {\left (x^{4} + 5 \, x^{3} - 8 \, x^{2} - {\left (x^{2} + 4 \, x\right )} e^{x} - 48 \, x\right )} \log \left (\log \left (x\right )\right )\right )}}{2 \, {\left (x + 4\right )} e^{x} \log \left (\log \left (x\right )\right ) + {\left (x^{2} + 8 \, x + 16\right )} \log \left (\log \left (x\right )\right )^{2} + e^{\left (2 \, x\right )}} \] Input:
integrate(((4*x^3+48*x^2+192*x+256)*log(x)*log(log(x))^3+((12*x^2+96*x+192 )*exp(x)-16*x^4-168*x^3-480*x^2+128*x+1536)*log(x)*log(log(x))^2+(((12*x+4 8)*exp(x)^2+(8*x^4-280*x^2-416*x+768)*exp(x)+12*x^5+120*x^4+288*x^3-384*x^ 2-1536*x)*log(x)+8*x^4+72*x^3+96*x^2-640*x-1536)*log(log(x))+(4*exp(x)^3+( 8*x^3-16*x^2-112*x+96)*exp(x)^2+(-8*x^5-20*x^4+144*x^3+288*x^2-384*x)*exp( x))*log(x)+(8*x^3+40*x^2-64*x-384)*exp(x)-8*x^5-72*x^4-96*x^3+640*x^2+1536 *x)/((x^3+12*x^2+48*x+64)*log(x)*log(log(x))^3+(3*x^2+24*x+48)*exp(x)*log( x)*log(log(x))^2+(3*x+12)*exp(x)^2*log(x)*log(log(x))+exp(x)^3*log(x)),x, algorithm="maxima")
Output:
4*(x^5 + 5*x^4 - 8*x^3 + (x^3 + 8*x^2 + 16*x)*log(log(x))^2 - 48*x^2 + x*e ^(2*x) - 2*(x^3 + x^2 - 12*x)*e^x - 2*(x^4 + 5*x^3 - 8*x^2 - (x^2 + 4*x)*e ^x - 48*x)*log(log(x)))/(2*(x + 4)*e^x*log(log(x)) + (x^2 + 8*x + 16)*log( log(x))^2 + e^(2*x))
Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (27) = 54\).
Time = 0.41 (sec) , antiderivative size = 167, normalized size of antiderivative = 5.96 \[ \int \frac {1536 x+640 x^2-96 x^3-72 x^4-8 x^5+e^x \left (-384-64 x+40 x^2+8 x^3\right )+\left (4 e^{3 x}+e^{2 x} \left (96-112 x-16 x^2+8 x^3\right )+e^x \left (-384 x+288 x^2+144 x^3-20 x^4-8 x^5\right )\right ) \log (x)+\left (-1536-640 x+96 x^2+72 x^3+8 x^4+\left (-1536 x-384 x^2+288 x^3+120 x^4+12 x^5+e^{2 x} (48+12 x)+e^x \left (768-416 x-280 x^2+8 x^4\right )\right ) \log (x)\right ) \log (\log (x))+\left (1536+128 x-480 x^2-168 x^3-16 x^4+e^x \left (192+96 x+12 x^2\right )\right ) \log (x) \log ^2(\log (x))+\left (256+192 x+48 x^2+4 x^3\right ) \log (x) \log ^3(\log (x))}{e^{3 x} \log (x)+e^{2 x} (12+3 x) \log (x) \log (\log (x))+e^x \left (48+24 x+3 x^2\right ) \log (x) \log ^2(\log (x))+\left (64+48 x+12 x^2+x^3\right ) \log (x) \log ^3(\log (x))} \, dx=\frac {4 \, {\left (x^{5} - 2 \, x^{4} \log \left (\log \left (x\right )\right ) + x^{3} \log \left (\log \left (x\right )\right )^{2} + 5 \, x^{4} - 2 \, x^{3} e^{x} - 10 \, x^{3} \log \left (\log \left (x\right )\right ) + 2 \, x^{2} e^{x} \log \left (\log \left (x\right )\right ) + 8 \, x^{2} \log \left (\log \left (x\right )\right )^{2} - 8 \, x^{3} - 2 \, x^{2} e^{x} + 16 \, x^{2} \log \left (\log \left (x\right )\right ) + 8 \, x e^{x} \log \left (\log \left (x\right )\right ) + 16 \, x \log \left (\log \left (x\right )\right )^{2} - 48 \, x^{2} + x e^{\left (2 \, x\right )} + 24 \, x e^{x} + 96 \, x \log \left (\log \left (x\right )\right )\right )}}{x^{2} \log \left (\log \left (x\right )\right )^{2} + 2 \, x e^{x} \log \left (\log \left (x\right )\right ) + 8 \, x \log \left (\log \left (x\right )\right )^{2} + 8 \, e^{x} \log \left (\log \left (x\right )\right ) + 16 \, \log \left (\log \left (x\right )\right )^{2} + e^{\left (2 \, x\right )}} \] Input:
integrate(((4*x^3+48*x^2+192*x+256)*log(x)*log(log(x))^3+((12*x^2+96*x+192 )*exp(x)-16*x^4-168*x^3-480*x^2+128*x+1536)*log(x)*log(log(x))^2+(((12*x+4 8)*exp(x)^2+(8*x^4-280*x^2-416*x+768)*exp(x)+12*x^5+120*x^4+288*x^3-384*x^ 2-1536*x)*log(x)+8*x^4+72*x^3+96*x^2-640*x-1536)*log(log(x))+(4*exp(x)^3+( 8*x^3-16*x^2-112*x+96)*exp(x)^2+(-8*x^5-20*x^4+144*x^3+288*x^2-384*x)*exp( x))*log(x)+(8*x^3+40*x^2-64*x-384)*exp(x)-8*x^5-72*x^4-96*x^3+640*x^2+1536 *x)/((x^3+12*x^2+48*x+64)*log(x)*log(log(x))^3+(3*x^2+24*x+48)*exp(x)*log( x)*log(log(x))^2+(3*x+12)*exp(x)^2*log(x)*log(log(x))+exp(x)^3*log(x)),x, algorithm="giac")
Output:
4*(x^5 - 2*x^4*log(log(x)) + x^3*log(log(x))^2 + 5*x^4 - 2*x^3*e^x - 10*x^ 3*log(log(x)) + 2*x^2*e^x*log(log(x)) + 8*x^2*log(log(x))^2 - 8*x^3 - 2*x^ 2*e^x + 16*x^2*log(log(x)) + 8*x*e^x*log(log(x)) + 16*x*log(log(x))^2 - 48 *x^2 + x*e^(2*x) + 24*x*e^x + 96*x*log(log(x)))/(x^2*log(log(x))^2 + 2*x*e ^x*log(log(x)) + 8*x*log(log(x))^2 + 8*e^x*log(log(x)) + 16*log(log(x))^2 + e^(2*x))
Time = 4.03 (sec) , antiderivative size = 121, normalized size of antiderivative = 4.32 \[ \int \frac {1536 x+640 x^2-96 x^3-72 x^4-8 x^5+e^x \left (-384-64 x+40 x^2+8 x^3\right )+\left (4 e^{3 x}+e^{2 x} \left (96-112 x-16 x^2+8 x^3\right )+e^x \left (-384 x+288 x^2+144 x^3-20 x^4-8 x^5\right )\right ) \log (x)+\left (-1536-640 x+96 x^2+72 x^3+8 x^4+\left (-1536 x-384 x^2+288 x^3+120 x^4+12 x^5+e^{2 x} (48+12 x)+e^x \left (768-416 x-280 x^2+8 x^4\right )\right ) \log (x)\right ) \log (\log (x))+\left (1536+128 x-480 x^2-168 x^3-16 x^4+e^x \left (192+96 x+12 x^2\right )\right ) \log (x) \log ^2(\log (x))+\left (256+192 x+48 x^2+4 x^3\right ) \log (x) \log ^3(\log (x))}{e^{3 x} \log (x)+e^{2 x} (12+3 x) \log (x) \log (\log (x))+e^x \left (48+24 x+3 x^2\right ) \log (x) \log ^2(\log (x))+\left (64+48 x+12 x^2+x^3\right ) \log (x) \log ^3(\log (x))} \, dx=\frac {4\,x\,\left ({\mathrm {e}}^{2\,x}-48\,x+96\,\ln \left (\ln \left (x\right )\right )+24\,{\mathrm {e}}^x-2\,x^2\,{\mathrm {e}}^x+16\,x\,\ln \left (\ln \left (x\right )\right )+8\,x\,{\ln \left (\ln \left (x\right )\right )}^2-10\,x^2\,\ln \left (\ln \left (x\right )\right )-2\,x^3\,\ln \left (\ln \left (x\right )\right )+8\,\ln \left (\ln \left (x\right )\right )\,{\mathrm {e}}^x+16\,{\ln \left (\ln \left (x\right )\right )}^2-2\,x\,{\mathrm {e}}^x-8\,x^2+5\,x^3+x^4+x^2\,{\ln \left (\ln \left (x\right )\right )}^2+2\,x\,\ln \left (\ln \left (x\right )\right )\,{\mathrm {e}}^x\right )}{{\left (4\,\ln \left (\ln \left (x\right )\right )+{\mathrm {e}}^x+x\,\ln \left (\ln \left (x\right )\right )\right )}^2} \] Input:
int((1536*x + log(log(x))*(log(x)*(exp(2*x)*(12*x + 48) - 1536*x - 384*x^2 + 288*x^3 + 120*x^4 + 12*x^5 - exp(x)*(416*x + 280*x^2 - 8*x^4 - 768)) - 640*x + 96*x^2 + 72*x^3 + 8*x^4 - 1536) - log(x)*(exp(2*x)*(112*x + 16*x^2 - 8*x^3 - 96) - 4*exp(3*x) + exp(x)*(384*x - 288*x^2 - 144*x^3 + 20*x^4 + 8*x^5)) + 640*x^2 - 96*x^3 - 72*x^4 - 8*x^5 - exp(x)*(64*x - 40*x^2 - 8*x ^3 + 384) + log(log(x))^2*log(x)*(128*x + exp(x)*(96*x + 12*x^2 + 192) - 4 80*x^2 - 168*x^3 - 16*x^4 + 1536) + log(log(x))^3*log(x)*(192*x + 48*x^2 + 4*x^3 + 256))/(exp(3*x)*log(x) + log(log(x))^3*log(x)*(48*x + 12*x^2 + x^ 3 + 64) + log(log(x))*exp(2*x)*log(x)*(3*x + 12) + log(log(x))^2*exp(x)*lo g(x)*(24*x + 3*x^2 + 48)),x)
Output:
(4*x*(exp(2*x) - 48*x + 96*log(log(x)) + 24*exp(x) - 2*x^2*exp(x) + 16*x*l og(log(x)) + 8*x*log(log(x))^2 - 10*x^2*log(log(x)) - 2*x^3*log(log(x)) + 8*log(log(x))*exp(x) + 16*log(log(x))^2 - 2*x*exp(x) - 8*x^2 + 5*x^3 + x^4 + x^2*log(log(x))^2 + 2*x*log(log(x))*exp(x)))/(4*log(log(x)) + exp(x) + x*log(log(x)))^2
Time = 0.19 (sec) , antiderivative size = 189, normalized size of antiderivative = 6.75 \[ \int \frac {1536 x+640 x^2-96 x^3-72 x^4-8 x^5+e^x \left (-384-64 x+40 x^2+8 x^3\right )+\left (4 e^{3 x}+e^{2 x} \left (96-112 x-16 x^2+8 x^3\right )+e^x \left (-384 x+288 x^2+144 x^3-20 x^4-8 x^5\right )\right ) \log (x)+\left (-1536-640 x+96 x^2+72 x^3+8 x^4+\left (-1536 x-384 x^2+288 x^3+120 x^4+12 x^5+e^{2 x} (48+12 x)+e^x \left (768-416 x-280 x^2+8 x^4\right )\right ) \log (x)\right ) \log (\log (x))+\left (1536+128 x-480 x^2-168 x^3-16 x^4+e^x \left (192+96 x+12 x^2\right )\right ) \log (x) \log ^2(\log (x))+\left (256+192 x+48 x^2+4 x^3\right ) \log (x) \log ^3(\log (x))}{e^{3 x} \log (x)+e^{2 x} (12+3 x) \log (x) \log (\log (x))+e^x \left (48+24 x+3 x^2\right ) \log (x) \log ^2(\log (x))+\left (64+48 x+12 x^2+x^3\right ) \log (x) \log ^3(\log (x))} \, dx=\frac {4 e^{2 x} x -16 e^{2 x}+8 e^{x} \mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x^{2}-128 e^{x} \mathrm {log}\left (\mathrm {log}\left (x \right )\right )-8 e^{x} x^{3}-8 e^{x} x^{2}+96 e^{x} x +4 \mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{2} x^{3}+16 \mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{2} x^{2}-64 \mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{2} x -256 \mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{2}-8 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x^{4}-40 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x^{3}+64 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x^{2}+384 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x +4 x^{5}+20 x^{4}-32 x^{3}-192 x^{2}}{e^{2 x}+2 e^{x} \mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x +8 e^{x} \mathrm {log}\left (\mathrm {log}\left (x \right )\right )+\mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{2} x^{2}+8 \mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{2} x +16 \mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{2}} \] Input:
int(((4*x^3+48*x^2+192*x+256)*log(x)*log(log(x))^3+((12*x^2+96*x+192)*exp( x)-16*x^4-168*x^3-480*x^2+128*x+1536)*log(x)*log(log(x))^2+(((12*x+48)*exp (x)^2+(8*x^4-280*x^2-416*x+768)*exp(x)+12*x^5+120*x^4+288*x^3-384*x^2-1536 *x)*log(x)+8*x^4+72*x^3+96*x^2-640*x-1536)*log(log(x))+(4*exp(x)^3+(8*x^3- 16*x^2-112*x+96)*exp(x)^2+(-8*x^5-20*x^4+144*x^3+288*x^2-384*x)*exp(x))*lo g(x)+(8*x^3+40*x^2-64*x-384)*exp(x)-8*x^5-72*x^4-96*x^3+640*x^2+1536*x)/(( x^3+12*x^2+48*x+64)*log(x)*log(log(x))^3+(3*x^2+24*x+48)*exp(x)*log(x)*log (log(x))^2+(3*x+12)*exp(x)^2*log(x)*log(log(x))+exp(x)^3*log(x)),x)
Output:
(4*(e**(2*x)*x - 4*e**(2*x) + 2*e**x*log(log(x))*x**2 - 32*e**x*log(log(x) ) - 2*e**x*x**3 - 2*e**x*x**2 + 24*e**x*x + log(log(x))**2*x**3 + 4*log(lo g(x))**2*x**2 - 16*log(log(x))**2*x - 64*log(log(x))**2 - 2*log(log(x))*x* *4 - 10*log(log(x))*x**3 + 16*log(log(x))*x**2 + 96*log(log(x))*x + x**5 + 5*x**4 - 8*x**3 - 48*x**2))/(e**(2*x) + 2*e**x*log(log(x))*x + 8*e**x*log (log(x)) + log(log(x))**2*x**2 + 8*log(log(x))**2*x + 16*log(log(x))**2)