Integrand size = 246, antiderivative size = 30 \[ \int \frac {2 e^{16}+2 e^{2 e^x}+8 e^{12} x+2 x^2+4 x^3+2 x^4+e^{e^x} \left (-4 e^8-4 x-8 e^4 x-4 x^2\right )+e^8 \left (4 x+12 x^2\right )+e^4 \left (8 x^2+8 x^3\right )+\left (2 e^{2 e^x+x} x \log \left (16 x^2\right )+e^{e^x} \left (-2 x-4 e^4 x-4 x^2+e^x \left (-2 e^8 x-2 x^2-4 e^4 x^2-2 x^3\right )\right ) \log \left (16 x^2\right )+\left (4 e^{12} x+2 x^2+6 x^3+4 x^4+e^8 \left (2 x+12 x^2\right )+e^4 \left (8 x^2+12 x^3\right )\right ) \log \left (16 x^2\right )\right ) \log \left (\frac {1}{4} \log \left (16 x^2\right )\right )}{x \log \left (16 x^2\right )} \, dx=\left (-e^{e^x}+x+\left (e^4+x\right )^2\right )^2 \log \left (\frac {1}{4} \log \left (16 x^2\right )\right ) \] Output:
ln(1/4*ln(16*x^2))*((x+exp(4))^2+x-exp(exp(x)))^2
\[ \int \frac {2 e^{16}+2 e^{2 e^x}+8 e^{12} x+2 x^2+4 x^3+2 x^4+e^{e^x} \left (-4 e^8-4 x-8 e^4 x-4 x^2\right )+e^8 \left (4 x+12 x^2\right )+e^4 \left (8 x^2+8 x^3\right )+\left (2 e^{2 e^x+x} x \log \left (16 x^2\right )+e^{e^x} \left (-2 x-4 e^4 x-4 x^2+e^x \left (-2 e^8 x-2 x^2-4 e^4 x^2-2 x^3\right )\right ) \log \left (16 x^2\right )+\left (4 e^{12} x+2 x^2+6 x^3+4 x^4+e^8 \left (2 x+12 x^2\right )+e^4 \left (8 x^2+12 x^3\right )\right ) \log \left (16 x^2\right )\right ) \log \left (\frac {1}{4} \log \left (16 x^2\right )\right )}{x \log \left (16 x^2\right )} \, dx=\int \frac {2 e^{16}+2 e^{2 e^x}+8 e^{12} x+2 x^2+4 x^3+2 x^4+e^{e^x} \left (-4 e^8-4 x-8 e^4 x-4 x^2\right )+e^8 \left (4 x+12 x^2\right )+e^4 \left (8 x^2+8 x^3\right )+\left (2 e^{2 e^x+x} x \log \left (16 x^2\right )+e^{e^x} \left (-2 x-4 e^4 x-4 x^2+e^x \left (-2 e^8 x-2 x^2-4 e^4 x^2-2 x^3\right )\right ) \log \left (16 x^2\right )+\left (4 e^{12} x+2 x^2+6 x^3+4 x^4+e^8 \left (2 x+12 x^2\right )+e^4 \left (8 x^2+12 x^3\right )\right ) \log \left (16 x^2\right )\right ) \log \left (\frac {1}{4} \log \left (16 x^2\right )\right )}{x \log \left (16 x^2\right )} \, dx \] Input:
Integrate[(2*E^16 + 2*E^(2*E^x) + 8*E^12*x + 2*x^2 + 4*x^3 + 2*x^4 + E^E^x *(-4*E^8 - 4*x - 8*E^4*x - 4*x^2) + E^8*(4*x + 12*x^2) + E^4*(8*x^2 + 8*x^ 3) + (2*E^(2*E^x + x)*x*Log[16*x^2] + E^E^x*(-2*x - 4*E^4*x - 4*x^2 + E^x* (-2*E^8*x - 2*x^2 - 4*E^4*x^2 - 2*x^3))*Log[16*x^2] + (4*E^12*x + 2*x^2 + 6*x^3 + 4*x^4 + E^8*(2*x + 12*x^2) + E^4*(8*x^2 + 12*x^3))*Log[16*x^2])*Lo g[Log[16*x^2]/4])/(x*Log[16*x^2]),x]
Output:
Integrate[(2*E^16 + 2*E^(2*E^x) + 8*E^12*x + 2*x^2 + 4*x^3 + 2*x^4 + E^E^x *(-4*E^8 - 4*x - 8*E^4*x - 4*x^2) + E^8*(4*x + 12*x^2) + E^4*(8*x^2 + 8*x^ 3) + (2*E^(2*E^x + x)*x*Log[16*x^2] + E^E^x*(-2*x - 4*E^4*x - 4*x^2 + E^x* (-2*E^8*x - 2*x^2 - 4*E^4*x^2 - 2*x^3))*Log[16*x^2] + (4*E^12*x + 2*x^2 + 6*x^3 + 4*x^4 + E^8*(2*x + 12*x^2) + E^4*(8*x^2 + 12*x^3))*Log[16*x^2])*Lo g[Log[16*x^2]/4])/(x*Log[16*x^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.
| \(\displaystyle \int \frac {2 x^4+4 x^3+2 x^2+e^{e^x} \left (-4 x^2-8 e^4 x-4 x-4 e^8\right )+e^8 \left (12 x^2+4 x\right )+e^4 \left (8 x^3+8 x^2\right )+\left (2 e^{x+2 e^x} x \log \left (16 x^2\right )+e^{e^x} \left (-4 x^2+e^x \left (-2 x^3-4 e^4 x^2-2 x^2-2 e^8 x\right )-4 e^4 x-2 x\right ) \log \left (16 x^2\right )+\left (4 x^4+6 x^3+2 x^2+e^8 \left (12 x^2+2 x\right )+e^4 \left (12 x^3+8 x^2\right )+4 e^{12} x\right ) \log \left (16 x^2\right )\right ) \log \left (\frac {1}{4} \log \left (16 x^2\right )\right )+8 e^{12} x+2 e^{2 e^x}+2 e^{16}}{x \log \left (16 x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 \left (x^2+\left (1+2 e^4\right ) x-e^{e^x}+e^8\right ) \left (x^2+\left (2 x-e^{x+e^x}+2 e^4+1\right ) x \log \left (16 x^2\right ) \log \left (\frac {\log \left (x^2\right )}{4}+\log (2)\right )+\left (1+2 e^4\right ) x-e^{e^x}+e^8\right )}{x \log \left (16 x^2\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int \frac {\left (x^2+\left (1+2 e^4\right ) x-e^{e^x}+e^8\right ) \left (x^2+\left (2 x-e^{x+e^x}+2 e^4+1\right ) \log \left (16 x^2\right ) \log \left (\frac {\log \left (x^2\right )}{4}+\log (2)\right ) x+\left (1+2 e^4\right ) x-e^{e^x}+e^8\right )}{x \log \left (16 x^2\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {\left (x^2+\left (1+2 e^4\right ) x-e^{e^x}+e^8\right ) \left (2 \log \left (16 x^2\right ) \log \left (\frac {\log \left (x^2\right )}{4}+\log (2)\right ) x^2+x^2+\left (1+2 e^4\right ) \log \left (16 x^2\right ) \log \left (\frac {\log \left (x^2\right )}{4}+\log (2)\right ) x+\left (1+2 e^4\right ) x-e^{e^x}+e^8\right )}{x \log \left (16 x^2\right )}+e^{x+e^x} \left (-x^2-\left (1+2 e^4\right ) x+e^{e^x}-e^8\right ) \log \left (\frac {1}{4} \log \left (16 x^2\right )\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (2 \left (1+2 e^4\right ) \int \frac {\operatorname {ExpIntegralEi}\left (e^x\right )}{x \log \left (16 x^2\right )}dx-2 \log \left (\frac {1}{4} \log \left (16 x^2\right )\right ) \int e^{e^x} xdx-\left (1+2 e^4\right ) \log \left (\frac {1}{4} \log \left (16 x^2\right )\right ) \int e^{x+e^x} xdx-\log \left (\frac {1}{4} \log \left (16 x^2\right )\right ) \int e^{x+e^x} x^2dx-2 \left (1+2 e^4\right ) \int \frac {e^{e^x}}{\log \left (16 x^2\right )}dx-2 \int \frac {e^{e^x} x}{\log \left (16 x^2\right )}dx+\int \frac {\left (x^2+\left (1+2 e^4\right ) x+e^8\right )^2}{x \log \left (16 x^2\right )}dx+4 \int \frac {\int e^{e^x} xdx}{x \log \left (16 x^2\right )}dx+2 \left (1+2 e^4\right ) \int \frac {\int e^{x+e^x} xdx}{x \log \left (16 x^2\right )}dx+2 \int \frac {\int e^{x+e^x} x^2dx}{x \log \left (16 x^2\right )}dx-\frac {e^8 \left (1+2 e^4\right ) x \operatorname {ExpIntegralEi}\left (\frac {1}{2} \log \left (16 x^2\right )\right )}{4 \sqrt {x^2}}-\frac {1}{512} \operatorname {ExpIntegralEi}\left (2 \log \left (16 x^2\right )\right )-\left (1+2 e^4\right ) \operatorname {ExpIntegralEi}\left (e^x\right ) \log \left (\frac {1}{4} \log \left (16 x^2\right )\right )-\frac {\left (1+2 e^4\right ) x^3 \operatorname {ExpIntegralEi}\left (\frac {3}{2} \log \left (16 x^2\right )\right )}{64 \left (x^2\right )^{3/2}}-\frac {1}{32} \left (1+4 e^4+6 e^8\right ) \operatorname {LogIntegral}\left (16 x^2\right )+\frac {1}{2} \left (1+4 e^4+6 e^8\right ) x^2 \log \left (\frac {1}{4} \log \left (16 x^2\right )\right )+e^8 \left (1+2 e^4\right ) x \log \left (\frac {1}{4} \log \left (16 x^2\right )\right )+\frac {1}{2} e^{2 e^x} \log \left (\frac {1}{4} \log \left (16 x^2\right )\right )-e^{e^x+8} \log \left (\frac {1}{4} \log \left (16 x^2\right )\right )+\frac {1}{2} x^4 \log \left (\frac {1}{4} \log \left (16 x^2\right )\right )+\left (1+2 e^4\right ) x^3 \log \left (\frac {1}{4} \log \left (16 x^2\right )\right )\right )\) |
Input:
Int[(2*E^16 + 2*E^(2*E^x) + 8*E^12*x + 2*x^2 + 4*x^3 + 2*x^4 + E^E^x*(-4*E ^8 - 4*x - 8*E^4*x - 4*x^2) + E^8*(4*x + 12*x^2) + E^4*(8*x^2 + 8*x^3) + ( 2*E^(2*E^x + x)*x*Log[16*x^2] + E^E^x*(-2*x - 4*E^4*x - 4*x^2 + E^x*(-2*E^ 8*x - 2*x^2 - 4*E^4*x^2 - 2*x^3))*Log[16*x^2] + (4*E^12*x + 2*x^2 + 6*x^3 + 4*x^4 + E^8*(2*x + 12*x^2) + E^4*(8*x^2 + 12*x^3))*Log[16*x^2])*Log[Log[ 16*x^2]/4])/(x*Log[16*x^2]),x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 125.62 (sec) , antiderivative size = 117, normalized size of antiderivative = 3.90
| method | result | size |
| risch | \(\left ({\mathrm e}^{16}+4 \,{\mathrm e}^{12} x +6 x^{2} {\mathrm e}^{8}+4 x^{3} {\mathrm e}^{4}+x^{4}+2 x \,{\mathrm e}^{8}-2 \,{\mathrm e}^{8+{\mathrm e}^{x}}+4 x^{2} {\mathrm e}^{4}-4 x \,{\mathrm e}^{{\mathrm e}^{x}+4}+2 x^{3}-2 \,{\mathrm e}^{{\mathrm e}^{x}} x^{2}+x^{2}-2 x \,{\mathrm e}^{{\mathrm e}^{x}}+{\mathrm e}^{2 \,{\mathrm e}^{x}}\right ) \ln \left (\ln \left (2\right )+\frac {\ln \left (x \right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{8}\right )\) | \(117\) |
| parallelrisch | \({\mathrm e}^{16} \ln \left (\frac {\ln \left (16 x^{2}\right )}{4}\right )+4 \,{\mathrm e}^{12} x \ln \left (\frac {\ln \left (16 x^{2}\right )}{4}\right )+6 \,{\mathrm e}^{8} x^{2} \ln \left (\frac {\ln \left (16 x^{2}\right )}{4}\right )+4 \ln \left (\frac {\ln \left (16 x^{2}\right )}{4}\right ) x^{3} {\mathrm e}^{4}+\ln \left (\frac {\ln \left (16 x^{2}\right )}{4}\right ) x^{4}+2 \,{\mathrm e}^{8} x \ln \left (\frac {\ln \left (16 x^{2}\right )}{4}\right )-2 \,{\mathrm e}^{8} \ln \left (\frac {\ln \left (16 x^{2}\right )}{4}\right ) {\mathrm e}^{{\mathrm e}^{x}}+4 \,{\mathrm e}^{4} x^{2} \ln \left (\frac {\ln \left (16 x^{2}\right )}{4}\right )-4 \,{\mathrm e}^{{\mathrm e}^{x}} \ln \left (\frac {\ln \left (16 x^{2}\right )}{4}\right ) x \,{\mathrm e}^{4}+2 \ln \left (\frac {\ln \left (16 x^{2}\right )}{4}\right ) x^{3}-2 \ln \left (\frac {\ln \left (16 x^{2}\right )}{4}\right ) x^{2} {\mathrm e}^{{\mathrm e}^{x}}+\ln \left (\frac {\ln \left (16 x^{2}\right )}{4}\right ) x^{2}-2 \,{\mathrm e}^{{\mathrm e}^{x}} \ln \left (\frac {\ln \left (16 x^{2}\right )}{4}\right ) x +\ln \left (\frac {\ln \left (16 x^{2}\right )}{4}\right ) {\mathrm e}^{2 \,{\mathrm e}^{x}}\) | \(220\) |
Input:
int(((2*x*exp(x)*ln(16*x^2)*exp(exp(x))^2+((-2*x*exp(4)^2-4*x^2*exp(4)-2*x ^3-2*x^2)*exp(x)-4*x*exp(4)-4*x^2-2*x)*ln(16*x^2)*exp(exp(x))+(4*x*exp(4)^ 3+(12*x^2+2*x)*exp(4)^2+(12*x^3+8*x^2)*exp(4)+4*x^4+6*x^3+2*x^2)*ln(16*x^2 ))*ln(1/4*ln(16*x^2))+2*exp(exp(x))^2+(-4*exp(4)^2-8*x*exp(4)-4*x^2-4*x)*e xp(exp(x))+2*exp(4)^4+8*x*exp(4)^3+(12*x^2+4*x)*exp(4)^2+(8*x^3+8*x^2)*exp (4)+2*x^4+4*x^3+2*x^2)/x/ln(16*x^2),x,method=_RETURNVERBOSE)
Output:
(exp(16)+4*exp(12)*x+6*x^2*exp(8)+4*x^3*exp(4)+x^4+2*x*exp(8)-2*exp(8+exp( x))+4*x^2*exp(4)-4*x*exp(exp(x)+4)+2*x^3-2*exp(exp(x))*x^2+x^2-2*x*exp(exp (x))+exp(2*exp(x)))*ln(ln(2)+1/2*ln(x)-1/8*I*Pi*csgn(I*x^2)*(-csgn(I*x^2)+ csgn(I*x))^2)
Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (25) = 50\).
Time = 0.08 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.43 \[ \int \frac {2 e^{16}+2 e^{2 e^x}+8 e^{12} x+2 x^2+4 x^3+2 x^4+e^{e^x} \left (-4 e^8-4 x-8 e^4 x-4 x^2\right )+e^8 \left (4 x+12 x^2\right )+e^4 \left (8 x^2+8 x^3\right )+\left (2 e^{2 e^x+x} x \log \left (16 x^2\right )+e^{e^x} \left (-2 x-4 e^4 x-4 x^2+e^x \left (-2 e^8 x-2 x^2-4 e^4 x^2-2 x^3\right )\right ) \log \left (16 x^2\right )+\left (4 e^{12} x+2 x^2+6 x^3+4 x^4+e^8 \left (2 x+12 x^2\right )+e^4 \left (8 x^2+12 x^3\right )\right ) \log \left (16 x^2\right )\right ) \log \left (\frac {1}{4} \log \left (16 x^2\right )\right )}{x \log \left (16 x^2\right )} \, dx={\left (x^{4} + 2 \, x^{3} + x^{2} + 4 \, x e^{12} + 2 \, {\left (3 \, x^{2} + x\right )} e^{8} + 4 \, {\left (x^{3} + x^{2}\right )} e^{4} - 2 \, {\left (x^{2} + 2 \, x e^{4} + x + e^{8}\right )} e^{\left (e^{x}\right )} + e^{16} + e^{\left (2 \, e^{x}\right )}\right )} \log \left (\frac {1}{4} \, \log \left (16 \, x^{2}\right )\right ) \] Input:
integrate(((2*x*exp(x)*log(16*x^2)*exp(exp(x))^2+((-2*x*exp(4)^2-4*x^2*exp (4)-2*x^3-2*x^2)*exp(x)-4*x*exp(4)-4*x^2-2*x)*log(16*x^2)*exp(exp(x))+(4*x *exp(4)^3+(12*x^2+2*x)*exp(4)^2+(12*x^3+8*x^2)*exp(4)+4*x^4+6*x^3+2*x^2)*l og(16*x^2))*log(1/4*log(16*x^2))+2*exp(exp(x))^2+(-4*exp(4)^2-8*x*exp(4)-4 *x^2-4*x)*exp(exp(x))+2*exp(4)^4+8*x*exp(4)^3+(12*x^2+4*x)*exp(4)^2+(8*x^3 +8*x^2)*exp(4)+2*x^4+4*x^3+2*x^2)/x/log(16*x^2),x, algorithm="fricas")
Output:
(x^4 + 2*x^3 + x^2 + 4*x*e^12 + 2*(3*x^2 + x)*e^8 + 4*(x^3 + x^2)*e^4 - 2* (x^2 + 2*x*e^4 + x + e^8)*e^(e^x) + e^16 + e^(2*e^x))*log(1/4*log(16*x^2))
Timed out. \[ \int \frac {2 e^{16}+2 e^{2 e^x}+8 e^{12} x+2 x^2+4 x^3+2 x^4+e^{e^x} \left (-4 e^8-4 x-8 e^4 x-4 x^2\right )+e^8 \left (4 x+12 x^2\right )+e^4 \left (8 x^2+8 x^3\right )+\left (2 e^{2 e^x+x} x \log \left (16 x^2\right )+e^{e^x} \left (-2 x-4 e^4 x-4 x^2+e^x \left (-2 e^8 x-2 x^2-4 e^4 x^2-2 x^3\right )\right ) \log \left (16 x^2\right )+\left (4 e^{12} x+2 x^2+6 x^3+4 x^4+e^8 \left (2 x+12 x^2\right )+e^4 \left (8 x^2+12 x^3\right )\right ) \log \left (16 x^2\right )\right ) \log \left (\frac {1}{4} \log \left (16 x^2\right )\right )}{x \log \left (16 x^2\right )} \, dx=\text {Timed out} \] Input:
integrate(((2*x*exp(x)*ln(16*x**2)*exp(exp(x))**2+((-2*x*exp(4)**2-4*x**2* exp(4)-2*x**3-2*x**2)*exp(x)-4*x*exp(4)-4*x**2-2*x)*ln(16*x**2)*exp(exp(x) )+(4*x*exp(4)**3+(12*x**2+2*x)*exp(4)**2+(12*x**3+8*x**2)*exp(4)+4*x**4+6* x**3+2*x**2)*ln(16*x**2))*ln(1/4*ln(16*x**2))+2*exp(exp(x))**2+(-4*exp(4)* *2-8*x*exp(4)-4*x**2-4*x)*exp(exp(x))+2*exp(4)**4+8*x*exp(4)**3+(12*x**2+4 *x)*exp(4)**2+(8*x**3+8*x**2)*exp(4)+2*x**4+4*x**3+2*x**2)/x/ln(16*x**2),x )
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (25) = 50\).
Time = 0.16 (sec) , antiderivative size = 168, normalized size of antiderivative = 5.60 \[ \int \frac {2 e^{16}+2 e^{2 e^x}+8 e^{12} x+2 x^2+4 x^3+2 x^4+e^{e^x} \left (-4 e^8-4 x-8 e^4 x-4 x^2\right )+e^8 \left (4 x+12 x^2\right )+e^4 \left (8 x^2+8 x^3\right )+\left (2 e^{2 e^x+x} x \log \left (16 x^2\right )+e^{e^x} \left (-2 x-4 e^4 x-4 x^2+e^x \left (-2 e^8 x-2 x^2-4 e^4 x^2-2 x^3\right )\right ) \log \left (16 x^2\right )+\left (4 e^{12} x+2 x^2+6 x^3+4 x^4+e^8 \left (2 x+12 x^2\right )+e^4 \left (8 x^2+12 x^3\right )\right ) \log \left (16 x^2\right )\right ) \log \left (\frac {1}{4} \log \left (16 x^2\right )\right )}{x \log \left (16 x^2\right )} \, dx=-x^{4} \log \left (2\right ) - 2 \, {\left (2 \, e^{4} \log \left (2\right ) + \log \left (2\right )\right )} x^{3} - {\left (6 \, e^{8} \log \left (2\right ) + 4 \, e^{4} \log \left (2\right ) + \log \left (2\right )\right )} x^{2} - 2 \, {\left (2 \, e^{12} \log \left (2\right ) + e^{8} \log \left (2\right )\right )} x + 2 \, {\left (x^{2} \log \left (2\right ) + {\left (2 \, e^{4} \log \left (2\right ) + \log \left (2\right )\right )} x + e^{8} \log \left (2\right )\right )} e^{\left (e^{x}\right )} - e^{\left (2 \, e^{x}\right )} \log \left (2\right ) + {\left (x^{4} + 2 \, x^{3} {\left (2 \, e^{4} + 1\right )} + x^{2} {\left (6 \, e^{8} + 4 \, e^{4} + 1\right )} + 2 \, x {\left (2 \, e^{12} + e^{8}\right )} - 2 \, {\left (x^{2} + x {\left (2 \, e^{4} + 1\right )} + e^{8}\right )} e^{\left (e^{x}\right )} + e^{16} + e^{\left (2 \, e^{x}\right )}\right )} \log \left (2 \, \log \left (2\right ) + \log \left (x\right )\right ) \] Input:
integrate(((2*x*exp(x)*log(16*x^2)*exp(exp(x))^2+((-2*x*exp(4)^2-4*x^2*exp (4)-2*x^3-2*x^2)*exp(x)-4*x*exp(4)-4*x^2-2*x)*log(16*x^2)*exp(exp(x))+(4*x *exp(4)^3+(12*x^2+2*x)*exp(4)^2+(12*x^3+8*x^2)*exp(4)+4*x^4+6*x^3+2*x^2)*l og(16*x^2))*log(1/4*log(16*x^2))+2*exp(exp(x))^2+(-4*exp(4)^2-8*x*exp(4)-4 *x^2-4*x)*exp(exp(x))+2*exp(4)^4+8*x*exp(4)^3+(12*x^2+4*x)*exp(4)^2+(8*x^3 +8*x^2)*exp(4)+2*x^4+4*x^3+2*x^2)/x/log(16*x^2),x, algorithm="maxima")
Output:
-x^4*log(2) - 2*(2*e^4*log(2) + log(2))*x^3 - (6*e^8*log(2) + 4*e^4*log(2) + log(2))*x^2 - 2*(2*e^12*log(2) + e^8*log(2))*x + 2*(x^2*log(2) + (2*e^4 *log(2) + log(2))*x + e^8*log(2))*e^(e^x) - e^(2*e^x)*log(2) + (x^4 + 2*x^ 3*(2*e^4 + 1) + x^2*(6*e^8 + 4*e^4 + 1) + 2*x*(2*e^12 + e^8) - 2*(x^2 + x* (2*e^4 + 1) + e^8)*e^(e^x) + e^16 + e^(2*e^x))*log(2*log(2) + log(x))
Exception generated. \[ \int \frac {2 e^{16}+2 e^{2 e^x}+8 e^{12} x+2 x^2+4 x^3+2 x^4+e^{e^x} \left (-4 e^8-4 x-8 e^4 x-4 x^2\right )+e^8 \left (4 x+12 x^2\right )+e^4 \left (8 x^2+8 x^3\right )+\left (2 e^{2 e^x+x} x \log \left (16 x^2\right )+e^{e^x} \left (-2 x-4 e^4 x-4 x^2+e^x \left (-2 e^8 x-2 x^2-4 e^4 x^2-2 x^3\right )\right ) \log \left (16 x^2\right )+\left (4 e^{12} x+2 x^2+6 x^3+4 x^4+e^8 \left (2 x+12 x^2\right )+e^4 \left (8 x^2+12 x^3\right )\right ) \log \left (16 x^2\right )\right ) \log \left (\frac {1}{4} \log \left (16 x^2\right )\right )}{x \log \left (16 x^2\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(((2*x*exp(x)*log(16*x^2)*exp(exp(x))^2+((-2*x*exp(4)^2-4*x^2*exp (4)-2*x^3-2*x^2)*exp(x)-4*x*exp(4)-4*x^2-2*x)*log(16*x^2)*exp(exp(x))+(4*x *exp(4)^3+(12*x^2+2*x)*exp(4)^2+(12*x^3+8*x^2)*exp(4)+4*x^4+6*x^3+2*x^2)*l og(16*x^2))*log(1/4*log(16*x^2))+2*exp(exp(x))^2+(-4*exp(4)^2-8*x*exp(4)-4 *x^2-4*x)*exp(exp(x))+2*exp(4)^4+8*x*exp(4)^3+(12*x^2+4*x)*exp(4)^2+(8*x^3 +8*x^2)*exp(4)+2*x^4+4*x^3+2*x^2)/x/log(16*x^2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:ln of unsigned or minus infinity Er ror: Bad Argument Value
Time = 2.29 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.17 \[ \int \frac {2 e^{16}+2 e^{2 e^x}+8 e^{12} x+2 x^2+4 x^3+2 x^4+e^{e^x} \left (-4 e^8-4 x-8 e^4 x-4 x^2\right )+e^8 \left (4 x+12 x^2\right )+e^4 \left (8 x^2+8 x^3\right )+\left (2 e^{2 e^x+x} x \log \left (16 x^2\right )+e^{e^x} \left (-2 x-4 e^4 x-4 x^2+e^x \left (-2 e^8 x-2 x^2-4 e^4 x^2-2 x^3\right )\right ) \log \left (16 x^2\right )+\left (4 e^{12} x+2 x^2+6 x^3+4 x^4+e^8 \left (2 x+12 x^2\right )+e^4 \left (8 x^2+12 x^3\right )\right ) \log \left (16 x^2\right )\right ) \log \left (\frac {1}{4} \log \left (16 x^2\right )\right )}{x \log \left (16 x^2\right )} \, dx=\ln \left (\frac {\ln \left (16\,x^2\right )}{4}\right )\,\left ({\mathrm {e}}^{2\,{\mathrm {e}}^x}-{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (2\,x+2\,{\mathrm {e}}^8+4\,x\,{\mathrm {e}}^4+2\,x^2\right )+\frac {x^5+\left (4\,{\mathrm {e}}^4+2\right )\,x^4+\left (4\,{\mathrm {e}}^4+6\,{\mathrm {e}}^8+1\right )\,x^3+2\,{\mathrm {e}}^8\,\left (2\,{\mathrm {e}}^4+1\right )\,x^2}{x}\right )+\ln \left (\ln \left (16\,x^2\right )\right )\,{\mathrm {e}}^{16} \] Input:
int((2*exp(16) + 2*exp(2*exp(x)) + log(log(16*x^2)/4)*(log(16*x^2)*(exp(8) *(2*x + 12*x^2) + 4*x*exp(12) + exp(4)*(8*x^2 + 12*x^3) + 2*x^2 + 6*x^3 + 4*x^4) - exp(exp(x))*log(16*x^2)*(2*x + 4*x*exp(4) + 4*x^2 + exp(x)*(2*x*e xp(8) + 4*x^2*exp(4) + 2*x^2 + 2*x^3)) + 2*x*exp(2*exp(x))*exp(x)*log(16*x ^2)) - exp(exp(x))*(4*x + 4*exp(8) + 8*x*exp(4) + 4*x^2) + exp(8)*(4*x + 1 2*x^2) + 8*x*exp(12) + exp(4)*(8*x^2 + 8*x^3) + 2*x^2 + 4*x^3 + 2*x^4)/(x* log(16*x^2)),x)
Output:
log(log(16*x^2)/4)*(exp(2*exp(x)) - exp(exp(x))*(2*x + 2*exp(8) + 4*x*exp( 4) + 2*x^2) + (x^3*(4*exp(4) + 6*exp(8) + 1) + x^4*(4*exp(4) + 2) + x^5 + 2*x^2*exp(8)*(2*exp(4) + 1))/x) + log(log(16*x^2))*exp(16)
Time = 0.15 (sec) , antiderivative size = 225, normalized size of antiderivative = 7.50 \[ \int \frac {2 e^{16}+2 e^{2 e^x}+8 e^{12} x+2 x^2+4 x^3+2 x^4+e^{e^x} \left (-4 e^8-4 x-8 e^4 x-4 x^2\right )+e^8 \left (4 x+12 x^2\right )+e^4 \left (8 x^2+8 x^3\right )+\left (2 e^{2 e^x+x} x \log \left (16 x^2\right )+e^{e^x} \left (-2 x-4 e^4 x-4 x^2+e^x \left (-2 e^8 x-2 x^2-4 e^4 x^2-2 x^3\right )\right ) \log \left (16 x^2\right )+\left (4 e^{12} x+2 x^2+6 x^3+4 x^4+e^8 \left (2 x+12 x^2\right )+e^4 \left (8 x^2+12 x^3\right )\right ) \log \left (16 x^2\right )\right ) \log \left (\frac {1}{4} \log \left (16 x^2\right )\right )}{x \log \left (16 x^2\right )} \, dx=e^{2 e^{x}} \mathrm {log}\left (\frac {\mathrm {log}\left (16 x^{2}\right )}{4}\right )-2 e^{e^{x}} \mathrm {log}\left (\frac {\mathrm {log}\left (16 x^{2}\right )}{4}\right ) e^{8}-4 e^{e^{x}} \mathrm {log}\left (\frac {\mathrm {log}\left (16 x^{2}\right )}{4}\right ) e^{4} x -2 e^{e^{x}} \mathrm {log}\left (\frac {\mathrm {log}\left (16 x^{2}\right )}{4}\right ) x^{2}-2 e^{e^{x}} \mathrm {log}\left (\frac {\mathrm {log}\left (16 x^{2}\right )}{4}\right ) x +\mathrm {log}\left (\mathrm {log}\left (16 x^{2}\right )\right ) e^{16}+4 \,\mathrm {log}\left (\frac {\mathrm {log}\left (16 x^{2}\right )}{4}\right ) e^{12} x +6 \,\mathrm {log}\left (\frac {\mathrm {log}\left (16 x^{2}\right )}{4}\right ) e^{8} x^{2}+2 \,\mathrm {log}\left (\frac {\mathrm {log}\left (16 x^{2}\right )}{4}\right ) e^{8} x +4 \,\mathrm {log}\left (\frac {\mathrm {log}\left (16 x^{2}\right )}{4}\right ) e^{4} x^{3}+4 \,\mathrm {log}\left (\frac {\mathrm {log}\left (16 x^{2}\right )}{4}\right ) e^{4} x^{2}+\mathrm {log}\left (\frac {\mathrm {log}\left (16 x^{2}\right )}{4}\right ) x^{4}+2 \,\mathrm {log}\left (\frac {\mathrm {log}\left (16 x^{2}\right )}{4}\right ) x^{3}+\mathrm {log}\left (\frac {\mathrm {log}\left (16 x^{2}\right )}{4}\right ) x^{2} \] Input:
int(((2*x*exp(x)*log(16*x^2)*exp(exp(x))^2+((-2*x*exp(4)^2-4*x^2*exp(4)-2* x^3-2*x^2)*exp(x)-4*x*exp(4)-4*x^2-2*x)*log(16*x^2)*exp(exp(x))+(4*x*exp(4 )^3+(12*x^2+2*x)*exp(4)^2+(12*x^3+8*x^2)*exp(4)+4*x^4+6*x^3+2*x^2)*log(16* x^2))*log(1/4*log(16*x^2))+2*exp(exp(x))^2+(-4*exp(4)^2-8*x*exp(4)-4*x^2-4 *x)*exp(exp(x))+2*exp(4)^4+8*x*exp(4)^3+(12*x^2+4*x)*exp(4)^2+(8*x^3+8*x^2 )*exp(4)+2*x^4+4*x^3+2*x^2)/x/log(16*x^2),x)
Output:
e**(2*e**x)*log(log(16*x**2)/4) - 2*e**(e**x)*log(log(16*x**2)/4)*e**8 - 4 *e**(e**x)*log(log(16*x**2)/4)*e**4*x - 2*e**(e**x)*log(log(16*x**2)/4)*x* *2 - 2*e**(e**x)*log(log(16*x**2)/4)*x + log(log(16*x**2))*e**16 + 4*log(l og(16*x**2)/4)*e**12*x + 6*log(log(16*x**2)/4)*e**8*x**2 + 2*log(log(16*x* *2)/4)*e**8*x + 4*log(log(16*x**2)/4)*e**4*x**3 + 4*log(log(16*x**2)/4)*e* *4*x**2 + log(log(16*x**2)/4)*x**4 + 2*log(log(16*x**2)/4)*x**3 + log(log( 16*x**2)/4)*x**2