Integrand size = 263, antiderivative size = 31 \[ \int \frac {e^3 \left (4 x^3+x^4\right )+e^3 \left (4 x^2+x^3\right ) \log (4+x)+\left (4 x^3+x^4+\left (4 x^2+x^3\right ) \log (4+x)\right ) \log (x+\log (4+x))+e^{e^{\frac {\log ^2\left (e^3+\log (x+\log (4+x))\right )}{x}}+\frac {\log ^2\left (e^3+\log (x+\log (4+x))\right )}{x}} \left (\left (-10 x-2 x^2\right ) \log \left (e^3+\log (x+\log (4+x))\right )+\left (e^3 \left (4 x+x^2\right )+e^3 (4+x) \log (4+x)+\left (4 x+x^2+(4+x) \log (4+x)\right ) \log (x+\log (4+x))\right ) \log ^2\left (e^3+\log (x+\log (4+x))\right )\right )}{e^3 \left (4 x^3+x^4\right ) \log (4)+e^3 \left (4 x^2+x^3\right ) \log (4) \log (4+x)+\left (\left (4 x^3+x^4\right ) \log (4)+\left (4 x^2+x^3\right ) \log (4) \log (4+x)\right ) \log (x+\log (4+x))} \, dx=\frac {-e^{e^{\frac {\log ^2\left (e^3+\log (x+\log (4+x))\right )}{x}}}+x}{\log (4)} \] Output:
1/2*(x-exp(exp(ln(ln(ln(4+x)+x)+exp(3))^2/x)))/ln(2)
Time = 0.15 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {e^3 \left (4 x^3+x^4\right )+e^3 \left (4 x^2+x^3\right ) \log (4+x)+\left (4 x^3+x^4+\left (4 x^2+x^3\right ) \log (4+x)\right ) \log (x+\log (4+x))+e^{e^{\frac {\log ^2\left (e^3+\log (x+\log (4+x))\right )}{x}}+\frac {\log ^2\left (e^3+\log (x+\log (4+x))\right )}{x}} \left (\left (-10 x-2 x^2\right ) \log \left (e^3+\log (x+\log (4+x))\right )+\left (e^3 \left (4 x+x^2\right )+e^3 (4+x) \log (4+x)+\left (4 x+x^2+(4+x) \log (4+x)\right ) \log (x+\log (4+x))\right ) \log ^2\left (e^3+\log (x+\log (4+x))\right )\right )}{e^3 \left (4 x^3+x^4\right ) \log (4)+e^3 \left (4 x^2+x^3\right ) \log (4) \log (4+x)+\left (\left (4 x^3+x^4\right ) \log (4)+\left (4 x^2+x^3\right ) \log (4) \log (4+x)\right ) \log (x+\log (4+x))} \, dx=\frac {-e^{e^{\frac {\log ^2\left (e^3+\log (x+\log (4+x))\right )}{x}}}+x}{\log (4)} \] Input:
Integrate[(E^3*(4*x^3 + x^4) + E^3*(4*x^2 + x^3)*Log[4 + x] + (4*x^3 + x^4 + (4*x^2 + x^3)*Log[4 + x])*Log[x + Log[4 + x]] + E^(E^(Log[E^3 + Log[x + Log[4 + x]]]^2/x) + Log[E^3 + Log[x + Log[4 + x]]]^2/x)*((-10*x - 2*x^2)* Log[E^3 + Log[x + Log[4 + x]]] + (E^3*(4*x + x^2) + E^3*(4 + x)*Log[4 + x] + (4*x + x^2 + (4 + x)*Log[4 + x])*Log[x + Log[4 + x]])*Log[E^3 + Log[x + Log[4 + x]]]^2))/(E^3*(4*x^3 + x^4)*Log[4] + E^3*(4*x^2 + x^3)*Log[4]*Log [4 + x] + ((4*x^3 + x^4)*Log[4] + (4*x^2 + x^3)*Log[4]*Log[4 + x])*Log[x + Log[4 + x]]),x]
Output:
(-E^E^(Log[E^3 + Log[x + Log[4 + x]]]^2/x) + x)/Log[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.
| \(\displaystyle \int \frac {\left (\left (e^3 \left (x^2+4 x\right )+\left (x^2+4 x+(x+4) \log (x+4)\right ) \log (x+\log (x+4))+e^3 (x+4) \log (x+4)\right ) \log ^2\left (\log (x+\log (x+4))+e^3\right )+\left (-2 x^2-10 x\right ) \log \left (\log (x+\log (x+4))+e^3\right )\right ) \exp \left (\frac {\log ^2\left (\log (x+\log (x+4))+e^3\right )}{x}+e^{\frac {\log ^2\left (\log (x+\log (x+4))+e^3\right )}{x}}\right )+e^3 \left (x^4+4 x^3\right )+e^3 \left (x^3+4 x^2\right ) \log (x+4)+\left (x^4+4 x^3+\left (x^3+4 x^2\right ) \log (x+4)\right ) \log (x+\log (x+4))}{e^3 \left (x^4+4 x^3\right ) \log (4)+e^3 \left (x^3+4 x^2\right ) \log (4) \log (x+4)+\left (\left (x^4+4 x^3\right ) \log (4)+\left (x^3+4 x^2\right ) \log (4) \log (x+4)\right ) \log (x+\log (x+4))} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (\left (e^3 \left (x^2+4 x\right )+\left (x^2+4 x+(x+4) \log (x+4)\right ) \log (x+\log (x+4))+e^3 (x+4) \log (x+4)\right ) \log ^2\left (\log (x+\log (x+4))+e^3\right )+\left (-2 x^2-10 x\right ) \log \left (\log (x+\log (x+4))+e^3\right )\right ) \exp \left (\frac {\log ^2\left (\log (x+\log (x+4))+e^3\right )}{x}+e^{\frac {\log ^2\left (\log (x+\log (x+4))+e^3\right )}{x}}\right )+e^3 \left (x^4+4 x^3\right )+e^3 \left (x^3+4 x^2\right ) \log (x+4)+\left (x^4+4 x^3+\left (x^3+4 x^2\right ) \log (x+4)\right ) \log (x+\log (x+4))}{x^2 (x+4) \log (4) (x+\log (x+4)) \left (\log (x+\log (x+4))+e^3\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {e^3 \left (x^4+4 x^3\right )+e^3 \left (x^3+4 x^2\right ) \log (x+4)+\left (x^4+4 x^3+\left (x^3+4 x^2\right ) \log (x+4)\right ) \log (x+\log (x+4))-\exp \left (\frac {\log ^2\left (\log (x+\log (x+4))+e^3\right )}{x}+e^{\frac {\log ^2\left (\log (x+\log (x+4))+e^3\right )}{x}}\right ) \left (2 \left (x^2+5 x\right ) \log \left (\log (x+\log (x+4))+e^3\right )-\left (e^3 \left (x^2+4 x\right )+e^3 (x+4) \log (x+4)+\left (x^2+4 x+(x+4) \log (x+4)\right ) \log (x+\log (x+4))\right ) \log ^2\left (\log (x+\log (x+4))+e^3\right )\right )}{x^2 (x+4) (x+\log (x+4)) \left (\log (x+\log (x+4))+e^3\right )}dx}{\log (4)}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\int \left (\frac {\exp \left (\frac {\log ^2\left (\log (x+\log (x+4))+e^3\right )}{x}+e^{\frac {\log ^2\left (\log (x+\log (x+4))+e^3\right )}{x}}\right ) \log \left (\log (x+\log (x+4))+e^3\right ) \left (\log (x+\log (x+4)) \log \left (\log (x+\log (x+4))+e^3\right ) x^2+e^3 \log \left (\log (x+\log (x+4))+e^3\right ) x^2-2 x^2+e^3 \log (x+4) \log \left (\log (x+\log (x+4))+e^3\right ) x+\log (x+4) \log (x+\log (x+4)) \log \left (\log (x+\log (x+4))+e^3\right ) x+4 \log (x+\log (x+4)) \log \left (\log (x+\log (x+4))+e^3\right ) x+4 e^3 \log \left (\log (x+\log (x+4))+e^3\right ) x-10 x+4 e^3 \log (x+4) \log \left (\log (x+\log (x+4))+e^3\right )+4 \log (x+4) \log (x+\log (x+4)) \log \left (\log (x+\log (x+4))+e^3\right )\right )}{x^2 (x+4) (x+\log (x+4)) \left (\log (x+\log (x+4))+e^3\right )}+1\right )dx}{\log (4)}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \left (\frac {\exp \left (\frac {\log ^2\left (\log (x+\log (x+4))+e^3\right )}{x}+e^{\frac {\log ^2\left (\log (x+\log (x+4))+e^3\right )}{x}}\right ) \log \left (\log (x+\log (x+4))+e^3\right ) \left ((x+4) (x+\log (x+4)) \left (\log (x+\log (x+4))+e^3\right ) \log \left (\log (x+\log (x+4))+e^3\right )-2 x (x+5)\right )}{x^2 (x+4) (x+\log (x+4)) \left (\log (x+\log (x+4))+e^3\right )}+1\right )dx}{\log (4)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\int \frac {\exp \left (\frac {\log ^2\left (\log (x+\log (x+4))+e^3\right )}{x}+e^{\frac {\log ^2\left (\log (x+\log (x+4))+e^3\right )}{x}}\right ) \log ^2\left (\log (x+\log (x+4))+e^3\right )}{x^2}dx-\frac {5}{2} \int \frac {\exp \left (\frac {\log ^2\left (\log (x+\log (x+4))+e^3\right )}{x}+e^{\frac {\log ^2\left (\log (x+\log (x+4))+e^3\right )}{x}}\right ) \log \left (\log (x+\log (x+4))+e^3\right )}{x (x+\log (x+4)) \left (\log (x+\log (x+4))+e^3\right )}dx+\frac {1}{2} \int \frac {\exp \left (\frac {\log ^2\left (\log (x+\log (x+4))+e^3\right )}{x}+e^{\frac {\log ^2\left (\log (x+\log (x+4))+e^3\right )}{x}}\right ) \log \left (\log (x+\log (x+4))+e^3\right )}{(x+4) (x+\log (x+4)) \left (\log (x+\log (x+4))+e^3\right )}dx+x}{\log (4)}\) |
Input:
Int[(E^3*(4*x^3 + x^4) + E^3*(4*x^2 + x^3)*Log[4 + x] + (4*x^3 + x^4 + (4* x^2 + x^3)*Log[4 + x])*Log[x + Log[4 + x]] + E^(E^(Log[E^3 + Log[x + Log[4 + x]]]^2/x) + Log[E^3 + Log[x + Log[4 + x]]]^2/x)*((-10*x - 2*x^2)*Log[E^ 3 + Log[x + Log[4 + x]]] + (E^3*(4*x + x^2) + E^3*(4 + x)*Log[4 + x] + (4* x + x^2 + (4 + x)*Log[4 + x])*Log[x + Log[4 + x]])*Log[E^3 + Log[x + Log[4 + x]]]^2))/(E^3*(4*x^3 + x^4)*Log[4] + E^3*(4*x^2 + x^3)*Log[4]*Log[4 + x ] + ((4*x^3 + x^4)*Log[4] + (4*x^2 + x^3)*Log[4]*Log[4 + x])*Log[x + Log[4 + x]]),x]
Output:
$Aborted
Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10
\[\frac {x}{2 \ln \left (2\right )}-\frac {{\mathrm e}^{{\mathrm e}^{\frac {\ln \left (\ln \left (\ln \left (4+x \right )+x \right )+{\mathrm e}^{3}\right )^{2}}{x}}}}{2 \ln \left (2\right )}\]
Input:
int((((((4+x)*ln(4+x)+x^2+4*x)*ln(ln(4+x)+x)+(4+x)*exp(3)*ln(4+x)+(x^2+4*x )*exp(3))*ln(ln(ln(4+x)+x)+exp(3))^2+(-2*x^2-10*x)*ln(ln(ln(4+x)+x)+exp(3) ))*exp(ln(ln(ln(4+x)+x)+exp(3))^2/x)*exp(exp(ln(ln(ln(4+x)+x)+exp(3))^2/x) )+((x^3+4*x^2)*ln(4+x)+x^4+4*x^3)*ln(ln(4+x)+x)+(x^3+4*x^2)*exp(3)*ln(4+x) +(x^4+4*x^3)*exp(3))/((2*(x^3+4*x^2)*ln(2)*ln(4+x)+2*(x^4+4*x^3)*ln(2))*ln (ln(4+x)+x)+2*(x^3+4*x^2)*exp(3)*ln(2)*ln(4+x)+2*(x^4+4*x^3)*exp(3)*ln(2)) ,x)
Output:
1/2*x/ln(2)-1/2/ln(2)*exp(exp(ln(ln(ln(4+x)+x)+exp(3))^2/x))
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (29) = 58\).
Time = 0.09 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.81 \[ \int \frac {e^3 \left (4 x^3+x^4\right )+e^3 \left (4 x^2+x^3\right ) \log (4+x)+\left (4 x^3+x^4+\left (4 x^2+x^3\right ) \log (4+x)\right ) \log (x+\log (4+x))+e^{e^{\frac {\log ^2\left (e^3+\log (x+\log (4+x))\right )}{x}}+\frac {\log ^2\left (e^3+\log (x+\log (4+x))\right )}{x}} \left (\left (-10 x-2 x^2\right ) \log \left (e^3+\log (x+\log (4+x))\right )+\left (e^3 \left (4 x+x^2\right )+e^3 (4+x) \log (4+x)+\left (4 x+x^2+(4+x) \log (4+x)\right ) \log (x+\log (4+x))\right ) \log ^2\left (e^3+\log (x+\log (4+x))\right )\right )}{e^3 \left (4 x^3+x^4\right ) \log (4)+e^3 \left (4 x^2+x^3\right ) \log (4) \log (4+x)+\left (\left (4 x^3+x^4\right ) \log (4)+\left (4 x^2+x^3\right ) \log (4) \log (4+x)\right ) \log (x+\log (4+x))} \, dx=\frac {{\left (x e^{\left (\frac {\log \left (e^{3} + \log \left (x + \log \left (x + 4\right )\right )\right )^{2}}{x}\right )} - e^{\left (\frac {x e^{\left (\frac {\log \left (e^{3} + \log \left (x + \log \left (x + 4\right )\right )\right )^{2}}{x}\right )} + \log \left (e^{3} + \log \left (x + \log \left (x + 4\right )\right )\right )^{2}}{x}\right )}\right )} e^{\left (-\frac {\log \left (e^{3} + \log \left (x + \log \left (x + 4\right )\right )\right )^{2}}{x}\right )}}{2 \, \log \left (2\right )} \] Input:
integrate((((((4+x)*log(4+x)+x^2+4*x)*log(log(4+x)+x)+(4+x)*exp(3)*log(4+x )+(x^2+4*x)*exp(3))*log(log(log(4+x)+x)+exp(3))^2+(-2*x^2-10*x)*log(log(lo g(4+x)+x)+exp(3)))*exp(log(log(log(4+x)+x)+exp(3))^2/x)*exp(exp(log(log(lo g(4+x)+x)+exp(3))^2/x))+((x^3+4*x^2)*log(4+x)+x^4+4*x^3)*log(log(4+x)+x)+( x^3+4*x^2)*exp(3)*log(4+x)+(x^4+4*x^3)*exp(3))/((2*(x^3+4*x^2)*log(2)*log( 4+x)+2*(x^4+4*x^3)*log(2))*log(log(4+x)+x)+2*(x^3+4*x^2)*exp(3)*log(2)*log (4+x)+2*(x^4+4*x^3)*exp(3)*log(2)),x, algorithm="fricas")
Output:
1/2*(x*e^(log(e^3 + log(x + log(x + 4)))^2/x) - e^((x*e^(log(e^3 + log(x + log(x + 4)))^2/x) + log(e^3 + log(x + log(x + 4)))^2)/x))*e^(-log(e^3 + l og(x + log(x + 4)))^2/x)/log(2)
Timed out. \[ \int \frac {e^3 \left (4 x^3+x^4\right )+e^3 \left (4 x^2+x^3\right ) \log (4+x)+\left (4 x^3+x^4+\left (4 x^2+x^3\right ) \log (4+x)\right ) \log (x+\log (4+x))+e^{e^{\frac {\log ^2\left (e^3+\log (x+\log (4+x))\right )}{x}}+\frac {\log ^2\left (e^3+\log (x+\log (4+x))\right )}{x}} \left (\left (-10 x-2 x^2\right ) \log \left (e^3+\log (x+\log (4+x))\right )+\left (e^3 \left (4 x+x^2\right )+e^3 (4+x) \log (4+x)+\left (4 x+x^2+(4+x) \log (4+x)\right ) \log (x+\log (4+x))\right ) \log ^2\left (e^3+\log (x+\log (4+x))\right )\right )}{e^3 \left (4 x^3+x^4\right ) \log (4)+e^3 \left (4 x^2+x^3\right ) \log (4) \log (4+x)+\left (\left (4 x^3+x^4\right ) \log (4)+\left (4 x^2+x^3\right ) \log (4) \log (4+x)\right ) \log (x+\log (4+x))} \, dx=\text {Timed out} \] Input:
integrate((((((4+x)*ln(4+x)+x**2+4*x)*ln(ln(4+x)+x)+(4+x)*exp(3)*ln(4+x)+( x**2+4*x)*exp(3))*ln(ln(ln(4+x)+x)+exp(3))**2+(-2*x**2-10*x)*ln(ln(ln(4+x) +x)+exp(3)))*exp(ln(ln(ln(4+x)+x)+exp(3))**2/x)*exp(exp(ln(ln(ln(4+x)+x)+e xp(3))**2/x))+((x**3+4*x**2)*ln(4+x)+x**4+4*x**3)*ln(ln(4+x)+x)+(x**3+4*x* *2)*exp(3)*ln(4+x)+(x**4+4*x**3)*exp(3))/((2*(x**3+4*x**2)*ln(2)*ln(4+x)+2 *(x**4+4*x**3)*ln(2))*ln(ln(4+x)+x)+2*(x**3+4*x**2)*exp(3)*ln(2)*ln(4+x)+2 *(x**4+4*x**3)*exp(3)*ln(2)),x)
Output:
Timed out
Time = 0.36 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {e^3 \left (4 x^3+x^4\right )+e^3 \left (4 x^2+x^3\right ) \log (4+x)+\left (4 x^3+x^4+\left (4 x^2+x^3\right ) \log (4+x)\right ) \log (x+\log (4+x))+e^{e^{\frac {\log ^2\left (e^3+\log (x+\log (4+x))\right )}{x}}+\frac {\log ^2\left (e^3+\log (x+\log (4+x))\right )}{x}} \left (\left (-10 x-2 x^2\right ) \log \left (e^3+\log (x+\log (4+x))\right )+\left (e^3 \left (4 x+x^2\right )+e^3 (4+x) \log (4+x)+\left (4 x+x^2+(4+x) \log (4+x)\right ) \log (x+\log (4+x))\right ) \log ^2\left (e^3+\log (x+\log (4+x))\right )\right )}{e^3 \left (4 x^3+x^4\right ) \log (4)+e^3 \left (4 x^2+x^3\right ) \log (4) \log (4+x)+\left (\left (4 x^3+x^4\right ) \log (4)+\left (4 x^2+x^3\right ) \log (4) \log (4+x)\right ) \log (x+\log (4+x))} \, dx=\frac {x - e^{\left (e^{\left (\frac {\log \left (e^{3} + \log \left (x + \log \left (x + 4\right )\right )\right )^{2}}{x}\right )}\right )}}{2 \, \log \left (2\right )} \] Input:
integrate((((((4+x)*log(4+x)+x^2+4*x)*log(log(4+x)+x)+(4+x)*exp(3)*log(4+x )+(x^2+4*x)*exp(3))*log(log(log(4+x)+x)+exp(3))^2+(-2*x^2-10*x)*log(log(lo g(4+x)+x)+exp(3)))*exp(log(log(log(4+x)+x)+exp(3))^2/x)*exp(exp(log(log(lo g(4+x)+x)+exp(3))^2/x))+((x^3+4*x^2)*log(4+x)+x^4+4*x^3)*log(log(4+x)+x)+( x^3+4*x^2)*exp(3)*log(4+x)+(x^4+4*x^3)*exp(3))/((2*(x^3+4*x^2)*log(2)*log( 4+x)+2*(x^4+4*x^3)*log(2))*log(log(4+x)+x)+2*(x^3+4*x^2)*exp(3)*log(2)*log (4+x)+2*(x^4+4*x^3)*exp(3)*log(2)),x, algorithm="maxima")
Output:
1/2*(x - e^(e^(log(e^3 + log(x + log(x + 4)))^2/x)))/log(2)
\[ \int \frac {e^3 \left (4 x^3+x^4\right )+e^3 \left (4 x^2+x^3\right ) \log (4+x)+\left (4 x^3+x^4+\left (4 x^2+x^3\right ) \log (4+x)\right ) \log (x+\log (4+x))+e^{e^{\frac {\log ^2\left (e^3+\log (x+\log (4+x))\right )}{x}}+\frac {\log ^2\left (e^3+\log (x+\log (4+x))\right )}{x}} \left (\left (-10 x-2 x^2\right ) \log \left (e^3+\log (x+\log (4+x))\right )+\left (e^3 \left (4 x+x^2\right )+e^3 (4+x) \log (4+x)+\left (4 x+x^2+(4+x) \log (4+x)\right ) \log (x+\log (4+x))\right ) \log ^2\left (e^3+\log (x+\log (4+x))\right )\right )}{e^3 \left (4 x^3+x^4\right ) \log (4)+e^3 \left (4 x^2+x^3\right ) \log (4) \log (4+x)+\left (\left (4 x^3+x^4\right ) \log (4)+\left (4 x^2+x^3\right ) \log (4) \log (4+x)\right ) \log (x+\log (4+x))} \, dx=\int { \frac {{\left (x^{3} + 4 \, x^{2}\right )} e^{3} \log \left (x + 4\right ) + {\left (x^{4} + 4 \, x^{3}\right )} e^{3} + {\left ({\left ({\left (x + 4\right )} e^{3} \log \left (x + 4\right ) + {\left (x^{2} + 4 \, x\right )} e^{3} + {\left (x^{2} + {\left (x + 4\right )} \log \left (x + 4\right ) + 4 \, x\right )} \log \left (x + \log \left (x + 4\right )\right )\right )} \log \left (e^{3} + \log \left (x + \log \left (x + 4\right )\right )\right )^{2} - 2 \, {\left (x^{2} + 5 \, x\right )} \log \left (e^{3} + \log \left (x + \log \left (x + 4\right )\right )\right )\right )} e^{\left (\frac {\log \left (e^{3} + \log \left (x + \log \left (x + 4\right )\right )\right )^{2}}{x} + e^{\left (\frac {\log \left (e^{3} + \log \left (x + \log \left (x + 4\right )\right )\right )^{2}}{x}\right )}\right )} + {\left (x^{4} + 4 \, x^{3} + {\left (x^{3} + 4 \, x^{2}\right )} \log \left (x + 4\right )\right )} \log \left (x + \log \left (x + 4\right )\right )}{2 \, {\left ({\left (x^{3} + 4 \, x^{2}\right )} e^{3} \log \left (2\right ) \log \left (x + 4\right ) + {\left (x^{4} + 4 \, x^{3}\right )} e^{3} \log \left (2\right ) + {\left ({\left (x^{3} + 4 \, x^{2}\right )} \log \left (2\right ) \log \left (x + 4\right ) + {\left (x^{4} + 4 \, x^{3}\right )} \log \left (2\right )\right )} \log \left (x + \log \left (x + 4\right )\right )\right )}} \,d x } \] Input:
integrate((((((4+x)*log(4+x)+x^2+4*x)*log(log(4+x)+x)+(4+x)*exp(3)*log(4+x )+(x^2+4*x)*exp(3))*log(log(log(4+x)+x)+exp(3))^2+(-2*x^2-10*x)*log(log(lo g(4+x)+x)+exp(3)))*exp(log(log(log(4+x)+x)+exp(3))^2/x)*exp(exp(log(log(lo g(4+x)+x)+exp(3))^2/x))+((x^3+4*x^2)*log(4+x)+x^4+4*x^3)*log(log(4+x)+x)+( x^3+4*x^2)*exp(3)*log(4+x)+(x^4+4*x^3)*exp(3))/((2*(x^3+4*x^2)*log(2)*log( 4+x)+2*(x^4+4*x^3)*log(2))*log(log(4+x)+x)+2*(x^3+4*x^2)*exp(3)*log(2)*log (4+x)+2*(x^4+4*x^3)*exp(3)*log(2)),x, algorithm="giac")
Output:
integrate(1/2*((x^3 + 4*x^2)*e^3*log(x + 4) + (x^4 + 4*x^3)*e^3 + (((x + 4 )*e^3*log(x + 4) + (x^2 + 4*x)*e^3 + (x^2 + (x + 4)*log(x + 4) + 4*x)*log( x + log(x + 4)))*log(e^3 + log(x + log(x + 4)))^2 - 2*(x^2 + 5*x)*log(e^3 + log(x + log(x + 4))))*e^(log(e^3 + log(x + log(x + 4)))^2/x + e^(log(e^3 + log(x + log(x + 4)))^2/x)) + (x^4 + 4*x^3 + (x^3 + 4*x^2)*log(x + 4))*l og(x + log(x + 4)))/((x^3 + 4*x^2)*e^3*log(2)*log(x + 4) + (x^4 + 4*x^3)*e ^3*log(2) + ((x^3 + 4*x^2)*log(2)*log(x + 4) + (x^4 + 4*x^3)*log(2))*log(x + log(x + 4))), x)
Time = 3.35 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {e^3 \left (4 x^3+x^4\right )+e^3 \left (4 x^2+x^3\right ) \log (4+x)+\left (4 x^3+x^4+\left (4 x^2+x^3\right ) \log (4+x)\right ) \log (x+\log (4+x))+e^{e^{\frac {\log ^2\left (e^3+\log (x+\log (4+x))\right )}{x}}+\frac {\log ^2\left (e^3+\log (x+\log (4+x))\right )}{x}} \left (\left (-10 x-2 x^2\right ) \log \left (e^3+\log (x+\log (4+x))\right )+\left (e^3 \left (4 x+x^2\right )+e^3 (4+x) \log (4+x)+\left (4 x+x^2+(4+x) \log (4+x)\right ) \log (x+\log (4+x))\right ) \log ^2\left (e^3+\log (x+\log (4+x))\right )\right )}{e^3 \left (4 x^3+x^4\right ) \log (4)+e^3 \left (4 x^2+x^3\right ) \log (4) \log (4+x)+\left (\left (4 x^3+x^4\right ) \log (4)+\left (4 x^2+x^3\right ) \log (4) \log (4+x)\right ) \log (x+\log (4+x))} \, dx=\frac {x-{\mathrm {e}}^{{\mathrm {e}}^{\frac {{\ln \left ({\mathrm {e}}^3+\ln \left (x+\ln \left (x+4\right )\right )\right )}^2}{x}}}}{2\,\ln \left (2\right )} \] Input:
int((exp(3)*(4*x^3 + x^4) + log(x + log(x + 4))*(log(x + 4)*(4*x^2 + x^3) + 4*x^3 + x^4) - exp(exp(log(exp(3) + log(x + log(x + 4)))^2/x))*exp(log(e xp(3) + log(x + log(x + 4)))^2/x)*(log(exp(3) + log(x + log(x + 4)))*(10*x + 2*x^2) - log(exp(3) + log(x + log(x + 4)))^2*(log(x + log(x + 4))*(4*x + log(x + 4)*(x + 4) + x^2) + exp(3)*(4*x + x^2) + log(x + 4)*exp(3)*(x + 4))) + log(x + 4)*exp(3)*(4*x^2 + x^3))/(log(x + log(x + 4))*(2*log(2)*(4* x^3 + x^4) + 2*log(x + 4)*log(2)*(4*x^2 + x^3)) + 2*exp(3)*log(2)*(4*x^3 + x^4) + 2*log(x + 4)*exp(3)*log(2)*(4*x^2 + x^3)),x)
Output:
(x - exp(exp(log(exp(3) + log(x + log(x + 4)))^2/x)))/(2*log(2))
Time = 0.30 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {e^3 \left (4 x^3+x^4\right )+e^3 \left (4 x^2+x^3\right ) \log (4+x)+\left (4 x^3+x^4+\left (4 x^2+x^3\right ) \log (4+x)\right ) \log (x+\log (4+x))+e^{e^{\frac {\log ^2\left (e^3+\log (x+\log (4+x))\right )}{x}}+\frac {\log ^2\left (e^3+\log (x+\log (4+x))\right )}{x}} \left (\left (-10 x-2 x^2\right ) \log \left (e^3+\log (x+\log (4+x))\right )+\left (e^3 \left (4 x+x^2\right )+e^3 (4+x) \log (4+x)+\left (4 x+x^2+(4+x) \log (4+x)\right ) \log (x+\log (4+x))\right ) \log ^2\left (e^3+\log (x+\log (4+x))\right )\right )}{e^3 \left (4 x^3+x^4\right ) \log (4)+e^3 \left (4 x^2+x^3\right ) \log (4) \log (4+x)+\left (\left (4 x^3+x^4\right ) \log (4)+\left (4 x^2+x^3\right ) \log (4) \log (4+x)\right ) \log (x+\log (4+x))} \, dx=\frac {-e^{e^{\frac {\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (x +4\right )+x \right )+e^{3}\right )^{2}}{x}}}+x}{2 \,\mathrm {log}\left (2\right )} \] Input:
int((((((4+x)*log(4+x)+x^2+4*x)*log(log(4+x)+x)+(4+x)*exp(3)*log(4+x)+(x^2 +4*x)*exp(3))*log(log(log(4+x)+x)+exp(3))^2+(-2*x^2-10*x)*log(log(log(4+x) +x)+exp(3)))*exp(log(log(log(4+x)+x)+exp(3))^2/x)*exp(exp(log(log(log(4+x) +x)+exp(3))^2/x))+((x^3+4*x^2)*log(4+x)+x^4+4*x^3)*log(log(4+x)+x)+(x^3+4* x^2)*exp(3)*log(4+x)+(x^4+4*x^3)*exp(3))/((2*(x^3+4*x^2)*log(2)*log(4+x)+2 *(x^4+4*x^3)*log(2))*log(log(4+x)+x)+2*(x^3+4*x^2)*exp(3)*log(2)*log(4+x)+ 2*(x^4+4*x^3)*exp(3)*log(2)),x)
Output:
( - e**(e**(log(log(log(x + 4) + x) + e**3)**2/x)) + x)/(2*log(2))