Integrand size = 103, antiderivative size = 28 \[ \int \frac {e^{-7-3 x-2 e^{3+x} x-2 x^2+\left (2 x+e^{3+x} x+x^2\right ) \log (\log (\log (x)))} \left (2+e^{3+x}+x+\left (-3+e^{3+x} (-2-2 x)-4 x\right ) \log (x) \log (\log (x))+\left (2+2 x+e^{3+x} (1+x)\right ) \log (x) \log (\log (x)) \log (\log (\log (x)))\right )}{\log (x) \log (\log (x))} \, dx=e^{-7+x-\left (2 x+x \left (e^{3+x}+x\right )\right ) (2-\log (\log (\log (x))))} \]
Time = 0.22 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25 \[ \int \frac {e^{-7-3 x-2 e^{3+x} x-2 x^2+\left (2 x+e^{3+x} x+x^2\right ) \log (\log (\log (x)))} \left (2+e^{3+x}+x+\left (-3+e^{3+x} (-2-2 x)-4 x\right ) \log (x) \log (\log (x))+\left (2+2 x+e^{3+x} (1+x)\right ) \log (x) \log (\log (x)) \log (\log (\log (x)))\right )}{\log (x) \log (\log (x))} \, dx=e^{-7-3 x-2 e^{3+x} x-2 x^2} \log ^{x \left (2+e^{3+x}+x\right )}(\log (x)) \]
Integrate[(E^(-7 - 3*x - 2*E^(3 + x)*x - 2*x^2 + (2*x + E^(3 + x)*x + x^2) *Log[Log[Log[x]]])*(2 + E^(3 + x) + x + (-3 + E^(3 + x)*(-2 - 2*x) - 4*x)* Log[x]*Log[Log[x]] + (2 + 2*x + E^(3 + x)*(1 + x))*Log[x]*Log[Log[x]]*Log[ Log[Log[x]]]))/(Log[x]*Log[Log[x]]),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.
| \(\displaystyle \int \frac {\left (x+e^{x+3}+\left (e^{x+3} (-2 x-2)-4 x-3\right ) \log (x) \log (\log (x))+\left (2 x+e^{x+3} (x+1)+2\right ) \log (x) \log (\log (x)) \log (\log (\log (x)))+2\right ) \exp \left (-2 x^2+\left (x^2+e^{x+3} x+2 x\right ) \log (\log (\log (x)))-2 e^{x+3} x-3 x-7\right )}{\log (x) \log (\log (x))} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(-2 x \log (x) \log (\log (x))-2 \log (x) \log (\log (x))+x \log (x) \log (\log (\log (x))) \log (\log (x))+\log (x) \log (\log (\log (x))) \log (\log (x))+1) \exp \left (-2 x^2+\left (x^2+e^{x+3} x+2 x\right ) \log (\log (\log (x)))-2 e^{x+3} x-2 x-4\right )}{\log (x) \log (\log (x))}+\frac {(x-4 x \log (x) \log (\log (x))+2 x \log (x) \log (\log (x)) \log (\log (\log (x)))-3 \log (x) \log (\log (x))+2 \log (x) \log (\log (x)) \log (\log (\log (x)))+2) \exp \left (-2 x^2+\left (x^2+e^{x+3} x+2 x\right ) \log (\log (\log (x)))-2 e^{x+3} x-3 x-7\right )}{\log (x) \log (\log (x))}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-2 x^2-2 e^{x+3} x-3 x-7} \log ^{x^2+\left (e^{x+3}+2\right ) x-1}(\log (x)) \left (x+e^{x+3}+\log (x) \log (\log (x)) \left (-4 x-2 e^{x+3} (x+1)+\left (e^{x+3}+2\right ) (x+1) \log (\log (\log (x)))-3\right )+2\right )}{\log (x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{-2 x^2-2 e^{x+3} x-2 x-4} (-2 x \log (x) \log (\log (x))-2 \log (x) \log (\log (x))+x \log (x) \log (\log (\log (x))) \log (\log (x))+\log (x) \log (\log (\log (x))) \log (\log (x))+1) \log ^{x^2+\left (e^{x+3}+2\right ) x-1}(\log (x))}{\log (x)}+\frac {e^{-2 x^2-2 e^{x+3} x-3 x-7} (x-4 x \log (x) \log (\log (x))+2 x \log (x) \log (\log (x)) \log (\log (\log (x)))-3 \log (x) \log (\log (x))+2 \log (x) \log (\log (x)) \log (\log (\log (x)))+2) \log ^{x^2+\left (e^{x+3}+2\right ) x-1}(\log (x))}{\log (x)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-2 x^2-2 e^{x+3} x-3 x-7} \log ^{x^2+\left (e^{x+3}+2\right ) x-1}(\log (x)) \left (x+e^{x+3}+\log (x) \log (\log (x)) \left (-4 x-2 e^{x+3} (x+1)+\left (e^{x+3}+2\right ) (x+1) \log (\log (\log (x)))-3\right )+2\right )}{\log (x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{-2 x^2-2 e^{x+3} x-2 x-4} (-2 x \log (x) \log (\log (x))-2 \log (x) \log (\log (x))+x \log (x) \log (\log (\log (x))) \log (\log (x))+\log (x) \log (\log (\log (x))) \log (\log (x))+1) \log ^{x^2+\left (e^{x+3}+2\right ) x-1}(\log (x))}{\log (x)}+\frac {e^{-2 x^2-2 e^{x+3} x-3 x-7} (x-4 x \log (x) \log (\log (x))+2 x \log (x) \log (\log (x)) \log (\log (\log (x)))-3 \log (x) \log (\log (x))+2 \log (x) \log (\log (x)) \log (\log (\log (x)))+2) \log ^{x^2+\left (e^{x+3}+2\right ) x-1}(\log (x))}{\log (x)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-2 x^2-2 e^{x+3} x-3 x-7} \log ^{x^2+\left (e^{x+3}+2\right ) x-1}(\log (x)) \left (x+e^{x+3}+\log (x) \log (\log (x)) \left (-4 x-2 e^{x+3} (x+1)+\left (e^{x+3}+2\right ) (x+1) \log (\log (\log (x)))-3\right )+2\right )}{\log (x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{-2 x^2-2 e^{x+3} x-2 x-4} (-2 x \log (x) \log (\log (x))-2 \log (x) \log (\log (x))+x \log (x) \log (\log (\log (x))) \log (\log (x))+\log (x) \log (\log (\log (x))) \log (\log (x))+1) \log ^{x^2+\left (e^{x+3}+2\right ) x-1}(\log (x))}{\log (x)}+\frac {e^{-2 x^2-2 e^{x+3} x-3 x-7} (x-4 x \log (x) \log (\log (x))+2 x \log (x) \log (\log (x)) \log (\log (\log (x)))-3 \log (x) \log (\log (x))+2 \log (x) \log (\log (x)) \log (\log (\log (x)))+2) \log ^{x^2+\left (e^{x+3}+2\right ) x-1}(\log (x))}{\log (x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \int \frac {e^{-2 x^2-2 e^{x+3} x-3 x-7} \log ^{x^2+\left (2+e^{x+3}\right ) x-1}(\log (x))}{\log (x)}dx+\int \frac {e^{-2 \left (x^2+e^{x+3} x+x+2\right )} \log ^{x^2+\left (2+e^{x+3}\right ) x-1}(\log (x))}{\log (x)}dx+\int \frac {e^{-2 x^2-2 e^{x+3} x-3 x-7} x \log ^{x^2+\left (2+e^{x+3}\right ) x-1}(\log (x))}{\log (x)}dx-3 \int e^{-2 x^2-2 e^{x+3} x-3 x-7} \log ^{x^2+\left (2+e^{x+3}\right ) x}(\log (x))dx-2 \int e^{-2 \left (x^2+e^{x+3} x+x+2\right )} \log ^{x^2+\left (2+e^{x+3}\right ) x}(\log (x))dx-4 \int e^{-2 x^2-2 e^{x+3} x-3 x-7} x \log ^{x^2+\left (2+e^{x+3}\right ) x}(\log (x))dx-2 \int e^{-2 \left (x^2+e^{x+3} x+x+2\right )} x \log ^{x^2+\left (2+e^{x+3}\right ) x}(\log (x))dx+2 \int e^{-2 x^2-2 e^{x+3} x-3 x-7} \log ^{x^2+\left (2+e^{x+3}\right ) x}(\log (x)) \log (\log (\log (x)))dx+\int e^{-2 \left (x^2+e^{x+3} x+x+2\right )} \log ^{x^2+\left (2+e^{x+3}\right ) x}(\log (x)) \log (\log (\log (x)))dx+2 \int e^{-2 x^2-2 e^{x+3} x-3 x-7} x \log ^{x^2+\left (2+e^{x+3}\right ) x}(\log (x)) \log (\log (\log (x)))dx+\int e^{-2 \left (x^2+e^{x+3} x+x+2\right )} x \log ^{x^2+\left (2+e^{x+3}\right ) x}(\log (x)) \log (\log (\log (x)))dx\) |
Int[(E^(-7 - 3*x - 2*E^(3 + x)*x - 2*x^2 + (2*x + E^(3 + x)*x + x^2)*Log[L og[Log[x]]])*(2 + E^(3 + x) + x + (-3 + E^(3 + x)*(-2 - 2*x) - 4*x)*Log[x] *Log[Log[x]] + (2 + 2*x + E^(3 + x)*(1 + x))*Log[x]*Log[Log[x]]*Log[Log[Lo g[x]]]))/(Log[x]*Log[Log[x]]),x]
3.13.79.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 31.36 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18
| method | result | size |
| risch | \(\ln \left (\ln \left (x \right )\right )^{\left ({\mathrm e}^{3+x}+x +2\right ) x} {\mathrm e}^{-7-2 \,{\mathrm e}^{3+x} x -2 x^{2}-3 x}\) | \(33\) |
| parallelrisch | \({\mathrm e}^{\left ({\mathrm e}^{3+x} x +x^{2}+2 x \right ) \ln \left (\ln \left (\ln \left (x \right )\right )\right )-2 \,{\mathrm e}^{3+x} x -2 x^{2}-3 x -7}\) | \(37\) |
int((((1+x)*exp(3+x)+2*x+2)*ln(x)*ln(ln(x))*ln(ln(ln(x)))+((-2-2*x)*exp(3+ x)-4*x-3)*ln(x)*ln(ln(x))+exp(3+x)+2+x)*exp((exp(3+x)*x+x^2+2*x)*ln(ln(ln( x)))-2*exp(3+x)*x-2*x^2-3*x-7)/ln(x)/ln(ln(x)),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {e^{-7-3 x-2 e^{3+x} x-2 x^2+\left (2 x+e^{3+x} x+x^2\right ) \log (\log (\log (x)))} \left (2+e^{3+x}+x+\left (-3+e^{3+x} (-2-2 x)-4 x\right ) \log (x) \log (\log (x))+\left (2+2 x+e^{3+x} (1+x)\right ) \log (x) \log (\log (x)) \log (\log (\log (x)))\right )}{\log (x) \log (\log (x))} \, dx=e^{\left (-2 \, x^{2} - 2 \, x e^{\left (x + 3\right )} + {\left (x^{2} + x e^{\left (x + 3\right )} + 2 \, x\right )} \log \left (\log \left (\log \left (x\right )\right )\right ) - 3 \, x - 7\right )} \]
integrate((((1+x)*exp(3+x)+2*x+2)*log(x)*log(log(x))*log(log(log(x)))+((-2 -2*x)*exp(3+x)-4*x-3)*log(x)*log(log(x))+exp(3+x)+2+x)*exp((exp(3+x)*x+x^2 +2*x)*log(log(log(x)))-2*exp(3+x)*x-2*x^2-3*x-7)/log(x)/log(log(x)),x, alg orithm=\
Time = 10.40 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {e^{-7-3 x-2 e^{3+x} x-2 x^2+\left (2 x+e^{3+x} x+x^2\right ) \log (\log (\log (x)))} \left (2+e^{3+x}+x+\left (-3+e^{3+x} (-2-2 x)-4 x\right ) \log (x) \log (\log (x))+\left (2+2 x+e^{3+x} (1+x)\right ) \log (x) \log (\log (x)) \log (\log (\log (x)))\right )}{\log (x) \log (\log (x))} \, dx=e^{- 2 x^{2} - 2 x e^{x + 3} - 3 x + \left (x^{2} + x e^{x + 3} + 2 x\right ) \log {\left (\log {\left (\log {\left (x \right )} \right )} \right )} - 7} \]
integrate((((1+x)*exp(3+x)+2*x+2)*ln(x)*ln(ln(x))*ln(ln(ln(x)))+((-2-2*x)* exp(3+x)-4*x-3)*ln(x)*ln(ln(x))+exp(3+x)+2+x)*exp((exp(3+x)*x+x**2+2*x)*ln (ln(ln(x)))-2*exp(3+x)*x-2*x**2-3*x-7)/ln(x)/ln(ln(x)),x)
Time = 0.32 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.54 \[ \int \frac {e^{-7-3 x-2 e^{3+x} x-2 x^2+\left (2 x+e^{3+x} x+x^2\right ) \log (\log (\log (x)))} \left (2+e^{3+x}+x+\left (-3+e^{3+x} (-2-2 x)-4 x\right ) \log (x) \log (\log (x))+\left (2+2 x+e^{3+x} (1+x)\right ) \log (x) \log (\log (x)) \log (\log (\log (x)))\right )}{\log (x) \log (\log (x))} \, dx=e^{\left (x^{2} \log \left (\log \left (\log \left (x\right )\right )\right ) + x e^{\left (x + 3\right )} \log \left (\log \left (\log \left (x\right )\right )\right ) - 2 \, x^{2} - 2 \, x e^{\left (x + 3\right )} + 2 \, x \log \left (\log \left (\log \left (x\right )\right )\right ) - 3 \, x - 7\right )} \]
integrate((((1+x)*exp(3+x)+2*x+2)*log(x)*log(log(x))*log(log(log(x)))+((-2 -2*x)*exp(3+x)-4*x-3)*log(x)*log(log(x))+exp(3+x)+2+x)*exp((exp(3+x)*x+x^2 +2*x)*log(log(log(x)))-2*exp(3+x)*x-2*x^2-3*x-7)/log(x)/log(log(x)),x, alg orithm=\
e^(x^2*log(log(log(x))) + x*e^(x + 3)*log(log(log(x))) - 2*x^2 - 2*x*e^(x + 3) + 2*x*log(log(log(x))) - 3*x - 7)
\[ \int \frac {e^{-7-3 x-2 e^{3+x} x-2 x^2+\left (2 x+e^{3+x} x+x^2\right ) \log (\log (\log (x)))} \left (2+e^{3+x}+x+\left (-3+e^{3+x} (-2-2 x)-4 x\right ) \log (x) \log (\log (x))+\left (2+2 x+e^{3+x} (1+x)\right ) \log (x) \log (\log (x)) \log (\log (\log (x)))\right )}{\log (x) \log (\log (x))} \, dx=\int { \frac {{\left ({\left ({\left (x + 1\right )} e^{\left (x + 3\right )} + 2 \, x + 2\right )} \log \left (x\right ) \log \left (\log \left (x\right )\right ) \log \left (\log \left (\log \left (x\right )\right )\right ) - {\left (2 \, {\left (x + 1\right )} e^{\left (x + 3\right )} + 4 \, x + 3\right )} \log \left (x\right ) \log \left (\log \left (x\right )\right ) + x + e^{\left (x + 3\right )} + 2\right )} e^{\left (-2 \, x^{2} - 2 \, x e^{\left (x + 3\right )} + {\left (x^{2} + x e^{\left (x + 3\right )} + 2 \, x\right )} \log \left (\log \left (\log \left (x\right )\right )\right ) - 3 \, x - 7\right )}}{\log \left (x\right ) \log \left (\log \left (x\right )\right )} \,d x } \]
integrate((((1+x)*exp(3+x)+2*x+2)*log(x)*log(log(x))*log(log(log(x)))+((-2 -2*x)*exp(3+x)-4*x-3)*log(x)*log(log(x))+exp(3+x)+2+x)*exp((exp(3+x)*x+x^2 +2*x)*log(log(log(x)))-2*exp(3+x)*x-2*x^2-3*x-7)/log(x)/log(log(x)),x, alg orithm=\
Time = 11.90 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {e^{-7-3 x-2 e^{3+x} x-2 x^2+\left (2 x+e^{3+x} x+x^2\right ) \log (\log (\log (x)))} \left (2+e^{3+x}+x+\left (-3+e^{3+x} (-2-2 x)-4 x\right ) \log (x) \log (\log (x))+\left (2+2 x+e^{3+x} (1+x)\right ) \log (x) \log (\log (x)) \log (\log (\log (x)))\right )}{\log (x) \log (\log (x))} \, dx={\ln \left (\ln \left (x\right )\right )}^{2\,x+x^2+x\,{\mathrm {e}}^3\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-3\,x}\,{\mathrm {e}}^{-7}\,{\mathrm {e}}^{-2\,x\,{\mathrm {e}}^3\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-2\,x^2} \]
int((exp(log(log(log(x)))*(2*x + x*exp(x + 3) + x^2) - 2*x*exp(x + 3) - 3* x - 2*x^2 - 7)*(x + exp(x + 3) - log(log(x))*log(x)*(4*x + exp(x + 3)*(2*x + 2) + 3) + log(log(x))*log(log(log(x)))*log(x)*(2*x + exp(x + 3)*(x + 1) + 2) + 2))/(log(log(x))*log(x)),x)