Integrand size = 76, antiderivative size = 31 \[ \int \frac {e^{5-e^{e^{\frac {e^{16} x}{3}}}-x} \left (-6-2 e^{16+e^{\frac {e^{16} x}{3}}+\frac {e^{16} x}{3}}-3 e^{-5+e^{e^{\frac {e^{16} x}{3}}}+x} \log (5)\right )}{3 \log (5)} \, dx=-x+\frac {2 e^{5-e^{e^{\frac {e^{16} x}{3}}}-x}}{\log (5)} \] Output:
2/ln(5)/exp(exp(exp(1/3*x*exp(16)))+x-5)-x
Time = 0.39 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {e^{5-e^{e^{\frac {e^{16} x}{3}}}-x} \left (-6-2 e^{16+e^{\frac {e^{16} x}{3}}+\frac {e^{16} x}{3}}-3 e^{-5+e^{e^{\frac {e^{16} x}{3}}}+x} \log (5)\right )}{3 \log (5)} \, dx=-\frac {-6 e^{5-e^{e^{\frac {e^{16} x}{3}}}-x}+x \log (125)}{\log (125)} \] Input:
Integrate[(E^(5 - E^E^((E^16*x)/3) - x)*(-6 - 2*E^(16 + E^((E^16*x)/3) + ( E^16*x)/3) - 3*E^(-5 + E^E^((E^16*x)/3) + x)*Log[5]))/(3*Log[5]),x]
Output:
-((-6*E^(5 - E^E^((E^16*x)/3) - x) + x*Log[125])/Log[125])
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.
\(\displaystyle \int \frac {e^{-x-e^{e^{\frac {e^{16} x}{3}}}+5} \left (-2 e^{\frac {e^{16} x}{3}+e^{\frac {e^{16} x}{3}}+16}-3 e^{x+e^{e^{\frac {e^{16} x}{3}}}-5} \log (5)-6\right )}{3 \log (5)} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int -e^{-x-e^{e^{\frac {e^{16} x}{3}}}+5} \left (6+2 e^{\frac {e^{16} x}{3}+e^{\frac {e^{16} x}{3}}+16}+3 e^{x+e^{e^{\frac {e^{16} x}{3}}}-5} \log (5)\right )dx}{3 \log (5)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int e^{-x-e^{e^{\frac {e^{16} x}{3}}}+5} \left (6+2 e^{\frac {e^{16} x}{3}+e^{\frac {e^{16} x}{3}}+16}+3 e^{x+e^{e^{\frac {e^{16} x}{3}}}-5} \log (5)\right )dx}{3 \log (5)}\) |
\(\Big \downarrow \) 7281 |
\(\displaystyle -\frac {\int e^{-x-e^{e^{\frac {e^{16} x}{3}}}+5} \left (6+2 e^{\frac {e^{16} x}{3}+e^{\frac {e^{16} x}{3}}+16}+3 e^{x+e^{e^{\frac {e^{16} x}{3}}}-5} \log (5)\right )d\frac {x}{3}}{\log (5)}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {\int \left (6 e^{-x-e^{e^{\frac {e^{16} x}{3}}}+5}+2 e^{\frac {e^{16} x}{3}-x-e^{e^{\frac {e^{16} x}{3}}}+e^{\frac {e^{16} x}{3}}+21}+3 \log (5)\right )d\frac {x}{3}}{\log (5)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \int \exp \left (-\left (\left (1-\frac {e^{16}}{3}\right ) x\right )-e^{e^{\frac {e^{16} x}{3}}}+e^{\frac {e^{16} x}{3}}+21\right )d\frac {x}{3}+6 \int e^{-x-e^{e^{\frac {e^{16} x}{3}}}+5}d\frac {x}{3}+x \log (5)}{\log (5)}\) |
Input:
Int[(E^(5 - E^E^((E^16*x)/3) - x)*(-6 - 2*E^(16 + E^((E^16*x)/3) + (E^16*x )/3) - 3*E^(-5 + E^E^((E^16*x)/3) + x)*Log[5]))/(3*Log[5]),x]
Output:
$Aborted
Time = 0.40 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84
method | result | size |
risch | \(\frac {2 \,{\mathrm e}^{-{\mathrm e}^{{\mathrm e}^{\frac {x \,{\mathrm e}^{16}}{3}}}-x +5}}{\ln \left (5\right )}-x\) | \(26\) |
norman | \(\left (\frac {2}{\ln \left (5\right )}-x \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{\frac {x \,{\mathrm e}^{16}}{3}}}+x -5}\right ) {\mathrm e}^{-{\mathrm e}^{{\mathrm e}^{\frac {x \,{\mathrm e}^{16}}{3}}}-x +5}\) | \(36\) |
parallelrisch | \(-\frac {\left (3 x \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{\frac {x \,{\mathrm e}^{16}}{3}}}+x -5} \ln \left (5\right )-6\right ) {\mathrm e}^{-{\mathrm e}^{{\mathrm e}^{\frac {x \,{\mathrm e}^{16}}{3}}}-x +5}}{3 \ln \left (5\right )}\) | \(38\) |
Input:
int(1/3*(-3*ln(5)*exp(exp(exp(1/3*x*exp(16)))+x-5)-2*exp(16)*exp(1/3*x*exp (16))*exp(exp(1/3*x*exp(16)))-6)/ln(5)/exp(exp(exp(1/3*x*exp(16)))+x-5),x, method=_RETURNVERBOSE)
Output:
2/ln(5)*exp(-exp(exp(1/3*x*exp(16)))-x+5)-x
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (25) = 50\).
Time = 0.09 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.81 \[ \int \frac {e^{5-e^{e^{\frac {e^{16} x}{3}}}-x} \left (-6-2 e^{16+e^{\frac {e^{16} x}{3}}+\frac {e^{16} x}{3}}-3 e^{-5+e^{e^{\frac {e^{16} x}{3}}}+x} \log (5)\right )}{3 \log (5)} \, dx=-\frac {{\left (x e^{\left ({\left ({\left (x - 5\right )} e^{\left (\frac {1}{3} \, x e^{16} + 16\right )} + e^{\left (\frac {1}{3} \, x e^{16} + e^{\left (\frac {1}{3} \, x e^{16}\right )} + 16\right )}\right )} e^{\left (-\frac {1}{3} \, x e^{16} - 16\right )}\right )} \log \left (5\right ) - 2\right )} e^{\left (-{\left ({\left (x - 5\right )} e^{\left (\frac {1}{3} \, x e^{16} + 16\right )} + e^{\left (\frac {1}{3} \, x e^{16} + e^{\left (\frac {1}{3} \, x e^{16}\right )} + 16\right )}\right )} e^{\left (-\frac {1}{3} \, x e^{16} - 16\right )}\right )}}{\log \left (5\right )} \] Input:
integrate(1/3*(-3*log(5)*exp(exp(exp(1/3*x*exp(16)))+x-5)-2*exp(16)*exp(1/ 3*x*exp(16))*exp(exp(1/3*x*exp(16)))-6)/log(5)/exp(exp(exp(1/3*x*exp(16))) +x-5),x, algorithm="fricas")
Output:
-(x*e^(((x - 5)*e^(1/3*x*e^16 + 16) + e^(1/3*x*e^16 + e^(1/3*x*e^16) + 16) )*e^(-1/3*x*e^16 - 16))*log(5) - 2)*e^(-((x - 5)*e^(1/3*x*e^16 + 16) + e^( 1/3*x*e^16 + e^(1/3*x*e^16) + 16))*e^(-1/3*x*e^16 - 16))/log(5)
Time = 0.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65 \[ \int \frac {e^{5-e^{e^{\frac {e^{16} x}{3}}}-x} \left (-6-2 e^{16+e^{\frac {e^{16} x}{3}}+\frac {e^{16} x}{3}}-3 e^{-5+e^{e^{\frac {e^{16} x}{3}}}+x} \log (5)\right )}{3 \log (5)} \, dx=- x + \frac {2 e^{- x - e^{e^{\frac {x e^{16}}{3}}} + 5}}{\log {\left (5 \right )}} \] Input:
integrate(1/3*(-3*ln(5)*exp(exp(exp(1/3*x*exp(16)))+x-5)-2*exp(16)*exp(1/3 *x*exp(16))*exp(exp(1/3*x*exp(16)))-6)/ln(5)/exp(exp(exp(1/3*x*exp(16)))+x -5),x)
Output:
-x + 2*exp(-x - exp(exp(x*exp(16)/3)) + 5)/log(5)
Time = 0.12 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int \frac {e^{5-e^{e^{\frac {e^{16} x}{3}}}-x} \left (-6-2 e^{16+e^{\frac {e^{16} x}{3}}+\frac {e^{16} x}{3}}-3 e^{-5+e^{e^{\frac {e^{16} x}{3}}}+x} \log (5)\right )}{3 \log (5)} \, dx=-\frac {x \log \left (5\right ) - 2 \, e^{\left (-x - e^{\left (e^{\left (\frac {1}{3} \, x e^{16}\right )}\right )} + 5\right )}}{\log \left (5\right )} \] Input:
integrate(1/3*(-3*log(5)*exp(exp(exp(1/3*x*exp(16)))+x-5)-2*exp(16)*exp(1/ 3*x*exp(16))*exp(exp(1/3*x*exp(16)))-6)/log(5)/exp(exp(exp(1/3*x*exp(16))) +x-5),x, algorithm="maxima")
Output:
-(x*log(5) - 2*e^(-x - e^(e^(1/3*x*e^16)) + 5))/log(5)
\[ \int \frac {e^{5-e^{e^{\frac {e^{16} x}{3}}}-x} \left (-6-2 e^{16+e^{\frac {e^{16} x}{3}}+\frac {e^{16} x}{3}}-3 e^{-5+e^{e^{\frac {e^{16} x}{3}}}+x} \log (5)\right )}{3 \log (5)} \, dx=\int { -\frac {{\left (3 \, e^{\left (x + e^{\left (e^{\left (\frac {1}{3} \, x e^{16}\right )}\right )} - 5\right )} \log \left (5\right ) + 2 \, e^{\left (\frac {1}{3} \, x e^{16} + e^{\left (\frac {1}{3} \, x e^{16}\right )} + 16\right )} + 6\right )} e^{\left (-x - e^{\left (e^{\left (\frac {1}{3} \, x e^{16}\right )}\right )} + 5\right )}}{3 \, \log \left (5\right )} \,d x } \] Input:
integrate(1/3*(-3*log(5)*exp(exp(exp(1/3*x*exp(16)))+x-5)-2*exp(16)*exp(1/ 3*x*exp(16))*exp(exp(1/3*x*exp(16)))-6)/log(5)/exp(exp(exp(1/3*x*exp(16))) +x-5),x, algorithm="giac")
Output:
integrate(-1/3*(3*e^(x + e^(e^(1/3*x*e^16)) - 5)*log(5) + 2*e^(1/3*x*e^16 + e^(1/3*x*e^16) + 16) + 6)*e^(-x - e^(e^(1/3*x*e^16)) + 5)/log(5), x)
Time = 0.13 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {e^{5-e^{e^{\frac {e^{16} x}{3}}}-x} \left (-6-2 e^{16+e^{\frac {e^{16} x}{3}}+\frac {e^{16} x}{3}}-3 e^{-5+e^{e^{\frac {e^{16} x}{3}}}+x} \log (5)\right )}{3 \log (5)} \, dx=\frac {2\,{\mathrm {e}}^{-{\mathrm {e}}^{{\left ({\mathrm {e}}^{x\,{\mathrm {e}}^{16}}\right )}^{1/3}}}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^5}{\ln \left (5\right )}-x \] Input:
int(-(exp(5 - exp(exp((x*exp(16))/3)) - x)*(exp(x + exp(exp((x*exp(16))/3) ) - 5)*log(5) + (2*exp(exp((x*exp(16))/3))*exp(16)*exp((x*exp(16))/3))/3 + 2))/log(5),x)
Output:
(2*exp(-exp(exp(x*exp(16))^(1/3)))*exp(-x)*exp(5))/log(5) - x
Time = 0.17 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48 \[ \int \frac {e^{5-e^{e^{\frac {e^{16} x}{3}}}-x} \left (-6-2 e^{16+e^{\frac {e^{16} x}{3}}+\frac {e^{16} x}{3}}-3 e^{-5+e^{e^{\frac {e^{16} x}{3}}}+x} \log (5)\right )}{3 \log (5)} \, dx=\frac {-e^{e^{e^{\frac {e^{16} x}{3}}}+x} \mathrm {log}\left (5\right ) x +2 e^{5}}{e^{e^{e^{\frac {e^{16} x}{3}}}+x} \mathrm {log}\left (5\right )} \] Input:
int(1/3*(-3*log(5)*exp(exp(exp(1/3*x*exp(16)))+x-5)-2*exp(16)*exp(1/3*x*ex p(16))*exp(exp(1/3*x*exp(16)))-6)/log(5)/exp(exp(exp(1/3*x*exp(16)))+x-5), x)
Output:
( - e**(e**(e**((e**16*x)/3)) + x)*log(5)*x + 2*e**5)/(e**(e**(e**((e**16* x)/3)) + x)*log(5))