Integrand size = 65, antiderivative size = 25 \[ \int \frac {e^{-2 x} \left (-3 e^{2 x}+e^{6+e^{e^{-2 x} \left (3+e^4\right )}} \left (e^{2 x}+e^{e^{-2 x} \left (3+e^4\right )} \left (6 x+2 e^4 x\right )\right )\right )}{x^2} \, dx=\frac {3-e^{6+e^{e^{-2 x} \left (3+e^4\right )}}}{x} \] Output:
(3-exp(exp((exp(4)+3)/exp(2*x))+6))/x
Time = 1.49 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-2 x} \left (-3 e^{2 x}+e^{6+e^{e^{-2 x} \left (3+e^4\right )}} \left (e^{2 x}+e^{e^{-2 x} \left (3+e^4\right )} \left (6 x+2 e^4 x\right )\right )\right )}{x^2} \, dx=\frac {3-e^{6+e^{e^{-2 x} \left (3+e^4\right )}}}{x} \] Input:
Integrate[(-3*E^(2*x) + E^(6 + E^((3 + E^4)/E^(2*x)))*(E^(2*x) + E^((3 + E ^4)/E^(2*x))*(6*x + 2*E^4*x)))/(E^(2*x)*x^2),x]
Output:
(3 - E^(6 + E^((3 + E^4)/E^(2*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.
\(\displaystyle \int \frac {e^{-2 x} \left (e^{e^{\left (3+e^4\right ) e^{-2 x}}+6} \left (e^{\left (3+e^4\right ) e^{-2 x}} \left (2 e^4 x+6 x\right )+e^{2 x}\right )-3 e^{2 x}\right )}{x^2} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 \left (3+e^4\right ) \exp \left (-2 x+e^{e^{4-2 x}+3 e^{-2 x}}+\left (1+\frac {3}{e^4}\right ) e^{4-2 x}+6\right )}{x}+\frac {e^{e^{e^{4-2 x}+3 e^{-2 x}}+6}-3}{x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (3+e^4\right ) \int \frac {\exp \left (-2 x+e^{e^{4-2 x}+3 e^{-2 x}}+\left (1+\frac {3}{e^4}\right ) e^{4-2 x}+6\right )}{x}dx+\int \frac {e^{6+e^{e^{4-2 x}+3 e^{-2 x}}}}{x^2}dx+\frac {3}{x}\) |
Input:
Int[(-3*E^(2*x) + E^(6 + E^((3 + E^4)/E^(2*x)))*(E^(2*x) + E^((3 + E^4)/E^ (2*x))*(6*x + 2*E^4*x)))/(E^(2*x)*x^2),x]
Output:
$Aborted
Time = 0.42 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96
method | result | size |
parallelrisch | \(\frac {3-{\mathrm e}^{{\mathrm e}^{\left ({\mathrm e}^{4}+3\right ) {\mathrm e}^{-2 x}}+6}}{x}\) | \(24\) |
risch | \(\frac {3}{x}-\frac {{\mathrm e}^{{\mathrm e}^{\left ({\mathrm e}^{4}+3\right ) {\mathrm e}^{-2 x}}+6}}{x}\) | \(25\) |
norman | \(\frac {\left (-{\mathrm e}^{{\mathrm e}^{\left ({\mathrm e}^{4}+3\right ) {\mathrm e}^{-2 x}}+6} {\mathrm e}^{2 x}+3 \,{\mathrm e}^{2 x}\right ) {\mathrm e}^{-2 x}}{x}\) | \(39\) |
Input:
int((((2*x*exp(4)+6*x)*exp((exp(4)+3)/exp(2*x))+exp(2*x))*exp(exp((exp(4)+ 3)/exp(2*x))+6)-3*exp(2*x))/exp(2*x)/x^2,x,method=_RETURNVERBOSE)
Output:
(3-exp(exp((exp(4)+3)/exp(2*x))+6))/x
Time = 0.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {e^{-2 x} \left (-3 e^{2 x}+e^{6+e^{e^{-2 x} \left (3+e^4\right )}} \left (e^{2 x}+e^{e^{-2 x} \left (3+e^4\right )} \left (6 x+2 e^4 x\right )\right )\right )}{x^2} \, dx=-\frac {e^{\left (e^{\left ({\left (e^{4} + 3\right )} e^{\left (-2 \, x\right )}\right )} + 6\right )} - 3}{x} \] Input:
integrate((((2*x*exp(4)+6*x)*exp((exp(4)+3)/exp(2*x))+exp(2*x))*exp(exp((e xp(4)+3)/exp(2*x))+6)-3*exp(2*x))/exp(2*x)/x^2,x, algorithm="fricas")
Output:
-(e^(e^((e^4 + 3)*e^(-2*x)) + 6) - 3)/x
Time = 0.15 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {e^{-2 x} \left (-3 e^{2 x}+e^{6+e^{e^{-2 x} \left (3+e^4\right )}} \left (e^{2 x}+e^{e^{-2 x} \left (3+e^4\right )} \left (6 x+2 e^4 x\right )\right )\right )}{x^2} \, dx=- \frac {e^{e^{\left (3 + e^{4}\right ) e^{- 2 x}} + 6}}{x} + \frac {3}{x} \] Input:
integrate((((2*x*exp(4)+6*x)*exp((exp(4)+3)/exp(2*x))+exp(2*x))*exp(exp((e xp(4)+3)/exp(2*x))+6)-3*exp(2*x))/exp(2*x)/x**2,x)
Output:
-exp(exp((3 + exp(4))*exp(-2*x)) + 6)/x + 3/x
Time = 0.16 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {e^{-2 x} \left (-3 e^{2 x}+e^{6+e^{e^{-2 x} \left (3+e^4\right )}} \left (e^{2 x}+e^{e^{-2 x} \left (3+e^4\right )} \left (6 x+2 e^4 x\right )\right )\right )}{x^2} \, dx=-\frac {e^{\left (e^{\left (3 \, e^{\left (-2 \, x\right )} + e^{\left (-2 \, x + 4\right )}\right )} + 6\right )}}{x} + \frac {3}{x} \] Input:
integrate((((2*x*exp(4)+6*x)*exp((exp(4)+3)/exp(2*x))+exp(2*x))*exp(exp((e xp(4)+3)/exp(2*x))+6)-3*exp(2*x))/exp(2*x)/x^2,x, algorithm="maxima")
Output:
-e^(e^(3*e^(-2*x) + e^(-2*x + 4)) + 6)/x + 3/x
\[ \int \frac {e^{-2 x} \left (-3 e^{2 x}+e^{6+e^{e^{-2 x} \left (3+e^4\right )}} \left (e^{2 x}+e^{e^{-2 x} \left (3+e^4\right )} \left (6 x+2 e^4 x\right )\right )\right )}{x^2} \, dx=\int { \frac {{\left ({\left (2 \, {\left (x e^{4} + 3 \, x\right )} e^{\left ({\left (e^{4} + 3\right )} e^{\left (-2 \, x\right )}\right )} + e^{\left (2 \, x\right )}\right )} e^{\left (e^{\left ({\left (e^{4} + 3\right )} e^{\left (-2 \, x\right )}\right )} + 6\right )} - 3 \, e^{\left (2 \, x\right )}\right )} e^{\left (-2 \, x\right )}}{x^{2}} \,d x } \] Input:
integrate((((2*x*exp(4)+6*x)*exp((exp(4)+3)/exp(2*x))+exp(2*x))*exp(exp((e xp(4)+3)/exp(2*x))+6)-3*exp(2*x))/exp(2*x)/x^2,x, algorithm="giac")
Output:
integrate(((2*(x*e^4 + 3*x)*e^((e^4 + 3)*e^(-2*x)) + e^(2*x))*e^(e^((e^4 + 3)*e^(-2*x)) + 6) - 3*e^(2*x))*e^(-2*x)/x^2, x)
Time = 2.84 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {e^{-2 x} \left (-3 e^{2 x}+e^{6+e^{e^{-2 x} \left (3+e^4\right )}} \left (e^{2 x}+e^{e^{-2 x} \left (3+e^4\right )} \left (6 x+2 e^4 x\right )\right )\right )}{x^2} \, dx=-\frac {{\mathrm {e}}^{{\mathrm {e}}^{3\,{\mathrm {e}}^{-2\,x}}\,{\mathrm {e}}^{{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^4}}\,{\mathrm {e}}^6-3}{x} \] Input:
int(-(exp(-2*x)*(3*exp(2*x) - exp(exp(exp(-2*x)*(exp(4) + 3)) + 6)*(exp(2* x) + exp(exp(-2*x)*(exp(4) + 3))*(6*x + 2*x*exp(4)))))/x^2,x)
Output:
-(exp(exp(3*exp(-2*x))*exp(exp(-2*x)*exp(4)))*exp(6) - 3)/x
\[ \int \frac {e^{-2 x} \left (-3 e^{2 x}+e^{6+e^{e^{-2 x} \left (3+e^4\right )}} \left (e^{2 x}+e^{e^{-2 x} \left (3+e^4\right )} \left (6 x+2 e^4 x\right )\right )\right )}{x^2} \, dx=\frac {\left (\int \frac {e^{e^{\frac {e^{4}+3}{e^{2 x}}}}}{x^{2}}d x \right ) e^{6} x +2 \left (\int \frac {e^{\frac {e^{\frac {2 e^{2 x} x +e^{4}+3}{e^{2 x}}}+e^{4}+3}{e^{2 x}}}}{e^{2 x} x}d x \right ) e^{10} x +6 \left (\int \frac {e^{\frac {e^{\frac {2 e^{2 x} x +e^{4}+3}{e^{2 x}}}+e^{4}+3}{e^{2 x}}}}{e^{2 x} x}d x \right ) e^{6} x +3}{x} \] Input:
int((((2*x*exp(4)+6*x)*exp((exp(4)+3)/exp(2*x))+exp(2*x))*exp(exp((exp(4)+ 3)/exp(2*x))+6)-3*exp(2*x))/exp(2*x)/x^2,x)
Output:
(int(e**(e**((e**4 + 3)/e**(2*x)))/x**2,x)*e**6*x + 2*int(e**((e**((2*e**( 2*x)*x + e**4 + 3)/e**(2*x)) + e**4 + 3)/e**(2*x))/(e**(2*x)*x),x)*e**10*x + 6*int(e**((e**((2*e**(2*x)*x + e**4 + 3)/e**(2*x)) + e**4 + 3)/e**(2*x) )/(e**(2*x)*x),x)*e**6*x + 3)/x