Integrand size = 113, antiderivative size = 33 \[ \int \frac {e^{\frac {-5 e^{-1+x}-14 x-5 x^2+e^x \left (e^{-1+x}+3 x+x^2\right )+\left (5-e^x\right ) \log \left (x^2\right )}{x}} \left (10+e^{-1+x} (5-5 x)-5 x^2+e^x \left (-2+4 x^2+x^3+e^{-1+x} (-1+2 x)\right )+\left (-5+e^x (1-x)\right ) \log \left (x^2\right )\right )}{x^2} \, dx=e^{1+\left (5-e^x\right ) \left (-3-x+\frac {-e^{-1+x}+\log \left (x^2\right )}{x}\right )} \] Output:
exp(1+(5-exp(x))*((ln(x^2)-exp(-1+x))/x-3-x))
Time = 0.13 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.55 \[ \int \frac {e^{\frac {-5 e^{-1+x}-14 x-5 x^2+e^x \left (e^{-1+x}+3 x+x^2\right )+\left (5-e^x\right ) \log \left (x^2\right )}{x}} \left (10+e^{-1+x} (5-5 x)-5 x^2+e^x \left (-2+4 x^2+x^3+e^{-1+x} (-1+2 x)\right )+\left (-5+e^x (1-x)\right ) \log \left (x^2\right )\right )}{x^2} \, dx=e^{-14-\frac {5 e^{-1+x}}{x}+\frac {e^{-1+2 x}}{x}-5 x+e^x (3+x)} \left (x^2\right )^{\frac {5-e^x}{x}} \] Input:
Integrate[(E^((-5*E^(-1 + x) - 14*x - 5*x^2 + E^x*(E^(-1 + x) + 3*x + x^2) + (5 - E^x)*Log[x^2])/x)*(10 + E^(-1 + x)*(5 - 5*x) - 5*x^2 + E^x*(-2 + 4 *x^2 + x^3 + E^(-1 + x)*(-1 + 2*x)) + (-5 + E^x*(1 - x))*Log[x^2]))/x^2,x]
Output:
E^(-14 - (5*E^(-1 + x))/x + E^(-1 + 2*x)/x - 5*x + E^x*(3 + x))*(x^2)^((5 - E^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.
| \(\displaystyle \int \frac {\left (-5 x^2+\left (e^x (1-x)-5\right ) \log \left (x^2\right )+e^x \left (x^3+4 x^2+e^{x-1} (2 x-1)-2\right )+e^{x-1} (5-5 x)+10\right ) \exp \left (\frac {-5 x^2+e^x \left (x^2+3 x+e^{x-1}\right )+\left (5-e^x\right ) \log \left (x^2\right )-14 x-5 e^{x-1}}{x}\right )}{x^2} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (-5 x^2+\left (e^x (1-x)-5\right ) \log \left (x^2\right )+e^x \left (x^3+4 x^2+e^{x-1} (2 x-1)-2\right )+e^{x-1} (5-5 x)+10\right ) \exp \left (\frac {e^x \left (x^2+3 x+e^{x-1}\right )}{x}+\frac {\left (5-e^x\right ) \log \left (x^2\right )}{x}-5 x-\frac {5 e^{x-1}}{x}-14\right )}{x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(2 x-1) \exp \left (\frac {e^x \left (x^2+3 x+e^{x-1}\right )}{x}+\frac {\left (5-e^x\right ) \log \left (x^2\right )}{x}-3 x-\frac {5 e^{x-1}}{x}-15\right )}{x^2}-\frac {5 \left (x^2+\log \left (x^2\right )-2\right ) \exp \left (\frac {e^x \left (x^2+3 x+e^{x-1}\right )}{x}+\frac {\left (5-e^x\right ) \log \left (x^2\right )}{x}-5 x-\frac {5 e^{x-1}}{x}-14\right )}{x^2}+\frac {\left (e x^3+4 e x^2-e x \log \left (x^2\right )+e \log \left (x^2\right )-5 x+5 \left (1-\frac {2 e}{5}\right )\right ) \exp \left (\frac {e^x \left (x^2+3 x+e^{x-1}\right )}{x}+\frac {\left (5-e^x\right ) \log \left (x^2\right )}{x}-4 x-\frac {5 e^{x-1}}{x}-15\right )}{x^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int e^{\frac {\left (e^x-5\right ) \left (e x (x+3)+e^x\right )}{e x}} \left (x^2\right )^{-\frac {e^x}{x}+\frac {5}{x}-1} \left (-5 e \left (x^2-2\right )-e \left (e^x (x-1)+5\right ) \log \left (x^2\right )+e^{x+1} \left (x^3+4 x^2-2\right )-5 e^x (x-1)+e^{2 x} (2 x-1)\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-5 (x-1) \left (x^2\right )^{-\frac {e^x}{x}+\frac {5}{x}-1} \exp \left (x+\frac {\left (e^x-5\right ) \left (e x (x+3)+e^x\right )}{e x}\right )+(2 x-1) \left (x^2\right )^{-\frac {e^x}{x}+\frac {5}{x}-1} \exp \left (2 x+\frac {\left (e^x-5\right ) \left (e x (x+3)+e^x\right )}{e x}\right )-5 \left (x^2-2\right ) \left (x^2\right )^{-\frac {e^x}{x}+\frac {5}{x}-1} \exp \left (\frac {\left (e^x-5\right ) \left (e x (x+3)+e^x\right )}{e x}+1\right )-\left (e^x x-e^x+5\right ) \left (x^2\right )^{-\frac {e^x}{x}+\frac {5}{x}-1} \exp \left (\frac {\left (e^x-5\right ) \left (e x (x+3)+e^x\right )}{e x}+1\right ) \log \left (x^2\right )+\left (x^3+4 x^2-2\right ) \left (x^2\right )^{-\frac {e^x}{x}+\frac {5}{x}-1} \exp \left (x+\frac {\left (e^x-5\right ) \left (e x (x+3)+e^x\right )}{e x}+1\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -5 \log \left (x^2\right ) \int e^{\frac {\left (-5+e^x\right ) \left (e x (x+3)+e^x\right )}{e x}+1} \left (x^2\right )^{-1-\frac {e^x}{x}+\frac {5}{x}}dx+10 \int e^{\frac {\left (-5+e^x\right ) \left (e x (x+3)+e^x\right )}{e x}+1} \left (x^2\right )^{-1-\frac {e^x}{x}+\frac {5}{x}}dx+5 \int e^{x+\frac {\left (-5+e^x\right ) \left (e x (x+3)+e^x\right )}{e x}} \left (x^2\right )^{-1-\frac {e^x}{x}+\frac {5}{x}}dx+\log \left (x^2\right ) \int e^{x+1+\frac {\left (-5+e^x\right ) \left (e x (x+3)+e^x\right )}{e x}} \left (x^2\right )^{-1-\frac {e^x}{x}+\frac {5}{x}}dx-2 \int e^{x+1+\frac {\left (-5+e^x\right ) \left (e x (x+3)+e^x\right )}{e x}} \left (x^2\right )^{-1-\frac {e^x}{x}+\frac {5}{x}}dx-\int e^{2 x+\frac {\left (-5+e^x\right ) \left (e x (x+3)+e^x\right )}{e x}} \left (x^2\right )^{-1-\frac {e^x}{x}+\frac {5}{x}}dx-5 \int e^{x+\frac {\left (-5+e^x\right ) \left (e x (x+3)+e^x\right )}{e x}} x \left (x^2\right )^{-1-\frac {e^x}{x}+\frac {5}{x}}dx-\log \left (x^2\right ) \int e^{x+1+\frac {\left (-5+e^x\right ) \left (e x (x+3)+e^x\right )}{e x}} x \left (x^2\right )^{-1-\frac {e^x}{x}+\frac {5}{x}}dx+2 \int e^{2 x+\frac {\left (-5+e^x\right ) \left (e x (x+3)+e^x\right )}{e x}} x \left (x^2\right )^{-1-\frac {e^x}{x}+\frac {5}{x}}dx+\int e^{x+1+\frac {\left (-5+e^x\right ) \left (e x (x+3)+e^x\right )}{e x}} x^3 \left (x^2\right )^{-1-\frac {e^x}{x}+\frac {5}{x}}dx-5 \int e^{\frac {\left (-5+e^x\right ) \left (e x (x+3)+e^x\right )}{e x}+1} \left (x^2\right )^{\frac {5}{x}-\frac {e^x}{x}}dx+4 \int e^{x+1+\frac {\left (-5+e^x\right ) \left (e x (x+3)+e^x\right )}{e x}} \left (x^2\right )^{\frac {5}{x}-\frac {e^x}{x}}dx+10 \int \frac {\int e^{\frac {\left (-5+e^x\right ) \left (e x (x+3)+e^x\right )}{e x}+1} \left (x^2\right )^{-1-\frac {e^x}{x}+\frac {5}{x}}dx}{x}dx-2 \int \frac {\int e^{x+1+\frac {\left (-5+e^x\right ) \left (e x (x+3)+e^x\right )}{e x}} \left (x^2\right )^{-1-\frac {e^x}{x}+\frac {5}{x}}dx}{x}dx+2 \int \frac {\int \frac {e^{x+1+\frac {\left (-5+e^x\right ) \left (e x (x+3)+e^x\right )}{e x}} \left (x^2\right )^{\frac {5}{x}-\frac {e^x}{x}}}{x}dx}{x}dx\) |
Input:
Int[(E^((-5*E^(-1 + x) - 14*x - 5*x^2 + E^x*(E^(-1 + x) + 3*x + x^2) + (5 - E^x)*Log[x^2])/x)*(10 + E^(-1 + x)*(5 - 5*x) - 5*x^2 + E^x*(-2 + 4*x^2 + x^3 + E^(-1 + x)*(-1 + 2*x)) + (-5 + E^x*(1 - x))*Log[x^2]))/x^2,x]
Output:
$Aborted
Time = 3.31 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.39
| method | result | size |
| parallelrisch | \({\mathrm e}^{\frac {\left (5-{\mathrm e}^{x}\right ) \ln \left (x^{2}\right )+\left ({\mathrm e}^{-1+x}+x^{2}+3 x \right ) {\mathrm e}^{x}-5 \,{\mathrm e}^{-1+x}-5 x^{2}-14 x}{x}}\) | \(46\) |
| risch | \(x^{-\frac {2 \,{\mathrm e}^{x}}{x}} x^{\frac {10}{x}} {\mathrm e}^{\frac {i {\mathrm e}^{x} \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-2 i {\mathrm e}^{x} \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )+i {\mathrm e}^{x} \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}-5 i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+10 i \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right ) \pi -5 i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}+2 \,{\mathrm e}^{x} x^{2}+6 \,{\mathrm e}^{x} x -10 x^{2}-10 \,{\mathrm e}^{-1+x}+2 \,{\mathrm e}^{-1+2 x}-28 x}{2 x}}\) | \(163\) |
Input:
int((((1-x)*exp(x)-5)*ln(x^2)+((-1+2*x)*exp(-1+x)+x^3+4*x^2-2)*exp(x)+(-5* x+5)*exp(-1+x)-5*x^2+10)*exp(((5-exp(x))*ln(x^2)+(exp(-1+x)+x^2+3*x)*exp(x )-5*exp(-1+x)-5*x^2-14*x)/x)/x^2,x,method=_RETURNVERBOSE)
Output:
exp(((5-exp(x))*ln(x^2)+(exp(-1+x)+x^2+3*x)*exp(x)-5*exp(-1+x)-5*x^2-14*x) /x)
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (26) = 52\).
Time = 0.11 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.82 \[ \int \frac {e^{\frac {-5 e^{-1+x}-14 x-5 x^2+e^x \left (e^{-1+x}+3 x+x^2\right )+\left (5-e^x\right ) \log \left (x^2\right )}{x}} \left (10+e^{-1+x} (5-5 x)-5 x^2+e^x \left (-2+4 x^2+x^3+e^{-1+x} (-1+2 x)\right )+\left (-5+e^x (1-x)\right ) \log \left (x^2\right )\right )}{x^2} \, dx=e^{\left (-\frac {{\left ({\left (5 \, x^{2} + 14 \, x\right )} e - {\left ({\left (x^{2} + 3 \, x\right )} e - 5\right )} e^{x} - {\left (5 \, e - e^{\left (x + 1\right )}\right )} \log \left (x^{2}\right ) - e^{\left (2 \, x\right )}\right )} e^{\left (-1\right )}}{x}\right )} \] Input:
integrate((((1-x)*exp(x)-5)*log(x^2)+((-1+2*x)*exp(-1+x)+x^3+4*x^2-2)*exp( x)+(-5*x+5)*exp(-1+x)-5*x^2+10)*exp(((5-exp(x))*log(x^2)+(exp(-1+x)+x^2+3* x)*exp(x)-5*exp(-1+x)-5*x^2-14*x)/x)/x^2,x, algorithm="fricas")
Output:
e^(-((5*x^2 + 14*x)*e - ((x^2 + 3*x)*e - 5)*e^x - (5*e - e^(x + 1))*log(x^ 2) - e^(2*x))*e^(-1)/x)
Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (22) = 44\).
Time = 0.48 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.39 \[ \int \frac {e^{\frac {-5 e^{-1+x}-14 x-5 x^2+e^x \left (e^{-1+x}+3 x+x^2\right )+\left (5-e^x\right ) \log \left (x^2\right )}{x}} \left (10+e^{-1+x} (5-5 x)-5 x^2+e^x \left (-2+4 x^2+x^3+e^{-1+x} (-1+2 x)\right )+\left (-5+e^x (1-x)\right ) \log \left (x^2\right )\right )}{x^2} \, dx=e^{\frac {- 5 x^{2} - 14 x + \left (5 - e^{x}\right ) \log {\left (x^{2} \right )} + \left (x^{2} + 3 x + \frac {e^{x}}{e}\right ) e^{x} - \frac {5 e^{x}}{e}}{x}} \] Input:
integrate((((1-x)*exp(x)-5)*ln(x**2)+((-1+2*x)*exp(-1+x)+x**3+4*x**2-2)*ex p(x)+(-5*x+5)*exp(-1+x)-5*x**2+10)*exp(((5-exp(x))*ln(x**2)+(exp(-1+x)+x** 2+3*x)*exp(x)-5*exp(-1+x)-5*x**2-14*x)/x)/x**2,x)
Output:
exp((-5*x**2 - 14*x + (5 - exp(x))*log(x**2) + (x**2 + 3*x + exp(-1)*exp(x ))*exp(x) - 5*exp(-1)*exp(x))/x)
Time = 0.44 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.48 \[ \int \frac {e^{\frac {-5 e^{-1+x}-14 x-5 x^2+e^x \left (e^{-1+x}+3 x+x^2\right )+\left (5-e^x\right ) \log \left (x^2\right )}{x}} \left (10+e^{-1+x} (5-5 x)-5 x^2+e^x \left (-2+4 x^2+x^3+e^{-1+x} (-1+2 x)\right )+\left (-5+e^x (1-x)\right ) \log \left (x^2\right )\right )}{x^2} \, dx=e^{\left (x e^{x} - 5 \, x - \frac {2 \, e^{x} \log \left (x\right )}{x} + \frac {e^{\left (2 \, x - 1\right )}}{x} - \frac {5 \, e^{\left (x - 1\right )}}{x} + \frac {10 \, \log \left (x\right )}{x} + 3 \, e^{x} - 14\right )} \] Input:
integrate((((1-x)*exp(x)-5)*log(x^2)+((-1+2*x)*exp(-1+x)+x^3+4*x^2-2)*exp( x)+(-5*x+5)*exp(-1+x)-5*x^2+10)*exp(((5-exp(x))*log(x^2)+(exp(-1+x)+x^2+3* x)*exp(x)-5*exp(-1+x)-5*x^2-14*x)/x)/x^2,x, algorithm="maxima")
Output:
e^(x*e^x - 5*x - 2*e^x*log(x)/x + e^(2*x - 1)/x - 5*e^(x - 1)/x + 10*log(x )/x + 3*e^x - 14)
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (26) = 52\).
Time = 0.38 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.61 \[ \int \frac {e^{\frac {-5 e^{-1+x}-14 x-5 x^2+e^x \left (e^{-1+x}+3 x+x^2\right )+\left (5-e^x\right ) \log \left (x^2\right )}{x}} \left (10+e^{-1+x} (5-5 x)-5 x^2+e^x \left (-2+4 x^2+x^3+e^{-1+x} (-1+2 x)\right )+\left (-5+e^x (1-x)\right ) \log \left (x^2\right )\right )}{x^2} \, dx=e^{\left (x e^{x} - 5 \, x - \frac {e^{x} \log \left (x^{2}\right )}{x} + \frac {e^{\left (2 \, x - 1\right )}}{x} - \frac {5 \, e^{\left (x - 1\right )}}{x} + \frac {5 \, \log \left (x^{2}\right )}{x} + 3 \, e^{x} - 14\right )} \] Input:
integrate((((1-x)*exp(x)-5)*log(x^2)+((-1+2*x)*exp(-1+x)+x^3+4*x^2-2)*exp( x)+(-5*x+5)*exp(-1+x)-5*x^2+10)*exp(((5-exp(x))*log(x^2)+(exp(-1+x)+x^2+3* x)*exp(x)-5*exp(-1+x)-5*x^2-14*x)/x)/x^2,x, algorithm="giac")
Output:
e^(x*e^x - 5*x - e^x*log(x^2)/x + e^(2*x - 1)/x - 5*e^(x - 1)/x + 5*log(x^ 2)/x + 3*e^x - 14)
Time = 4.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.52 \[ \int \frac {e^{\frac {-5 e^{-1+x}-14 x-5 x^2+e^x \left (e^{-1+x}+3 x+x^2\right )+\left (5-e^x\right ) \log \left (x^2\right )}{x}} \left (10+e^{-1+x} (5-5 x)-5 x^2+e^x \left (-2+4 x^2+x^3+e^{-1+x} (-1+2 x)\right )+\left (-5+e^x (1-x)\right ) \log \left (x^2\right )\right )}{x^2} \, dx={\mathrm {e}}^{x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-5\,x}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-1}}{x}}\,{\mathrm {e}}^{-14}\,{\mathrm {e}}^{3\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-\frac {5\,{\mathrm {e}}^{-1}\,{\mathrm {e}}^x}{x}}\,{\left (\frac {1}{x^2}\right )}^{\frac {{\mathrm {e}}^x-5}{x}} \] Input:
int(-(exp(-(14*x + 5*exp(x - 1) - exp(x)*(3*x + exp(x - 1) + x^2) + 5*x^2 + log(x^2)*(exp(x) - 5))/x)*(log(x^2)*(exp(x)*(x - 1) + 5) - exp(x)*(exp(x - 1)*(2*x - 1) + 4*x^2 + x^3 - 2) + exp(x - 1)*(5*x - 5) + 5*x^2 - 10))/x ^2,x)
Output:
exp(x*exp(x))*exp(-5*x)*exp((exp(2*x)*exp(-1))/x)*exp(-14)*exp(3*exp(x))*e xp(-(5*exp(-1)*exp(x))/x)*(1/x^2)^((exp(x) - 5)/x)
Time = 0.19 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.21 \[ \int \frac {e^{\frac {-5 e^{-1+x}-14 x-5 x^2+e^x \left (e^{-1+x}+3 x+x^2\right )+\left (5-e^x\right ) \log \left (x^2\right )}{x}} \left (10+e^{-1+x} (5-5 x)-5 x^2+e^x \left (-2+4 x^2+x^3+e^{-1+x} (-1+2 x)\right )+\left (-5+e^x (1-x)\right ) \log \left (x^2\right )\right )}{x^2} \, dx=\frac {e^{\frac {e^{2 x}+e^{x} e \,x^{2}+3 e^{x} e x +5 \,\mathrm {log}\left (x^{2}\right ) e}{e x}}}{e^{\frac {e^{x} \mathrm {log}\left (x^{2}\right ) e +5 e^{x}+5 e \,x^{2}}{e x}} e^{14}} \] Input:
int((((1-x)*exp(x)-5)*log(x^2)+((-1+2*x)*exp(-1+x)+x^3+4*x^2-2)*exp(x)+(-5 *x+5)*exp(-1+x)-5*x^2+10)*exp(((5-exp(x))*log(x^2)+(exp(-1+x)+x^2+3*x)*exp (x)-5*exp(-1+x)-5*x^2-14*x)/x)/x^2,x)
Output:
e**((e**(2*x) + e**x*e*x**2 + 3*e**x*e*x + 5*log(x**2)*e)/(e*x))/(e**((e** x*log(x**2)*e + 5*e**x + 5*e*x**2)/(e*x))*e**14)