Integrand size = 89, antiderivative size = 31 \[ \int \frac {e^{-\frac {e^x}{x}} \left (e^{2 x} (9-9 x)+24 e^x x+24 x^2+\left (-16 e^x x-16 x^2+e^{2 x} (-12+12 x)\right ) \log \left (\frac {4}{x^2}\right )+e^{2 x} (4-4 x) \log ^2\left (\frac {4}{x^2}\right )\right )}{4 x^3} \, dx=\frac {e^{-\frac {e^x}{x}} \left (e^x+x\right ) \left (-\frac {3}{2}+\log \left (\frac {4}{x^2}\right )\right )^2}{x} \]
Time = 3.23 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {e^{-\frac {e^x}{x}} \left (e^{2 x} (9-9 x)+24 e^x x+24 x^2+\left (-16 e^x x-16 x^2+e^{2 x} (-12+12 x)\right ) \log \left (\frac {4}{x^2}\right )+e^{2 x} (4-4 x) \log ^2\left (\frac {4}{x^2}\right )\right )}{4 x^3} \, dx=\frac {e^{-\frac {e^x}{x}} \left (e^x+x\right ) \left (3-2 \log \left (\frac {4}{x^2}\right )\right )^2}{4 x} \]
Integrate[(E^(2*x)*(9 - 9*x) + 24*E^x*x + 24*x^2 + (-16*E^x*x - 16*x^2 + E ^(2*x)*(-12 + 12*x))*Log[4/x^2] + E^(2*x)*(4 - 4*x)*Log[4/x^2]^2)/(4*E^(E^ x/x)*x^3),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.
| \(\displaystyle \int \frac {e^{-\frac {e^x}{x}} \left (24 x^2+e^{2 x} (4-4 x) \log ^2\left (\frac {4}{x^2}\right )+\left (-16 x^2-16 e^x x+e^{2 x} (12 x-12)\right ) \log \left (\frac {4}{x^2}\right )+24 e^x x+e^{2 x} (9-9 x)\right )}{4 x^3} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \int \frac {e^{-\frac {e^x}{x}} \left (24 x^2+24 e^x x+4 e^{2 x} (1-x) \log ^2\left (\frac {4}{x^2}\right )+9 e^{2 x} (1-x)-4 \left (4 x^2+4 e^x x+3 e^{2 x} (1-x)\right ) \log \left (\frac {4}{x^2}\right )\right )}{x^3}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {1}{4} \int \frac {e^{-\frac {e^x}{x}} \left (3-2 \log \left (\frac {4}{x^2}\right )\right ) \left (8 x^2+8 e^x x-3 e^{2 x} x+2 e^{2 x} \log \left (\frac {4}{x^2}\right ) x+3 e^{2 x}-2 e^{2 x} \log \left (\frac {4}{x^2}\right )\right )}{x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{4} \int \left (-\frac {e^{2 x-\frac {e^x}{x}} (x-1) \left (2 \log \left (\frac {4}{x^2}\right )-3\right )^2}{x^3}-\frac {8 e^{-\frac {e^x}{x}} \left (2 \log \left (\frac {4}{x^2}\right )-3\right )}{x}-\frac {8 e^{x-\frac {e^x}{x}} \left (2 \log \left (\frac {4}{x^2}\right )-3\right )}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{4} \left (9 \int \frac {e^{2 x-\frac {e^x}{x}}}{x^3}dx-24 \int \frac {\int \frac {e^{2 x-\frac {e^x}{x}}}{x^3}dx}{x}dx+24 \int \frac {e^{x-\frac {e^x}{x}}}{x^2}dx-9 \int \frac {e^{2 x-\frac {e^x}{x}}}{x^2}dx-32 \int \frac {\int \frac {e^{x-\frac {e^x}{x}}}{x^2}dx}{x}dx+24 \int \frac {\int \frac {e^{2 x-\frac {e^x}{x}}}{x^2}dx}{x}dx-4 \int \frac {e^{2 x-\frac {e^x}{x}} \log ^2\left (\frac {4}{x^2}\right )}{x^2}dx-16 \log \left (\frac {4}{x^2}\right ) \int \frac {e^{x-\frac {e^x}{x}}}{x^2}dx+12 \log \left (\frac {4}{x^2}\right ) \int \frac {e^{2 x-\frac {e^x}{x}}}{x^2}dx-16 \log \left (\frac {4}{x^2}\right ) \int \frac {e^{-\frac {e^x}{x}}}{x}dx+4 \int \frac {e^{2 x-\frac {e^x}{x}} \log ^2\left (\frac {4}{x^2}\right )}{x^3}dx-12 \log \left (\frac {4}{x^2}\right ) \int \frac {e^{2 x-\frac {e^x}{x}}}{x^3}dx+24 \int \frac {e^{-\frac {e^x}{x}}}{x}dx-32 \int \frac {\int \frac {e^{-\frac {e^x}{x}}}{x}dx}{x}dx\right )\) |
Int[(E^(2*x)*(9 - 9*x) + 24*E^x*x + 24*x^2 + (-16*E^x*x - 16*x^2 + E^(2*x) *(-12 + 12*x))*Log[4/x^2] + E^(2*x)*(4 - 4*x)*Log[4/x^2]^2)/(4*E^(E^x/x)*x ^3),x]
3.19.49.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(64\) vs. \(2(27)=54\).
Time = 0.63 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.10
| method | result | size |
| parallelrisch | \(\frac {\left (4 x \ln \left (\frac {4}{x^{2}}\right )^{2}+4 \ln \left (\frac {4}{x^{2}}\right )^{2} {\mathrm e}^{x}-12 x \ln \left (\frac {4}{x^{2}}\right )-12 \,{\mathrm e}^{x} \ln \left (\frac {4}{x^{2}}\right )+9 x +9 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-\frac {{\mathrm e}^{x}}{x}}}{4 x}\) | \(65\) |
| risch | \(\frac {\left (9 x +16 \,{\mathrm e}^{x} \ln \left (x \right )^{2}-24 \,{\mathrm e}^{x} \ln \left (2\right )+16 \ln \left (2\right )^{2} {\mathrm e}^{x}+16 x \ln \left (2\right )^{2}+24 \,{\mathrm e}^{x} \ln \left (x \right )+16 x \ln \left (x \right )^{2}-24 x \ln \left (2\right )+24 x \ln \left (x \right )-32 x \ln \left (2\right ) \ln \left (x \right )+9 \,{\mathrm e}^{x}+8 i \pi \ln \left (2\right ) x \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-16 i \pi \ln \left (2\right ) x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-8 i \pi x \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right ) \ln \left (x \right )-x \,\pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{6}-\pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{6} {\mathrm e}^{x}+4 x \,\pi ^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{5}-6 i \pi x \operatorname {csgn}\left (i x^{2}\right )^{3}-6 i \pi \operatorname {csgn}\left (i x^{2}\right )^{3} {\mathrm e}^{x}-x \,\pi ^{2} \operatorname {csgn}\left (i x \right )^{4} \operatorname {csgn}\left (i x^{2}\right )^{2}+4 x \,\pi ^{2} \operatorname {csgn}\left (i x \right )^{3} \operatorname {csgn}\left (i x^{2}\right )^{3}-6 x \,\pi ^{2} \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )^{4}-\pi ^{2} \operatorname {csgn}\left (i x \right )^{4} \operatorname {csgn}\left (i x^{2}\right )^{2} {\mathrm e}^{x}+4 \pi ^{2} \operatorname {csgn}\left (i x \right )^{3} \operatorname {csgn}\left (i x^{2}\right )^{3} {\mathrm e}^{x}-6 \pi ^{2} \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )^{4} {\mathrm e}^{x}+4 \pi ^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{5} {\mathrm e}^{x}+8 i \pi \ln \left (2\right ) x \operatorname {csgn}\left (i x^{2}\right )^{3}+8 i \pi \ln \left (2\right ) \operatorname {csgn}\left (i x^{2}\right )^{3} {\mathrm e}^{x}-8 i \pi x \operatorname {csgn}\left (i x^{2}\right )^{3} \ln \left (x \right )+12 i \pi x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-32 \ln \left (2\right ) {\mathrm e}^{x} \ln \left (x \right )-6 i \pi x \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-6 i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right ) {\mathrm e}^{x}+12 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2} {\mathrm e}^{x}-8 i \pi \operatorname {csgn}\left (i x^{2}\right )^{3} {\mathrm e}^{x} \ln \left (x \right )+8 i \pi \ln \left (2\right ) \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right ) {\mathrm e}^{x}-8 i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right ) {\mathrm e}^{x} \ln \left (x \right )+16 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2} {\mathrm e}^{x} \ln \left (x \right )+16 i \pi x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2} \ln \left (x \right )-16 i \pi \ln \left (2\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2} {\mathrm e}^{x}\right ) {\mathrm e}^{-\frac {{\mathrm e}^{x}}{x}}}{4 x}\) | \(641\) |
int(1/4*((4-4*x)*exp(x)^2*ln(4/x^2)^2+((12*x-12)*exp(x)^2-16*exp(x)*x-16*x ^2)*ln(4/x^2)+(-9*x+9)*exp(x)^2+24*exp(x)*x+24*x^2)/x^3/exp(exp(x)/x),x,me thod=_RETURNVERBOSE)
1/4/x*(4*x*ln(4/x^2)^2+4*ln(4/x^2)^2*exp(x)-12*x*ln(4/x^2)-12*exp(x)*ln(4/ x^2)+9*x+9*exp(x))/exp(exp(x)/x)
Time = 0.25 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.52 \[ \int \frac {e^{-\frac {e^x}{x}} \left (e^{2 x} (9-9 x)+24 e^x x+24 x^2+\left (-16 e^x x-16 x^2+e^{2 x} (-12+12 x)\right ) \log \left (\frac {4}{x^2}\right )+e^{2 x} (4-4 x) \log ^2\left (\frac {4}{x^2}\right )\right )}{4 x^3} \, dx=\frac {{\left (4 \, {\left (x + e^{x}\right )} \log \left (\frac {4}{x^{2}}\right )^{2} - 12 \, {\left (x + e^{x}\right )} \log \left (\frac {4}{x^{2}}\right ) + 9 \, x + 9 \, e^{x}\right )} e^{\left (-\frac {e^{x}}{x}\right )}}{4 \, x} \]
integrate(1/4*((4-4*x)*exp(x)^2*log(4/x^2)^2+((12*x-12)*exp(x)^2-16*exp(x) *x-16*x^2)*log(4/x^2)+(-9*x+9)*exp(x)^2+24*exp(x)*x+24*x^2)/x^3/exp(exp(x) /x),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (24) = 48\).
Time = 0.16 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.10 \[ \int \frac {e^{-\frac {e^x}{x}} \left (e^{2 x} (9-9 x)+24 e^x x+24 x^2+\left (-16 e^x x-16 x^2+e^{2 x} (-12+12 x)\right ) \log \left (\frac {4}{x^2}\right )+e^{2 x} (4-4 x) \log ^2\left (\frac {4}{x^2}\right )\right )}{4 x^3} \, dx=\frac {\left (4 x \log {\left (\frac {4}{x^{2}} \right )}^{2} - 12 x \log {\left (\frac {4}{x^{2}} \right )} + 9 x + 4 e^{x} \log {\left (\frac {4}{x^{2}} \right )}^{2} - 12 e^{x} \log {\left (\frac {4}{x^{2}} \right )} + 9 e^{x}\right ) e^{- \frac {e^{x}}{x}}}{4 x} \]
integrate(1/4*((4-4*x)*exp(x)**2*ln(4/x**2)**2+((12*x-12)*exp(x)**2-16*exp (x)*x-16*x**2)*ln(4/x**2)+(-9*x+9)*exp(x)**2+24*exp(x)*x+24*x**2)/x**3/exp (exp(x)/x),x)
(4*x*log(4/x**2)**2 - 12*x*log(4/x**2) + 9*x + 4*exp(x)*log(4/x**2)**2 - 1 2*exp(x)*log(4/x**2) + 9*exp(x))*exp(-exp(x)/x)/(4*x)
Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (29) = 58\).
Time = 0.33 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.55 \[ \int \frac {e^{-\frac {e^x}{x}} \left (e^{2 x} (9-9 x)+24 e^x x+24 x^2+\left (-16 e^x x-16 x^2+e^{2 x} (-12+12 x)\right ) \log \left (\frac {4}{x^2}\right )+e^{2 x} (4-4 x) \log ^2\left (\frac {4}{x^2}\right )\right )}{4 x^3} \, dx=-\frac {{\left (8 \, x {\left (4 \, \log \left (2\right ) - 3\right )} \log \left (x\right ) - 16 \, x \log \left (x\right )^{2} - {\left (16 \, \log \left (2\right )^{2} - 24 \, \log \left (2\right ) + 9\right )} x - {\left (16 \, \log \left (2\right )^{2} - 8 \, {\left (4 \, \log \left (2\right ) - 3\right )} \log \left (x\right ) + 16 \, \log \left (x\right )^{2} - 24 \, \log \left (2\right ) + 9\right )} e^{x}\right )} e^{\left (-\frac {e^{x}}{x}\right )}}{4 \, x} \]
integrate(1/4*((4-4*x)*exp(x)^2*log(4/x^2)^2+((12*x-12)*exp(x)^2-16*exp(x) *x-16*x^2)*log(4/x^2)+(-9*x+9)*exp(x)^2+24*exp(x)*x+24*x^2)/x^3/exp(exp(x) /x),x, algorithm=\
-1/4*(8*x*(4*log(2) - 3)*log(x) - 16*x*log(x)^2 - (16*log(2)^2 - 24*log(2) + 9)*x - (16*log(2)^2 - 8*(4*log(2) - 3)*log(x) + 16*log(x)^2 - 24*log(2) + 9)*e^x)*e^(-e^x/x)/x
\[ \int \frac {e^{-\frac {e^x}{x}} \left (e^{2 x} (9-9 x)+24 e^x x+24 x^2+\left (-16 e^x x-16 x^2+e^{2 x} (-12+12 x)\right ) \log \left (\frac {4}{x^2}\right )+e^{2 x} (4-4 x) \log ^2\left (\frac {4}{x^2}\right )\right )}{4 x^3} \, dx=\int { -\frac {{\left (4 \, {\left (x - 1\right )} e^{\left (2 \, x\right )} \log \left (\frac {4}{x^{2}}\right )^{2} - 24 \, x^{2} + 9 \, {\left (x - 1\right )} e^{\left (2 \, x\right )} - 24 \, x e^{x} + 4 \, {\left (4 \, x^{2} - 3 \, {\left (x - 1\right )} e^{\left (2 \, x\right )} + 4 \, x e^{x}\right )} \log \left (\frac {4}{x^{2}}\right )\right )} e^{\left (-\frac {e^{x}}{x}\right )}}{4 \, x^{3}} \,d x } \]
integrate(1/4*((4-4*x)*exp(x)^2*log(4/x^2)^2+((12*x-12)*exp(x)^2-16*exp(x) *x-16*x^2)*log(4/x^2)+(-9*x+9)*exp(x)^2+24*exp(x)*x+24*x^2)/x^3/exp(exp(x) /x),x, algorithm=\
integrate(-1/4*(4*(x - 1)*e^(2*x)*log(4/x^2)^2 - 24*x^2 + 9*(x - 1)*e^(2*x ) - 24*x*e^x + 4*(4*x^2 - 3*(x - 1)*e^(2*x) + 4*x*e^x)*log(4/x^2))*e^(-e^x /x)/x^3, x)
Timed out. \[ \int \frac {e^{-\frac {e^x}{x}} \left (e^{2 x} (9-9 x)+24 e^x x+24 x^2+\left (-16 e^x x-16 x^2+e^{2 x} (-12+12 x)\right ) \log \left (\frac {4}{x^2}\right )+e^{2 x} (4-4 x) \log ^2\left (\frac {4}{x^2}\right )\right )}{4 x^3} \, dx=\int -\frac {{\mathrm {e}}^{-\frac {{\mathrm {e}}^x}{x}}\,\left (\frac {\ln \left (\frac {4}{x^2}\right )\,\left (16\,x\,{\mathrm {e}}^x-{\mathrm {e}}^{2\,x}\,\left (12\,x-12\right )+16\,x^2\right )}{4}-6\,x\,{\mathrm {e}}^x+\frac {{\mathrm {e}}^{2\,x}\,\left (9\,x-9\right )}{4}-6\,x^2+\frac {{\mathrm {e}}^{2\,x}\,{\ln \left (\frac {4}{x^2}\right )}^2\,\left (4\,x-4\right )}{4}\right )}{x^3} \,d x \]
int(-(exp(-exp(x)/x)*((log(4/x^2)*(16*x*exp(x) - exp(2*x)*(12*x - 12) + 16 *x^2))/4 - 6*x*exp(x) + (exp(2*x)*(9*x - 9))/4 - 6*x^2 + (exp(2*x)*log(4/x ^2)^2*(4*x - 4))/4))/x^3,x)