Integrand size = 67, antiderivative size = 22 \[ \int \frac {144+e^{e^{2 x}} \left (-4+\left (4-8 e^{2 x} x\right ) \log (x)\right )}{1296-288 x+16 x^2+e^{e^{2 x}} (72-8 x) \log (x)+e^{2 e^{2 x}} \log ^2(x)} \, dx=\frac {x}{9-x+\frac {1}{4} e^{e^{2 x}} \log (x)} \] Output:
x/(9-x+1/4*ln(x)*exp(exp(x)^2))
Time = 0.19 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {144+e^{e^{2 x}} \left (-4+\left (4-8 e^{2 x} x\right ) \log (x)\right )}{1296-288 x+16 x^2+e^{e^{2 x}} (72-8 x) \log (x)+e^{2 e^{2 x}} \log ^2(x)} \, dx=\frac {4 x}{36-4 x+e^{e^{2 x}} \log (x)} \] Input:
Integrate[(144 + E^E^(2*x)*(-4 + (4 - 8*E^(2*x)*x)*Log[x]))/(1296 - 288*x + 16*x^2 + E^E^(2*x)*(72 - 8*x)*Log[x] + E^(2*E^(2*x))*Log[x]^2),x]
Output:
(4*x)/(36 - 4*x + E^E^(2*x)*Log[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.
\(\displaystyle \int \frac {e^{e^{2 x}} \left (\left (4-8 e^{2 x} x\right ) \log (x)-4\right )+144}{16 x^2-288 x+e^{2 e^{2 x}} \log ^2(x)+e^{e^{2 x}} (72-8 x) \log (x)+1296} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{e^{2 x}} \left (\left (4-8 e^{2 x} x\right ) \log (x)-4\right )+144}{\left (-4 x+e^{e^{2 x}} \log (x)+36\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {4 \left (-e^{e^{2 x}}+e^{e^{2 x}} \log (x)+36\right )}{\left (4 x-e^{e^{2 x}} \log (x)-36\right )^2}-\frac {8 e^{2 x+e^{2 x}} x \log (x)}{\left (-4 x+e^{e^{2 x}} \log (x)+36\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 16 \int \frac {x}{\left (4 x-e^{e^{2 x}} \log (x)-36\right )^2}dx-16 \int \frac {x}{\log (x) \left (4 x-e^{e^{2 x}} \log (x)-36\right )^2}dx-4 \int \frac {1}{4 x-e^{e^{2 x}} \log (x)-36}dx+144 \int \frac {1}{\log (x) \left (-4 x+e^{e^{2 x}} \log (x)+36\right )^2}dx-8 \int \frac {e^{2 x+e^{2 x}} x \log (x)}{\left (-4 x+e^{e^{2 x}} \log (x)+36\right )^2}dx-4 \int \frac {1}{\log (x) \left (-4 x+e^{e^{2 x}} \log (x)+36\right )}dx\) |
Input:
Int[(144 + E^E^(2*x)*(-4 + (4 - 8*E^(2*x)*x)*Log[x]))/(1296 - 288*x + 16*x ^2 + E^E^(2*x)*(72 - 8*x)*Log[x] + E^(2*E^(2*x))*Log[x]^2),x]
Output:
$Aborted
Time = 0.74 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91
method | result | size |
risch | \(-\frac {4 x}{-{\mathrm e}^{{\mathrm e}^{2 x}} \ln \left (x \right )+4 x -36}\) | \(20\) |
parallelrisch | \(-\frac {4 x}{-{\mathrm e}^{{\mathrm e}^{2 x}} \ln \left (x \right )+4 x -36}\) | \(20\) |
Input:
int((((-8*x*exp(x)^2+4)*ln(x)-4)*exp(exp(x)^2)+144)/(ln(x)^2*exp(exp(x)^2) ^2+(-8*x+72)*ln(x)*exp(exp(x)^2)+16*x^2-288*x+1296),x,method=_RETURNVERBOS E)
Output:
-4*x/(-exp(exp(2*x))*ln(x)+4*x-36)
Time = 0.08 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {144+e^{e^{2 x}} \left (-4+\left (4-8 e^{2 x} x\right ) \log (x)\right )}{1296-288 x+16 x^2+e^{e^{2 x}} (72-8 x) \log (x)+e^{2 e^{2 x}} \log ^2(x)} \, dx=\frac {4 \, x}{e^{\left (e^{\left (2 \, x\right )}\right )} \log \left (x\right ) - 4 \, x + 36} \] Input:
integrate((((-8*x*exp(x)^2+4)*log(x)-4)*exp(exp(x)^2)+144)/(log(x)^2*exp(e xp(x)^2)^2+(-8*x+72)*log(x)*exp(exp(x)^2)+16*x^2-288*x+1296),x, algorithm= "fricas")
Output:
4*x/(e^(e^(2*x))*log(x) - 4*x + 36)
Time = 0.12 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {144+e^{e^{2 x}} \left (-4+\left (4-8 e^{2 x} x\right ) \log (x)\right )}{1296-288 x+16 x^2+e^{e^{2 x}} (72-8 x) \log (x)+e^{2 e^{2 x}} \log ^2(x)} \, dx=\frac {4 x}{- 4 x + e^{e^{2 x}} \log {\left (x \right )} + 36} \] Input:
integrate((((-8*x*exp(x)**2+4)*ln(x)-4)*exp(exp(x)**2)+144)/(ln(x)**2*exp( exp(x)**2)**2+(-8*x+72)*ln(x)*exp(exp(x)**2)+16*x**2-288*x+1296),x)
Output:
4*x/(-4*x + exp(exp(2*x))*log(x) + 36)
Time = 0.09 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {144+e^{e^{2 x}} \left (-4+\left (4-8 e^{2 x} x\right ) \log (x)\right )}{1296-288 x+16 x^2+e^{e^{2 x}} (72-8 x) \log (x)+e^{2 e^{2 x}} \log ^2(x)} \, dx=\frac {4 \, x}{e^{\left (e^{\left (2 \, x\right )}\right )} \log \left (x\right ) - 4 \, x + 36} \] Input:
integrate((((-8*x*exp(x)^2+4)*log(x)-4)*exp(exp(x)^2)+144)/(log(x)^2*exp(e xp(x)^2)^2+(-8*x+72)*log(x)*exp(exp(x)^2)+16*x^2-288*x+1296),x, algorithm= "maxima")
Output:
4*x/(e^(e^(2*x))*log(x) - 4*x + 36)
Time = 0.14 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {144+e^{e^{2 x}} \left (-4+\left (4-8 e^{2 x} x\right ) \log (x)\right )}{1296-288 x+16 x^2+e^{e^{2 x}} (72-8 x) \log (x)+e^{2 e^{2 x}} \log ^2(x)} \, dx=\frac {4 \, x}{e^{\left (e^{\left (2 \, x\right )}\right )} \log \left (x\right ) - 4 \, x + 36} \] Input:
integrate((((-8*x*exp(x)^2+4)*log(x)-4)*exp(exp(x)^2)+144)/(log(x)^2*exp(e xp(x)^2)^2+(-8*x+72)*log(x)*exp(exp(x)^2)+16*x^2-288*x+1296),x, algorithm= "giac")
Output:
4*x/(e^(e^(2*x))*log(x) - 4*x + 36)
Timed out. \[ \int \frac {144+e^{e^{2 x}} \left (-4+\left (4-8 e^{2 x} x\right ) \log (x)\right )}{1296-288 x+16 x^2+e^{e^{2 x}} (72-8 x) \log (x)+e^{2 e^{2 x}} \log ^2(x)} \, dx=\int -\frac {{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}\,\left (\ln \left (x\right )\,\left (8\,x\,{\mathrm {e}}^{2\,x}-4\right )+4\right )-144}{{\mathrm {e}}^{2\,{\mathrm {e}}^{2\,x}}\,{\ln \left (x\right )}^2-288\,x+16\,x^2-{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}\,\ln \left (x\right )\,\left (8\,x-72\right )+1296} \,d x \] Input:
int(-(exp(exp(2*x))*(log(x)*(8*x*exp(2*x) - 4) + 4) - 144)/(exp(2*exp(2*x) )*log(x)^2 - 288*x + 16*x^2 - exp(exp(2*x))*log(x)*(8*x - 72) + 1296),x)
Output:
int(-(exp(exp(2*x))*(log(x)*(8*x*exp(2*x) - 4) + 4) - 144)/(exp(2*exp(2*x) )*log(x)^2 - 288*x + 16*x^2 - exp(exp(2*x))*log(x)*(8*x - 72) + 1296), x)
Time = 0.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {144+e^{e^{2 x}} \left (-4+\left (4-8 e^{2 x} x\right ) \log (x)\right )}{1296-288 x+16 x^2+e^{e^{2 x}} (72-8 x) \log (x)+e^{2 e^{2 x}} \log ^2(x)} \, dx=\frac {4 x}{e^{e^{2 x}} \mathrm {log}\left (x \right )-4 x +36} \] Input:
int((((-8*x*exp(x)^2+4)*log(x)-4)*exp(exp(x)^2)+144)/(log(x)^2*exp(exp(x)^ 2)^2+(-8*x+72)*log(x)*exp(exp(x)^2)+16*x^2-288*x+1296),x)
Output:
(4*x)/(e**(e**(2*x))*log(x) - 4*x + 36)