Integrand size = 99, antiderivative size = 25 \[ \int \frac {4+8 e^{2 x}+e^x (12+4 x)-750 \log ^2(5)+\left (-2250 \log ^2(5)-3000 e^x \log ^2(5)\right ) \log (x)+281250 \log ^4(5) \log ^2(x)}{4+8 e^x+4 e^{2 x}+\left (-1500 \log ^2(5)-1500 e^x \log ^2(5)\right ) \log (x)+140625 \log ^4(5) \log ^2(x)} \, dx=2 x+\frac {x}{-1-e^x+\frac {375}{2} \log ^2(5) \log (x)} \] Output:
x/(375/2*ln(x)*ln(5)^2-1-exp(x))+2*x
Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {4+8 e^{2 x}+e^x (12+4 x)-750 \log ^2(5)+\left (-2250 \log ^2(5)-3000 e^x \log ^2(5)\right ) \log (x)+281250 \log ^4(5) \log ^2(x)}{4+8 e^x+4 e^{2 x}+\left (-1500 \log ^2(5)-1500 e^x \log ^2(5)\right ) \log (x)+140625 \log ^4(5) \log ^2(x)} \, dx=2 x+\frac {2 x}{-2-2 e^x+375 \log ^2(5) \log (x)} \] Input:
Integrate[(4 + 8*E^(2*x) + E^x*(12 + 4*x) - 750*Log[5]^2 + (-2250*Log[5]^2 - 3000*E^x*Log[5]^2)*Log[x] + 281250*Log[5]^4*Log[x]^2)/(4 + 8*E^x + 4*E^ (2*x) + (-1500*Log[5]^2 - 1500*E^x*Log[5]^2)*Log[x] + 140625*Log[5]^4*Log[ x]^2),x]
Output:
2*x + (2*x)/(-2 - 2*E^x + 375*Log[5]^2*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.
| \(\displaystyle \int \frac {8 e^{2 x}+e^x (4 x+12)+\left (-3000 e^x \log ^2(5)-2250 \log ^2(5)\right ) \log (x)+281250 \log ^4(5) \log ^2(x)+4-750 \log ^2(5)}{8 e^x+4 e^{2 x}+\left (-1500 e^x \log ^2(5)-1500 \log ^2(5)\right ) \log (x)+140625 \log ^4(5) \log ^2(x)+4} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {8 e^{2 x}+e^x (4 x+12)+\left (-3000 e^x \log ^2(5)-2250 \log ^2(5)\right ) \log (x)+281250 \log ^4(5) \log ^2(x)+4 \left (1-\frac {375 \log ^2(5)}{2}\right )}{\left (2 e^x-375 \log ^2(5) \log (x)+2\right )^2}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {2 \left (6 e^x+4 e^{2 x}+2 e^x x-1500 e^x \log ^2(5) \log (x)-1125 \log ^2(5) \log (x)+140625 \log ^4(5) \log ^2(x)+2 \left (1-\frac {375 \log ^2(5)}{2}\right )\right )}{\left (2 e^x-375 \log ^2(5) \log (x)+2\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int \frac {140625 \log ^4(5) \log ^2(x)-1500 e^x \log ^2(5) \log (x)-1125 \log ^2(5) \log (x)+6 e^x+4 e^{2 x}+2 e^x x-375 \log ^2(5)+2}{\left (-375 \log ^2(5) \log (x)+2 e^x+2\right )^2}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle 2 \int \frac {140625 \log ^4(5) \log ^2(x)-1500 e^x \log ^2(5) \log (x)-1125 \log ^2(5) \log (x)+6 e^x+4 e^{2 x}+2 e^x x+2 \left (1-\frac {375 \log ^2(5)}{2}\right )}{\left (-375 \log ^2(5) \log (x)+2 e^x+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {x-1}{-375 \log ^2(5) \log (x)+2 e^x+2}+\frac {375 \log ^2(5) \log (x) x-2 x-375 \log ^2(5)}{\left (-375 \log ^2(5) \log (x)+2 e^x+2\right )^2}+1\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (-2 \int \frac {x}{\left (-375 \log ^2(5) \log (x)+2 e^x+2\right )^2}dx-\int \frac {1}{-375 \log ^2(5) \log (x)+2 e^x+2}dx+\int \frac {x}{-375 \log ^2(5) \log (x)+2 e^x+2}dx-375 \log ^2(5) \int \frac {1}{\left (375 \log ^2(5) \log (x)-2 e^x-2\right )^2}dx+375 \log ^2(5) \int \frac {x \log (x)}{\left (375 \log ^2(5) \log (x)-2 e^x-2\right )^2}dx+x\right )\) |
Input:
Int[(4 + 8*E^(2*x) + E^x*(12 + 4*x) - 750*Log[5]^2 + (-2250*Log[5]^2 - 300 0*E^x*Log[5]^2)*Log[x] + 281250*Log[5]^4*Log[x]^2)/(4 + 8*E^x + 4*E^(2*x) + (-1500*Log[5]^2 - 1500*E^x*Log[5]^2)*Log[x] + 140625*Log[5]^4*Log[x]^2), x]
Output:
$Aborted
Time = 0.58 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96
| method | result | size |
| risch | \(2 x +\frac {2 x}{375 \ln \left (x \right ) \ln \left (5\right )^{2}-2 \,{\mathrm e}^{x}-2}\) | \(24\) |
| norman | \(\frac {-2 x -4 \,{\mathrm e}^{x} x +750 \ln \left (x \right ) \ln \left (5\right )^{2} x}{375 \ln \left (x \right ) \ln \left (5\right )^{2}-2 \,{\mathrm e}^{x}-2}\) | \(36\) |
| parallelrisch | \(\frac {1500 \ln \left (x \right ) \ln \left (5\right )^{2} x -8 \,{\mathrm e}^{x} x -4 x}{750 \ln \left (x \right ) \ln \left (5\right )^{2}-4 \,{\mathrm e}^{x}-4}\) | \(37\) |
Input:
int((281250*ln(5)^4*ln(x)^2+(-3000*ln(5)^2*exp(x)-2250*ln(5)^2)*ln(x)+8*ex p(x)^2+(4*x+12)*exp(x)-750*ln(5)^2+4)/(140625*ln(5)^4*ln(x)^2+(-1500*ln(5) ^2*exp(x)-1500*ln(5)^2)*ln(x)+4*exp(x)^2+8*exp(x)+4),x,method=_RETURNVERBO SE)
Output:
2*x+2*x/(375*ln(x)*ln(5)^2-2*exp(x)-2)
Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {4+8 e^{2 x}+e^x (12+4 x)-750 \log ^2(5)+\left (-2250 \log ^2(5)-3000 e^x \log ^2(5)\right ) \log (x)+281250 \log ^4(5) \log ^2(x)}{4+8 e^x+4 e^{2 x}+\left (-1500 \log ^2(5)-1500 e^x \log ^2(5)\right ) \log (x)+140625 \log ^4(5) \log ^2(x)} \, dx=\frac {2 \, {\left (375 \, x \log \left (5\right )^{2} \log \left (x\right ) - 2 \, x e^{x} - x\right )}}{375 \, \log \left (5\right )^{2} \log \left (x\right ) - 2 \, e^{x} - 2} \] Input:
integrate((281250*log(5)^4*log(x)^2+(-3000*log(5)^2*exp(x)-2250*log(5)^2)* log(x)+8*exp(x)^2+(4*x+12)*exp(x)-750*log(5)^2+4)/(140625*log(5)^4*log(x)^ 2+(-1500*log(5)^2*exp(x)-1500*log(5)^2)*log(x)+4*exp(x)^2+8*exp(x)+4),x, a lgorithm="fricas")
Output:
2*(375*x*log(5)^2*log(x) - 2*x*e^x - x)/(375*log(5)^2*log(x) - 2*e^x - 2)
Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {4+8 e^{2 x}+e^x (12+4 x)-750 \log ^2(5)+\left (-2250 \log ^2(5)-3000 e^x \log ^2(5)\right ) \log (x)+281250 \log ^4(5) \log ^2(x)}{4+8 e^x+4 e^{2 x}+\left (-1500 \log ^2(5)-1500 e^x \log ^2(5)\right ) \log (x)+140625 \log ^4(5) \log ^2(x)} \, dx=2 x - \frac {x}{e^{x} - \frac {375 \log {\left (5 \right )}^{2} \log {\left (x \right )}}{2} + 1} \] Input:
integrate((281250*ln(5)**4*ln(x)**2+(-3000*ln(5)**2*exp(x)-2250*ln(5)**2)* ln(x)+8*exp(x)**2+(4*x+12)*exp(x)-750*ln(5)**2+4)/(140625*ln(5)**4*ln(x)** 2+(-1500*ln(5)**2*exp(x)-1500*ln(5)**2)*ln(x)+4*exp(x)**2+8*exp(x)+4),x)
Output:
2*x - x/(exp(x) - 375*log(5)**2*log(x)/2 + 1)
Time = 0.15 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {4+8 e^{2 x}+e^x (12+4 x)-750 \log ^2(5)+\left (-2250 \log ^2(5)-3000 e^x \log ^2(5)\right ) \log (x)+281250 \log ^4(5) \log ^2(x)}{4+8 e^x+4 e^{2 x}+\left (-1500 \log ^2(5)-1500 e^x \log ^2(5)\right ) \log (x)+140625 \log ^4(5) \log ^2(x)} \, dx=\frac {2 \, {\left (375 \, x \log \left (5\right )^{2} \log \left (x\right ) - 2 \, x e^{x} - x\right )}}{375 \, \log \left (5\right )^{2} \log \left (x\right ) - 2 \, e^{x} - 2} \] Input:
integrate((281250*log(5)^4*log(x)^2+(-3000*log(5)^2*exp(x)-2250*log(5)^2)* log(x)+8*exp(x)^2+(4*x+12)*exp(x)-750*log(5)^2+4)/(140625*log(5)^4*log(x)^ 2+(-1500*log(5)^2*exp(x)-1500*log(5)^2)*log(x)+4*exp(x)^2+8*exp(x)+4),x, a lgorithm="maxima")
Output:
2*(375*x*log(5)^2*log(x) - 2*x*e^x - x)/(375*log(5)^2*log(x) - 2*e^x - 2)
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (23) = 46\).
Time = 0.13 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.96 \[ \int \frac {4+8 e^{2 x}+e^x (12+4 x)-750 \log ^2(5)+\left (-2250 \log ^2(5)-3000 e^x \log ^2(5)\right ) \log (x)+281250 \log ^4(5) \log ^2(x)}{4+8 e^x+4 e^{2 x}+\left (-1500 \log ^2(5)-1500 e^x \log ^2(5)\right ) \log (x)+140625 \log ^4(5) \log ^2(x)} \, dx=\frac {2 \, {\left (375 \, x \log \left (5\right )^{2} \log \left (x\right ) + 375 \, \log \left (5\right )^{2} \log \left (x\right ) - 2 \, x e^{x} - x - 2 \, e^{x} - 2\right )}}{375 \, \log \left (5\right )^{2} \log \left (x\right ) - 2 \, e^{x} - 2} \] Input:
integrate((281250*log(5)^4*log(x)^2+(-3000*log(5)^2*exp(x)-2250*log(5)^2)* log(x)+8*exp(x)^2+(4*x+12)*exp(x)-750*log(5)^2+4)/(140625*log(5)^4*log(x)^ 2+(-1500*log(5)^2*exp(x)-1500*log(5)^2)*log(x)+4*exp(x)^2+8*exp(x)+4),x, a lgorithm="giac")
Output:
2*(375*x*log(5)^2*log(x) + 375*log(5)^2*log(x) - 2*x*e^x - x - 2*e^x - 2)/ (375*log(5)^2*log(x) - 2*e^x - 2)
Timed out. \[ \int \frac {4+8 e^{2 x}+e^x (12+4 x)-750 \log ^2(5)+\left (-2250 \log ^2(5)-3000 e^x \log ^2(5)\right ) \log (x)+281250 \log ^4(5) \log ^2(x)}{4+8 e^x+4 e^{2 x}+\left (-1500 \log ^2(5)-1500 e^x \log ^2(5)\right ) \log (x)+140625 \log ^4(5) \log ^2(x)} \, dx=\int \frac {281250\,{\ln \left (5\right )}^4\,{\ln \left (x\right )}^2+\left (-3000\,{\mathrm {e}}^x\,{\ln \left (5\right )}^2-2250\,{\ln \left (5\right )}^2\right )\,\ln \left (x\right )+8\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\left (4\,x+12\right )-750\,{\ln \left (5\right )}^2+4}{140625\,{\ln \left (5\right )}^4\,{\ln \left (x\right )}^2+\left (-1500\,{\mathrm {e}}^x\,{\ln \left (5\right )}^2-1500\,{\ln \left (5\right )}^2\right )\,\ln \left (x\right )+4\,{\mathrm {e}}^{2\,x}+8\,{\mathrm {e}}^x+4} \,d x \] Input:
int((8*exp(2*x) + 281250*log(5)^4*log(x)^2 + exp(x)*(4*x + 12) - 750*log(5 )^2 - log(x)*(3000*exp(x)*log(5)^2 + 2250*log(5)^2) + 4)/(4*exp(2*x) + 8*e xp(x) + 140625*log(5)^4*log(x)^2 - log(x)*(1500*exp(x)*log(5)^2 + 1500*log (5)^2) + 4),x)
Output:
int((8*exp(2*x) + 281250*log(5)^4*log(x)^2 + exp(x)*(4*x + 12) - 750*log(5 )^2 - log(x)*(3000*exp(x)*log(5)^2 + 2250*log(5)^2) + 4)/(4*exp(2*x) + 8*e xp(x) + 140625*log(5)^4*log(x)^2 - log(x)*(1500*exp(x)*log(5)^2 + 1500*log (5)^2) + 4), x)
Time = 0.21 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int \frac {4+8 e^{2 x}+e^x (12+4 x)-750 \log ^2(5)+\left (-2250 \log ^2(5)-3000 e^x \log ^2(5)\right ) \log (x)+281250 \log ^4(5) \log ^2(x)}{4+8 e^x+4 e^{2 x}+\left (-1500 \log ^2(5)-1500 e^x \log ^2(5)\right ) \log (x)+140625 \log ^4(5) \log ^2(x)} \, dx=\frac {2 x \left (2 e^{x}-375 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (5\right )^{2}+1\right )}{2 e^{x}-375 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (5\right )^{2}+2} \] Input:
int((281250*log(5)^4*log(x)^2+(-3000*log(5)^2*exp(x)-2250*log(5)^2)*log(x) +8*exp(x)^2+(4*x+12)*exp(x)-750*log(5)^2+4)/(140625*log(5)^4*log(x)^2+(-15 00*log(5)^2*exp(x)-1500*log(5)^2)*log(x)+4*exp(x)^2+8*exp(x)+4),x)
Output:
(2*x*(2*e**x - 375*log(x)*log(5)**2 + 1))/(2*e**x - 375*log(x)*log(5)**2 + 2)