Integrand size = 180, antiderivative size = 31 \[ \int \frac {-16+8 x-x^2+e^{\frac {-5+x}{-4+x}} \left (32-14 x+2 x^2\right )+\left (32-16 x+2 x^2\right ) \log (16)}{512 x^2-256 x^3+32 x^4+e^{\frac {2 (-5+x)}{-4+x}} \left (2048 x^2-1024 x^3+128 x^4\right )+\left (-2048 x^2+1024 x^3-128 x^4\right ) \log (16)+\left (2048 x^2-1024 x^3+128 x^4\right ) \log ^2(16)+e^{\frac {-5+x}{-4+x}} \left (-2048 x^2+1024 x^3-128 x^4+\left (4096 x^2-2048 x^3+256 x^4\right ) \log (16)\right )} \, dx=\frac {1}{16 x \left (2-4 \left (e^{\frac {x}{x+\frac {x}{-5+x}}}+\log (16)\right )\right )} \] Output:
1/16/x/(2-4*exp(x/(x+x/(-5+x)))-16*ln(2))
Time = 5.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {-16+8 x-x^2+e^{\frac {-5+x}{-4+x}} \left (32-14 x+2 x^2\right )+\left (32-16 x+2 x^2\right ) \log (16)}{512 x^2-256 x^3+32 x^4+e^{\frac {2 (-5+x)}{-4+x}} \left (2048 x^2-1024 x^3+128 x^4\right )+\left (-2048 x^2+1024 x^3-128 x^4\right ) \log (16)+\left (2048 x^2-1024 x^3+128 x^4\right ) \log ^2(16)+e^{\frac {-5+x}{-4+x}} \left (-2048 x^2+1024 x^3-128 x^4+\left (4096 x^2-2048 x^3+256 x^4\right ) \log (16)\right )} \, dx=-\frac {e^{\frac {1}{-4+x}}}{32 x \left (2 e+e^{\frac {1}{-4+x}} (-1+\log (256))\right )} \] Input:
Integrate[(-16 + 8*x - x^2 + E^((-5 + x)/(-4 + x))*(32 - 14*x + 2*x^2) + ( 32 - 16*x + 2*x^2)*Log[16])/(512*x^2 - 256*x^3 + 32*x^4 + E^((2*(-5 + x))/ (-4 + x))*(2048*x^2 - 1024*x^3 + 128*x^4) + (-2048*x^2 + 1024*x^3 - 128*x^ 4)*Log[16] + (2048*x^2 - 1024*x^3 + 128*x^4)*Log[16]^2 + E^((-5 + x)/(-4 + x))*(-2048*x^2 + 1024*x^3 - 128*x^4 + (4096*x^2 - 2048*x^3 + 256*x^4)*Log [16])),x]
Output:
-1/32*E^(-4 + x)^(-1)/(x*(2*E + E^(-4 + x)^(-1)*(-1 + Log[256])))
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.
| \(\displaystyle \int \frac {-x^2+e^{\frac {x-5}{x-4}} \left (2 x^2-14 x+32\right )+\left (2 x^2-16 x+32\right ) \log (16)+8 x-16}{32 x^4-256 x^3+512 x^2+e^{\frac {2 (x-5)}{x-4}} \left (128 x^4-1024 x^3+2048 x^2\right )+\left (128 x^4-1024 x^3+2048 x^2\right ) \log ^2(16)+e^{\frac {x-5}{x-4}} \left (-128 x^4+1024 x^3-2048 x^2+\left (256 x^4-2048 x^3+4096 x^2\right ) \log (16)\right )+\left (-128 x^4+1024 x^3-2048 x^2\right ) \log (16)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 e^{\frac {x+5}{x-4}} \left (x^2-7 x+16\right )+e^{\frac {10}{x-4}} (x-4)^2 (\log (256)-1)}{32 (4-x)^2 x^2 \left (2 e^{\frac {x}{x-4}}+e^{\frac {5}{x-4}} (\log (256)-1)\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{32} \int \frac {2 e^{-\frac {x+5}{4-x}} \left (x^2-7 x+16\right )-e^{-\frac {10}{4-x}} (4-x)^2 (1-\log (256))}{(4-x)^2 x^2 \left (2 e^{-\frac {x}{4-x}}-e^{-\frac {5}{4-x}} (1-\log (256))\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{32} \int \left (\frac {2 e^{\frac {x}{x-4}} \left (x^2-9 x+16\right )}{(4-x)^2 x^2 \left (2 e^{\frac {x}{x-4}}-e^{\frac {5}{x-4}} (1-\log (256))\right ) (1-\log (256))}+\frac {1}{x^2 (-1+\log (256))}+\frac {4 e^{\frac {2 x}{x-4}}}{(4-x)^2 x \left (2 e^{\frac {x}{x-4}}-e^{\frac {5}{x-4}} (1-\log (256))\right )^2 (1-\log (256))}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{32} \left (\frac {2 \int \frac {e^{\frac {x}{x-4}}}{x^2 \left (2 e^{\frac {x}{x-4}}-e^{\frac {5}{x-4}} (1-\log (256))\right )}dx}{1-\log (256)}+\frac {\int \frac {e^{\frac {2 x}{x-4}}}{(4-x)^2 \left (2 e^{\frac {x}{x-4}}-e^{\frac {5}{x-4}} (1-\log (256))\right )^2}dx}{1-\log (256)}+\frac {\int \frac {e^{\frac {2 x}{x-4}}}{(4-x) \left (2 e^{\frac {x}{x-4}}-e^{\frac {5}{x-4}} (1-\log (256))\right )^2}dx}{4 (1-\log (256))}+\frac {\int \frac {e^{\frac {2 x}{x-4}}}{x \left (2 e^{\frac {x}{x-4}}-e^{\frac {5}{x-4}} (1-\log (256))\right )^2}dx}{4 (1-\log (256))}+\frac {\int \frac {e^{\frac {x}{x-4}}}{(x-4) \left (2 e^{\frac {x}{x-4}}-e^{\frac {5}{x-4}} (1-\log (256))\right )}dx}{8 (1-\log (256))}+\frac {\int \frac {e^{\frac {x}{x-4}}}{(4-x)^2 \left (-2 e^{\frac {x}{x-4}}+e^{\frac {5}{x-4}} (1-\log (256))\right )}dx}{2 (1-\log (256))}+\frac {\int \frac {e^{\frac {x}{x-4}}}{x \left (-2 e^{\frac {x}{x-4}}+e^{\frac {5}{x-4}} (1-\log (256))\right )}dx}{8 (1-\log (256))}+\frac {1}{x (1-\log (256))}\right )\) |
Input:
Int[(-16 + 8*x - x^2 + E^((-5 + x)/(-4 + x))*(32 - 14*x + 2*x^2) + (32 - 1 6*x + 2*x^2)*Log[16])/(512*x^2 - 256*x^3 + 32*x^4 + E^((2*(-5 + x))/(-4 + x))*(2048*x^2 - 1024*x^3 + 128*x^4) + (-2048*x^2 + 1024*x^3 - 128*x^4)*Log [16] + (2048*x^2 - 1024*x^3 + 128*x^4)*Log[16]^2 + E^((-5 + x)/(-4 + x))*( -2048*x^2 + 1024*x^3 - 128*x^4 + (4096*x^2 - 2048*x^3 + 256*x^4)*Log[16])) ,x]
Output:
$Aborted
Time = 1.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84
| method | result | size |
| risch | \(-\frac {1}{32 x \left (-1+2 \,{\mathrm e}^{\frac {-5+x}{x -4}}+8 \ln \left (2\right )\right )}\) | \(26\) |
| parallelrisch | \(-\frac {1}{32 x \left (-1+2 \,{\mathrm e}^{\frac {-5+x}{x -4}}+8 \ln \left (2\right )\right )}\) | \(26\) |
| norman | \(\frac {\frac {1}{8}-\frac {x}{32}}{x \left (x -4\right ) \left (-1+2 \,{\mathrm e}^{\frac {-5+x}{x -4}}+8 \ln \left (2\right )\right )}\) | \(35\) |
Input:
int(((2*x^2-14*x+32)*exp((-5+x)/(x-4))+4*(2*x^2-16*x+32)*ln(2)-x^2+8*x-16) /((128*x^4-1024*x^3+2048*x^2)*exp((-5+x)/(x-4))^2+(4*(256*x^4-2048*x^3+409 6*x^2)*ln(2)-128*x^4+1024*x^3-2048*x^2)*exp((-5+x)/(x-4))+16*(128*x^4-1024 *x^3+2048*x^2)*ln(2)^2+4*(-128*x^4+1024*x^3-2048*x^2)*ln(2)+32*x^4-256*x^3 +512*x^2),x,method=_RETURNVERBOSE)
Output:
-1/32/x/(-1+2*exp((-5+x)/(x-4))+8*ln(2))
Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {-16+8 x-x^2+e^{\frac {-5+x}{-4+x}} \left (32-14 x+2 x^2\right )+\left (32-16 x+2 x^2\right ) \log (16)}{512 x^2-256 x^3+32 x^4+e^{\frac {2 (-5+x)}{-4+x}} \left (2048 x^2-1024 x^3+128 x^4\right )+\left (-2048 x^2+1024 x^3-128 x^4\right ) \log (16)+\left (2048 x^2-1024 x^3+128 x^4\right ) \log ^2(16)+e^{\frac {-5+x}{-4+x}} \left (-2048 x^2+1024 x^3-128 x^4+\left (4096 x^2-2048 x^3+256 x^4\right ) \log (16)\right )} \, dx=-\frac {1}{32 \, {\left (2 \, x e^{\left (\frac {x - 5}{x - 4}\right )} + 8 \, x \log \left (2\right ) - x\right )}} \] Input:
integrate(((2*x^2-14*x+32)*exp((-5+x)/(-4+x))+4*(2*x^2-16*x+32)*log(2)-x^2 +8*x-16)/((128*x^4-1024*x^3+2048*x^2)*exp((-5+x)/(-4+x))^2+(4*(256*x^4-204 8*x^3+4096*x^2)*log(2)-128*x^4+1024*x^3-2048*x^2)*exp((-5+x)/(-4+x))+16*(1 28*x^4-1024*x^3+2048*x^2)*log(2)^2+4*(-128*x^4+1024*x^3-2048*x^2)*log(2)+3 2*x^4-256*x^3+512*x^2),x, algorithm="fricas")
Output:
-1/32/(2*x*e^((x - 5)/(x - 4)) + 8*x*log(2) - x)
Time = 0.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \frac {-16+8 x-x^2+e^{\frac {-5+x}{-4+x}} \left (32-14 x+2 x^2\right )+\left (32-16 x+2 x^2\right ) \log (16)}{512 x^2-256 x^3+32 x^4+e^{\frac {2 (-5+x)}{-4+x}} \left (2048 x^2-1024 x^3+128 x^4\right )+\left (-2048 x^2+1024 x^3-128 x^4\right ) \log (16)+\left (2048 x^2-1024 x^3+128 x^4\right ) \log ^2(16)+e^{\frac {-5+x}{-4+x}} \left (-2048 x^2+1024 x^3-128 x^4+\left (4096 x^2-2048 x^3+256 x^4\right ) \log (16)\right )} \, dx=- \frac {1}{64 x e^{\frac {x - 5}{x - 4}} - 32 x + 256 x \log {\left (2 \right )}} \] Input:
integrate(((2*x**2-14*x+32)*exp((-5+x)/(-4+x))+4*(2*x**2-16*x+32)*ln(2)-x* *2+8*x-16)/((128*x**4-1024*x**3+2048*x**2)*exp((-5+x)/(-4+x))**2+(4*(256*x **4-2048*x**3+4096*x**2)*ln(2)-128*x**4+1024*x**3-2048*x**2)*exp((-5+x)/(- 4+x))+16*(128*x**4-1024*x**3+2048*x**2)*ln(2)**2+4*(-128*x**4+1024*x**3-20 48*x**2)*ln(2)+32*x**4-256*x**3+512*x**2),x)
Output:
-1/(64*x*exp((x - 5)/(x - 4)) - 32*x + 256*x*log(2))
Time = 0.18 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \frac {-16+8 x-x^2+e^{\frac {-5+x}{-4+x}} \left (32-14 x+2 x^2\right )+\left (32-16 x+2 x^2\right ) \log (16)}{512 x^2-256 x^3+32 x^4+e^{\frac {2 (-5+x)}{-4+x}} \left (2048 x^2-1024 x^3+128 x^4\right )+\left (-2048 x^2+1024 x^3-128 x^4\right ) \log (16)+\left (2048 x^2-1024 x^3+128 x^4\right ) \log ^2(16)+e^{\frac {-5+x}{-4+x}} \left (-2048 x^2+1024 x^3-128 x^4+\left (4096 x^2-2048 x^3+256 x^4\right ) \log (16)\right )} \, dx=-\frac {e^{\left (\frac {1}{x - 4}\right )}}{32 \, {\left (x {\left (8 \, \log \left (2\right ) - 1\right )} e^{\left (\frac {1}{x - 4}\right )} + 2 \, x e\right )}} \] Input:
integrate(((2*x^2-14*x+32)*exp((-5+x)/(-4+x))+4*(2*x^2-16*x+32)*log(2)-x^2 +8*x-16)/((128*x^4-1024*x^3+2048*x^2)*exp((-5+x)/(-4+x))^2+(4*(256*x^4-204 8*x^3+4096*x^2)*log(2)-128*x^4+1024*x^3-2048*x^2)*exp((-5+x)/(-4+x))+16*(1 28*x^4-1024*x^3+2048*x^2)*log(2)^2+4*(-128*x^4+1024*x^3-2048*x^2)*log(2)+3 2*x^4-256*x^3+512*x^2),x, algorithm="maxima")
Output:
-1/32*e^(1/(x - 4))/(x*(8*log(2) - 1)*e^(1/(x - 4)) + 2*x*e)
Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {-16+8 x-x^2+e^{\frac {-5+x}{-4+x}} \left (32-14 x+2 x^2\right )+\left (32-16 x+2 x^2\right ) \log (16)}{512 x^2-256 x^3+32 x^4+e^{\frac {2 (-5+x)}{-4+x}} \left (2048 x^2-1024 x^3+128 x^4\right )+\left (-2048 x^2+1024 x^3-128 x^4\right ) \log (16)+\left (2048 x^2-1024 x^3+128 x^4\right ) \log ^2(16)+e^{\frac {-5+x}{-4+x}} \left (-2048 x^2+1024 x^3-128 x^4+\left (4096 x^2-2048 x^3+256 x^4\right ) \log (16)\right )} \, dx=-\frac {1}{32 \, {\left (2 \, x e^{\left (-\frac {x}{4 \, {\left (x - 4\right )}} + \frac {5}{4}\right )} + 8 \, x \log \left (2\right ) - x\right )}} \] Input:
integrate(((2*x^2-14*x+32)*exp((-5+x)/(-4+x))+4*(2*x^2-16*x+32)*log(2)-x^2 +8*x-16)/((128*x^4-1024*x^3+2048*x^2)*exp((-5+x)/(-4+x))^2+(4*(256*x^4-204 8*x^3+4096*x^2)*log(2)-128*x^4+1024*x^3-2048*x^2)*exp((-5+x)/(-4+x))+16*(1 28*x^4-1024*x^3+2048*x^2)*log(2)^2+4*(-128*x^4+1024*x^3-2048*x^2)*log(2)+3 2*x^4-256*x^3+512*x^2),x, algorithm="giac")
Output:
-1/32/(2*x*e^(-1/4*x/(x - 4) + 5/4) + 8*x*log(2) - x)
Timed out. \[ \int \frac {-16+8 x-x^2+e^{\frac {-5+x}{-4+x}} \left (32-14 x+2 x^2\right )+\left (32-16 x+2 x^2\right ) \log (16)}{512 x^2-256 x^3+32 x^4+e^{\frac {2 (-5+x)}{-4+x}} \left (2048 x^2-1024 x^3+128 x^4\right )+\left (-2048 x^2+1024 x^3-128 x^4\right ) \log (16)+\left (2048 x^2-1024 x^3+128 x^4\right ) \log ^2(16)+e^{\frac {-5+x}{-4+x}} \left (-2048 x^2+1024 x^3-128 x^4+\left (4096 x^2-2048 x^3+256 x^4\right ) \log (16)\right )} \, dx=\int \frac {8\,x+4\,\ln \left (2\right )\,\left (2\,x^2-16\,x+32\right )+{\mathrm {e}}^{\frac {x-5}{x-4}}\,\left (2\,x^2-14\,x+32\right )-x^2-16}{{\mathrm {e}}^{\frac {x-5}{x-4}}\,\left (4\,\ln \left (2\right )\,\left (256\,x^4-2048\,x^3+4096\,x^2\right )-2048\,x^2+1024\,x^3-128\,x^4\right )+16\,{\ln \left (2\right )}^2\,\left (128\,x^4-1024\,x^3+2048\,x^2\right )+{\mathrm {e}}^{\frac {2\,\left (x-5\right )}{x-4}}\,\left (128\,x^4-1024\,x^3+2048\,x^2\right )-4\,\ln \left (2\right )\,\left (128\,x^4-1024\,x^3+2048\,x^2\right )+512\,x^2-256\,x^3+32\,x^4} \,d x \] Input:
int((8*x + 4*log(2)*(2*x^2 - 16*x + 32) + exp((x - 5)/(x - 4))*(2*x^2 - 14 *x + 32) - x^2 - 16)/(exp((x - 5)/(x - 4))*(4*log(2)*(4096*x^2 - 2048*x^3 + 256*x^4) - 2048*x^2 + 1024*x^3 - 128*x^4) + 16*log(2)^2*(2048*x^2 - 1024 *x^3 + 128*x^4) + exp((2*(x - 5))/(x - 4))*(2048*x^2 - 1024*x^3 + 128*x^4) - 4*log(2)*(2048*x^2 - 1024*x^3 + 128*x^4) + 512*x^2 - 256*x^3 + 32*x^4), x)
Output:
int((8*x + 4*log(2)*(2*x^2 - 16*x + 32) + exp((x - 5)/(x - 4))*(2*x^2 - 14 *x + 32) - x^2 - 16)/(exp((x - 5)/(x - 4))*(4*log(2)*(4096*x^2 - 2048*x^3 + 256*x^4) - 2048*x^2 + 1024*x^3 - 128*x^4) + 16*log(2)^2*(2048*x^2 - 1024 *x^3 + 128*x^4) + exp((2*(x - 5))/(x - 4))*(2048*x^2 - 1024*x^3 + 128*x^4) - 4*log(2)*(2048*x^2 - 1024*x^3 + 128*x^4) + 512*x^2 - 256*x^3 + 32*x^4), x)
Time = 0.16 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.23 \[ \int \frac {-16+8 x-x^2+e^{\frac {-5+x}{-4+x}} \left (32-14 x+2 x^2\right )+\left (32-16 x+2 x^2\right ) \log (16)}{512 x^2-256 x^3+32 x^4+e^{\frac {2 (-5+x)}{-4+x}} \left (2048 x^2-1024 x^3+128 x^4\right )+\left (-2048 x^2+1024 x^3-128 x^4\right ) \log (16)+\left (2048 x^2-1024 x^3+128 x^4\right ) \log ^2(16)+e^{\frac {-5+x}{-4+x}} \left (-2048 x^2+1024 x^3-128 x^4+\left (4096 x^2-2048 x^3+256 x^4\right ) \log (16)\right )} \, dx=-\frac {e^{\frac {1}{x -4}}}{32 x \left (8 e^{\frac {1}{x -4}} \mathrm {log}\left (2\right )-e^{\frac {1}{x -4}}+2 e \right )} \] Input:
int(((2*x^2-14*x+32)*exp((-5+x)/(-4+x))+4*(2*x^2-16*x+32)*log(2)-x^2+8*x-1 6)/((128*x^4-1024*x^3+2048*x^2)*exp((-5+x)/(-4+x))^2+(4*(256*x^4-2048*x^3+ 4096*x^2)*log(2)-128*x^4+1024*x^3-2048*x^2)*exp((-5+x)/(-4+x))+16*(128*x^4 -1024*x^3+2048*x^2)*log(2)^2+4*(-128*x^4+1024*x^3-2048*x^2)*log(2)+32*x^4- 256*x^3+512*x^2),x)
Output:
( - e**(1/(x - 4)))/(32*x*(8*e**(1/(x - 4))*log(2) - e**(1/(x - 4)) + 2*e) )