Integrand size = 107, antiderivative size = 26 \[ \int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} \left (4-x+97 x^2+56 x^3+256 x^4+254 x^5+96 x^6+16 x^7+x^8\right )}{\left (4+4 x-63 x^2-64 x^3+236 x^4+254 x^5+96 x^6+16 x^7+x^8\right ) \log (2)} \, dx=\frac {e^{-\frac {5}{-2-x+x^2 (4+x)^2}} x}{\log (2)} \] Output:
x/ln(2)/exp(5/((4+x)^2*x^2-2-x))
Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} \left (4-x+97 x^2+56 x^3+256 x^4+254 x^5+96 x^6+16 x^7+x^8\right )}{\left (4+4 x-63 x^2-64 x^3+236 x^4+254 x^5+96 x^6+16 x^7+x^8\right ) \log (2)} \, dx=\frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} x}{\log (2)} \] Input:
Integrate[(4 - x + 97*x^2 + 56*x^3 + 256*x^4 + 254*x^5 + 96*x^6 + 16*x^7 + x^8)/(E^(5/(-2 - x + 16*x^2 + 8*x^3 + x^4))*(4 + 4*x - 63*x^2 - 64*x^3 + 236*x^4 + 254*x^5 + 96*x^6 + 16*x^7 + x^8)*Log[2]),x]
Output:
x/(E^(5/(-2 - x + 16*x^2 + 8*x^3 + x^4))*Log[2])
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.
\(\displaystyle \int \frac {e^{-\frac {5}{x^4+8 x^3+16 x^2-x-2}} \left (x^8+16 x^7+96 x^6+254 x^5+256 x^4+56 x^3+97 x^2-x+4\right )}{\left (x^8+16 x^7+96 x^6+254 x^5+236 x^4-64 x^3-63 x^2+4 x+4\right ) \log (2)} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {e^{\frac {5}{-x^4-8 x^3-16 x^2+x+2}} \left (x^8+16 x^7+96 x^6+254 x^5+256 x^4+56 x^3+97 x^2-x+4\right )}{x^8+16 x^7+96 x^6+254 x^5+236 x^4-64 x^3-63 x^2+4 x+4}dx}{\log (2)}\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \frac {\int \frac {e^{\frac {5}{-x^4-8 x^3-16 x^2+x+2}} \left (x^8+16 x^7+96 x^6+254 x^5+256 x^4+56 x^3+97 x^2-x+4\right )}{\left (x^4+8 x^3+16 x^2-x-2\right )^2}dx}{\log (2)}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {\int \left (-\frac {5 e^{\frac {5}{-x^4-8 x^3-16 x^2+x+2}} \left (8 x^3+32 x^2-3 x-8\right )}{\left (x^4+8 x^3+16 x^2-x-2\right )^2}+e^{\frac {5}{-x^4-8 x^3-16 x^2+x+2}}+\frac {20 e^{\frac {5}{-x^4-8 x^3-16 x^2+x+2}}}{x^4+8 x^3+16 x^2-x-2}\right )dx}{\log (2)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\int e^{\frac {5}{-x^4-8 x^3-16 x^2+x+2}}dx+40 \int \frac {e^{\frac {5}{-x^4-8 x^3-16 x^2+x+2}}}{\left (x^4+8 x^3+16 x^2-x-2\right )^2}dx+15 \int \frac {e^{\frac {5}{-x^4-8 x^3-16 x^2+x+2}} x}{\left (x^4+8 x^3+16 x^2-x-2\right )^2}dx-160 \int \frac {e^{\frac {5}{-x^4-8 x^3-16 x^2+x+2}} x^2}{\left (x^4+8 x^3+16 x^2-x-2\right )^2}dx-40 \int \frac {e^{\frac {5}{-x^4-8 x^3-16 x^2+x+2}} x^3}{\left (x^4+8 x^3+16 x^2-x-2\right )^2}dx+20 \int \frac {e^{\frac {5}{-x^4-8 x^3-16 x^2+x+2}}}{x^4+8 x^3+16 x^2-x-2}dx}{\log (2)}\) |
Input:
Int[(4 - x + 97*x^2 + 56*x^3 + 256*x^4 + 254*x^5 + 96*x^6 + 16*x^7 + x^8)/ (E^(5/(-2 - x + 16*x^2 + 8*x^3 + x^4))*(4 + 4*x - 63*x^2 - 64*x^3 + 236*x^ 4 + 254*x^5 + 96*x^6 + 16*x^7 + x^8)*Log[2]),x]
Output:
$Aborted
Time = 0.44 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15
method | result | size |
risch | \(\frac {x \,{\mathrm e}^{-\frac {5}{x^{4}+8 x^{3}+16 x^{2}-x -2}}}{\ln \left (2\right )}\) | \(30\) |
gosper | \(\frac {x \,{\mathrm e}^{-\frac {5}{x^{4}+8 x^{3}+16 x^{2}-x -2}}}{\ln \left (2\right )}\) | \(32\) |
parallelrisch | \(\frac {\left (11224 x^{5}+89792 x^{4}+179584 x^{3}-11224 x^{2}-22448 x \right ) {\mathrm e}^{-\frac {5}{x^{4}+8 x^{3}+16 x^{2}-x -2}}}{11224 \ln \left (2\right ) \left (x^{4}+8 x^{3}+16 x^{2}-x -2\right )}\) | \(76\) |
norman | \(\frac {\left (\frac {x^{5}}{\ln \left (2\right )}-\frac {2 x}{\ln \left (2\right )}-\frac {x^{2}}{\ln \left (2\right )}+\frac {16 x^{3}}{\ln \left (2\right )}+\frac {8 x^{4}}{\ln \left (2\right )}\right ) {\mathrm e}^{-\frac {5}{x^{4}+8 x^{3}+16 x^{2}-x -2}}}{x^{4}+8 x^{3}+16 x^{2}-x -2}\) | \(90\) |
orering | \(\frac {\left (x^{4}+8 x^{3}+16 x^{2}-x -2\right )^{2} x \,{\mathrm e}^{-\frac {5}{x^{4}+8 x^{3}+16 x^{2}-x -2}}}{\left (x^{8}+16 x^{7}+96 x^{6}+254 x^{5}+236 x^{4}-64 x^{3}-63 x^{2}+4 x +4\right ) \ln \left (2\right )}\) | \(92\) |
Input:
int((x^8+16*x^7+96*x^6+254*x^5+256*x^4+56*x^3+97*x^2-x+4)/(x^8+16*x^7+96*x ^6+254*x^5+236*x^4-64*x^3-63*x^2+4*x+4)/ln(2)/exp(5/(x^4+8*x^3+16*x^2-x-2) ),x,method=_RETURNVERBOSE)
Output:
x*exp(-5/(x^4+8*x^3+16*x^2-x-2))/ln(2)
Time = 0.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} \left (4-x+97 x^2+56 x^3+256 x^4+254 x^5+96 x^6+16 x^7+x^8\right )}{\left (4+4 x-63 x^2-64 x^3+236 x^4+254 x^5+96 x^6+16 x^7+x^8\right ) \log (2)} \, dx=\frac {x e^{\left (-\frac {5}{x^{4} + 8 \, x^{3} + 16 \, x^{2} - x - 2}\right )}}{\log \left (2\right )} \] Input:
integrate((x^8+16*x^7+96*x^6+254*x^5+256*x^4+56*x^3+97*x^2-x+4)/(x^8+16*x^ 7+96*x^6+254*x^5+236*x^4-64*x^3-63*x^2+4*x+4)/log(2)/exp(5/(x^4+8*x^3+16*x ^2-x-2)),x, algorithm="fricas")
Output:
x*e^(-5/(x^4 + 8*x^3 + 16*x^2 - x - 2))/log(2)
Time = 0.83 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} \left (4-x+97 x^2+56 x^3+256 x^4+254 x^5+96 x^6+16 x^7+x^8\right )}{\left (4+4 x-63 x^2-64 x^3+236 x^4+254 x^5+96 x^6+16 x^7+x^8\right ) \log (2)} \, dx=\frac {x e^{- \frac {5}{x^{4} + 8 x^{3} + 16 x^{2} - x - 2}}}{\log {\left (2 \right )}} \] Input:
integrate((x**8+16*x**7+96*x**6+254*x**5+256*x**4+56*x**3+97*x**2-x+4)/(x* *8+16*x**7+96*x**6+254*x**5+236*x**4-64*x**3-63*x**2+4*x+4)/ln(2)/exp(5/(x **4+8*x**3+16*x**2-x-2)),x)
Output:
x*exp(-5/(x**4 + 8*x**3 + 16*x**2 - x - 2))/log(2)
Time = 0.12 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} \left (4-x+97 x^2+56 x^3+256 x^4+254 x^5+96 x^6+16 x^7+x^8\right )}{\left (4+4 x-63 x^2-64 x^3+236 x^4+254 x^5+96 x^6+16 x^7+x^8\right ) \log (2)} \, dx=\frac {x e^{\left (-\frac {5}{x^{4} + 8 \, x^{3} + 16 \, x^{2} - x - 2}\right )}}{\log \left (2\right )} \] Input:
integrate((x^8+16*x^7+96*x^6+254*x^5+256*x^4+56*x^3+97*x^2-x+4)/(x^8+16*x^ 7+96*x^6+254*x^5+236*x^4-64*x^3-63*x^2+4*x+4)/log(2)/exp(5/(x^4+8*x^3+16*x ^2-x-2)),x, algorithm="maxima")
Output:
x*e^(-5/(x^4 + 8*x^3 + 16*x^2 - x - 2))/log(2)
Time = 0.12 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.85 \[ \int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} \left (4-x+97 x^2+56 x^3+256 x^4+254 x^5+96 x^6+16 x^7+x^8\right )}{\left (4+4 x-63 x^2-64 x^3+236 x^4+254 x^5+96 x^6+16 x^7+x^8\right ) \log (2)} \, dx=\frac {x e^{\left (-\frac {5 \, {\left (x^{4} + 8 \, x^{3} + 16 \, x^{2} - x\right )}}{2 \, {\left (x^{4} + 8 \, x^{3} + 16 \, x^{2} - x - 2\right )}} + \frac {5}{2}\right )}}{\log \left (2\right )} \] Input:
integrate((x^8+16*x^7+96*x^6+254*x^5+256*x^4+56*x^3+97*x^2-x+4)/(x^8+16*x^ 7+96*x^6+254*x^5+236*x^4-64*x^3-63*x^2+4*x+4)/log(2)/exp(5/(x^4+8*x^3+16*x ^2-x-2)),x, algorithm="giac")
Output:
x*e^(-5/2*(x^4 + 8*x^3 + 16*x^2 - x)/(x^4 + 8*x^3 + 16*x^2 - x - 2) + 5/2) /log(2)
Time = 2.17 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} \left (4-x+97 x^2+56 x^3+256 x^4+254 x^5+96 x^6+16 x^7+x^8\right )}{\left (4+4 x-63 x^2-64 x^3+236 x^4+254 x^5+96 x^6+16 x^7+x^8\right ) \log (2)} \, dx=\frac {x\,{\mathrm {e}}^{-\frac {5}{x^4+8\,x^3+16\,x^2-x-2}}}{\ln \left (2\right )} \] Input:
int((exp(-5/(16*x^2 - x + 8*x^3 + x^4 - 2))*(97*x^2 - x + 56*x^3 + 256*x^4 + 254*x^5 + 96*x^6 + 16*x^7 + x^8 + 4))/(log(2)*(4*x - 63*x^2 - 64*x^3 + 236*x^4 + 254*x^5 + 96*x^6 + 16*x^7 + x^8 + 4)),x)
Output:
(x*exp(-5/(16*x^2 - x + 8*x^3 + x^4 - 2)))/log(2)
Time = 0.19 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} \left (4-x+97 x^2+56 x^3+256 x^4+254 x^5+96 x^6+16 x^7+x^8\right )}{\left (4+4 x-63 x^2-64 x^3+236 x^4+254 x^5+96 x^6+16 x^7+x^8\right ) \log (2)} \, dx=\frac {x}{e^{\frac {5}{x^{4}+8 x^{3}+16 x^{2}-x -2}} \mathrm {log}\left (2\right )} \] Input:
int((x^8+16*x^7+96*x^6+254*x^5+256*x^4+56*x^3+97*x^2-x+4)/(x^8+16*x^7+96*x ^6+254*x^5+236*x^4-64*x^3-63*x^2+4*x+4)/log(2)/exp(5/(x^4+8*x^3+16*x^2-x-2 )),x)
Output:
x/(e**(5/(x**4 + 8*x**3 + 16*x**2 - x - 2))*log(2))