\(\int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} (4-x+97 x^2+56 x^3+256 x^4+254 x^5+96 x^6+16 x^7+x^8)}{(4+4 x-63 x^2-64 x^3+236 x^4+254 x^5+96 x^6+16 x^7+x^8) \log (2)} \, dx\) [2039]

Optimal result

Integrand size = 107, antiderivative size = 26 \[ \int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} \left (4-x+97 x^2+56 x^3+256 x^4+254 x^5+96 x^6+16 x^7+x^8\right )}{\left (4+4 x-63 x^2-64 x^3+236 x^4+254 x^5+96 x^6+16 x^7+x^8\right ) \log (2)} \, dx=\frac {e^{-\frac {5}{-2-x+x^2 (4+x)^2}} x}{\log (2)} \]

[Out]

x/ln(2)/exp(5/((4+x)^2*x^2-2-x))

Rubi [F]

\[ \int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} \left (4-x+97 x^2+56 x^3+256 x^4+254 x^5+96 x^6+16 x^7+x^8\right )}{\left (4+4 x-63 x^2-64 x^3+236 x^4+254 x^5+96 x^6+16 x^7+x^8\right ) \log (2)} \, dx=\int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} \left (4-x+97 x^2+56 x^3+256 x^4+254 x^5+96 x^6+16 x^7+x^8\right )}{\left (4+4 x-63 x^2-64 x^3+236 x^4+254 x^5+96 x^6+16 x^7+x^8\right ) \log (2)} \, dx \]

[In]

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]

[Out]

Defer[Int][E^(-5/(-2 - x + 16*x^2 + 8*x^3 + x^4)), x]/Log[2] + (40*Defer[Int][1/(E^(5/(-2 - x + 16*x^2 + 8*x^3
+ x^4))*(-2 - x + 16*x^2 + 8*x^3 + x^4)^2), x])/Log[2] + (15*Defer[Int][x/(E^(5/(-2 - x + 16*x^2 + 8*x^3 + x^
4))*(-2 - x + 16*x^2 + 8*x^3 + x^4)^2), x])/Log[2] - (160*Defer[Int][x^2/(E^(5/(-2 - x + 16*x^2 + 8*x^3 + x^4)
)*(-2 - x + 16*x^2 + 8*x^3 + x^4)^2), x])/Log[2] - (40*Defer[Int][x^3/(E^(5/(-2 - x + 16*x^2 + 8*x^3 + x^4))*(
-2 - x + 16*x^2 + 8*x^3 + x^4)^2), x])/Log[2] + (20*Defer[Int][1/(E^(5/(-2 - x + 16*x^2 + 8*x^3 + x^4))*(-2 -
x + 16*x^2 + 8*x^3 + x^4)), x])/Log[2]

Rubi steps \begin{align*} \text {integral}& = \frac {\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 )}{4+4 x-63 x^2-64 x^3+236 x^4+254 x^5+96 x^6+16 x^7+x^8} \, dx}{\log (2)} \\ & = \frac {\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 (2+x-16 x^2-8 x^3-x^4\right )^2} \, dx}{\log (2)} \\ & = \frac {\int \left (e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}}-\frac {5 e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} \left (-8-3 x+32 x^2+8 x^3\right )}{\left (-2-x+16 x^2+8 x^3+x^4\right )^2}+\frac {20 e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}}}{-2-x+16 x^2+8 x^3+x^4}\right ) \, dx}{\log (2)} \\ & = \frac {\int e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} \, dx}{\log (2)}-\frac {5 \int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} \left (-8-3 x+32 x^2+8 x^3\right )}{\left (-2-x+16 x^2+8 x^3+x^4\right )^2} \, dx}{\log (2)}+\frac {20 \int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}}}{-2-x+16 x^2+8 x^3+x^4} \, dx}{\log (2)} \\ & = \frac {\int e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} \, dx}{\log (2)}-\frac {5 \int \left (-\frac {8 e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}}}{\left (-2-x+16 x^2+8 x^3+x^4\right )^2}-\frac {3 e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} x}{\left (-2-x+16 x^2+8 x^3+x^4\right )^2}+\frac {32 e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} x^2}{\left (-2-x+16 x^2+8 x^3+x^4\right )^2}+\frac {8 e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} x^3}{\left (-2-x+16 x^2+8 x^3+x^4\right )^2}\right ) \, dx}{\log (2)}+\frac {20 \int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}}}{-2-x+16 x^2+8 x^3+x^4} \, dx}{\log (2)} \\ & = \frac {\int e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} \, dx}{\log (2)}+\frac {15 \int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} x}{\left (-2-x+16 x^2+8 x^3+x^4\right )^2} \, dx}{\log (2)}+\frac {20 \int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}}}{-2-x+16 x^2+8 x^3+x^4} \, dx}{\log (2)}+\frac {40 \int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}}}{\left (-2-x+16 x^2+8 x^3+x^4\right )^2} \, dx}{\log (2)}-\frac {40 \int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} x^3}{\left (-2-x+16 x^2+8 x^3+x^4\right )^2} \, dx}{\log (2)}-\frac {160 \int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} x^2}{\left (-2-x+16 x^2+8 x^3+x^4\right )^2} \, dx}{\log (2)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} \left (4-x+97 x^2+56 x^3+256 x^4+254 x^5+96 x^6+16 x^7+x^8\right )}{\left (4+4 x-63 x^2-64 x^3+236 x^4+254 x^5+96 x^6+16 x^7+x^8\right ) \log (2)} \, dx=\frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} x}{\log (2)} \]

[In]

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]

[Out]

x/(E^(5/(-2 - x + 16*x^2 + 8*x^3 + x^4))*Log[2])

Maple [A] (verified)

Time = 0.31 (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	\(\frac {\left (\frac {x^{5}}{\ln \left (2\right )}-\frac {2 x}{\ln \left (2\right )}-\frac {x^{2}}{\ln \left (2\right )}+\frac {16 x^{3}}{\ln \left (2\right )}+\frac {8 x^{4}}{\ln \left (2\right )}\right ) {\mathrm e}^{-\frac {5}{x^{4}+8 x^{3}+16 x^{2}-x -2}}}{x^{4}+8 x^{3}+16 x^{2}-x -2}\)	\(90\)

[In]

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)

[Out]

x*exp(-5/(x^4+8*x^3+16*x^2-x-2))/ln(2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} \left (4-x+97 x^2+56 x^3+256 x^4+254 x^5+96 x^6+16 x^7+x^8\right )}{\left (4+4 x-63 x^2-64 x^3+236 x^4+254 x^5+96 x^6+16 x^7+x^8\right ) \log (2)} \, dx=\frac {x e^{\left (-\frac {5}{x^{4} + 8 \, x^{3} + 16 \, x^{2} - x - 2}\right )}}{\log \left (2\right )} \]

[In]

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")

[Out]

x*e^(-5/(x^4 + 8*x^3 + 16*x^2 - x - 2))/log(2)

Sympy [A] (verification not implemented)

Time = 0.70 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} \left (4-x+97 x^2+56 x^3+256 x^4+254 x^5+96 x^6+16 x^7+x^8\right )}{\left (4+4 x-63 x^2-64 x^3+236 x^4+254 x^5+96 x^6+16 x^7+x^8\right ) \log (2)} \, dx=\frac {x e^{- \frac {5}{x^{4} + 8 x^{3} + 16 x^{2} - x - 2}}}{\log {\left (2 \right )}} \]

[In]

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)

[Out]

x*exp(-5/(x**4 + 8*x**3 + 16*x**2 - x - 2))/log(2)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} \left (4-x+97 x^2+56 x^3+256 x^4+254 x^5+96 x^6+16 x^7+x^8\right )}{\left (4+4 x-63 x^2-64 x^3+236 x^4+254 x^5+96 x^6+16 x^7+x^8\right ) \log (2)} \, dx=\frac {x e^{\left (-\frac {5}{x^{4} + 8 \, x^{3} + 16 \, x^{2} - x - 2}\right )}}{\log \left (2\right )} \]

[In]

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")

[Out]

x*e^(-5/(x^4 + 8*x^3 + 16*x^2 - x - 2))/log(2)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.85 \[ \int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} \left (4-x+97 x^2+56 x^3+256 x^4+254 x^5+96 x^6+16 x^7+x^8\right )}{\left (4+4 x-63 x^2-64 x^3+236 x^4+254 x^5+96 x^6+16 x^7+x^8\right ) \log (2)} \, dx=\frac {x e^{\left (-\frac {5 \, {\left (x^{4} + 8 \, x^{3} + 16 \, x^{2} - x\right )}}{2 \, {\left (x^{4} + 8 \, x^{3} + 16 \, x^{2} - x - 2\right )}} + \frac {5}{2}\right )}}{\log \left (2\right )} \]

[In]

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")

[Out]

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)

Mupad [B] (verification not implemented)

Time = 9.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} \left (4-x+97 x^2+56 x^3+256 x^4+254 x^5+96 x^6+16 x^7+x^8\right )}{\left (4+4 x-63 x^2-64 x^3+236 x^4+254 x^5+96 x^6+16 x^7+x^8\right ) \log (2)} \, dx=\frac {x\,{\mathrm {e}}^{-\frac {5}{x^4+8\,x^3+16\,x^2-x-2}}}{\ln \left (2\right )} \]

[In]

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)

[Out]

(x*exp(-5/(16*x^2 - x + 8*x^3 + x^4 - 2)))/log(2)