\(\int \frac {e^{\frac {-5-6 x+6 \log (\frac {16}{x^2})}{-x+\log (\frac {16}{x^2})}} (-40 x-30 x^2+3 x^3+x^4+e^x (10 x+5 x^2-2 x^3+x^4)+(-16 x^2-2 x^3+e^x (4 x^2-2 x^3)) \log (\frac {16}{x^2})+(8 x+x^2+e^x (-2 x+x^2)) \log ^2(\frac {16}{x^2}))}{16 x^2+e^{2 x} x^2+8 x^3+x^4+e^x (-8 x^2-2 x^3)+(-32 x-2 e^{2 x} x-16 x^2-2 x^3+e^x (16 x+4 x^2)) \log (\frac {16}{x^2})+(16+e^{2 x}+e^x (-8-2 x)+8 x+x^2) \log ^2(\frac {16}{x^2})} \, dx\) [8356]

Optimal result

Integrand size = 240, antiderivative size = 32 \[ \int \frac {e^{\frac {-5-6 x+6 \log \left (\frac {16}{x^2}\right )}{-x+\log \left (\frac {16}{x^2}\right )}} \left (-40 x-30 x^2+3 x^3+x^4+e^x \left (10 x+5 x^2-2 x^3+x^4\right )+\left (-16 x^2-2 x^3+e^x \left (4 x^2-2 x^3\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (8 x+x^2+e^x \left (-2 x+x^2\right )\right ) \log ^2\left (\frac {16}{x^2}\right )\right )}{16 x^2+e^{2 x} x^2+8 x^3+x^4+e^x \left (-8 x^2-2 x^3\right )+\left (-32 x-2 e^{2 x} x-16 x^2-2 x^3+e^x \left (16 x+4 x^2\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (16+e^{2 x}+e^x (-8-2 x)+8 x+x^2\right ) \log ^2\left (\frac {16}{x^2}\right )} \, dx=\frac {e^{6-\frac {5}{-x+\log \left (\frac {16}{x^2}\right )}} x^2}{4-e^x+x} \]

[Out]

x^2*exp(6-5/(ln(16/x^2)-x))/(x-exp(x)+4)

Rubi [F(-1)]

Timed out. \[ \int \frac {e^{\frac {-5-6 x+6 \log \left (\frac {16}{x^2}\right )}{-x+\log \left (\frac {16}{x^2}\right )}} \left (-40 x-30 x^2+3 x^3+x^4+e^x \left (10 x+5 x^2-2 x^3+x^4\right )+\left (-16 x^2-2 x^3+e^x \left (4 x^2-2 x^3\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (8 x+x^2+e^x \left (-2 x+x^2\right )\right ) \log ^2\left (\frac {16}{x^2}\right )\right )}{16 x^2+e^{2 x} x^2+8 x^3+x^4+e^x \left (-8 x^2-2 x^3\right )+\left (-32 x-2 e^{2 x} x-16 x^2-2 x^3+e^x \left (16 x+4 x^2\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (16+e^{2 x}+e^x (-8-2 x)+8 x+x^2\right ) \log ^2\left (\frac {16}{x^2}\right )} \, dx=\text {\$Aborted} \]

[In]

Int[(E^((-5 - 6*x + 6*Log[16/x^2])/(-x + Log[16/x^2]))*(-40*x - 30*x^2 + 3*x^3 + x^4 + E^x*(10*x + 5*x^2 - 2*x
^3 + x^4) + (-16*x^2 - 2*x^3 + E^x*(4*x^2 - 2*x^3))*Log[16/x^2] + (8*x + x^2 + E^x*(-2*x + x^2))*Log[16/x^2]^2
))/(16*x^2 + E^(2*x)*x^2 + 8*x^3 + x^4 + E^x*(-8*x^2 - 2*x^3) + (-32*x - 2*E^(2*x)*x - 16*x^2 - 2*x^3 + E^x*(1
6*x + 4*x^2))*Log[16/x^2] + (16 + E^(2*x) + E^x*(-8 - 2*x) + 8*x + x^2)*Log[16/x^2]^2),x]

[Out]

$Aborted

Rubi steps Aborted

Mathematica [F]

\[ \int \frac {e^{\frac {-5-6 x+6 \log \left (\frac {16}{x^2}\right )}{-x+\log \left (\frac {16}{x^2}\right )}} \left (-40 x-30 x^2+3 x^3+x^4+e^x \left (10 x+5 x^2-2 x^3+x^4\right )+\left (-16 x^2-2 x^3+e^x \left (4 x^2-2 x^3\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (8 x+x^2+e^x \left (-2 x+x^2\right )\right ) \log ^2\left (\frac {16}{x^2}\right )\right )}{16 x^2+e^{2 x} x^2+8 x^3+x^4+e^x \left (-8 x^2-2 x^3\right )+\left (-32 x-2 e^{2 x} x-16 x^2-2 x^3+e^x \left (16 x+4 x^2\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (16+e^{2 x}+e^x (-8-2 x)+8 x+x^2\right ) \log ^2\left (\frac {16}{x^2}\right )} \, dx=\int \frac {e^{\frac {-5-6 x+6 \log \left (\frac {16}{x^2}\right )}{-x+\log \left (\frac {16}{x^2}\right )}} \left (-40 x-30 x^2+3 x^3+x^4+e^x \left (10 x+5 x^2-2 x^3+x^4\right )+\left (-16 x^2-2 x^3+e^x \left (4 x^2-2 x^3\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (8 x+x^2+e^x \left (-2 x+x^2\right )\right ) \log ^2\left (\frac {16}{x^2}\right )\right )}{16 x^2+e^{2 x} x^2+8 x^3+x^4+e^x \left (-8 x^2-2 x^3\right )+\left (-32 x-2 e^{2 x} x-16 x^2-2 x^3+e^x \left (16 x+4 x^2\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (16+e^{2 x}+e^x (-8-2 x)+8 x+x^2\right ) \log ^2\left (\frac {16}{x^2}\right )} \, dx \]

[In]

Integrate[(E^((-5 - 6*x + 6*Log[16/x^2])/(-x + Log[16/x^2]))*(-40*x - 30*x^2 + 3*x^3 + x^4 + E^x*(10*x + 5*x^2
- 2*x^3 + x^4) + (-16*x^2 - 2*x^3 + E^x*(4*x^2 - 2*x^3))*Log[16/x^2] + (8*x + x^2 + E^x*(-2*x + x^2))*Log[16/
x^2]^2))/(16*x^2 + E^(2*x)*x^2 + 8*x^3 + x^4 + E^x*(-8*x^2 - 2*x^3) + (-32*x - 2*E^(2*x)*x - 16*x^2 - 2*x^3 +
E^x*(16*x + 4*x^2))*Log[16/x^2] + (16 + E^(2*x) + E^x*(-8 - 2*x) + 8*x + x^2)*Log[16/x^2]^2),x]

[Out]

Integrate[(E^((-5 - 6*x + 6*Log[16/x^2])/(-x + Log[16/x^2]))*(-40*x - 30*x^2 + 3*x^3 + x^4 + E^x*(10*x + 5*x^2
- 2*x^3 + x^4) + (-16*x^2 - 2*x^3 + E^x*(4*x^2 - 2*x^3))*Log[16/x^2] + (8*x + x^2 + E^x*(-2*x + x^2))*Log[16/
x^2]^2))/(16*x^2 + E^(2*x)*x^2 + 8*x^3 + x^4 + E^x*(-8*x^2 - 2*x^3) + (-32*x - 2*E^(2*x)*x - 16*x^2 - 2*x^3 +
E^x*(16*x + 4*x^2))*Log[16/x^2] + (16 + E^(2*x) + E^x*(-8 - 2*x) + 8*x + x^2)*Log[16/x^2]^2), x]

Maple [A] (verified)

Time = 19.14 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28

method	result	size
parallelrisch	\(\frac {{\mathrm e}^{\frac {6 \ln \left (\frac {16}{x^{2}}\right )-5-6 x}{\ln \left (\frac {16}{x^{2}}\right )-x}} x^{2}}{x -{\mathrm e}^{x}+4}\)	\(41\)
risch	\(\frac {x^{2} {\mathrm e}^{\frac {6 i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-12 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+6 i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-24 \ln \left (x \right )+48 \ln \left (2\right )-12 x -10}{i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-2 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-4 \ln \left (x \right )+8 \ln \left (2\right )-2 x}}}{x -{\mathrm e}^{x}+4}\)	\(142\)

[In]

int((((x^2-2*x)*exp(x)+x^2+8*x)*ln(16/x^2)^2+((-2*x^3+4*x^2)*exp(x)-2*x^3-16*x^2)*ln(16/x^2)+(x^4-2*x^3+5*x^2+
10*x)*exp(x)+x^4+3*x^3-30*x^2-40*x)*exp((6*ln(16/x^2)-5-6*x)/(ln(16/x^2)-x))/((exp(x)^2+(-2*x-8)*exp(x)+x^2+8*
x+16)*ln(16/x^2)^2+(-2*x*exp(x)^2+(4*x^2+16*x)*exp(x)-2*x^3-16*x^2-32*x)*ln(16/x^2)+exp(x)^2*x^2+(-2*x^3-8*x^2
)*exp(x)+x^4+8*x^3+16*x^2),x,method=_RETURNVERBOSE)

[Out]

exp((6*ln(16/x^2)-5-6*x)/(ln(16/x^2)-x))*x^2/(x-exp(x)+4)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.25 \[ \int \frac {e^{\frac {-5-6 x+6 \log \left (\frac {16}{x^2}\right )}{-x+\log \left (\frac {16}{x^2}\right )}} \left (-40 x-30 x^2+3 x^3+x^4+e^x \left (10 x+5 x^2-2 x^3+x^4\right )+\left (-16 x^2-2 x^3+e^x \left (4 x^2-2 x^3\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (8 x+x^2+e^x \left (-2 x+x^2\right )\right ) \log ^2\left (\frac {16}{x^2}\right )\right )}{16 x^2+e^{2 x} x^2+8 x^3+x^4+e^x \left (-8 x^2-2 x^3\right )+\left (-32 x-2 e^{2 x} x-16 x^2-2 x^3+e^x \left (16 x+4 x^2\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (16+e^{2 x}+e^x (-8-2 x)+8 x+x^2\right ) \log ^2\left (\frac {16}{x^2}\right )} \, dx=\frac {x^{2} e^{\left (\frac {6 \, x - 6 \, \log \left (\frac {16}{x^{2}}\right ) + 5}{x - \log \left (\frac {16}{x^{2}}\right )}\right )}}{x - e^{x} + 4} \]

[In]

integrate((((x^2-2*x)*exp(x)+x^2+8*x)*log(16/x^2)^2+((-2*x^3+4*x^2)*exp(x)-2*x^3-16*x^2)*log(16/x^2)+(x^4-2*x^
3+5*x^2+10*x)*exp(x)+x^4+3*x^3-30*x^2-40*x)*exp((6*log(16/x^2)-5-6*x)/(log(16/x^2)-x))/((exp(x)^2+(-2*x-8)*exp
(x)+x^2+8*x+16)*log(16/x^2)^2+(-2*x*exp(x)^2+(4*x^2+16*x)*exp(x)-2*x^3-16*x^2-32*x)*log(16/x^2)+exp(x)^2*x^2+(
-2*x^3-8*x^2)*exp(x)+x^4+8*x^3+16*x^2),x, algorithm="fricas")

[Out]

x^2*e^((6*x - 6*log(16/x^2) + 5)/(x - log(16/x^2)))/(x - e^x + 4)

Sympy [A] (verification not implemented)

Time = 0.73 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {-5-6 x+6 \log \left (\frac {16}{x^2}\right )}{-x+\log \left (\frac {16}{x^2}\right )}} \left (-40 x-30 x^2+3 x^3+x^4+e^x \left (10 x+5 x^2-2 x^3+x^4\right )+\left (-16 x^2-2 x^3+e^x \left (4 x^2-2 x^3\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (8 x+x^2+e^x \left (-2 x+x^2\right )\right ) \log ^2\left (\frac {16}{x^2}\right )\right )}{16 x^2+e^{2 x} x^2+8 x^3+x^4+e^x \left (-8 x^2-2 x^3\right )+\left (-32 x-2 e^{2 x} x-16 x^2-2 x^3+e^x \left (16 x+4 x^2\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (16+e^{2 x}+e^x (-8-2 x)+8 x+x^2\right ) \log ^2\left (\frac {16}{x^2}\right )} \, dx=\frac {x^{2} e^{\frac {- 6 x + 6 \log {\left (\frac {16}{x^{2}} \right )} - 5}{- x + \log {\left (\frac {16}{x^{2}} \right )}}}}{x - e^{x} + 4} \]

[In]

integrate((((x**2-2*x)*exp(x)+x**2+8*x)*ln(16/x**2)**2+((-2*x**3+4*x**2)*exp(x)-2*x**3-16*x**2)*ln(16/x**2)+(x
**4-2*x**3+5*x**2+10*x)*exp(x)+x**4+3*x**3-30*x**2-40*x)*exp((6*ln(16/x**2)-5-6*x)/(ln(16/x**2)-x))/((exp(x)**
2+(-2*x-8)*exp(x)+x**2+8*x+16)*ln(16/x**2)**2+(-2*x*exp(x)**2+(4*x**2+16*x)*exp(x)-2*x**3-16*x**2-32*x)*ln(16/
x**2)+exp(x)**2*x**2+(-2*x**3-8*x**2)*exp(x)+x**4+8*x**3+16*x**2),x)

[Out]

x**2*exp((-6*x + 6*log(16/x**2) - 5)/(-x + log(16/x**2)))/(x - exp(x) + 4)

Maxima [A] (verification not implemented)

none

Time = 0.47 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {e^{\frac {-5-6 x+6 \log \left (\frac {16}{x^2}\right )}{-x+\log \left (\frac {16}{x^2}\right )}} \left (-40 x-30 x^2+3 x^3+x^4+e^x \left (10 x+5 x^2-2 x^3+x^4\right )+\left (-16 x^2-2 x^3+e^x \left (4 x^2-2 x^3\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (8 x+x^2+e^x \left (-2 x+x^2\right )\right ) \log ^2\left (\frac {16}{x^2}\right )\right )}{16 x^2+e^{2 x} x^2+8 x^3+x^4+e^x \left (-8 x^2-2 x^3\right )+\left (-32 x-2 e^{2 x} x-16 x^2-2 x^3+e^x \left (16 x+4 x^2\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (16+e^{2 x}+e^x (-8-2 x)+8 x+x^2\right ) \log ^2\left (\frac {16}{x^2}\right )} \, dx=\frac {x^{2} e^{\left (\frac {5}{x - 4 \, \log \left (2\right ) + 2 \, \log \left (x\right )} + 6\right )}}{x - e^{x} + 4} \]

[In]

integrate((((x^2-2*x)*exp(x)+x^2+8*x)*log(16/x^2)^2+((-2*x^3+4*x^2)*exp(x)-2*x^3-16*x^2)*log(16/x^2)+(x^4-2*x^
3+5*x^2+10*x)*exp(x)+x^4+3*x^3-30*x^2-40*x)*exp((6*log(16/x^2)-5-6*x)/(log(16/x^2)-x))/((exp(x)^2+(-2*x-8)*exp
(x)+x^2+8*x+16)*log(16/x^2)^2+(-2*x*exp(x)^2+(4*x^2+16*x)*exp(x)-2*x^3-16*x^2-32*x)*log(16/x^2)+exp(x)^2*x^2+(
-2*x^3-8*x^2)*exp(x)+x^4+8*x^3+16*x^2),x, algorithm="maxima")

[Out]

x^2*e^(5/(x - 4*log(2) + 2*log(x)) + 6)/(x - e^x + 4)

Giac [A] (verification not implemented)

none

Time = 2.06 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.25 \[ \int \frac {e^{\frac {-5-6 x+6 \log \left (\frac {16}{x^2}\right )}{-x+\log \left (\frac {16}{x^2}\right )}} \left (-40 x-30 x^2+3 x^3+x^4+e^x \left (10 x+5 x^2-2 x^3+x^4\right )+\left (-16 x^2-2 x^3+e^x \left (4 x^2-2 x^3\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (8 x+x^2+e^x \left (-2 x+x^2\right )\right ) \log ^2\left (\frac {16}{x^2}\right )\right )}{16 x^2+e^{2 x} x^2+8 x^3+x^4+e^x \left (-8 x^2-2 x^3\right )+\left (-32 x-2 e^{2 x} x-16 x^2-2 x^3+e^x \left (16 x+4 x^2\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (16+e^{2 x}+e^x (-8-2 x)+8 x+x^2\right ) \log ^2\left (\frac {16}{x^2}\right )} \, dx=\frac {x^{2} e^{\left (\frac {6 \, x - 6 \, \log \left (\frac {16}{x^{2}}\right ) + 5}{x - \log \left (\frac {16}{x^{2}}\right )}\right )}}{x - e^{x} + 4} \]

[In]

integrate((((x^2-2*x)*exp(x)+x^2+8*x)*log(16/x^2)^2+((-2*x^3+4*x^2)*exp(x)-2*x^3-16*x^2)*log(16/x^2)+(x^4-2*x^
3+5*x^2+10*x)*exp(x)+x^4+3*x^3-30*x^2-40*x)*exp((6*log(16/x^2)-5-6*x)/(log(16/x^2)-x))/((exp(x)^2+(-2*x-8)*exp
(x)+x^2+8*x+16)*log(16/x^2)^2+(-2*x*exp(x)^2+(4*x^2+16*x)*exp(x)-2*x^3-16*x^2-32*x)*log(16/x^2)+exp(x)^2*x^2+(
-2*x^3-8*x^2)*exp(x)+x^4+8*x^3+16*x^2),x, algorithm="giac")

[Out]

x^2*e^((6*x - 6*log(16/x^2) + 5)/(x - log(16/x^2)))/(x - e^x + 4)

Mupad [B] (verification not implemented)

Time = 15.44 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.12 \[ \int \frac {e^{\frac {-5-6 x+6 \log \left (\frac {16}{x^2}\right )}{-x+\log \left (\frac {16}{x^2}\right )}} \left (-40 x-30 x^2+3 x^3+x^4+e^x \left (10 x+5 x^2-2 x^3+x^4\right )+\left (-16 x^2-2 x^3+e^x \left (4 x^2-2 x^3\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (8 x+x^2+e^x \left (-2 x+x^2\right )\right ) \log ^2\left (\frac {16}{x^2}\right )\right )}{16 x^2+e^{2 x} x^2+8 x^3+x^4+e^x \left (-8 x^2-2 x^3\right )+\left (-32 x-2 e^{2 x} x-16 x^2-2 x^3+e^x \left (16 x+4 x^2\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (16+e^{2 x}+e^x (-8-2 x)+8 x+x^2\right ) \log ^2\left (\frac {16}{x^2}\right )} \, dx=\frac {x^2\,{\mathrm {e}}^{-\frac {5}{\ln \left (\frac {1}{x^2}\right )-x+4\,\ln \left (2\right )}}\,{\mathrm {e}}^{-\frac {6\,x}{\ln \left (\frac {1}{x^2}\right )-x+4\,\ln \left (2\right )}}\,{\left (\frac {16777216}{x^{12}}\right )}^{\frac {1}{\ln \left (\frac {1}{x^2}\right )-x+4\,\ln \left (2\right )}}}{x-{\mathrm {e}}^x+4} \]

[In]

int((exp((6*x - 6*log(16/x^2) + 5)/(x - log(16/x^2)))*(log(16/x^2)^2*(8*x - exp(x)*(2*x - x^2) + x^2) - 40*x +
exp(x)*(10*x + 5*x^2 - 2*x^3 + x^4) - 30*x^2 + 3*x^3 + x^4 - log(16/x^2)*(16*x^2 - exp(x)*(4*x^2 - 2*x^3) + 2
*x^3)))/(log(16/x^2)^2*(8*x + exp(2*x) - exp(x)*(2*x + 8) + x^2 + 16) - exp(x)*(8*x^2 + 2*x^3) - log(16/x^2)*(
32*x + 2*x*exp(2*x) - exp(x)*(16*x + 4*x^2) + 16*x^2 + 2*x^3) + x^2*exp(2*x) + 16*x^2 + 8*x^3 + x^4),x)

[Out]

(x^2*exp(-5/(log(1/x^2) - x + 4*log(2)))*exp(-(6*x)/(log(1/x^2) - x + 4*log(2)))*(16777216/x^12)^(1/(log(1/x^2
) - x + 4*log(2))))/(x - exp(x) + 4)