Integrand size = 240, antiderivative size = 32 \[ \int \frac {e^{\frac {-5-6 x+6 \log \left (\frac {16}{x^2}\right )}{-x+\log \left (\frac {16}{x^2}\right )}} \left (-40 x-30 x^2+3 x^3+x^4+e^x \left (10 x+5 x^2-2 x^3+x^4\right )+\left (-16 x^2-2 x^3+e^x \left (4 x^2-2 x^3\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (8 x+x^2+e^x \left (-2 x+x^2\right )\right ) \log ^2\left (\frac {16}{x^2}\right )\right )}{16 x^2+e^{2 x} x^2+8 x^3+x^4+e^x \left (-8 x^2-2 x^3\right )+\left (-32 x-2 e^{2 x} x-16 x^2-2 x^3+e^x \left (16 x+4 x^2\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (16+e^{2 x}+e^x (-8-2 x)+8 x+x^2\right ) \log ^2\left (\frac {16}{x^2}\right )} \, dx=\frac {e^{6-\frac {5}{-x+\log \left (\frac {16}{x^2}\right )}} x^2}{4-e^x+x} \]
\[ \int \frac {e^{\frac {-5-6 x+6 \log \left (\frac {16}{x^2}\right )}{-x+\log \left (\frac {16}{x^2}\right )}} \left (-40 x-30 x^2+3 x^3+x^4+e^x \left (10 x+5 x^2-2 x^3+x^4\right )+\left (-16 x^2-2 x^3+e^x \left (4 x^2-2 x^3\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (8 x+x^2+e^x \left (-2 x+x^2\right )\right ) \log ^2\left (\frac {16}{x^2}\right )\right )}{16 x^2+e^{2 x} x^2+8 x^3+x^4+e^x \left (-8 x^2-2 x^3\right )+\left (-32 x-2 e^{2 x} x-16 x^2-2 x^3+e^x \left (16 x+4 x^2\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (16+e^{2 x}+e^x (-8-2 x)+8 x+x^2\right ) \log ^2\left (\frac {16}{x^2}\right )} \, dx=\int \frac {e^{\frac {-5-6 x+6 \log \left (\frac {16}{x^2}\right )}{-x+\log \left (\frac {16}{x^2}\right )}} \left (-40 x-30 x^2+3 x^3+x^4+e^x \left (10 x+5 x^2-2 x^3+x^4\right )+\left (-16 x^2-2 x^3+e^x \left (4 x^2-2 x^3\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (8 x+x^2+e^x \left (-2 x+x^2\right )\right ) \log ^2\left (\frac {16}{x^2}\right )\right )}{16 x^2+e^{2 x} x^2+8 x^3+x^4+e^x \left (-8 x^2-2 x^3\right )+\left (-32 x-2 e^{2 x} x-16 x^2-2 x^3+e^x \left (16 x+4 x^2\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (16+e^{2 x}+e^x (-8-2 x)+8 x+x^2\right ) \log ^2\left (\frac {16}{x^2}\right )} \, dx \]
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]
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]
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.
| \(\displaystyle \int \frac {e^{\frac {6 \log \left (\frac {16}{x^2}\right )-6 x-5}{\log \left (\frac {16}{x^2}\right )-x}} \left (x^4+3 x^3-30 x^2+\left (x^2+e^x \left (x^2-2 x\right )+8 x\right ) \log ^2\left (\frac {16}{x^2}\right )+\left (-2 x^3-16 x^2+e^x \left (4 x^2-2 x^3\right )\right ) \log \left (\frac {16}{x^2}\right )+e^x \left (x^4-2 x^3+5 x^2+10 x\right )-40 x\right )}{x^4+8 x^3+e^{2 x} x^2+16 x^2+\left (x^2+8 x+e^{2 x}+e^x (-2 x-8)+16\right ) \log ^2\left (\frac {16}{x^2}\right )+e^x \left (-2 x^3-8 x^2\right )+\left (-2 x^3-16 x^2+e^x \left (4 x^2+16 x\right )-2 e^{2 x} x-32 x\right ) \log \left (\frac {16}{x^2}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{\frac {6 \log \left (\frac {16}{x^2}\right )-6 x-5}{\log \left (\frac {16}{x^2}\right )-x}} \left (x^4+3 x^3-30 x^2+\left (x^2+e^x \left (x^2-2 x\right )+8 x\right ) \log ^2\left (\frac {16}{x^2}\right )+\left (-2 x^3-16 x^2+e^x \left (4 x^2-2 x^3\right )\right ) \log \left (\frac {16}{x^2}\right )+e^x \left (x^4-2 x^3+5 x^2+10 x\right )-40 x\right )}{\left (x-e^x+4\right )^2 \left (x-\log \left (\frac {16}{x^2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(x+3) x^2 e^{\frac {6 \log \left (\frac {16}{x^2}\right )-6 x-5}{\log \left (\frac {16}{x^2}\right )-x}}}{\left (-x+e^x-4\right )^2}+\frac {x e^{\frac {6 \log \left (\frac {16}{x^2}\right )-6 x-5}{\log \left (\frac {16}{x^2}\right )-x}} \left (x^3-2 x^2+x \log ^2\left (\frac {16}{x^2}\right )-2 \log ^2\left (\frac {16}{x^2}\right )-2 x^2 \log \left (\frac {16}{x^2}\right )+4 x \log \left (\frac {16}{x^2}\right )+5 x+10\right )}{\left (-x+e^x-4\right ) \left (x-\log \left (\frac {16}{x^2}\right )\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(x+3) 16^{-\frac {6}{x-\log \left (\frac {16}{x^2}\right )}} e^{\frac {6 x+5}{x-\log \left (\frac {16}{x^2}\right )}} \left (\frac {1}{x^2}\right )^{-\frac {6}{x-\log \left (\frac {16}{x^2}\right )}-1}}{\left (x-e^x+4\right )^2}+\frac {x 16^{-\frac {6}{x-\log \left (\frac {16}{x^2}\right )}} e^{\frac {6 x+5}{x-\log \left (\frac {16}{x^2}\right )}} \left (-x^3+2 x^2-x \log ^2\left (\frac {16}{x^2}\right )+2 \log ^2\left (\frac {16}{x^2}\right )+2 x^2 \log \left (\frac {16}{x^2}\right )-4 x \log \left (\frac {16}{x^2}\right )-5 x-10\right ) \left (\frac {1}{x^2}\right )^{-\frac {6}{x-\log \left (\frac {16}{x^2}\right )}}}{\left (x-e^x+4\right ) \left (x-\log \left (\frac {16}{x^2}\right )\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {(x+3) 16^{-\frac {6}{x-\log \left (\frac {16}{x^2}\right )}} e^{\frac {6 x+5}{x-\log \left (\frac {16}{x^2}\right )}} \left (\frac {1}{x^2}\right )^{-\frac {6}{x-\log \left (\frac {16}{x^2}\right )}-1}}{\left (x-e^x+4\right )^2}+\frac {x 16^{-\frac {6}{x-\log \left (\frac {16}{x^2}\right )}} e^{\frac {6 x+5}{x-\log \left (\frac {16}{x^2}\right )}} \left (-x^3+2 x^2-x \log ^2\left (\frac {16}{x^2}\right )+2 \log ^2\left (\frac {16}{x^2}\right )+2 x^2 \log \left (\frac {16}{x^2}\right )-4 x \log \left (\frac {16}{x^2}\right )-5 x-10\right ) \left (\frac {1}{x^2}\right )^{-\frac {6}{x-\log \left (\frac {16}{x^2}\right )}}}{\left (x-e^x+4\right ) \left (x-\log \left (\frac {16}{x^2}\right )\right )^2}\right )dx\) |
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*(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]
3.24.56.3.1 Defintions of rubi rules used
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 | \(\frac {x^{2} {\mathrm e}^{\frac {6 i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-12 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+6 i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-24 \ln \left (x \right )+48 \ln \left (2\right )-12 x -10}{i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-2 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-4 \ln \left (x \right )+8 \ln \left (2\right )-2 x}}}{x -{\mathrm e}^{x}+4}\) | \(142\) |
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)*ex p((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=_RET URNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.25 \[ \int \frac {e^{\frac {-5-6 x+6 \log \left (\frac {16}{x^2}\right )}{-x+\log \left (\frac {16}{x^2}\right )}} \left (-40 x-30 x^2+3 x^3+x^4+e^x \left (10 x+5 x^2-2 x^3+x^4\right )+\left (-16 x^2-2 x^3+e^x \left (4 x^2-2 x^3\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (8 x+x^2+e^x \left (-2 x+x^2\right )\right ) \log ^2\left (\frac {16}{x^2}\right )\right )}{16 x^2+e^{2 x} x^2+8 x^3+x^4+e^x \left (-8 x^2-2 x^3\right )+\left (-32 x-2 e^{2 x} x-16 x^2-2 x^3+e^x \left (16 x+4 x^2\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (16+e^{2 x}+e^x (-8-2 x)+8 x+x^2\right ) \log ^2\left (\frac {16}{x^2}\right )} \, dx=\frac {x^{2} e^{\left (\frac {6 \, x - 6 \, \log \left (\frac {16}{x^{2}}\right ) + 5}{x - \log \left (\frac {16}{x^{2}}\right )}\right )}}{x - e^{x} + 4} \]
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=\
Time = 0.73 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {-5-6 x+6 \log \left (\frac {16}{x^2}\right )}{-x+\log \left (\frac {16}{x^2}\right )}} \left (-40 x-30 x^2+3 x^3+x^4+e^x \left (10 x+5 x^2-2 x^3+x^4\right )+\left (-16 x^2-2 x^3+e^x \left (4 x^2-2 x^3\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (8 x+x^2+e^x \left (-2 x+x^2\right )\right ) \log ^2\left (\frac {16}{x^2}\right )\right )}{16 x^2+e^{2 x} x^2+8 x^3+x^4+e^x \left (-8 x^2-2 x^3\right )+\left (-32 x-2 e^{2 x} x-16 x^2-2 x^3+e^x \left (16 x+4 x^2\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (16+e^{2 x}+e^x (-8-2 x)+8 x+x^2\right ) \log ^2\left (\frac {16}{x^2}\right )} \, dx=\frac {x^{2} e^{\frac {- 6 x + 6 \log {\left (\frac {16}{x^{2}} \right )} - 5}{- x + \log {\left (\frac {16}{x^{2}} \right )}}}}{x - e^{x} + 4} \]
integrate((((x**2-2*x)*exp(x)+x**2+8*x)*ln(16/x**2)**2+((-2*x**3+4*x**2)*e xp(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)*ex p(x)+x**4+8*x**3+16*x**2),x)
Time = 0.47 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {e^{\frac {-5-6 x+6 \log \left (\frac {16}{x^2}\right )}{-x+\log \left (\frac {16}{x^2}\right )}} \left (-40 x-30 x^2+3 x^3+x^4+e^x \left (10 x+5 x^2-2 x^3+x^4\right )+\left (-16 x^2-2 x^3+e^x \left (4 x^2-2 x^3\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (8 x+x^2+e^x \left (-2 x+x^2\right )\right ) \log ^2\left (\frac {16}{x^2}\right )\right )}{16 x^2+e^{2 x} x^2+8 x^3+x^4+e^x \left (-8 x^2-2 x^3\right )+\left (-32 x-2 e^{2 x} x-16 x^2-2 x^3+e^x \left (16 x+4 x^2\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (16+e^{2 x}+e^x (-8-2 x)+8 x+x^2\right ) \log ^2\left (\frac {16}{x^2}\right )} \, dx=\frac {x^{2} e^{\left (\frac {5}{x - 4 \, \log \left (2\right ) + 2 \, \log \left (x\right )} + 6\right )}}{x - e^{x} + 4} \]
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=\
Time = 2.06 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.25 \[ \int \frac {e^{\frac {-5-6 x+6 \log \left (\frac {16}{x^2}\right )}{-x+\log \left (\frac {16}{x^2}\right )}} \left (-40 x-30 x^2+3 x^3+x^4+e^x \left (10 x+5 x^2-2 x^3+x^4\right )+\left (-16 x^2-2 x^3+e^x \left (4 x^2-2 x^3\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (8 x+x^2+e^x \left (-2 x+x^2\right )\right ) \log ^2\left (\frac {16}{x^2}\right )\right )}{16 x^2+e^{2 x} x^2+8 x^3+x^4+e^x \left (-8 x^2-2 x^3\right )+\left (-32 x-2 e^{2 x} x-16 x^2-2 x^3+e^x \left (16 x+4 x^2\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (16+e^{2 x}+e^x (-8-2 x)+8 x+x^2\right ) \log ^2\left (\frac {16}{x^2}\right )} \, dx=\frac {x^{2} e^{\left (\frac {6 \, x - 6 \, \log \left (\frac {16}{x^{2}}\right ) + 5}{x - \log \left (\frac {16}{x^{2}}\right )}\right )}}{x - e^{x} + 4} \]
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=\
Time = 15.44 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.12 \[ \int \frac {e^{\frac {-5-6 x+6 \log \left (\frac {16}{x^2}\right )}{-x+\log \left (\frac {16}{x^2}\right )}} \left (-40 x-30 x^2+3 x^3+x^4+e^x \left (10 x+5 x^2-2 x^3+x^4\right )+\left (-16 x^2-2 x^3+e^x \left (4 x^2-2 x^3\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (8 x+x^2+e^x \left (-2 x+x^2\right )\right ) \log ^2\left (\frac {16}{x^2}\right )\right )}{16 x^2+e^{2 x} x^2+8 x^3+x^4+e^x \left (-8 x^2-2 x^3\right )+\left (-32 x-2 e^{2 x} x-16 x^2-2 x^3+e^x \left (16 x+4 x^2\right )\right ) \log \left (\frac {16}{x^2}\right )+\left (16+e^{2 x}+e^x (-8-2 x)+8 x+x^2\right ) \log ^2\left (\frac {16}{x^2}\right )} \, dx=\frac {x^2\,{\mathrm {e}}^{-\frac {5}{\ln \left (\frac {1}{x^2}\right )-x+4\,\ln \left (2\right )}}\,{\mathrm {e}}^{-\frac {6\,x}{\ln \left (\frac {1}{x^2}\right )-x+4\,\ln \left (2\right )}}\,{\left (\frac {16777216}{x^{12}}\right )}^{\frac {1}{\ln \left (\frac {1}{x^2}\right )-x+4\,\ln \left (2\right )}}}{x-{\mathrm {e}}^x+4} \]
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)