Integrand size = 90, antiderivative size = 23 \[ \int \frac {e^{\frac {4096 x^2-x^3+8192 x \log (4)+4096 \log ^2(4)+(-8192 x-8192 \log (4)) \log (x)+4096 \log ^2(x)}{x^2}} \left (-24576 x-3 x^3+(-24576-24576 x) \log (4)-24576 \log ^2(4)+(24576+24576 x+49152 \log (4)) \log (x)-24576 \log ^2(x)\right )}{x^3} \, dx=3 e^{-x+\frac {4096 (x+\log (4)-\log (x))^2}{x^2}} \] Output:
3*exp(4096/x^2*(2*ln(2)+x-ln(x))^2-x)
\[ \int \frac {e^{\frac {4096 x^2-x^3+8192 x \log (4)+4096 \log ^2(4)+(-8192 x-8192 \log (4)) \log (x)+4096 \log ^2(x)}{x^2}} \left (-24576 x-3 x^3+(-24576-24576 x) \log (4)-24576 \log ^2(4)+(24576+24576 x+49152 \log (4)) \log (x)-24576 \log ^2(x)\right )}{x^3} \, dx=\int \frac {e^{\frac {4096 x^2-x^3+8192 x \log (4)+4096 \log ^2(4)+(-8192 x-8192 \log (4)) \log (x)+4096 \log ^2(x)}{x^2}} \left (-24576 x-3 x^3+(-24576-24576 x) \log (4)-24576 \log ^2(4)+(24576+24576 x+49152 \log (4)) \log (x)-24576 \log ^2(x)\right )}{x^3} \, dx \] Input:
Integrate[(E^((4096*x^2 - x^3 + 8192*x*Log[4] + 4096*Log[4]^2 + (-8192*x - 8192*Log[4])*Log[x] + 4096*Log[x]^2)/x^2)*(-24576*x - 3*x^3 + (-24576 - 2 4576*x)*Log[4] - 24576*Log[4]^2 + (24576 + 24576*x + 49152*Log[4])*Log[x] - 24576*Log[x]^2))/x^3,x]
Output:
Integrate[(E^((4096*x^2 - x^3 + 8192*x*Log[4] + 4096*Log[4]^2 + (-8192*x - 8192*Log[4])*Log[x] + 4096*Log[x]^2)/x^2)*(-24576*x - 3*x^3 + (-24576 - 2 4576*x)*Log[4] - 24576*Log[4]^2 + (24576 + 24576*x + 49152*Log[4])*Log[x] - 24576*Log[x]^2))/x^3, 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.
| \(\displaystyle \int \frac {\left (-3 x^3-24576 x-24576 \log ^2(x)+(24576 x+24576+49152 \log (4)) \log (x)+(-24576 x-24576) \log (4)-24576 \log ^2(4)\right ) \exp \left (\frac {-x^3+4096 x^2+4096 \log ^2(x)+8192 x \log (4)+(-8192 x-8192 \log (4)) \log (x)+4096 \log ^2(4)}{x^2}\right )}{x^3} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (-3 x^3-24576 x-24576 \log ^2(x)+(24576 x+24576+49152 \log (4)) \log (x)+(-24576 x-24576) \log (4)-24576 \log ^2(4)\right ) \exp \left (\frac {4096 \log ^2(x)}{x^2}+\frac {4096 \log ^2(4)}{x^2}+\frac {(-8192 x-8192 \log (4)) \log (x)}{x^2}-x+\frac {8192 \log (4)}{x}+4096\right )}{x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {24576 \log ^2(x) \exp \left (\frac {4096 \log ^2(x)}{x^2}+\frac {4096 \log ^2(4)}{x^2}+\frac {(-8192 x-8192 \log (4)) \log (x)}{x^2}-x+\frac {8192 \log (4)}{x}+4096\right )}{x^3}+\frac {24576 (x+1+\log (16)) \log (x) \exp \left (\frac {4096 \log ^2(x)}{x^2}+\frac {4096 \log ^2(4)}{x^2}+\frac {(-8192 x-8192 \log (4)) \log (x)}{x^2}-x+\frac {8192 \log (4)}{x}+4096\right )}{x^3}+\frac {3 \left (-x^3-8192 x (1+\log (4))-8192 \log (4) (1+\log (4))\right ) \exp \left (\frac {4096 \log ^2(x)}{x^2}+\frac {4096 \log ^2(4)}{x^2}+\frac {(-8192 x-8192 \log (4)) \log (x)}{x^2}-x+\frac {8192 \log (4)}{x}+4096\right )}{x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 \int \exp \left (\frac {4096 \log ^2(x)}{x^2}+\frac {(-8192 x-8192 \log (4)) \log (x)}{x^2}-x+\frac {4096 \log ^2(4)}{x^2}+\frac {8192 \log (4)}{x}+4096\right )dx-24576 (1+\log (4)) \int \frac {\exp \left (\frac {4096 \log ^2(x)}{x^2}+\frac {(-8192 x-8192 \log (4)) \log (x)}{x^2}-x+\frac {4096 \log ^2(4)}{x^2}+\frac {8192 \log (4)}{x}+4096\right )}{x^2}dx+24576 \int \frac {\exp \left (\frac {4096 \log ^2(x)}{x^2}+\frac {(-8192 x-8192 \log (4)) \log (x)}{x^2}-x+\frac {4096 \log ^2(4)}{x^2}+\frac {8192 \log (4)}{x}+4096\right ) \log (x)}{x^2}dx-24576 \log (4) (1+\log (4)) \int \frac {\exp \left (\frac {4096 \log ^2(x)}{x^2}+\frac {(-8192 x-8192 \log (4)) \log (x)}{x^2}-x+\frac {4096 \log ^2(4)}{x^2}+\frac {8192 \log (4)}{x}+4096\right )}{x^3}dx+24576 (1+\log (16)) \int \frac {\exp \left (\frac {4096 \log ^2(x)}{x^2}+\frac {(-8192 x-8192 \log (4)) \log (x)}{x^2}-x+\frac {4096 \log ^2(4)}{x^2}+\frac {8192 \log (4)}{x}+4096\right ) \log (x)}{x^3}dx-24576 \int \frac {\exp \left (\frac {4096 \log ^2(x)}{x^2}+\frac {(-8192 x-8192 \log (4)) \log (x)}{x^2}-x+\frac {4096 \log ^2(4)}{x^2}+\frac {8192 \log (4)}{x}+4096\right ) \log ^2(x)}{x^3}dx\) |
Input:
Int[(E^((4096*x^2 - x^3 + 8192*x*Log[4] + 4096*Log[4]^2 + (-8192*x - 8192* Log[4])*Log[x] + 4096*Log[x]^2)/x^2)*(-24576*x - 3*x^3 + (-24576 - 24576*x )*Log[4] - 24576*Log[4]^2 + (24576 + 24576*x + 49152*Log[4])*Log[x] - 2457 6*Log[x]^2))/x^3,x]
Output:
$Aborted
Time = 0.95 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.04
| method | result | size |
| parallelrisch | \(3 \,{\mathrm e}^{\frac {4096 \ln \left (x \right )^{2}+\left (-16384 \ln \left (2\right )-8192 x \right ) \ln \left (x \right )+16384 \ln \left (2\right )^{2}+16384 x \ln \left (2\right )-x^{3}+4096 x^{2}}{x^{2}}}\) | \(47\) |
| risch | \(3 x^{-\frac {16384 \ln \left (2\right )}{x^{2}}} x^{-\frac {8192}{x}} {\mathrm e}^{\frac {-x^{3}+16384 \ln \left (2\right )^{2}+16384 x \ln \left (2\right )+4096 \ln \left (x \right )^{2}+4096 x^{2}}{x^{2}}}\) | \(52\) |
Input:
int((-24576*ln(x)^2+(98304*ln(2)+24576*x+24576)*ln(x)-98304*ln(2)^2+2*(-24 576*x-24576)*ln(2)-3*x^3-24576*x)*exp((4096*ln(x)^2+(-16384*ln(2)-8192*x)* ln(x)+16384*ln(2)^2+16384*x*ln(2)-x^3+4096*x^2)/x^2)/x^3,x,method=_RETURNV ERBOSE)
Output:
3*exp((4096*ln(x)^2+(-16384*ln(2)-8192*x)*ln(x)+16384*ln(2)^2+16384*x*ln(2 )-x^3+4096*x^2)/x^2)
Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.91 \[ \int \frac {e^{\frac {4096 x^2-x^3+8192 x \log (4)+4096 \log ^2(4)+(-8192 x-8192 \log (4)) \log (x)+4096 \log ^2(x)}{x^2}} \left (-24576 x-3 x^3+(-24576-24576 x) \log (4)-24576 \log ^2(4)+(24576+24576 x+49152 \log (4)) \log (x)-24576 \log ^2(x)\right )}{x^3} \, dx=3 \, e^{\left (-\frac {x^{3} - 4096 \, x^{2} - 16384 \, x \log \left (2\right ) - 16384 \, \log \left (2\right )^{2} + 8192 \, {\left (x + 2 \, \log \left (2\right )\right )} \log \left (x\right ) - 4096 \, \log \left (x\right )^{2}}{x^{2}}\right )} \] Input:
integrate((-24576*log(x)^2+(98304*log(2)+24576*x+24576)*log(x)-98304*log(2 )^2+2*(-24576*x-24576)*log(2)-3*x^3-24576*x)*exp((4096*log(x)^2+(-16384*lo g(2)-8192*x)*log(x)+16384*log(2)^2+16384*x*log(2)-x^3+4096*x^2)/x^2)/x^3,x , algorithm="fricas")
Output:
3*e^(-(x^3 - 4096*x^2 - 16384*x*log(2) - 16384*log(2)^2 + 8192*(x + 2*log( 2))*log(x) - 4096*log(x)^2)/x^2)
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (20) = 40\).
Time = 0.24 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.09 \[ \int \frac {e^{\frac {4096 x^2-x^3+8192 x \log (4)+4096 \log ^2(4)+(-8192 x-8192 \log (4)) \log (x)+4096 \log ^2(x)}{x^2}} \left (-24576 x-3 x^3+(-24576-24576 x) \log (4)-24576 \log ^2(4)+(24576+24576 x+49152 \log (4)) \log (x)-24576 \log ^2(x)\right )}{x^3} \, dx=3 e^{\frac {- x^{3} + 4096 x^{2} + 16384 x \log {\left (2 \right )} + \left (- 8192 x - 16384 \log {\left (2 \right )}\right ) \log {\left (x \right )} + 4096 \log {\left (x \right )}^{2} + 16384 \log {\left (2 \right )}^{2}}{x^{2}}} \] Input:
integrate((-24576*ln(x)**2+(98304*ln(2)+24576*x+24576)*ln(x)-98304*ln(2)** 2+2*(-24576*x-24576)*ln(2)-3*x**3-24576*x)*exp((4096*ln(x)**2+(-16384*ln(2 )-8192*x)*ln(x)+16384*ln(2)**2+16384*x*ln(2)-x**3+4096*x**2)/x**2)/x**3,x)
Output:
3*exp((-x**3 + 4096*x**2 + 16384*x*log(2) + (-8192*x - 16384*log(2))*log(x ) + 4096*log(x)**2 + 16384*log(2)**2)/x**2)
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (24) = 48\).
Time = 1.88 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.13 \[ \int \frac {e^{\frac {4096 x^2-x^3+8192 x \log (4)+4096 \log ^2(4)+(-8192 x-8192 \log (4)) \log (x)+4096 \log ^2(x)}{x^2}} \left (-24576 x-3 x^3+(-24576-24576 x) \log (4)-24576 \log ^2(4)+(24576+24576 x+49152 \log (4)) \log (x)-24576 \log ^2(x)\right )}{x^3} \, dx=3 \, e^{\left (-x + \frac {16384 \, \log \left (2\right )}{x} + \frac {16384 \, \log \left (2\right )^{2}}{x^{2}} - \frac {8192 \, \log \left (x\right )}{x} - \frac {16384 \, \log \left (2\right ) \log \left (x\right )}{x^{2}} + \frac {4096 \, \log \left (x\right )^{2}}{x^{2}} + 4096\right )} \] Input:
integrate((-24576*log(x)^2+(98304*log(2)+24576*x+24576)*log(x)-98304*log(2 )^2+2*(-24576*x-24576)*log(2)-3*x^3-24576*x)*exp((4096*log(x)^2+(-16384*lo g(2)-8192*x)*log(x)+16384*log(2)^2+16384*x*log(2)-x^3+4096*x^2)/x^2)/x^3,x , algorithm="maxima")
Output:
3*e^(-x + 16384*log(2)/x + 16384*log(2)^2/x^2 - 8192*log(x)/x - 16384*log( 2)*log(x)/x^2 + 4096*log(x)^2/x^2 + 4096)
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (24) = 48\).
Time = 0.14 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.13 \[ \int \frac {e^{\frac {4096 x^2-x^3+8192 x \log (4)+4096 \log ^2(4)+(-8192 x-8192 \log (4)) \log (x)+4096 \log ^2(x)}{x^2}} \left (-24576 x-3 x^3+(-24576-24576 x) \log (4)-24576 \log ^2(4)+(24576+24576 x+49152 \log (4)) \log (x)-24576 \log ^2(x)\right )}{x^3} \, dx=3 \, e^{\left (-x + \frac {16384 \, \log \left (2\right )}{x} + \frac {16384 \, \log \left (2\right )^{2}}{x^{2}} - \frac {8192 \, \log \left (x\right )}{x} - \frac {16384 \, \log \left (2\right ) \log \left (x\right )}{x^{2}} + \frac {4096 \, \log \left (x\right )^{2}}{x^{2}} + 4096\right )} \] Input:
integrate((-24576*log(x)^2+(98304*log(2)+24576*x+24576)*log(x)-98304*log(2 )^2+2*(-24576*x-24576)*log(2)-3*x^3-24576*x)*exp((4096*log(x)^2+(-16384*lo g(2)-8192*x)*log(x)+16384*log(2)^2+16384*x*log(2)-x^3+4096*x^2)/x^2)/x^3,x , algorithm="giac")
Output:
3*e^(-x + 16384*log(2)/x + 16384*log(2)^2/x^2 - 8192*log(x)/x - 16384*log( 2)*log(x)/x^2 + 4096*log(x)^2/x^2 + 4096)
Time = 1.48 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.26 \[ \int \frac {e^{\frac {4096 x^2-x^3+8192 x \log (4)+4096 \log ^2(4)+(-8192 x-8192 \log (4)) \log (x)+4096 \log ^2(x)}{x^2}} \left (-24576 x-3 x^3+(-24576-24576 x) \log (4)-24576 \log ^2(4)+(24576+24576 x+49152 \log (4)) \log (x)-24576 \log ^2(x)\right )}{x^3} \, dx=\frac {3\,2^{16384/x}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{4096}\,{\mathrm {e}}^{\frac {16384\,{\ln \left (2\right )}^2}{x^2}}\,{\mathrm {e}}^{\frac {4096\,{\ln \left (x\right )}^2}{x^2}}}{x^{\frac {16384\,\ln \left (2\right )}{x^2}+\frac {8192}{x}}} \] Input:
int(-(exp((16384*x*log(2) + 4096*log(x)^2 - log(x)*(8192*x + 16384*log(2)) + 16384*log(2)^2 + 4096*x^2 - x^3)/x^2)*(24576*x + 2*log(2)*(24576*x + 24 576) + 24576*log(x)^2 - log(x)*(24576*x + 98304*log(2) + 24576) + 98304*lo g(2)^2 + 3*x^3))/x^3,x)
Output:
(3*2^(16384/x)*exp(-x)*exp(4096)*exp((16384*log(2)^2)/x^2)*exp((4096*log(x )^2)/x^2))/x^((16384*log(2))/x^2 + 8192/x)
\[ \int \frac {e^{\frac {4096 x^2-x^3+8192 x \log (4)+4096 \log ^2(4)+(-8192 x-8192 \log (4)) \log (x)+4096 \log ^2(x)}{x^2}} \left (-24576 x-3 x^3+(-24576-24576 x) \log (4)-24576 \log ^2(4)+(24576+24576 x+49152 \log (4)) \log (x)-24576 \log ^2(x)\right )}{x^3} \, dx=\int \frac {\left (-24576 \mathrm {log}\left (x \right )^{2}+\left (98304 \,\mathrm {log}\left (2\right )+24576 x +24576\right ) \mathrm {log}\left (x \right )-98304 \mathrm {log}\left (2\right )^{2}+2 \left (-24576 x -24576\right ) \mathrm {log}\left (2\right )-3 x^{3}-24576 x \right ) {\mathrm e}^{\frac {4096 \mathrm {log}\left (x \right )^{2}+\left (-16384 \,\mathrm {log}\left (2\right )-8192 x \right ) \mathrm {log}\left (x \right )+16384 \mathrm {log}\left (2\right )^{2}+16384 \,\mathrm {log}\left (2\right ) x -x^{3}+4096 x^{2}}{x^{2}}}}{x^{3}}d x \] Input:
int((-24576*log(x)^2+(98304*log(2)+24576*x+24576)*log(x)-98304*log(2)^2+2* (-24576*x-24576)*log(2)-3*x^3-24576*x)*exp((4096*log(x)^2+(-16384*log(2)-8 192*x)*log(x)+16384*log(2)^2+16384*x*log(2)-x^3+4096*x^2)/x^2)/x^3,x)
Output:
int((-24576*log(x)^2+(98304*log(2)+24576*x+24576)*log(x)-98304*log(2)^2+2* (-24576*x-24576)*log(2)-3*x^3-24576*x)*exp((4096*log(x)^2+(-16384*log(2)-8 192*x)*log(x)+16384*log(2)^2+16384*x*log(2)-x^3+4096*x^2)/x^2)/x^3,x)