\(\int \frac {e^{-2+\frac {e^{-2-2 x} (7744 e^2+e^x (-704 e-704 e^{2+\frac {2}{x}})+e^{2 x} (16+32 e^{1+\frac {2}{x}}+16 e^{2+\frac {4}{x}}))}{x^2}-2 x} (e^{2 x} (e^{1+\frac {2}{x}} (-64-64 x)+e^{2+\frac {4}{x}} (-64-32 x)-32 x)+e^2 (-15488 x-15488 x^2)+e^x (e (1408 x+704 x^2)+e^{2+\frac {2}{x}} (1408+1408 x+704 x^2)))}{x^4} \, dx\) [841]

Optimal result

Integrand size = 167, antiderivative size = 31 \[ \int \frac {e^{-2+\frac {e^{-2-2 x} \left (7744 e^2+e^x \left (-704 e-704 e^{2+\frac {2}{x}}\right )+e^{2 x} \left (16+32 e^{1+\frac {2}{x}}+16 e^{2+\frac {4}{x}}\right )\right )}{x^2}-2 x} \left (e^{2 x} \left (e^{1+\frac {2}{x}} (-64-64 x)+e^{2+\frac {4}{x}} (-64-32 x)-32 x\right )+e^2 \left (-15488 x-15488 x^2\right )+e^x \left (e \left (1408 x+704 x^2\right )+e^{2+\frac {2}{x}} \left (1408+1408 x+704 x^2\right )\right )\right )}{x^4} \, dx=e^{\frac {16 \left (-\frac {1}{e}-e^{2/x}+22 e^{-x}\right )^2}{x^2}} \] Output:

exp(16*(22/exp(x)-1/exp(1)-exp(2/x))^2/x^2)
 

Mathematica [F]

\[ \int \frac {e^{-2+\frac {e^{-2-2 x} \left (7744 e^2+e^x \left (-704 e-704 e^{2+\frac {2}{x}}\right )+e^{2 x} \left (16+32 e^{1+\frac {2}{x}}+16 e^{2+\frac {4}{x}}\right )\right )}{x^2}-2 x} \left (e^{2 x} \left (e^{1+\frac {2}{x}} (-64-64 x)+e^{2+\frac {4}{x}} (-64-32 x)-32 x\right )+e^2 \left (-15488 x-15488 x^2\right )+e^x \left (e \left (1408 x+704 x^2\right )+e^{2+\frac {2}{x}} \left (1408+1408 x+704 x^2\right )\right )\right )}{x^4} \, dx=\int \frac {e^{-2+\frac {e^{-2-2 x} \left (7744 e^2+e^x \left (-704 e-704 e^{2+\frac {2}{x}}\right )+e^{2 x} \left (16+32 e^{1+\frac {2}{x}}+16 e^{2+\frac {4}{x}}\right )\right )}{x^2}-2 x} \left (e^{2 x} \left (e^{1+\frac {2}{x}} (-64-64 x)+e^{2+\frac {4}{x}} (-64-32 x)-32 x\right )+e^2 \left (-15488 x-15488 x^2\right )+e^x \left (e \left (1408 x+704 x^2\right )+e^{2+\frac {2}{x}} \left (1408+1408 x+704 x^2\right )\right )\right )}{x^4} \, dx \] Input:

Integrate[(E^(-2 + (E^(-2 - 2*x)*(7744*E^2 + E^x*(-704*E - 704*E^(2 + 2/x) 
) + E^(2*x)*(16 + 32*E^(1 + 2/x) + 16*E^(2 + 4/x))))/x^2 - 2*x)*(E^(2*x)*( 
E^(1 + 2/x)*(-64 - 64*x) + E^(2 + 4/x)*(-64 - 32*x) - 32*x) + E^2*(-15488* 
x - 15488*x^2) + E^x*(E*(1408*x + 704*x^2) + E^(2 + 2/x)*(1408 + 1408*x + 
704*x^2))))/x^4,x]
 

Output:

Integrate[(E^(-2 + (E^(-2 - 2*x)*(7744*E^2 + E^x*(-704*E - 704*E^(2 + 2/x) 
) + E^(2*x)*(16 + 32*E^(1 + 2/x) + 16*E^(2 + 4/x))))/x^2 - 2*x)*(E^(2*x)*( 
E^(1 + 2/x)*(-64 - 64*x) + E^(2 + 4/x)*(-64 - 32*x) - 32*x) + E^2*(-15488* 
x - 15488*x^2) + E^x*(E*(1408*x + 704*x^2) + E^(2 + 2/x)*(1408 + 1408*x + 
704*x^2))))/x^4, x]
 

Rubi [F]

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.

\(\displaystyle \int \frac {\left (e^2 \left (-15488 x^2-15488 x\right )+e^x \left (e \left (704 x^2+1408 x\right )+e^{\frac {2}{x}+2} \left (704 x^2+1408 x+1408\right )\right )+e^{2 x} \left (e^{\frac {2}{x}+1} (-64 x-64)+e^{\frac {4}{x}+2} (-32 x-64)-32 x\right )\right ) \exp \left (\frac {e^{-2 x-2} \left (e^x \left (-704 e^{\frac {2}{x}+2}-704 e\right )+e^{2 x} \left (32 e^{\frac {2}{x}+1}+16 e^{\frac {4}{x}+2}+16\right )+7744 e^2\right )}{x^2}-2 x-2\right )}{x^4} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {32 \left (-e^x-e^{x+\frac {2}{x}+1}+22 e\right ) \left (-22 e x^2+e^x x+e^{x+\frac {2}{x}+1} x-22 e x+2 e^{x+\frac {2}{x}+1}\right ) \exp \left (\frac {e^{-2 x-2} \left (e^x \left (-704 e^{\frac {2}{x}+2}-704 e\right )+e^{2 x} \left (32 e^{\frac {2}{x}+1}+16 e^{\frac {4}{x}+2}+16\right )+7744 e^2\right )}{x^2}-2 x-2\right )}{x^4}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 32 \int \frac {\exp \left (\frac {16 e^{-2 x-2} \left (-44 e^x \left (e+e^{2+\frac {2}{x}}\right )+e^{2 x} \left (1+2 e^{1+\frac {2}{x}}+e^{2+\frac {4}{x}}\right )+484 e^2\right )}{x^2}-2 x-2\right ) \left (22 e-e^x-e^{x+1+\frac {2}{x}}\right ) \left (-22 e x^2+e^x x+e^{x+1+\frac {2}{x}} x-22 e x+2 e^{x+1+\frac {2}{x}}\right )}{x^4}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 32 \int \left (-\frac {484 \exp \left (\frac {16 e^{-2 x-2} \left (-44 e^x \left (e+e^{2+\frac {2}{x}}\right )+e^{2 x} \left (1+2 e^{1+\frac {2}{x}}+e^{2+\frac {4}{x}}\right )+484 e^2\right )}{x^2}-2 x\right ) (x+1)}{x^3}-\frac {\exp \left (\frac {16 e^{-2 x-2} \left (-44 e^x \left (e+e^{2+\frac {2}{x}}\right )+e^{2 x} \left (1+2 e^{1+\frac {2}{x}}+e^{2+\frac {4}{x}}\right )+484 e^2\right )}{x^2}-2\right ) \left (1+e^{1+\frac {2}{x}}\right ) \left (e^{1+\frac {2}{x}} x+x+2 e^{1+\frac {2}{x}}\right )}{x^4}+\frac {22 \exp \left (\frac {16 e^{-2 x-2} \left (-44 e^x \left (e+e^{2+\frac {2}{x}}\right )+e^{2 x} \left (1+2 e^{1+\frac {2}{x}}+e^{2+\frac {4}{x}}\right )+484 e^2\right )}{x^2}-x-1\right ) \left (e^{1+\frac {2}{x}} x^2+x^2+2 e^{1+\frac {2}{x}} x+2 x+2 e^{1+\frac {2}{x}}\right )}{x^4}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 32 \int \frac {\exp \left (-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (22 e-e^x-e^{x+1+\frac {2}{x}}\right ) \left (e^x x-22 e (x+1) x+e^{x+1+\frac {2}{x}} (x+2)\right )}{x^4}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 32 \int \left (-\frac {484 \exp \left (-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}+2\right ) (x+1)}{x^3}-\frac {\exp \left (2 x-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (1+e^{1+\frac {2}{x}}\right ) \left (e^{1+\frac {2}{x}} x+x+2 e^{1+\frac {2}{x}}\right )}{x^4}+\frac {22 \exp \left (x-2 (x+1)+1+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (e^{1+\frac {2}{x}} x^2+x^2+2 e^{1+\frac {2}{x}} x+2 x+2 e^{1+\frac {2}{x}}\right )}{x^4}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 32 \int \frac {\exp \left (-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (22 e-e^x-e^{x+1+\frac {2}{x}}\right ) \left (e^x x-22 e (x+1) x+e^{x+1+\frac {2}{x}} (x+2)\right )}{x^4}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 32 \int \left (-\frac {484 \exp \left (-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}+2\right ) (x+1)}{x^3}-\frac {\exp \left (2 x-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (1+e^{1+\frac {2}{x}}\right ) \left (e^{1+\frac {2}{x}} x+x+2 e^{1+\frac {2}{x}}\right )}{x^4}+\frac {22 \exp \left (x-2 (x+1)+1+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (e^{1+\frac {2}{x}} x^2+x^2+2 e^{1+\frac {2}{x}} x+2 x+2 e^{1+\frac {2}{x}}\right )}{x^4}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 32 \int \frac {\exp \left (-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (22 e-e^x-e^{x+1+\frac {2}{x}}\right ) \left (e^x x-22 e (x+1) x+e^{x+1+\frac {2}{x}} (x+2)\right )}{x^4}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 32 \int \left (-\frac {484 \exp \left (-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}+2\right ) (x+1)}{x^3}-\frac {\exp \left (2 x-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (1+e^{1+\frac {2}{x}}\right ) \left (e^{1+\frac {2}{x}} x+x+2 e^{1+\frac {2}{x}}\right )}{x^4}+\frac {22 \exp \left (x-2 (x+1)+1+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (e^{1+\frac {2}{x}} x^2+x^2+2 e^{1+\frac {2}{x}} x+2 x+2 e^{1+\frac {2}{x}}\right )}{x^4}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 32 \int \frac {\exp \left (-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (22 e-e^x-e^{x+1+\frac {2}{x}}\right ) \left (e^x x-22 e (x+1) x+e^{x+1+\frac {2}{x}} (x+2)\right )}{x^4}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 32 \int \left (-\frac {484 \exp \left (-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}+2\right ) (x+1)}{x^3}-\frac {\exp \left (2 x-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (1+e^{1+\frac {2}{x}}\right ) \left (e^{1+\frac {2}{x}} x+x+2 e^{1+\frac {2}{x}}\right )}{x^4}+\frac {22 \exp \left (x-2 (x+1)+1+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (e^{1+\frac {2}{x}} x^2+x^2+2 e^{1+\frac {2}{x}} x+2 x+2 e^{1+\frac {2}{x}}\right )}{x^4}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 32 \int \frac {\exp \left (-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (22 e-e^x-e^{x+1+\frac {2}{x}}\right ) \left (e^x x-22 e (x+1) x+e^{x+1+\frac {2}{x}} (x+2)\right )}{x^4}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 32 \int \left (-\frac {484 \exp \left (-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}+2\right ) (x+1)}{x^3}-\frac {\exp \left (2 x-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (1+e^{1+\frac {2}{x}}\right ) \left (e^{1+\frac {2}{x}} x+x+2 e^{1+\frac {2}{x}}\right )}{x^4}+\frac {22 \exp \left (x-2 (x+1)+1+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (e^{1+\frac {2}{x}} x^2+x^2+2 e^{1+\frac {2}{x}} x+2 x+2 e^{1+\frac {2}{x}}\right )}{x^4}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 32 \int \frac {\exp \left (-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (22 e-e^x-e^{x+1+\frac {2}{x}}\right ) \left (e^x x-22 e (x+1) x+e^{x+1+\frac {2}{x}} (x+2)\right )}{x^4}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 32 \int \left (-\frac {484 \exp \left (-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}+2\right ) (x+1)}{x^3}-\frac {\exp \left (2 x-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (1+e^{1+\frac {2}{x}}\right ) \left (e^{1+\frac {2}{x}} x+x+2 e^{1+\frac {2}{x}}\right )}{x^4}+\frac {22 \exp \left (x-2 (x+1)+1+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (e^{1+\frac {2}{x}} x^2+x^2+2 e^{1+\frac {2}{x}} x+2 x+2 e^{1+\frac {2}{x}}\right )}{x^4}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 32 \int \frac {\exp \left (-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (22 e-e^x-e^{x+1+\frac {2}{x}}\right ) \left (e^x x-22 e (x+1) x+e^{x+1+\frac {2}{x}} (x+2)\right )}{x^4}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 32 \int \left (-\frac {484 \exp \left (-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}+2\right ) (x+1)}{x^3}-\frac {\exp \left (2 x-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (1+e^{1+\frac {2}{x}}\right ) \left (e^{1+\frac {2}{x}} x+x+2 e^{1+\frac {2}{x}}\right )}{x^4}+\frac {22 \exp \left (x-2 (x+1)+1+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (e^{1+\frac {2}{x}} x^2+x^2+2 e^{1+\frac {2}{x}} x+2 x+2 e^{1+\frac {2}{x}}\right )}{x^4}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 32 \int \frac {\exp \left (-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (22 e-e^x-e^{x+1+\frac {2}{x}}\right ) \left (e^x x-22 e (x+1) x+e^{x+1+\frac {2}{x}} (x+2)\right )}{x^4}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 32 \int \left (-\frac {484 \exp \left (-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}+2\right ) (x+1)}{x^3}-\frac {\exp \left (2 x-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (1+e^{1+\frac {2}{x}}\right ) \left (e^{1+\frac {2}{x}} x+x+2 e^{1+\frac {2}{x}}\right )}{x^4}+\frac {22 \exp \left (x-2 (x+1)+1+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (e^{1+\frac {2}{x}} x^2+x^2+2 e^{1+\frac {2}{x}} x+2 x+2 e^{1+\frac {2}{x}}\right )}{x^4}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 32 \int \frac {\exp \left (-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (22 e-e^x-e^{x+1+\frac {2}{x}}\right ) \left (e^x x-22 e (x+1) x+e^{x+1+\frac {2}{x}} (x+2)\right )}{x^4}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 32 \int \left (-\frac {484 \exp \left (-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}+2\right ) (x+1)}{x^3}-\frac {\exp \left (2 x-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (1+e^{1+\frac {2}{x}}\right ) \left (e^{1+\frac {2}{x}} x+x+2 e^{1+\frac {2}{x}}\right )}{x^4}+\frac {22 \exp \left (x-2 (x+1)+1+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (e^{1+\frac {2}{x}} x^2+x^2+2 e^{1+\frac {2}{x}} x+2 x+2 e^{1+\frac {2}{x}}\right )}{x^4}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 32 \int \frac {\exp \left (-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (22 e-e^x-e^{x+1+\frac {2}{x}}\right ) \left (e^x x-22 e (x+1) x+e^{x+1+\frac {2}{x}} (x+2)\right )}{x^4}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 32 \int \left (-\frac {484 \exp \left (-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}+2\right ) (x+1)}{x^3}-\frac {\exp \left (2 x-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (1+e^{1+\frac {2}{x}}\right ) \left (e^{1+\frac {2}{x}} x+x+2 e^{1+\frac {2}{x}}\right )}{x^4}+\frac {22 \exp \left (x-2 (x+1)+1+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (e^{1+\frac {2}{x}} x^2+x^2+2 e^{1+\frac {2}{x}} x+2 x+2 e^{1+\frac {2}{x}}\right )}{x^4}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 32 \int \frac {\exp \left (-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (22 e-e^x-e^{x+1+\frac {2}{x}}\right ) \left (e^x x-22 e (x+1) x+e^{x+1+\frac {2}{x}} (x+2)\right )}{x^4}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 32 \int \left (-\frac {484 \exp \left (-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}+2\right ) (x+1)}{x^3}-\frac {\exp \left (2 x-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (1+e^{1+\frac {2}{x}}\right ) \left (e^{1+\frac {2}{x}} x+x+2 e^{1+\frac {2}{x}}\right )}{x^4}+\frac {22 \exp \left (x-2 (x+1)+1+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (e^{1+\frac {2}{x}} x^2+x^2+2 e^{1+\frac {2}{x}} x+2 x+2 e^{1+\frac {2}{x}}\right )}{x^4}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 32 \int \frac {\exp \left (-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (22 e-e^x-e^{x+1+\frac {2}{x}}\right ) \left (e^x x-22 e (x+1) x+e^{x+1+\frac {2}{x}} (x+2)\right )}{x^4}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 32 \int \left (-\frac {484 \exp \left (-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}+2\right ) (x+1)}{x^3}-\frac {\exp \left (2 x-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (1+e^{1+\frac {2}{x}}\right ) \left (e^{1+\frac {2}{x}} x+x+2 e^{1+\frac {2}{x}}\right )}{x^4}+\frac {22 \exp \left (x-2 (x+1)+1+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (e^{1+\frac {2}{x}} x^2+x^2+2 e^{1+\frac {2}{x}} x+2 x+2 e^{1+\frac {2}{x}}\right )}{x^4}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 32 \int \frac {\exp \left (-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (22 e-e^x-e^{x+1+\frac {2}{x}}\right ) \left (e^x x-22 e (x+1) x+e^{x+1+\frac {2}{x}} (x+2)\right )}{x^4}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 32 \int \left (-\frac {484 \exp \left (-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}+2\right ) (x+1)}{x^3}-\frac {\exp \left (2 x-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (1+e^{1+\frac {2}{x}}\right ) \left (e^{1+\frac {2}{x}} x+x+2 e^{1+\frac {2}{x}}\right )}{x^4}+\frac {22 \exp \left (x-2 (x+1)+1+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (e^{1+\frac {2}{x}} x^2+x^2+2 e^{1+\frac {2}{x}} x+2 x+2 e^{1+\frac {2}{x}}\right )}{x^4}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 32 \int \frac {\exp \left (-2 (x+1)+\frac {32 e^{\frac {2}{x}-1}}{x^2}-\frac {704 e^{-x-1}}{x^2}-\frac {704 e^{\frac {2}{x}-x}}{x^2}+\frac {16 e^{4/x}}{x^2}+\frac {7744 e^{-2 x}}{x^2}+\frac {16}{e^2 x^2}\right ) \left (22 e-e^x-e^{x+1+\frac {2}{x}}\right ) \left (e^x x-22 e (x+1) x+e^{x+1+\frac {2}{x}} (x+2)\right )}{x^4}dx\)

Input:

Int[(E^(-2 + (E^(-2 - 2*x)*(7744*E^2 + E^x*(-704*E - 704*E^(2 + 2/x)) + E^ 
(2*x)*(16 + 32*E^(1 + 2/x) + 16*E^(2 + 4/x))))/x^2 - 2*x)*(E^(2*x)*(E^(1 + 
 2/x)*(-64 - 64*x) + E^(2 + 4/x)*(-64 - 32*x) - 32*x) + E^2*(-15488*x - 15 
488*x^2) + E^x*(E*(1408*x + 704*x^2) + E^(2 + 2/x)*(1408 + 1408*x + 704*x^ 
2))))/x^4,x]
 

Output:

$Aborted
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(69\) vs. \(2(29)=58\).

Time = 132.88 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.26

method	result	size
risch	\({\mathrm e}^{\frac {16 \left (-44 \,{\mathrm e}^{1+x}-44 \,{\mathrm e}^{\frac {x^{2}+2 x +2}{x}}+{\mathrm e}^{\frac {2 x^{2}+2 x +4}{x}}+2 \,{\mathrm e}^{\frac {2 x^{2}+x +2}{x}}+{\mathrm e}^{2 x}+484 \,{\mathrm e}^{2}\right ) {\mathrm e}^{-2-2 x}}{x^{2}}}\)	\(70\)
parallelrisch	\({\mathrm e}^{\frac {\left (\left (16 \,{\mathrm e}^{2} {\mathrm e}^{\frac {4}{x}}+32 \,{\mathrm e} \,{\mathrm e}^{\frac {2}{x}}+16\right ) {\mathrm e}^{2 x}+\left (-704 \,{\mathrm e}^{2} {\mathrm e}^{\frac {2}{x}}-704 \,{\mathrm e}\right ) {\mathrm e}^{x}+7744 \,{\mathrm e}^{2}\right ) {\mathrm e}^{-2} {\mathrm e}^{-2 x}}{x^{2}}}\)	\(72\)

Input:

int((((-32*x-64)*exp(1)^2*exp(2/x)^2+(-64*x-64)*exp(1)*exp(2/x)-32*x)*exp( 
x)^2+((704*x^2+1408*x+1408)*exp(1)^2*exp(2/x)+(704*x^2+1408*x)*exp(1))*exp 
(x)+(-15488*x^2-15488*x)*exp(1)^2)*exp(((16*exp(1)^2*exp(2/x)^2+32*exp(1)* 
exp(2/x)+16)*exp(x)^2+(-704*exp(1)^2*exp(2/x)-704*exp(1))*exp(x)+7744*exp( 
1)^2)/x^2/exp(1)^2/exp(x)^2)/x^4/exp(1)^2/exp(x)^2,x,method=_RETURNVERBOSE 
)
 

Output:

exp(16*(-44*exp(1+x)-44*exp((x^2+2*x+2)/x)+exp(2*(x^2+x+2)/x)+2*exp((2*x^2 
+x+2)/x)+exp(2*x)+484*exp(2))*exp(-2-2*x)/x^2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (23) = 46\).

Time = 0.12 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.71 \[ \int \frac {e^{-2+\frac {e^{-2-2 x} \left (7744 e^2+e^x \left (-704 e-704 e^{2+\frac {2}{x}}\right )+e^{2 x} \left (16+32 e^{1+\frac {2}{x}}+16 e^{2+\frac {4}{x}}\right )\right )}{x^2}-2 x} \left (e^{2 x} \left (e^{1+\frac {2}{x}} (-64-64 x)+e^{2+\frac {4}{x}} (-64-32 x)-32 x\right )+e^2 \left (-15488 x-15488 x^2\right )+e^x \left (e \left (1408 x+704 x^2\right )+e^{2+\frac {2}{x}} \left (1408+1408 x+704 x^2\right )\right )\right )}{x^4} \, dx=e^{\left (2 \, x - \frac {2 \, {\left ({\left ({\left (x^{3} + x^{2}\right )} e^{4} - 8 \, e^{2} - 8 \, e^{\left (\frac {4 \, {\left (x + 1\right )}}{x}\right )} - 16 \, e^{\left (\frac {2 \, {\left (x + 1\right )}}{x} + 1\right )}\right )} e^{\left (2 \, x\right )} + 352 \, {\left (e^{3} + e^{\left (\frac {2 \, {\left (x + 1\right )}}{x} + 2\right )}\right )} e^{x} - 3872 \, e^{4}\right )} e^{\left (-2 \, x - 4\right )}}{x^{2}} + 2\right )} \] Input:

integrate((((-32*x-64)*exp(1)^2*exp(2/x)^2+(-64*x-64)*exp(1)*exp(2/x)-32*x 
)*exp(x)^2+((704*x^2+1408*x+1408)*exp(1)^2*exp(2/x)+(704*x^2+1408*x)*exp(1 
))*exp(x)+(-15488*x^2-15488*x)*exp(1)^2)*exp(((16*exp(1)^2*exp(2/x)^2+32*e 
xp(1)*exp(2/x)+16)*exp(x)^2+(-704*exp(1)^2*exp(2/x)-704*exp(1))*exp(x)+774 
4*exp(1)^2)/x^2/exp(1)^2/exp(x)^2)/x^4/exp(1)^2/exp(x)^2,x, algorithm="fri 
cas")
 

Output:

e^(2*x - 2*(((x^3 + x^2)*e^4 - 8*e^2 - 8*e^(4*(x + 1)/x) - 16*e^(2*(x + 1) 
/x + 1))*e^(2*x) + 352*(e^3 + e^(2*(x + 1)/x + 2))*e^x - 3872*e^4)*e^(-2*x 
 - 4)/x^2 + 2)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (22) = 44\).

Time = 6.00 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.10 \[ \int \frac {e^{-2+\frac {e^{-2-2 x} \left (7744 e^2+e^x \left (-704 e-704 e^{2+\frac {2}{x}}\right )+e^{2 x} \left (16+32 e^{1+\frac {2}{x}}+16 e^{2+\frac {4}{x}}\right )\right )}{x^2}-2 x} \left (e^{2 x} \left (e^{1+\frac {2}{x}} (-64-64 x)+e^{2+\frac {4}{x}} (-64-32 x)-32 x\right )+e^2 \left (-15488 x-15488 x^2\right )+e^x \left (e \left (1408 x+704 x^2\right )+e^{2+\frac {2}{x}} \left (1408+1408 x+704 x^2\right )\right )\right )}{x^4} \, dx=e^{\frac {\left (\left (- 704 e^{2} e^{\frac {2}{x}} - 704 e\right ) e^{x} + \left (16 e^{2} e^{\frac {4}{x}} + 32 e e^{\frac {2}{x}} + 16\right ) e^{2 x} + 7744 e^{2}\right ) e^{- 2 x}}{x^{2} e^{2}}} \] Input:

integrate((((-32*x-64)*exp(1)**2*exp(2/x)**2+(-64*x-64)*exp(1)*exp(2/x)-32 
*x)*exp(x)**2+((704*x**2+1408*x+1408)*exp(1)**2*exp(2/x)+(704*x**2+1408*x) 
*exp(1))*exp(x)+(-15488*x**2-15488*x)*exp(1)**2)*exp(((16*exp(1)**2*exp(2/ 
x)**2+32*exp(1)*exp(2/x)+16)*exp(x)**2+(-704*exp(1)**2*exp(2/x)-704*exp(1) 
)*exp(x)+7744*exp(1)**2)/x**2/exp(1)**2/exp(x)**2)/x**4/exp(1)**2/exp(x)** 
2,x)
 

Output:

exp(((-704*exp(2)*exp(2/x) - 704*E)*exp(x) + (16*exp(2)*exp(4/x) + 32*E*ex 
p(2/x) + 16)*exp(2*x) + 7744*exp(2))*exp(-2)*exp(-2*x)/x**2)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (23) = 46\).

Time = 1.38 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.19 \[ \int \frac {e^{-2+\frac {e^{-2-2 x} \left (7744 e^2+e^x \left (-704 e-704 e^{2+\frac {2}{x}}\right )+e^{2 x} \left (16+32 e^{1+\frac {2}{x}}+16 e^{2+\frac {4}{x}}\right )\right )}{x^2}-2 x} \left (e^{2 x} \left (e^{1+\frac {2}{x}} (-64-64 x)+e^{2+\frac {4}{x}} (-64-32 x)-32 x\right )+e^2 \left (-15488 x-15488 x^2\right )+e^x \left (e \left (1408 x+704 x^2\right )+e^{2+\frac {2}{x}} \left (1408+1408 x+704 x^2\right )\right )\right )}{x^4} \, dx=e^{\left (\frac {16 \, e^{\left (-2\right )}}{x^{2}} + \frac {7744 \, e^{\left (-2 \, x\right )}}{x^{2}} - \frac {704 \, e^{\left (-x + \frac {2}{x}\right )}}{x^{2}} - \frac {704 \, e^{\left (-x - 1\right )}}{x^{2}} + \frac {16 \, e^{\frac {4}{x}}}{x^{2}} + \frac {32 \, e^{\left (\frac {2}{x} - 1\right )}}{x^{2}}\right )} \] Input:

integrate((((-32*x-64)*exp(1)^2*exp(2/x)^2+(-64*x-64)*exp(1)*exp(2/x)-32*x 
)*exp(x)^2+((704*x^2+1408*x+1408)*exp(1)^2*exp(2/x)+(704*x^2+1408*x)*exp(1 
))*exp(x)+(-15488*x^2-15488*x)*exp(1)^2)*exp(((16*exp(1)^2*exp(2/x)^2+32*e 
xp(1)*exp(2/x)+16)*exp(x)^2+(-704*exp(1)^2*exp(2/x)-704*exp(1))*exp(x)+774 
4*exp(1)^2)/x^2/exp(1)^2/exp(x)^2)/x^4/exp(1)^2/exp(x)^2,x, algorithm="max 
ima")
 

Output:

e^(16*e^(-2)/x^2 + 7744*e^(-2*x)/x^2 - 704*e^(-x + 2/x)/x^2 - 704*e^(-x - 
1)/x^2 + 16*e^(4/x)/x^2 + 32*e^(2/x - 1)/x^2)
 

Giac [F]

\[ \int \frac {e^{-2+\frac {e^{-2-2 x} \left (7744 e^2+e^x \left (-704 e-704 e^{2+\frac {2}{x}}\right )+e^{2 x} \left (16+32 e^{1+\frac {2}{x}}+16 e^{2+\frac {4}{x}}\right )\right )}{x^2}-2 x} \left (e^{2 x} \left (e^{1+\frac {2}{x}} (-64-64 x)+e^{2+\frac {4}{x}} (-64-32 x)-32 x\right )+e^2 \left (-15488 x-15488 x^2\right )+e^x \left (e \left (1408 x+704 x^2\right )+e^{2+\frac {2}{x}} \left (1408+1408 x+704 x^2\right )\right )\right )}{x^4} \, dx=\int { -\frac {32 \, {\left (484 \, {\left (x^{2} + x\right )} e^{2} + {\left ({\left (x + 2\right )} e^{\left (\frac {4}{x} + 2\right )} + 2 \, {\left (x + 1\right )} e^{\left (\frac {2}{x} + 1\right )} + x\right )} e^{\left (2 \, x\right )} - 22 \, {\left ({\left (x^{2} + 2 \, x\right )} e + {\left (x^{2} + 2 \, x + 2\right )} e^{\left (\frac {2}{x} + 2\right )}\right )} e^{x}\right )} e^{\left (-2 \, x + \frac {16 \, {\left ({\left (e^{\left (\frac {4}{x} + 2\right )} + 2 \, e^{\left (\frac {2}{x} + 1\right )} + 1\right )} e^{\left (2 \, x\right )} - 44 \, {\left (e + e^{\left (\frac {2}{x} + 2\right )}\right )} e^{x} + 484 \, e^{2}\right )} e^{\left (-2 \, x - 2\right )}}{x^{2}} - 2\right )}}{x^{4}} \,d x } \] Input:

integrate((((-32*x-64)*exp(1)^2*exp(2/x)^2+(-64*x-64)*exp(1)*exp(2/x)-32*x 
)*exp(x)^2+((704*x^2+1408*x+1408)*exp(1)^2*exp(2/x)+(704*x^2+1408*x)*exp(1 
))*exp(x)+(-15488*x^2-15488*x)*exp(1)^2)*exp(((16*exp(1)^2*exp(2/x)^2+32*e 
xp(1)*exp(2/x)+16)*exp(x)^2+(-704*exp(1)^2*exp(2/x)-704*exp(1))*exp(x)+774 
4*exp(1)^2)/x^2/exp(1)^2/exp(x)^2)/x^4/exp(1)^2/exp(x)^2,x, algorithm="gia 
c")
 

Output:

integrate(-32*(484*(x^2 + x)*e^2 + ((x + 2)*e^(4/x + 2) + 2*(x + 1)*e^(2/x 
 + 1) + x)*e^(2*x) - 22*((x^2 + 2*x)*e + (x^2 + 2*x + 2)*e^(2/x + 2))*e^x) 
*e^(-2*x + 16*((e^(4/x + 2) + 2*e^(2/x + 1) + 1)*e^(2*x) - 44*(e + e^(2/x 
+ 2))*e^x + 484*e^2)*e^(-2*x - 2)/x^2 - 2)/x^4, x)
 

Mupad [B] (verification not implemented)

Time = 3.51 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.35 \[ \int \frac {e^{-2+\frac {e^{-2-2 x} \left (7744 e^2+e^x \left (-704 e-704 e^{2+\frac {2}{x}}\right )+e^{2 x} \left (16+32 e^{1+\frac {2}{x}}+16 e^{2+\frac {4}{x}}\right )\right )}{x^2}-2 x} \left (e^{2 x} \left (e^{1+\frac {2}{x}} (-64-64 x)+e^{2+\frac {4}{x}} (-64-32 x)-32 x\right )+e^2 \left (-15488 x-15488 x^2\right )+e^x \left (e \left (1408 x+704 x^2\right )+e^{2+\frac {2}{x}} \left (1408+1408 x+704 x^2\right )\right )\right )}{x^4} \, dx={\mathrm {e}}^{\frac {16\,{\mathrm {e}}^{-2}}{x^2}}\,{\mathrm {e}}^{\frac {16\,{\mathrm {e}}^{4/x}}{x^2}}\,{\mathrm {e}}^{-\frac {704\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-1}}{x^2}}\,{\mathrm {e}}^{-\frac {704\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{2/x}}{x^2}}\,{\mathrm {e}}^{\frac {32\,{\mathrm {e}}^{-1}\,{\mathrm {e}}^{2/x}}{x^2}}\,{\mathrm {e}}^{\frac {7744\,{\mathrm {e}}^{-2\,x}}{x^2}} \] Input:

int(-(exp(-2*x)*exp(-2)*exp((exp(-2*x)*exp(-2)*(7744*exp(2) + exp(2*x)*(32 
*exp(1)*exp(2/x) + 16*exp(2)*exp(4/x) + 16) - exp(x)*(704*exp(1) + 704*exp 
(2)*exp(2/x))))/x^2)*(exp(2*x)*(32*x + exp(2)*exp(4/x)*(32*x + 64) + exp(1 
)*exp(2/x)*(64*x + 64)) + exp(2)*(15488*x + 15488*x^2) - exp(x)*(exp(1)*(1 
408*x + 704*x^2) + exp(2)*exp(2/x)*(1408*x + 704*x^2 + 1408))))/x^4,x)
 

Output:

exp((16*exp(-2))/x^2)*exp((16*exp(4/x))/x^2)*exp(-(704*exp(-x)*exp(-1))/x^ 
2)*exp(-(704*exp(-x)*exp(2/x))/x^2)*exp((32*exp(-1)*exp(2/x))/x^2)*exp((77 
44*exp(-2*x))/x^2)
 

Reduce [F]

\[ \int \frac {e^{-2+\frac {e^{-2-2 x} \left (7744 e^2+e^x \left (-704 e-704 e^{2+\frac {2}{x}}\right )+e^{2 x} \left (16+32 e^{1+\frac {2}{x}}+16 e^{2+\frac {4}{x}}\right )\right )}{x^2}-2 x} \left (e^{2 x} \left (e^{1+\frac {2}{x}} (-64-64 x)+e^{2+\frac {4}{x}} (-64-32 x)-32 x\right )+e^2 \left (-15488 x-15488 x^2\right )+e^x \left (e \left (1408 x+704 x^2\right )+e^{2+\frac {2}{x}} \left (1408+1408 x+704 x^2\right )\right )\right )}{x^4} \, dx=\int \frac {\left (\left (\left (-32 x -64\right ) \left ({\mathrm e}\right )^{2} \left ({\mathrm e}^{\frac {2}{x}}\right )^{2}+\left (-64 x -64\right ) {\mathrm e} \,{\mathrm e}^{\frac {2}{x}}-32 x \right ) \left ({\mathrm e}^{x}\right )^{2}+\left (\left (704 x^{2}+1408 x +1408\right ) \left ({\mathrm e}\right )^{2} {\mathrm e}^{\frac {2}{x}}+\left (704 x^{2}+1408 x \right ) {\mathrm e}\right ) {\mathrm e}^{x}+\left (-15488 x^{2}-15488 x \right ) \left ({\mathrm e}\right )^{2}\right ) {\mathrm e}^{\frac {\left (16 \left ({\mathrm e}\right )^{2} \left ({\mathrm e}^{\frac {2}{x}}\right )^{2}+32 \,{\mathrm e} \,{\mathrm e}^{\frac {2}{x}}+16\right ) \left ({\mathrm e}^{x}\right )^{2}+\left (-704 \left ({\mathrm e}\right )^{2} {\mathrm e}^{\frac {2}{x}}-704 \,{\mathrm e}\right ) {\mathrm e}^{x}+7744 \left ({\mathrm e}\right )^{2}}{x^{2} \left ({\mathrm e}\right )^{2} \left ({\mathrm e}^{x}\right )^{2}}}}{x^{4} \left ({\mathrm e}\right )^{2} \left ({\mathrm e}^{x}\right )^{2}}d x \] Input:

int((((-32*x-64)*exp(1)^2*exp(2/x)^2+(-64*x-64)*exp(1)*exp(2/x)-32*x)*exp( 
x)^2+((704*x^2+1408*x+1408)*exp(1)^2*exp(2/x)+(704*x^2+1408*x)*exp(1))*exp 
(x)+(-15488*x^2-15488*x)*exp(1)^2)*exp(((16*exp(1)^2*exp(2/x)^2+32*exp(1)* 
exp(2/x)+16)*exp(x)^2+(-704*exp(1)^2*exp(2/x)-704*exp(1))*exp(x)+7744*exp( 
1)^2)/x^2/exp(1)^2/exp(x)^2)/x^4/exp(1)^2/exp(x)^2,x)
 

Output:

int((((-32*x-64)*exp(1)^2*exp(2/x)^2+(-64*x-64)*exp(1)*exp(2/x)-32*x)*exp( 
x)^2+((704*x^2+1408*x+1408)*exp(1)^2*exp(2/x)+(704*x^2+1408*x)*exp(1))*exp 
(x)+(-15488*x^2-15488*x)*exp(1)^2)*exp(((16*exp(1)^2*exp(2/x)^2+32*exp(1)* 
exp(2/x)+16)*exp(x)^2+(-704*exp(1)^2*exp(2/x)-704*exp(1))*exp(x)+7744*exp( 
1)^2)/x^2/exp(1)^2/exp(x)^2)/x^4/exp(1)^2/exp(x)^2,x)