\(\int \frac {(16-8 x) \log (x)+4 x \log ^2(x)+\sqrt {\frac {e^{x/2}}{x^2}} (-8 \log (x)+(-4+x) \log ^2(x))}{\frac {4 e^{x/2}}{x}+16 x-16 x^2+4 x^3+\sqrt {\frac {e^{x/2}}{x^2}} (-16 x+8 x^2)} \, dx\) [949]

Optimal result

Integrand size = 99, antiderivative size = 31 \[ \int \frac {(16-8 x) \log (x)+4 x \log ^2(x)+\sqrt {\frac {e^{x/2}}{x^2}} \left (-8 \log (x)+(-4+x) \log ^2(x)\right )}{\frac {4 e^{x/2}}{x}+16 x-16 x^2+4 x^3+\sqrt {\frac {e^{x/2}}{x^2}} \left (-16 x+8 x^2\right )} \, dx=3+\frac {\log ^2(x)}{2-\sqrt {\frac {e^{x/2}}{x^2}}-x} \] Output:

ln(x)^2/(2-x-(exp(1/2*x)/x^2)^(1/2))+3
 

Mathematica [F]

\[ \int \frac {(16-8 x) \log (x)+4 x \log ^2(x)+\sqrt {\frac {e^{x/2}}{x^2}} \left (-8 \log (x)+(-4+x) \log ^2(x)\right )}{\frac {4 e^{x/2}}{x}+16 x-16 x^2+4 x^3+\sqrt {\frac {e^{x/2}}{x^2}} \left (-16 x+8 x^2\right )} \, dx=\int \frac {(16-8 x) \log (x)+4 x \log ^2(x)+\sqrt {\frac {e^{x/2}}{x^2}} \left (-8 \log (x)+(-4+x) \log ^2(x)\right )}{\frac {4 e^{x/2}}{x}+16 x-16 x^2+4 x^3+\sqrt {\frac {e^{x/2}}{x^2}} \left (-16 x+8 x^2\right )} \, dx \] Input:

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

Output:

Integrate[((16 - 8*x)*Log[x] + 4*x*Log[x]^2 + Sqrt[E^(x/2)/x^2]*(-8*Log[x] 
 + (-4 + x)*Log[x]^2))/((4*E^(x/2))/x + 16*x - 16*x^2 + 4*x^3 + Sqrt[E^(x/ 
2)/x^2]*(-16*x + 8*x^2)), 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.

\(\displaystyle \int \frac {\sqrt {\frac {e^{x/2}}{x^2}} \left ((x-4) \log ^2(x)-8 \log (x)\right )+4 x \log ^2(x)+(16-8 x) \log (x)}{4 x^3-16 x^2+\sqrt {\frac {e^{x/2}}{x^2}} \left (8 x^2-16 x\right )+16 x+\frac {4 e^{x/2}}{x}} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {14 \sqrt {\frac {e^{x/2}}{x^2}} x^4 \log ^2(x)}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}-\frac {8 \sqrt {\frac {e^{x/2}}{x^2}} x^3 \log ^2(x)}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}-\frac {(x-2)^2 \left (x^2-10 x+8\right ) x^3 \log ^2(x)}{2 \left (x^4-4 x^3+4 x^2-e^{x/2}\right )^2}+\frac {\sqrt {\frac {e^{x/2}}{x^2}} x^2 \log ^2(x)}{4 \left (-x^4+4 x^3-4 x^2+e^{x/2}\right )}-\frac {\sqrt {\frac {e^{x/2}}{x^2}} x \log ^2(x)}{-x^4+4 x^3-4 x^2+e^{x/2}}-\frac {2 \sqrt {\frac {e^{x/2}}{x^2}} x \log (x)}{-x^4+4 x^3-4 x^2+e^{x/2}}+\frac {x \log (x) \left (x^2 \log (x)-4 x-8 x \log (x)+8 \log (x)+8\right )}{2 \left (x^4-4 x^3+4 x^2-e^{x/2}\right )}+\frac {\sqrt {\frac {e^{x/2}}{x^2}} x^6 \log ^2(x)}{2 \left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}-\frac {6 \sqrt {\frac {e^{x/2}}{x^2}} x^5 \log ^2(x)}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \log (x) \left (\left ((x-2) \left (\left (\sqrt {\frac {e^{x/2}}{x^2}}+4\right ) x^2-2 \left (7 \sqrt {\frac {e^{x/2}}{x^2}}+4\right ) x+8 \sqrt {\frac {e^{x/2}}{x^2}}\right ) x^2+e^{x/2} \left (-2 x^2+\left (\sqrt {\frac {e^{x/2}}{x^2}}+16\right ) x-4 \left (\sqrt {\frac {e^{x/2}}{x^2}}+4\right )\right )\right ) \log (x)-8 \left (\sqrt {\frac {e^{x/2}}{x^2}}-x+2\right ) \left (e^{x/2}-(x-2)^2 x^2\right )\right )}{4 \left (e^{x/2}-(x-2)^2 x^2\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \int -\frac {x \log (x) \left (8 \left (-x+\sqrt {\frac {e^{x/2}}{x^2}}+2\right ) \left (e^{x/2}-(2-x)^2 x^2\right )+\left ((2-x) \left (\left (\sqrt {\frac {e^{x/2}}{x^2}}+4\right ) x^2-2 \left (7 \sqrt {\frac {e^{x/2}}{x^2}}+4\right ) x+8 \sqrt {\frac {e^{x/2}}{x^2}}\right ) x^2+e^{x/2} \left (2 x^2-\left (\sqrt {\frac {e^{x/2}}{x^2}}+16\right ) x+4 \left (\sqrt {\frac {e^{x/2}}{x^2}}+4\right )\right )\right ) \log (x)\right )}{\left (e^{x/2}-(2-x)^2 x^2\right )^2}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1}{4} \int \frac {x \log (x) \left (8 \left (-x+\sqrt {\frac {e^{x/2}}{x^2}}+2\right ) \left (e^{x/2}-(2-x)^2 x^2\right )+\left ((2-x) \left (\left (\sqrt {\frac {e^{x/2}}{x^2}}+4\right ) x^2-2 \left (7 \sqrt {\frac {e^{x/2}}{x^2}}+4\right ) x+8 \sqrt {\frac {e^{x/2}}{x^2}}\right ) x^2+e^{x/2} \left (2 x^2-\left (\sqrt {\frac {e^{x/2}}{x^2}}+16\right ) x+4 \left (\sqrt {\frac {e^{x/2}}{x^2}}+4\right )\right )\right ) \log (x)\right )}{\left (e^{x/2}-(2-x)^2 x^2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{4} \int \left (\frac {16 \left (\frac {e^{x/2}}{x^2}\right )^{3/2} \log ^2(x) x^3}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {32 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^3}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {\sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^2}{-x^4+4 x^3-4 x^2+e^{x/2}}+\frac {16 \left (\frac {e^{x/2}}{x^2}\right )^{3/2} \log ^2(x) x^2}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}-\frac {64 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^2}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}-\frac {12 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x}{-x^4+4 x^3-4 x^2+e^{x/2}}+\frac {2 e^{x/2} \left (x^2-10 x+8\right ) \log ^2(x) x}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {8 \sqrt {\frac {e^{x/2}}{x^2}} \log (x) x}{-x^4+4 x^3-4 x^2+e^{x/2}}-\frac {4 \log (x) (\log (x) x-2 x+4) x}{x^4-4 x^3+4 x^2-e^{x/2}}-\frac {16 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x)}{-x^4+4 x^3-4 x^2+e^{x/2}}-\frac {2 e^x \log ^2(x)}{\sqrt {\frac {e^{x/2}}{x^2}} \left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{4} \int \frac {x \log (x) \left (8 \left (-x+\sqrt {\frac {e^{x/2}}{x^2}}+2\right ) \left (e^{x/2}-(x-2)^2 x^2\right )-\left ((x-2) \left (\left (\sqrt {\frac {e^{x/2}}{x^2}}+4\right ) x^2-2 \left (7 \sqrt {\frac {e^{x/2}}{x^2}}+4\right ) x+8 \sqrt {\frac {e^{x/2}}{x^2}}\right ) x^2+e^{x/2} \left (-2 x^2+\left (\sqrt {\frac {e^{x/2}}{x^2}}+16\right ) x-4 \left (\sqrt {\frac {e^{x/2}}{x^2}}+4\right )\right )\right ) \log (x)\right )}{\left (e^{x/2}-(x-2)^2 x^2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{4} \int \left (\frac {16 \left (\frac {e^{x/2}}{x^2}\right )^{3/2} \log ^2(x) x^3}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {32 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^3}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {\sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^2}{-x^4+4 x^3-4 x^2+e^{x/2}}+\frac {16 \left (\frac {e^{x/2}}{x^2}\right )^{3/2} \log ^2(x) x^2}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}-\frac {64 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^2}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}-\frac {12 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x}{-x^4+4 x^3-4 x^2+e^{x/2}}+\frac {2 e^{x/2} \left (x^2-10 x+8\right ) \log ^2(x) x}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {8 \sqrt {\frac {e^{x/2}}{x^2}} \log (x) x}{-x^4+4 x^3-4 x^2+e^{x/2}}-\frac {4 \log (x) (\log (x) x-2 x+4) x}{x^4-4 x^3+4 x^2-e^{x/2}}-\frac {16 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x)}{-x^4+4 x^3-4 x^2+e^{x/2}}-\frac {2 e^x \log ^2(x)}{\sqrt {\frac {e^{x/2}}{x^2}} \left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{4} \int \frac {x \log (x) \left (8 \left (-x+\sqrt {\frac {e^{x/2}}{x^2}}+2\right ) \left (e^{x/2}-(x-2)^2 x^2\right )-\left ((x-2) \left (\left (\sqrt {\frac {e^{x/2}}{x^2}}+4\right ) x^2-2 \left (7 \sqrt {\frac {e^{x/2}}{x^2}}+4\right ) x+8 \sqrt {\frac {e^{x/2}}{x^2}}\right ) x^2+e^{x/2} \left (-2 x^2+\left (\sqrt {\frac {e^{x/2}}{x^2}}+16\right ) x-4 \left (\sqrt {\frac {e^{x/2}}{x^2}}+4\right )\right )\right ) \log (x)\right )}{\left (e^{x/2}-(x-2)^2 x^2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{4} \int \left (\frac {16 \left (\frac {e^{x/2}}{x^2}\right )^{3/2} \log ^2(x) x^3}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {32 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^3}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {\sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^2}{-x^4+4 x^3-4 x^2+e^{x/2}}+\frac {16 \left (\frac {e^{x/2}}{x^2}\right )^{3/2} \log ^2(x) x^2}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}-\frac {64 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^2}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}-\frac {12 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x}{-x^4+4 x^3-4 x^2+e^{x/2}}+\frac {2 e^{x/2} \left (x^2-10 x+8\right ) \log ^2(x) x}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {8 \sqrt {\frac {e^{x/2}}{x^2}} \log (x) x}{-x^4+4 x^3-4 x^2+e^{x/2}}-\frac {4 \log (x) (\log (x) x-2 x+4) x}{x^4-4 x^3+4 x^2-e^{x/2}}-\frac {16 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x)}{-x^4+4 x^3-4 x^2+e^{x/2}}-\frac {2 e^x \log ^2(x)}{\sqrt {\frac {e^{x/2}}{x^2}} \left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{4} \int \frac {x \log (x) \left (8 \left (-x+\sqrt {\frac {e^{x/2}}{x^2}}+2\right ) \left (e^{x/2}-(x-2)^2 x^2\right )-\left ((x-2) \left (\left (\sqrt {\frac {e^{x/2}}{x^2}}+4\right ) x^2-2 \left (7 \sqrt {\frac {e^{x/2}}{x^2}}+4\right ) x+8 \sqrt {\frac {e^{x/2}}{x^2}}\right ) x^2+e^{x/2} \left (-2 x^2+\left (\sqrt {\frac {e^{x/2}}{x^2}}+16\right ) x-4 \left (\sqrt {\frac {e^{x/2}}{x^2}}+4\right )\right )\right ) \log (x)\right )}{\left (e^{x/2}-(x-2)^2 x^2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{4} \int \left (\frac {16 \left (\frac {e^{x/2}}{x^2}\right )^{3/2} \log ^2(x) x^3}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {32 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^3}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {\sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^2}{-x^4+4 x^3-4 x^2+e^{x/2}}+\frac {16 \left (\frac {e^{x/2}}{x^2}\right )^{3/2} \log ^2(x) x^2}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}-\frac {64 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^2}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}-\frac {12 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x}{-x^4+4 x^3-4 x^2+e^{x/2}}+\frac {2 e^{x/2} \left (x^2-10 x+8\right ) \log ^2(x) x}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {8 \sqrt {\frac {e^{x/2}}{x^2}} \log (x) x}{-x^4+4 x^3-4 x^2+e^{x/2}}-\frac {4 \log (x) (\log (x) x-2 x+4) x}{x^4-4 x^3+4 x^2-e^{x/2}}-\frac {16 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x)}{-x^4+4 x^3-4 x^2+e^{x/2}}-\frac {2 e^x \log ^2(x)}{\sqrt {\frac {e^{x/2}}{x^2}} \left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{4} \int \frac {x \log (x) \left (8 \left (-x+\sqrt {\frac {e^{x/2}}{x^2}}+2\right ) \left (e^{x/2}-(x-2)^2 x^2\right )-\left ((x-2) \left (\left (\sqrt {\frac {e^{x/2}}{x^2}}+4\right ) x^2-2 \left (7 \sqrt {\frac {e^{x/2}}{x^2}}+4\right ) x+8 \sqrt {\frac {e^{x/2}}{x^2}}\right ) x^2+e^{x/2} \left (-2 x^2+\left (\sqrt {\frac {e^{x/2}}{x^2}}+16\right ) x-4 \left (\sqrt {\frac {e^{x/2}}{x^2}}+4\right )\right )\right ) \log (x)\right )}{\left (e^{x/2}-(x-2)^2 x^2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{4} \int \left (\frac {16 \left (\frac {e^{x/2}}{x^2}\right )^{3/2} \log ^2(x) x^3}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {32 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^3}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {\sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^2}{-x^4+4 x^3-4 x^2+e^{x/2}}+\frac {16 \left (\frac {e^{x/2}}{x^2}\right )^{3/2} \log ^2(x) x^2}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}-\frac {64 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^2}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}-\frac {12 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x}{-x^4+4 x^3-4 x^2+e^{x/2}}+\frac {2 e^{x/2} \left (x^2-10 x+8\right ) \log ^2(x) x}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {8 \sqrt {\frac {e^{x/2}}{x^2}} \log (x) x}{-x^4+4 x^3-4 x^2+e^{x/2}}-\frac {4 \log (x) (\log (x) x-2 x+4) x}{x^4-4 x^3+4 x^2-e^{x/2}}-\frac {16 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x)}{-x^4+4 x^3-4 x^2+e^{x/2}}-\frac {2 e^x \log ^2(x)}{\sqrt {\frac {e^{x/2}}{x^2}} \left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{4} \int \frac {x \log (x) \left (8 \left (-x+\sqrt {\frac {e^{x/2}}{x^2}}+2\right ) \left (e^{x/2}-(x-2)^2 x^2\right )-\left ((x-2) \left (\left (\sqrt {\frac {e^{x/2}}{x^2}}+4\right ) x^2-2 \left (7 \sqrt {\frac {e^{x/2}}{x^2}}+4\right ) x+8 \sqrt {\frac {e^{x/2}}{x^2}}\right ) x^2+e^{x/2} \left (-2 x^2+\left (\sqrt {\frac {e^{x/2}}{x^2}}+16\right ) x-4 \left (\sqrt {\frac {e^{x/2}}{x^2}}+4\right )\right )\right ) \log (x)\right )}{\left (e^{x/2}-(x-2)^2 x^2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{4} \int \left (\frac {16 \left (\frac {e^{x/2}}{x^2}\right )^{3/2} \log ^2(x) x^3}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {32 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^3}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {\sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^2}{-x^4+4 x^3-4 x^2+e^{x/2}}+\frac {16 \left (\frac {e^{x/2}}{x^2}\right )^{3/2} \log ^2(x) x^2}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}-\frac {64 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^2}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}-\frac {12 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x}{-x^4+4 x^3-4 x^2+e^{x/2}}+\frac {2 e^{x/2} \left (x^2-10 x+8\right ) \log ^2(x) x}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {8 \sqrt {\frac {e^{x/2}}{x^2}} \log (x) x}{-x^4+4 x^3-4 x^2+e^{x/2}}-\frac {4 \log (x) (\log (x) x-2 x+4) x}{x^4-4 x^3+4 x^2-e^{x/2}}-\frac {16 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x)}{-x^4+4 x^3-4 x^2+e^{x/2}}-\frac {2 e^x \log ^2(x)}{\sqrt {\frac {e^{x/2}}{x^2}} \left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{4} \int \frac {x \log (x) \left (8 \left (-x+\sqrt {\frac {e^{x/2}}{x^2}}+2\right ) \left (e^{x/2}-(x-2)^2 x^2\right )-\left ((x-2) \left (\left (\sqrt {\frac {e^{x/2}}{x^2}}+4\right ) x^2-2 \left (7 \sqrt {\frac {e^{x/2}}{x^2}}+4\right ) x+8 \sqrt {\frac {e^{x/2}}{x^2}}\right ) x^2+e^{x/2} \left (-2 x^2+\left (\sqrt {\frac {e^{x/2}}{x^2}}+16\right ) x-4 \left (\sqrt {\frac {e^{x/2}}{x^2}}+4\right )\right )\right ) \log (x)\right )}{\left (e^{x/2}-(x-2)^2 x^2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{4} \int \left (\frac {16 \left (\frac {e^{x/2}}{x^2}\right )^{3/2} \log ^2(x) x^3}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {32 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^3}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {\sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^2}{-x^4+4 x^3-4 x^2+e^{x/2}}+\frac {16 \left (\frac {e^{x/2}}{x^2}\right )^{3/2} \log ^2(x) x^2}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}-\frac {64 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^2}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}-\frac {12 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x}{-x^4+4 x^3-4 x^2+e^{x/2}}+\frac {2 e^{x/2} \left (x^2-10 x+8\right ) \log ^2(x) x}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {8 \sqrt {\frac {e^{x/2}}{x^2}} \log (x) x}{-x^4+4 x^3-4 x^2+e^{x/2}}-\frac {4 \log (x) (\log (x) x-2 x+4) x}{x^4-4 x^3+4 x^2-e^{x/2}}-\frac {16 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x)}{-x^4+4 x^3-4 x^2+e^{x/2}}-\frac {2 e^x \log ^2(x)}{\sqrt {\frac {e^{x/2}}{x^2}} \left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{4} \int \frac {x \log (x) \left (8 \left (-x+\sqrt {\frac {e^{x/2}}{x^2}}+2\right ) \left (e^{x/2}-(x-2)^2 x^2\right )-\left ((x-2) \left (\left (\sqrt {\frac {e^{x/2}}{x^2}}+4\right ) x^2-2 \left (7 \sqrt {\frac {e^{x/2}}{x^2}}+4\right ) x+8 \sqrt {\frac {e^{x/2}}{x^2}}\right ) x^2+e^{x/2} \left (-2 x^2+\left (\sqrt {\frac {e^{x/2}}{x^2}}+16\right ) x-4 \left (\sqrt {\frac {e^{x/2}}{x^2}}+4\right )\right )\right ) \log (x)\right )}{\left (e^{x/2}-(x-2)^2 x^2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{4} \int \left (\frac {16 \left (\frac {e^{x/2}}{x^2}\right )^{3/2} \log ^2(x) x^3}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {32 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^3}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {\sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^2}{-x^4+4 x^3-4 x^2+e^{x/2}}+\frac {16 \left (\frac {e^{x/2}}{x^2}\right )^{3/2} \log ^2(x) x^2}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}-\frac {64 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^2}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}-\frac {12 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x}{-x^4+4 x^3-4 x^2+e^{x/2}}+\frac {2 e^{x/2} \left (x^2-10 x+8\right ) \log ^2(x) x}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {8 \sqrt {\frac {e^{x/2}}{x^2}} \log (x) x}{-x^4+4 x^3-4 x^2+e^{x/2}}-\frac {4 \log (x) (\log (x) x-2 x+4) x}{x^4-4 x^3+4 x^2-e^{x/2}}-\frac {16 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x)}{-x^4+4 x^3-4 x^2+e^{x/2}}-\frac {2 e^x \log ^2(x)}{\sqrt {\frac {e^{x/2}}{x^2}} \left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{4} \int \frac {x \log (x) \left (8 \left (-x+\sqrt {\frac {e^{x/2}}{x^2}}+2\right ) \left (e^{x/2}-(x-2)^2 x^2\right )-\left ((x-2) \left (\left (\sqrt {\frac {e^{x/2}}{x^2}}+4\right ) x^2-2 \left (7 \sqrt {\frac {e^{x/2}}{x^2}}+4\right ) x+8 \sqrt {\frac {e^{x/2}}{x^2}}\right ) x^2+e^{x/2} \left (-2 x^2+\left (\sqrt {\frac {e^{x/2}}{x^2}}+16\right ) x-4 \left (\sqrt {\frac {e^{x/2}}{x^2}}+4\right )\right )\right ) \log (x)\right )}{\left (e^{x/2}-(x-2)^2 x^2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{4} \int \left (\frac {16 \left (\frac {e^{x/2}}{x^2}\right )^{3/2} \log ^2(x) x^3}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {32 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^3}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {\sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^2}{-x^4+4 x^3-4 x^2+e^{x/2}}+\frac {16 \left (\frac {e^{x/2}}{x^2}\right )^{3/2} \log ^2(x) x^2}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}-\frac {64 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^2}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}-\frac {12 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x}{-x^4+4 x^3-4 x^2+e^{x/2}}+\frac {2 e^{x/2} \left (x^2-10 x+8\right ) \log ^2(x) x}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {8 \sqrt {\frac {e^{x/2}}{x^2}} \log (x) x}{-x^4+4 x^3-4 x^2+e^{x/2}}-\frac {4 \log (x) (\log (x) x-2 x+4) x}{x^4-4 x^3+4 x^2-e^{x/2}}-\frac {16 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x)}{-x^4+4 x^3-4 x^2+e^{x/2}}-\frac {2 e^x \log ^2(x)}{\sqrt {\frac {e^{x/2}}{x^2}} \left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{4} \int \frac {x \log (x) \left (8 \left (-x+\sqrt {\frac {e^{x/2}}{x^2}}+2\right ) \left (e^{x/2}-(x-2)^2 x^2\right )-\left ((x-2) \left (\left (\sqrt {\frac {e^{x/2}}{x^2}}+4\right ) x^2-2 \left (7 \sqrt {\frac {e^{x/2}}{x^2}}+4\right ) x+8 \sqrt {\frac {e^{x/2}}{x^2}}\right ) x^2+e^{x/2} \left (-2 x^2+\left (\sqrt {\frac {e^{x/2}}{x^2}}+16\right ) x-4 \left (\sqrt {\frac {e^{x/2}}{x^2}}+4\right )\right )\right ) \log (x)\right )}{\left (e^{x/2}-(x-2)^2 x^2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{4} \int \left (\frac {16 \left (\frac {e^{x/2}}{x^2}\right )^{3/2} \log ^2(x) x^3}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {32 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^3}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {\sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^2}{-x^4+4 x^3-4 x^2+e^{x/2}}+\frac {16 \left (\frac {e^{x/2}}{x^2}\right )^{3/2} \log ^2(x) x^2}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}-\frac {64 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^2}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}-\frac {12 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x}{-x^4+4 x^3-4 x^2+e^{x/2}}+\frac {2 e^{x/2} \left (x^2-10 x+8\right ) \log ^2(x) x}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {8 \sqrt {\frac {e^{x/2}}{x^2}} \log (x) x}{-x^4+4 x^3-4 x^2+e^{x/2}}-\frac {4 \log (x) (\log (x) x-2 x+4) x}{x^4-4 x^3+4 x^2-e^{x/2}}-\frac {16 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x)}{-x^4+4 x^3-4 x^2+e^{x/2}}-\frac {2 e^x \log ^2(x)}{\sqrt {\frac {e^{x/2}}{x^2}} \left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{4} \int \frac {x \log (x) \left (8 \left (-x+\sqrt {\frac {e^{x/2}}{x^2}}+2\right ) \left (e^{x/2}-(x-2)^2 x^2\right )-\left ((x-2) \left (\left (\sqrt {\frac {e^{x/2}}{x^2}}+4\right ) x^2-2 \left (7 \sqrt {\frac {e^{x/2}}{x^2}}+4\right ) x+8 \sqrt {\frac {e^{x/2}}{x^2}}\right ) x^2+e^{x/2} \left (-2 x^2+\left (\sqrt {\frac {e^{x/2}}{x^2}}+16\right ) x-4 \left (\sqrt {\frac {e^{x/2}}{x^2}}+4\right )\right )\right ) \log (x)\right )}{\left (e^{x/2}-(x-2)^2 x^2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{4} \int \left (\frac {16 \left (\frac {e^{x/2}}{x^2}\right )^{3/2} \log ^2(x) x^3}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {32 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^3}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {\sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^2}{-x^4+4 x^3-4 x^2+e^{x/2}}+\frac {16 \left (\frac {e^{x/2}}{x^2}\right )^{3/2} \log ^2(x) x^2}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}-\frac {64 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^2}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}-\frac {12 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x}{-x^4+4 x^3-4 x^2+e^{x/2}}+\frac {2 e^{x/2} \left (x^2-10 x+8\right ) \log ^2(x) x}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {8 \sqrt {\frac {e^{x/2}}{x^2}} \log (x) x}{-x^4+4 x^3-4 x^2+e^{x/2}}-\frac {4 \log (x) (\log (x) x-2 x+4) x}{x^4-4 x^3+4 x^2-e^{x/2}}-\frac {16 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x)}{-x^4+4 x^3-4 x^2+e^{x/2}}-\frac {2 e^x \log ^2(x)}{\sqrt {\frac {e^{x/2}}{x^2}} \left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{4} \int \frac {x \log (x) \left (8 \left (-x+\sqrt {\frac {e^{x/2}}{x^2}}+2\right ) \left (e^{x/2}-(x-2)^2 x^2\right )-\left ((x-2) \left (\left (\sqrt {\frac {e^{x/2}}{x^2}}+4\right ) x^2-2 \left (7 \sqrt {\frac {e^{x/2}}{x^2}}+4\right ) x+8 \sqrt {\frac {e^{x/2}}{x^2}}\right ) x^2+e^{x/2} \left (-2 x^2+\left (\sqrt {\frac {e^{x/2}}{x^2}}+16\right ) x-4 \left (\sqrt {\frac {e^{x/2}}{x^2}}+4\right )\right )\right ) \log (x)\right )}{\left (e^{x/2}-(x-2)^2 x^2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{4} \int \left (\frac {16 \left (\frac {e^{x/2}}{x^2}\right )^{3/2} \log ^2(x) x^3}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {32 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^3}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {\sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^2}{-x^4+4 x^3-4 x^2+e^{x/2}}+\frac {16 \left (\frac {e^{x/2}}{x^2}\right )^{3/2} \log ^2(x) x^2}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}-\frac {64 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^2}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}-\frac {12 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x}{-x^4+4 x^3-4 x^2+e^{x/2}}+\frac {2 e^{x/2} \left (x^2-10 x+8\right ) \log ^2(x) x}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {8 \sqrt {\frac {e^{x/2}}{x^2}} \log (x) x}{-x^4+4 x^3-4 x^2+e^{x/2}}-\frac {4 \log (x) (\log (x) x-2 x+4) x}{x^4-4 x^3+4 x^2-e^{x/2}}-\frac {16 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x)}{-x^4+4 x^3-4 x^2+e^{x/2}}-\frac {2 e^x \log ^2(x)}{\sqrt {\frac {e^{x/2}}{x^2}} \left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{4} \int \frac {x \log (x) \left (8 \left (-x+\sqrt {\frac {e^{x/2}}{x^2}}+2\right ) \left (e^{x/2}-(x-2)^2 x^2\right )-\left ((x-2) \left (\left (\sqrt {\frac {e^{x/2}}{x^2}}+4\right ) x^2-2 \left (7 \sqrt {\frac {e^{x/2}}{x^2}}+4\right ) x+8 \sqrt {\frac {e^{x/2}}{x^2}}\right ) x^2+e^{x/2} \left (-2 x^2+\left (\sqrt {\frac {e^{x/2}}{x^2}}+16\right ) x-4 \left (\sqrt {\frac {e^{x/2}}{x^2}}+4\right )\right )\right ) \log (x)\right )}{\left (e^{x/2}-(x-2)^2 x^2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{4} \int \left (\frac {16 \left (\frac {e^{x/2}}{x^2}\right )^{3/2} \log ^2(x) x^3}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {32 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^3}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {\sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^2}{-x^4+4 x^3-4 x^2+e^{x/2}}+\frac {16 \left (\frac {e^{x/2}}{x^2}\right )^{3/2} \log ^2(x) x^2}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}-\frac {64 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x^2}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}-\frac {12 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x) x}{-x^4+4 x^3-4 x^2+e^{x/2}}+\frac {2 e^{x/2} \left (x^2-10 x+8\right ) \log ^2(x) x}{\left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}+\frac {8 \sqrt {\frac {e^{x/2}}{x^2}} \log (x) x}{-x^4+4 x^3-4 x^2+e^{x/2}}-\frac {4 \log (x) (\log (x) x-2 x+4) x}{x^4-4 x^3+4 x^2-e^{x/2}}-\frac {16 \sqrt {\frac {e^{x/2}}{x^2}} \log ^2(x)}{-x^4+4 x^3-4 x^2+e^{x/2}}-\frac {2 e^x \log ^2(x)}{\sqrt {\frac {e^{x/2}}{x^2}} \left (-x^4+4 x^3-4 x^2+e^{x/2}\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{4} \int \frac {x \log (x) \left (8 \left (-x+\sqrt {\frac {e^{x/2}}{x^2}}+2\right ) \left (e^{x/2}-(x-2)^2 x^2\right )-\left ((x-2) \left (\left (\sqrt {\frac {e^{x/2}}{x^2}}+4\right ) x^2-2 \left (7 \sqrt {\frac {e^{x/2}}{x^2}}+4\right ) x+8 \sqrt {\frac {e^{x/2}}{x^2}}\right ) x^2+e^{x/2} \left (-2 x^2+\left (\sqrt {\frac {e^{x/2}}{x^2}}+16\right ) x-4 \left (\sqrt {\frac {e^{x/2}}{x^2}}+4\right )\right )\right ) \log (x)\right )}{\left (e^{x/2}-(x-2)^2 x^2\right )^2}dx\)

Input:

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

Output:

$Aborted
 

Maple [F]

\[\int \frac {\left (\left (x -4\right ) \ln \left (x \right )^{2}-8 \ln \left (x \right )\right ) \sqrt {\frac {{\mathrm e}^{\frac {x}{2}}}{x^{2}}}+4 x \ln \left (x \right )^{2}+\left (-8 x +16\right ) \ln \left (x \right )}{\frac {4 \,{\mathrm e}^{\frac {x}{2}}}{x}+\left (8 x^{2}-16 x \right ) \sqrt {\frac {{\mathrm e}^{\frac {x}{2}}}{x^{2}}}+4 x^{3}-16 x^{2}+16 x}d x\]

Input:

int((((x-4)*ln(x)^2-8*ln(x))*(exp(1/2*x)/x^2)^(1/2)+4*x*ln(x)^2+(-8*x+16)* 
ln(x))/(4*exp(1/2*x)/x+(8*x^2-16*x)*(exp(1/2*x)/x^2)^(1/2)+4*x^3-16*x^2+16 
*x),x)
 

Output:

int((((x-4)*ln(x)^2-8*ln(x))*(exp(1/2*x)/x^2)^(1/2)+4*x*ln(x)^2+(-8*x+16)* 
ln(x))/(4*exp(1/2*x)/x+(8*x^2-16*x)*(exp(1/2*x)/x^2)^(1/2)+4*x^3-16*x^2+16 
*x),x)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65 \[ \int \frac {(16-8 x) \log (x)+4 x \log ^2(x)+\sqrt {\frac {e^{x/2}}{x^2}} \left (-8 \log (x)+(-4+x) \log ^2(x)\right )}{\frac {4 e^{x/2}}{x}+16 x-16 x^2+4 x^3+\sqrt {\frac {e^{x/2}}{x^2}} \left (-16 x+8 x^2\right )} \, dx=-\frac {x \log \left (x\right )^{2}}{x^{2} - 2 \, x + e^{\left (\frac {1}{4} \, x\right )}} \] Input:

integrate((((-4+x)*log(x)^2-8*log(x))*(exp(1/2*x)/x^2)^(1/2)+4*x*log(x)^2+ 
(-8*x+16)*log(x))/(4*exp(1/2*x)/x+(8*x^2-16*x)*(exp(1/2*x)/x^2)^(1/2)+4*x^ 
3-16*x^2+16*x),x, algorithm="fricas")
 

Output:

-x*log(x)^2/(x^2 - 2*x + e^(1/4*x))
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(16-8 x) \log (x)+4 x \log ^2(x)+\sqrt {\frac {e^{x/2}}{x^2}} \left (-8 \log (x)+(-4+x) \log ^2(x)\right )}{\frac {4 e^{x/2}}{x}+16 x-16 x^2+4 x^3+\sqrt {\frac {e^{x/2}}{x^2}} \left (-16 x+8 x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((((-4+x)*ln(x)**2-8*ln(x))*(exp(1/2*x)/x**2)**(1/2)+4*x*ln(x)**2 
+(-8*x+16)*ln(x))/(4*exp(1/2*x)/x+(8*x**2-16*x)*(exp(1/2*x)/x**2)**(1/2)+4 
*x**3-16*x**2+16*x),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65 \[ \int \frac {(16-8 x) \log (x)+4 x \log ^2(x)+\sqrt {\frac {e^{x/2}}{x^2}} \left (-8 \log (x)+(-4+x) \log ^2(x)\right )}{\frac {4 e^{x/2}}{x}+16 x-16 x^2+4 x^3+\sqrt {\frac {e^{x/2}}{x^2}} \left (-16 x+8 x^2\right )} \, dx=-\frac {x \log \left (x\right )^{2}}{x^{2} - 2 \, x + e^{\left (\frac {1}{4} \, x\right )}} \] Input:

integrate((((-4+x)*log(x)^2-8*log(x))*(exp(1/2*x)/x^2)^(1/2)+4*x*log(x)^2+ 
(-8*x+16)*log(x))/(4*exp(1/2*x)/x+(8*x^2-16*x)*(exp(1/2*x)/x^2)^(1/2)+4*x^ 
3-16*x^2+16*x),x, algorithm="maxima")
 

Output:

-x*log(x)^2/(x^2 - 2*x + e^(1/4*x))
 

Giac [F]

\[ \int \frac {(16-8 x) \log (x)+4 x \log ^2(x)+\sqrt {\frac {e^{x/2}}{x^2}} \left (-8 \log (x)+(-4+x) \log ^2(x)\right )}{\frac {4 e^{x/2}}{x}+16 x-16 x^2+4 x^3+\sqrt {\frac {e^{x/2}}{x^2}} \left (-16 x+8 x^2\right )} \, dx=\int { \frac {4 \, x \log \left (x\right )^{2} + {\left ({\left (x - 4\right )} \log \left (x\right )^{2} - 8 \, \log \left (x\right )\right )} \sqrt {\frac {1}{x^{2}}} e^{\left (\frac {1}{4} \, x\right )} - 8 \, {\left (x - 2\right )} \log \left (x\right )}{4 \, {\left (x^{3} + 2 \, {\left (x^{2} - 2 \, x\right )} \sqrt {\frac {1}{x^{2}}} e^{\left (\frac {1}{4} \, x\right )} - 4 \, x^{2} + 4 \, x + \frac {e^{\left (\frac {1}{2} \, x\right )}}{x}\right )}} \,d x } \] Input:

integrate((((-4+x)*log(x)^2-8*log(x))*(exp(1/2*x)/x^2)^(1/2)+4*x*log(x)^2+ 
(-8*x+16)*log(x))/(4*exp(1/2*x)/x+(8*x^2-16*x)*(exp(1/2*x)/x^2)^(1/2)+4*x^ 
3-16*x^2+16*x),x, algorithm="giac")
 

Output:

integrate(1/4*(4*x*log(x)^2 + ((x - 4)*log(x)^2 - 8*log(x))*sqrt(x^(-2))*e 
^(1/4*x) - 8*(x - 2)*log(x))/(x^3 + 2*(x^2 - 2*x)*sqrt(x^(-2))*e^(1/4*x) - 
 4*x^2 + 4*x + e^(1/2*x)/x), x)
 

Mupad [B] (verification not implemented)

Time = 3.61 (sec) , antiderivative size = 564, normalized size of antiderivative = 18.19 \[ \int \frac {(16-8 x) \log (x)+4 x \log ^2(x)+\sqrt {\frac {e^{x/2}}{x^2}} \left (-8 \log (x)+(-4+x) \log ^2(x)\right )}{\frac {4 e^{x/2}}{x}+16 x-16 x^2+4 x^3+\sqrt {\frac {e^{x/2}}{x^2}} \left (-16 x+8 x^2\right )} \, dx=\frac {x^2\,{\left (x-2\right )}^2\,\left (\frac {x^3\,\ln \left (x\right )\,\left (x^3-24\,x^2+68\,x-48\right )\,\left (8\,\ln \left (x\right )-4\,x+x^2\,\ln \left (x\right )-8\,x\,\ln \left (x\right )+8\right )-2\,x^3\,\ln \left (x\right )\,{\left (x-2\right )}^2\,\left (x+4\right )\,\left (x\,\ln \left (x\right )-2\,x+4\right )}{2\,x^9-48\,x^8+400\,x^7-1408\,x^6+2336\,x^5-1792\,x^4+512\,x^3}+\frac {x^3\,\ln \left (x\right )\,{\left (x-2\right )}^2\,\left (x+4\right )\,\left (8\,\ln \left (x\right )-4\,x+x^2\,\ln \left (x\right )-8\,x\,\ln \left (x\right )+8\right )}{2\,x^9-48\,x^8+400\,x^7-1408\,x^6+2336\,x^5-1792\,x^4+512\,x^3}\right )}{{\mathrm {e}}^{x/2}-x^2\,{\left (x-2\right )}^2}-\frac {\sqrt {\frac {{\mathrm {e}}^{x/2}}{x^2}}\,\left (x^2\,\left (\frac {x^4\,\ln \left (x\right )\,\left (x-2\right )\,\left (8\,x+8\,\ln \left (x\right )+x^2\,\ln \left (x\right )-14\,x\,\ln \left (x\right )-16\right )+8\,x^3\,\ln \left (x\right )\,\left (x^2-3\,x+2\right )\,\left (4\,\ln \left (x\right )-x\,\ln \left (x\right )+8\right )}{2\,x^9-48\,x^8+400\,x^7-1408\,x^6+2336\,x^5-1792\,x^4+512\,x^3}-\frac {x^4\,\ln \left (x\right )\,{\left (x-2\right )}^2\,\left (4\,\ln \left (x\right )-x\,\ln \left (x\right )+8\right )}{2\,x^9-48\,x^8+400\,x^7-1408\,x^6+2336\,x^5-1792\,x^4+512\,x^3}\right )\,{\left (x-2\right )}^2-\frac {8\,x^5\,\ln \left (x\right )\,\left (x-2\right )\,\left (x^2-3\,x+2\right )\,\left (8\,x+8\,\ln \left (x\right )+x^2\,\ln \left (x\right )-14\,x\,\ln \left (x\right )-16\right )}{2\,x^9-48\,x^8+400\,x^7-1408\,x^6+2336\,x^5-1792\,x^4+512\,x^3}\right )}{{\mathrm {e}}^{x/2}-x^2\,{\left (x-2\right )}^2}-\frac {2\,x^5\,\ln \left (x\right )\,{\left (x-2\right )}^2\,\left (x\,\ln \left (x\right )-2\,x+4\right )\,\left (x^3-24\,x^2+68\,x-48\right )}{\left ({\mathrm {e}}^{x/2}-x^2\,{\left (x-2\right )}^2\right )\,\left (2\,x^9-48\,x^8+400\,x^7-1408\,x^6+2336\,x^5-1792\,x^4+512\,x^3\right )} \] Input:

int(-((8*log(x) - log(x)^2*(x - 4))*(exp(x/2)/x^2)^(1/2) - 4*x*log(x)^2 + 
log(x)*(8*x - 16))/(16*x + (4*exp(x/2))/x - (16*x - 8*x^2)*(exp(x/2)/x^2)^ 
(1/2) - 16*x^2 + 4*x^3),x)
 

Output:

(x^2*(x - 2)^2*((x^3*log(x)*(68*x - 24*x^2 + x^3 - 48)*(8*log(x) - 4*x + x 
^2*log(x) - 8*x*log(x) + 8) - 2*x^3*log(x)*(x - 2)^2*(x + 4)*(x*log(x) - 2 
*x + 4))/(512*x^3 - 1792*x^4 + 2336*x^5 - 1408*x^6 + 400*x^7 - 48*x^8 + 2* 
x^9) + (x^3*log(x)*(x - 2)^2*(x + 4)*(8*log(x) - 4*x + x^2*log(x) - 8*x*lo 
g(x) + 8))/(512*x^3 - 1792*x^4 + 2336*x^5 - 1408*x^6 + 400*x^7 - 48*x^8 + 
2*x^9)))/(exp(x/2) - x^2*(x - 2)^2) - ((exp(x/2)/x^2)^(1/2)*(x^2*((x^4*log 
(x)*(x - 2)*(8*x + 8*log(x) + x^2*log(x) - 14*x*log(x) - 16) + 8*x^3*log(x 
)*(x^2 - 3*x + 2)*(4*log(x) - x*log(x) + 8))/(512*x^3 - 1792*x^4 + 2336*x^ 
5 - 1408*x^6 + 400*x^7 - 48*x^8 + 2*x^9) - (x^4*log(x)*(x - 2)^2*(4*log(x) 
 - x*log(x) + 8))/(512*x^3 - 1792*x^4 + 2336*x^5 - 1408*x^6 + 400*x^7 - 48 
*x^8 + 2*x^9))*(x - 2)^2 - (8*x^5*log(x)*(x - 2)*(x^2 - 3*x + 2)*(8*x + 8* 
log(x) + x^2*log(x) - 14*x*log(x) - 16))/(512*x^3 - 1792*x^4 + 2336*x^5 - 
1408*x^6 + 400*x^7 - 48*x^8 + 2*x^9)))/(exp(x/2) - x^2*(x - 2)^2) - (2*x^5 
*log(x)*(x - 2)^2*(x*log(x) - 2*x + 4)*(68*x - 24*x^2 + x^3 - 48))/((exp(x 
/2) - x^2*(x - 2)^2)*(512*x^3 - 1792*x^4 + 2336*x^5 - 1408*x^6 + 400*x^7 - 
 48*x^8 + 2*x^9))
 

Reduce [F]

\[ \int \frac {(16-8 x) \log (x)+4 x \log ^2(x)+\sqrt {\frac {e^{x/2}}{x^2}} \left (-8 \log (x)+(-4+x) \log ^2(x)\right )}{\frac {4 e^{x/2}}{x}+16 x-16 x^2+4 x^3+\sqrt {\frac {e^{x/2}}{x^2}} \left (-16 x+8 x^2\right )} \, dx=\int \frac {\mathrm {log}\left (x \right )^{2} x^{2}}{2 e^{\frac {x}{4}} x^{2}-4 e^{\frac {x}{4}} x +e^{\frac {x}{2}}+x^{4}-4 x^{3}+4 x^{2}}d x +\frac {\left (\int \frac {e^{\frac {x}{4}} \mathrm {log}\left (x \right )^{2} x}{2 e^{\frac {x}{4}} x^{2}-4 e^{\frac {x}{4}} x +e^{\frac {x}{2}}+x^{4}-4 x^{3}+4 x^{2}}d x \right )}{4}-\left (\int \frac {e^{\frac {x}{4}} \mathrm {log}\left (x \right )^{2}}{2 e^{\frac {x}{4}} x^{2}-4 e^{\frac {x}{4}} x +e^{\frac {x}{2}}+x^{4}-4 x^{3}+4 x^{2}}d x \right )-2 \left (\int \frac {e^{\frac {x}{4}} \mathrm {log}\left (x \right )}{2 e^{\frac {x}{4}} x^{2}-4 e^{\frac {x}{4}} x +e^{\frac {x}{2}}+x^{4}-4 x^{3}+4 x^{2}}d x \right )-2 \left (\int \frac {\mathrm {log}\left (x \right ) x^{2}}{2 e^{\frac {x}{4}} x^{2}-4 e^{\frac {x}{4}} x +e^{\frac {x}{2}}+x^{4}-4 x^{3}+4 x^{2}}d x \right )+4 \left (\int \frac {\mathrm {log}\left (x \right ) x}{2 e^{\frac {x}{4}} x^{2}-4 e^{\frac {x}{4}} x +e^{\frac {x}{2}}+x^{4}-4 x^{3}+4 x^{2}}d x \right ) \] Input:

int((((-4+x)*log(x)^2-8*log(x))*(exp(1/2*x)/x^2)^(1/2)+4*x*log(x)^2+(-8*x+ 
16)*log(x))/(4*exp(1/2*x)/x+(8*x^2-16*x)*(exp(1/2*x)/x^2)^(1/2)+4*x^3-16*x 
^2+16*x),x)
 

Output:

(4*int((log(x)**2*x**2)/(2*e**(x/4)*x**2 - 4*e**(x/4)*x + e**(x/2) + x**4 
- 4*x**3 + 4*x**2),x) + int((e**(x/4)*log(x)**2*x)/(2*e**(x/4)*x**2 - 4*e* 
*(x/4)*x + e**(x/2) + x**4 - 4*x**3 + 4*x**2),x) - 4*int((e**(x/4)*log(x)* 
*2)/(2*e**(x/4)*x**2 - 4*e**(x/4)*x + e**(x/2) + x**4 - 4*x**3 + 4*x**2),x 
) - 8*int((e**(x/4)*log(x))/(2*e**(x/4)*x**2 - 4*e**(x/4)*x + e**(x/2) + x 
**4 - 4*x**3 + 4*x**2),x) - 8*int((log(x)*x**2)/(2*e**(x/4)*x**2 - 4*e**(x 
/4)*x + e**(x/2) + x**4 - 4*x**3 + 4*x**2),x) + 16*int((log(x)*x)/(2*e**(x 
/4)*x**2 - 4*e**(x/4)*x + e**(x/2) + x**4 - 4*x**3 + 4*x**2),x))/4