\(\int \frac {e^{-\frac {\log ^2(x)+2 e^x x \log (x) \log (x^2)+(e^{2 x} x^2+4 x^7) \log ^2(x^2)}{4 x^6 \log ^2(x^2)}} (2 \log ^2(x)+((-1+2 e^x x) \log (x)+3 \log ^2(x)) \log (x^2)+(-e^x x+e^x (5 x-x^2) \log (x)) \log ^2(x^2)+(2 x^6-2 x^7+e^{2 x} (2 x^2-x^3)) \log ^3(x^2))}{2 x^6 \log ^3(x^2)} \, dx\) [1941]

Optimal result

Integrand size = 161, antiderivative size = 33 \[ \int \frac {e^{-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}} \left (2 \log ^2(x)+\left (\left (-1+2 e^x x\right ) \log (x)+3 \log ^2(x)\right ) \log \left (x^2\right )+\left (-e^x x+e^x \left (5 x-x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (2 x^6-2 x^7+e^{2 x} \left (2 x^2-x^3\right )\right ) \log ^3\left (x^2\right )\right )}{2 x^6 \log ^3\left (x^2\right )} \, dx=e^{-x-\frac {\left (e^x+\frac {\log (x)}{x \log \left (x^2\right )}\right )^2}{4 x^4}} x \] Output:

x/exp(x+1/4*(ln(x)/x/ln(x^2)+exp(x))^2/x^4)
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.70 \[ \int \frac {e^{-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}} \left (2 \log ^2(x)+\left (\left (-1+2 e^x x\right ) \log (x)+3 \log ^2(x)\right ) \log \left (x^2\right )+\left (-e^x x+e^x \left (5 x-x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (2 x^6-2 x^7+e^{2 x} \left (2 x^2-x^3\right )\right ) \log ^3\left (x^2\right )\right )}{2 x^6 \log ^3\left (x^2\right )} \, dx=e^{-\frac {e^{2 x}}{4 x^4}-x-\frac {\log ^2(x)}{4 x^6 \log ^2\left (x^2\right )}} x^{1-\frac {e^x}{2 x^5 \log \left (x^2\right )}} \] Input:

Integrate[(2*Log[x]^2 + ((-1 + 2*E^x*x)*Log[x] + 3*Log[x]^2)*Log[x^2] + (- 
(E^x*x) + E^x*(5*x - x^2)*Log[x])*Log[x^2]^2 + (2*x^6 - 2*x^7 + E^(2*x)*(2 
*x^2 - x^3))*Log[x^2]^3)/(2*E^((Log[x]^2 + 2*E^x*x*Log[x]*Log[x^2] + (E^(2 
*x)*x^2 + 4*x^7)*Log[x^2]^2)/(4*x^6*Log[x^2]^2))*x^6*Log[x^2]^3),x]
 

Output:

E^(-1/4*E^(2*x)/x^4 - x - Log[x]^2/(4*x^6*Log[x^2]^2))*x^(1 - E^x/(2*x^5*L 
og[x^2]))
 

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.

\(\displaystyle \int \frac {\left (\left (e^x \left (5 x-x^2\right ) \log (x)-e^x x\right ) \log ^2\left (x^2\right )+\left (3 \log ^2(x)+\left (2 e^x x-1\right ) \log (x)\right ) \log \left (x^2\right )+\left (-2 x^7+2 x^6+e^{2 x} \left (2 x^2-x^3\right )\right ) \log ^3\left (x^2\right )+2 \log ^2(x)\right ) \exp \left (-\frac {2 e^x x \log \left (x^2\right ) \log (x)+\left (4 x^7+e^{2 x} x^2\right ) \log ^2\left (x^2\right )+\log ^2(x)}{4 x^6 \log ^2\left (x^2\right )}\right )}{2 x^6 \log ^3\left (x^2\right )} \, dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \int \frac {\exp \left (-\frac {\log ^2(x)+\left (4 x^7+e^{2 x} x^2\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) x^{-6-\frac {e^x}{2 x^5 \log \left (x^2\right )}} \left (\left (-2 x^7+2 x^6+e^{2 x} \left (2 x^2-x^3\right )\right ) \log ^3\left (x^2\right )-\left (e^x x-e^x \left (5 x-x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )-\left (\left (1-2 e^x x\right ) \log (x)-3 \log ^2(x)\right ) \log \left (x^2\right )+2 \log ^2(x)\right )}{\log ^3\left (x^2\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{2} \int \left (\frac {\exp \left (-\frac {\log ^2(x)+\left (4 x^7+e^{2 x} x^2\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \left (-2 \log ^3\left (x^2\right ) x^7+2 \log ^3\left (x^2\right ) x^6+2 \log ^2(x)+3 \log ^2(x) \log \left (x^2\right )-\log (x) \log \left (x^2\right )\right ) x^{-6-\frac {e^x}{2 x^5 \log \left (x^2\right )}}}{\log ^3\left (x^2\right )}-\frac {\exp \left (x-\frac {\log ^2(x)+\left (4 x^7+e^{2 x} x^2\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \left (x \log \left (x^2\right ) \log (x)-5 \log \left (x^2\right ) \log (x)-2 \log (x)+\log \left (x^2\right )\right ) x^{-5-\frac {e^x}{2 x^5 \log \left (x^2\right )}}}{\log ^2\left (x^2\right )}-\exp \left (2 x-\frac {\log ^2(x)+\left (4 x^7+e^{2 x} x^2\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) (x-2) x^{-4-\frac {e^x}{2 x^5 \log \left (x^2\right )}}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {1}{2} \int \frac {e^{-\frac {\log ^2(x)}{4 x^6 \log ^2\left (x^2\right )}-x-\frac {e^{2 x}}{4 x^4}} x^{-6-\frac {e^x}{2 x^5 \log \left (x^2\right )}} \left (\left (3 \log \left (x^2\right )+2\right ) \log ^2(x)-\log \left (x^2\right ) \left (-2 e^x x+e^x (x-5) \log \left (x^2\right ) x+1\right ) \log (x)-x \log ^2\left (x^2\right ) \left (x \left (2 (x-1) x^4+e^{2 x} (x-2)\right ) \log \left (x^2\right )+e^x\right )\right )}{\log ^3\left (x^2\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{2} \int \left (\frac {e^{-\frac {\log ^2(x)}{4 x^6 \log ^2\left (x^2\right )}-x-\frac {e^{2 x}}{4 x^4}} \left (-2 \log ^3\left (x^2\right ) x^7+2 \log ^3\left (x^2\right ) x^6+2 \log ^2(x)+3 \log ^2(x) \log \left (x^2\right )-\log (x) \log \left (x^2\right )\right ) x^{-6-\frac {e^x}{2 x^5 \log \left (x^2\right )}}}{\log ^3\left (x^2\right )}-\frac {e^{-\frac {\log ^2(x)}{4 x^6 \log ^2\left (x^2\right )}-\frac {e^{2 x}}{4 x^4}} \left (x \log \left (x^2\right ) \log (x)-5 \log \left (x^2\right ) \log (x)-2 \log (x)+\log \left (x^2\right )\right ) x^{-5-\frac {e^x}{2 x^5 \log \left (x^2\right )}}}{\log ^2\left (x^2\right )}-e^{-\frac {\log ^2(x)}{4 x^6 \log ^2\left (x^2\right )}+x-\frac {e^{2 x}}{4 x^4}} (x-2) x^{-4-\frac {e^x}{2 x^5 \log \left (x^2\right )}}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {1}{2} \int \frac {e^{-\frac {\log ^2(x)}{4 x^6 \log ^2\left (x^2\right )}-x-\frac {e^{2 x}}{4 x^4}} x^{-6-\frac {e^x}{2 x^5 \log \left (x^2\right )}} \left (\left (3 \log \left (x^2\right )+2\right ) \log ^2(x)-\log \left (x^2\right ) \left (-2 e^x x+e^x (x-5) \log \left (x^2\right ) x+1\right ) \log (x)-x \log ^2\left (x^2\right ) \left (x \left (2 (x-1) x^4+e^{2 x} (x-2)\right ) \log \left (x^2\right )+e^x\right )\right )}{\log ^3\left (x^2\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{2} \int \left (\frac {e^{-\frac {\log ^2(x)}{4 x^6 \log ^2\left (x^2\right )}-x-\frac {e^{2 x}}{4 x^4}} \left (-2 \log ^3\left (x^2\right ) x^7+2 \log ^3\left (x^2\right ) x^6+2 \log ^2(x)+3 \log ^2(x) \log \left (x^2\right )-\log (x) \log \left (x^2\right )\right ) x^{-6-\frac {e^x}{2 x^5 \log \left (x^2\right )}}}{\log ^3\left (x^2\right )}-\frac {e^{-\frac {\log ^2(x)}{4 x^6 \log ^2\left (x^2\right )}-\frac {e^{2 x}}{4 x^4}} \left (x \log \left (x^2\right ) \log (x)-5 \log \left (x^2\right ) \log (x)-2 \log (x)+\log \left (x^2\right )\right ) x^{-5-\frac {e^x}{2 x^5 \log \left (x^2\right )}}}{\log ^2\left (x^2\right )}-e^{-\frac {\log ^2(x)}{4 x^6 \log ^2\left (x^2\right )}+x-\frac {e^{2 x}}{4 x^4}} (x-2) x^{-4-\frac {e^x}{2 x^5 \log \left (x^2\right )}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (2 \int e^{-\frac {\log ^2(x)}{4 x^6 \log ^2\left (x^2\right )}+x-\frac {e^{2 x}}{4 x^4}} x^{-4-\frac {e^x}{2 x^5 \log \left (x^2\right )}}dx-\int e^{-\frac {\log ^2(x)}{4 x^6 \log ^2\left (x^2\right )}+x-\frac {e^{2 x}}{4 x^4}} x^{-3-\frac {e^x}{2 x^5 \log \left (x^2\right )}}dx-2 \int e^{-\frac {\log ^2(x)}{4 x^6 \log ^2\left (x^2\right )}-x-\frac {e^{2 x}}{4 x^4}} x^{1-\frac {e^x}{2 x^5 \log \left (x^2\right )}}dx+2 \int e^{-\frac {\log ^2(x)}{4 x^6 \log ^2\left (x^2\right )}-x-\frac {e^{2 x}}{4 x^4}} x^{-\frac {e^x}{2 x^5 \log \left (x^2\right )}}dx-\int \frac {e^{-\frac {\log ^2(x)}{4 x^6 \log ^2\left (x^2\right )}-x-\frac {e^{2 x}}{4 x^4}} x^{-6-\frac {e^x}{2 x^5 \log \left (x^2\right )}} \log (x)}{\log ^2\left (x^2\right )}dx+2 \int \frac {e^{-\frac {\log ^2(x)}{4 x^6 \log ^2\left (x^2\right )}-\frac {e^{2 x}}{4 x^4}} x^{-5-\frac {e^x}{2 x^5 \log \left (x^2\right )}} \log (x)}{\log ^2\left (x^2\right )}dx+3 \int \frac {e^{-\frac {\log ^2(x)}{4 x^6 \log ^2\left (x^2\right )}-x-\frac {e^{2 x}}{4 x^4}} x^{-6-\frac {e^x}{2 x^5 \log \left (x^2\right )}} \log ^2(x)}{\log ^2\left (x^2\right )}dx-\int \frac {e^{-\frac {\log ^2(x)}{4 x^6 \log ^2\left (x^2\right )}-\frac {e^{2 x}}{4 x^4}} x^{-5-\frac {e^x}{2 x^5 \log \left (x^2\right )}}}{\log \left (x^2\right )}dx+5 \int \frac {e^{-\frac {\log ^2(x)}{4 x^6 \log ^2\left (x^2\right )}-\frac {e^{2 x}}{4 x^4}} x^{-5-\frac {e^x}{2 x^5 \log \left (x^2\right )}} \log (x)}{\log \left (x^2\right )}dx-\int \frac {e^{-\frac {\log ^2(x)}{4 x^6 \log ^2\left (x^2\right )}-\frac {e^{2 x}}{4 x^4}} x^{-4-\frac {e^x}{2 x^5 \log \left (x^2\right )}} \log (x)}{\log \left (x^2\right )}dx+2 \int \frac {e^{-\frac {\log ^2(x)}{4 x^6 \log ^2\left (x^2\right )}-x-\frac {e^{2 x}}{4 x^4}} x^{-6-\frac {e^x}{2 x^5 \log \left (x^2\right )}} \log ^2(x)}{\log ^3\left (x^2\right )}dx\right )\)

Input:

Int[(2*Log[x]^2 + ((-1 + 2*E^x*x)*Log[x] + 3*Log[x]^2)*Log[x^2] + (-(E^x*x 
) + E^x*(5*x - x^2)*Log[x])*Log[x^2]^2 + (2*x^6 - 2*x^7 + E^(2*x)*(2*x^2 - 
 x^3))*Log[x^2]^3)/(2*E^((Log[x]^2 + 2*E^x*x*Log[x]*Log[x^2] + (E^(2*x)*x^ 
2 + 4*x^7)*Log[x^2]^2)/(4*x^6*Log[x^2]^2))*x^6*Log[x^2]^3),x]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.53 (sec) , antiderivative size = 549, normalized size of antiderivative = 16.64

\[x \,{\mathrm e}^{-\frac {-4 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{6} x^{7}+16 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{5} \operatorname {csgn}\left (i x \right ) x^{7}-24 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{4} \operatorname {csgn}\left (i x \right )^{2} x^{7}+16 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{3} \operatorname {csgn}\left (i x \right )^{3} x^{7}-4 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )^{4} x^{7}-8 i \ln \left (x \right ) \pi \operatorname {csgn}\left (i x^{2}\right )^{3} {\mathrm e}^{2 x} x^{2}+64 i \ln \left (x \right ) \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right ) x^{7}+8 i \ln \left (x \right ) {\mathrm e}^{x} x \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )-\pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{6} {\mathrm e}^{2 x} x^{2}+4 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{5} \operatorname {csgn}\left (i x \right ) {\mathrm e}^{2 x} x^{2}-6 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{4} \operatorname {csgn}\left (i x \right )^{2} {\mathrm e}^{2 x} x^{2}+4 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{3} \operatorname {csgn}\left (i x \right )^{3} {\mathrm e}^{2 x} x^{2}-\pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )^{4} {\mathrm e}^{2 x} x^{2}+64 \ln \left (x \right )^{2} x^{7}-8 i \ln \left (x \right ) \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2} {\mathrm e}^{2 x} x^{2}-32 i \ln \left (x \right ) \pi \operatorname {csgn}\left (i x^{2}\right )^{3} x^{7}-4 i \ln \left (x \right ) {\mathrm e}^{x} x \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-4 i \ln \left (x \right ) {\mathrm e}^{x} x \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}-32 i \ln \left (x \right ) \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2} x^{7}+16 i \ln \left (x \right ) \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right ) {\mathrm e}^{2 x} x^{2}+16 \ln \left (x \right )^{2} {\mathrm e}^{2 x} x^{2}+16 \,{\mathrm e}^{x} \ln \left (x \right )^{2} x +4 \ln \left (x \right )^{2}}{4 x^{6} {\left (-i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+2 i \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )-i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+4 \ln \left (x \right )\right )}^{2}}}\]

Input:

int(1/2*(((-x^3+2*x^2)*exp(x)^2-2*x^7+2*x^6)*ln(x^2)^3+((-x^2+5*x)*exp(x)* 
ln(x)-exp(x)*x)*ln(x^2)^2+(3*ln(x)^2+(2*exp(x)*x-1)*ln(x))*ln(x^2)+2*ln(x) 
^2)/x^6/ln(x^2)^3/exp(1/4*((exp(x)^2*x^2+4*x^7)*ln(x^2)^2+2*x*exp(x)*ln(x) 
*ln(x^2)+ln(x)^2)/x^6/ln(x^2)^2),x)
 

Output:

x*exp(-1/4*(-4*Pi^2*csgn(I*x^2)^6*x^7+16*Pi^2*csgn(I*x^2)^5*csgn(I*x)*x^7- 
24*Pi^2*csgn(I*x^2)^4*csgn(I*x)^2*x^7+16*Pi^2*csgn(I*x^2)^3*csgn(I*x)^3*x^ 
7-4*Pi^2*csgn(I*x^2)^2*csgn(I*x)^4*x^7-8*I*ln(x)*Pi*csgn(I*x^2)^3*exp(2*x) 
*x^2+64*I*ln(x)*Pi*csgn(I*x^2)^2*csgn(I*x)*x^7+8*I*ln(x)*exp(x)*x*Pi*csgn( 
I*x^2)^2*csgn(I*x)-Pi^2*csgn(I*x^2)^6*exp(2*x)*x^2+4*Pi^2*csgn(I*x^2)^5*cs 
gn(I*x)*exp(2*x)*x^2-6*Pi^2*csgn(I*x^2)^4*csgn(I*x)^2*exp(2*x)*x^2+4*Pi^2* 
csgn(I*x^2)^3*csgn(I*x)^3*exp(2*x)*x^2-Pi^2*csgn(I*x^2)^2*csgn(I*x)^4*exp( 
2*x)*x^2+64*ln(x)^2*x^7-8*I*ln(x)*Pi*csgn(I*x^2)*csgn(I*x)^2*exp(2*x)*x^2- 
32*I*ln(x)*Pi*csgn(I*x^2)^3*x^7-4*I*ln(x)*exp(x)*x*Pi*csgn(I*x^2)^3-4*I*ln 
(x)*exp(x)*x*Pi*csgn(I*x^2)*csgn(I*x)^2-32*I*ln(x)*Pi*csgn(I*x^2)*csgn(I*x 
)^2*x^7+16*I*ln(x)*Pi*csgn(I*x^2)^2*csgn(I*x)*exp(2*x)*x^2+16*ln(x)^2*exp( 
2*x)*x^2+16*exp(x)*ln(x)^2*x+4*ln(x)^2)/x^6/(-I*Pi*csgn(I*x^2)^3+2*I*Pi*cs 
gn(I*x^2)^2*csgn(I*x)-I*Pi*csgn(I*x)^2*csgn(I*x^2)+4*ln(x))^2)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int \frac {e^{-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}} \left (2 \log ^2(x)+\left (\left (-1+2 e^x x\right ) \log (x)+3 \log ^2(x)\right ) \log \left (x^2\right )+\left (-e^x x+e^x \left (5 x-x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (2 x^6-2 x^7+e^{2 x} \left (2 x^2-x^3\right )\right ) \log ^3\left (x^2\right )\right )}{2 x^6 \log ^3\left (x^2\right )} \, dx=x e^{\left (-\frac {16 \, x^{7} + 4 \, x^{2} e^{\left (2 \, x\right )} + 4 \, x e^{x} + 1}{16 \, x^{6}}\right )} \] Input:

integrate(1/2*(((-x^3+2*x^2)*exp(x)^2-2*x^7+2*x^6)*log(x^2)^3+((-x^2+5*x)* 
exp(x)*log(x)-exp(x)*x)*log(x^2)^2+(3*log(x)^2+(2*exp(x)*x-1)*log(x))*log( 
x^2)+2*log(x)^2)/x^6/log(x^2)^3/exp(1/4*((exp(x)^2*x^2+4*x^7)*log(x^2)^2+2 
*x*exp(x)*log(x)*log(x^2)+log(x)^2)/x^6/log(x^2)^2),x, algorithm="fricas")
 

Output:

x*e^(-1/16*(16*x^7 + 4*x^2*e^(2*x) + 4*x*e^x + 1)/x^6)
 

Sympy [A] (verification not implemented)

Time = 1.75 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.45 \[ \int \frac {e^{-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}} \left (2 \log ^2(x)+\left (\left (-1+2 e^x x\right ) \log (x)+3 \log ^2(x)\right ) \log \left (x^2\right )+\left (-e^x x+e^x \left (5 x-x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (2 x^6-2 x^7+e^{2 x} \left (2 x^2-x^3\right )\right ) \log ^3\left (x^2\right )\right )}{2 x^6 \log ^3\left (x^2\right )} \, dx=x e^{- \frac {x e^{x} \log {\left (x \right )}^{2} + \left (4 x^{7} + x^{2} e^{2 x}\right ) \log {\left (x \right )}^{2} + \frac {\log {\left (x \right )}^{2}}{4}}{4 x^{6} \log {\left (x \right )}^{2}}} \] Input:

integrate(1/2*(((-x**3+2*x**2)*exp(x)**2-2*x**7+2*x**6)*ln(x**2)**3+((-x** 
2+5*x)*exp(x)*ln(x)-exp(x)*x)*ln(x**2)**2+(3*ln(x)**2+(2*exp(x)*x-1)*ln(x) 
)*ln(x**2)+2*ln(x)**2)/x**6/ln(x**2)**3/exp(1/4*((exp(x)**2*x**2+4*x**7)*l 
n(x**2)**2+2*x*exp(x)*ln(x)*ln(x**2)+ln(x)**2)/x**6/ln(x**2)**2),x)
 

Output:

x*exp(-(x*exp(x)*log(x)**2 + (4*x**7 + x**2*exp(2*x))*log(x)**2 + log(x)** 
2/4)/(4*x**6*log(x)**2))
 

Maxima [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int \frac {e^{-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}} \left (2 \log ^2(x)+\left (\left (-1+2 e^x x\right ) \log (x)+3 \log ^2(x)\right ) \log \left (x^2\right )+\left (-e^x x+e^x \left (5 x-x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (2 x^6-2 x^7+e^{2 x} \left (2 x^2-x^3\right )\right ) \log ^3\left (x^2\right )\right )}{2 x^6 \log ^3\left (x^2\right )} \, dx=x e^{\left (-x - \frac {e^{\left (2 \, x\right )}}{4 \, x^{4}} - \frac {e^{x}}{4 \, x^{5}} - \frac {1}{16 \, x^{6}}\right )} \] Input:

integrate(1/2*(((-x^3+2*x^2)*exp(x)^2-2*x^7+2*x^6)*log(x^2)^3+((-x^2+5*x)* 
exp(x)*log(x)-exp(x)*x)*log(x^2)^2+(3*log(x)^2+(2*exp(x)*x-1)*log(x))*log( 
x^2)+2*log(x)^2)/x^6/log(x^2)^3/exp(1/4*((exp(x)^2*x^2+4*x^7)*log(x^2)^2+2 
*x*exp(x)*log(x)*log(x^2)+log(x)^2)/x^6/log(x^2)^2),x, algorithm="maxima")
 

Output:

x*e^(-x - 1/4*e^(2*x)/x^4 - 1/4*e^x/x^5 - 1/16/x^6)
 

Giac [F]

\[ \int \frac {e^{-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}} \left (2 \log ^2(x)+\left (\left (-1+2 e^x x\right ) \log (x)+3 \log ^2(x)\right ) \log \left (x^2\right )+\left (-e^x x+e^x \left (5 x-x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (2 x^6-2 x^7+e^{2 x} \left (2 x^2-x^3\right )\right ) \log ^3\left (x^2\right )\right )}{2 x^6 \log ^3\left (x^2\right )} \, dx=\int { -\frac {{\left ({\left (2 \, x^{7} - 2 \, x^{6} + {\left (x^{3} - 2 \, x^{2}\right )} e^{\left (2 \, x\right )}\right )} \log \left (x^{2}\right )^{3} + {\left ({\left (x^{2} - 5 \, x\right )} e^{x} \log \left (x\right ) + x e^{x}\right )} \log \left (x^{2}\right )^{2} - {\left ({\left (2 \, x e^{x} - 1\right )} \log \left (x\right ) + 3 \, \log \left (x\right )^{2}\right )} \log \left (x^{2}\right ) - 2 \, \log \left (x\right )^{2}\right )} e^{\left (-\frac {2 \, x e^{x} \log \left (x^{2}\right ) \log \left (x\right ) + {\left (4 \, x^{7} + x^{2} e^{\left (2 \, x\right )}\right )} \log \left (x^{2}\right )^{2} + \log \left (x\right )^{2}}{4 \, x^{6} \log \left (x^{2}\right )^{2}}\right )}}{2 \, x^{6} \log \left (x^{2}\right )^{3}} \,d x } \] Input:

integrate(1/2*(((-x^3+2*x^2)*exp(x)^2-2*x^7+2*x^6)*log(x^2)^3+((-x^2+5*x)* 
exp(x)*log(x)-exp(x)*x)*log(x^2)^2+(3*log(x)^2+(2*exp(x)*x-1)*log(x))*log( 
x^2)+2*log(x)^2)/x^6/log(x^2)^3/exp(1/4*((exp(x)^2*x^2+4*x^7)*log(x^2)^2+2 
*x*exp(x)*log(x)*log(x^2)+log(x)^2)/x^6/log(x^2)^2),x, algorithm="giac")
 

Output:

integrate(-1/2*((2*x^7 - 2*x^6 + (x^3 - 2*x^2)*e^(2*x))*log(x^2)^3 + ((x^2 
 - 5*x)*e^x*log(x) + x*e^x)*log(x^2)^2 - ((2*x*e^x - 1)*log(x) + 3*log(x)^ 
2)*log(x^2) - 2*log(x)^2)*e^(-1/4*(2*x*e^x*log(x^2)*log(x) + (4*x^7 + x^2* 
e^(2*x))*log(x^2)^2 + log(x)^2)/(x^6*log(x^2)^2))/(x^6*log(x^2)^3), x)
 

Mupad [B] (verification not implemented)

Time = 2.71 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.45 \[ \int \frac {e^{-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}} \left (2 \log ^2(x)+\left (\left (-1+2 e^x x\right ) \log (x)+3 \log ^2(x)\right ) \log \left (x^2\right )+\left (-e^x x+e^x \left (5 x-x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (2 x^6-2 x^7+e^{2 x} \left (2 x^2-x^3\right )\right ) \log ^3\left (x^2\right )\right )}{2 x^6 \log ^3\left (x^2\right )} \, dx=x\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^x\,\ln \left (x\right )}{2\,x^5\,\ln \left (x^2\right )}}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-\frac {{\ln \left (x\right )}^2}{4\,x^6\,{\ln \left (x^2\right )}^2}}\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{2\,x}}{4\,x^4}} \] Input:

int((exp(-((log(x^2)^2*(x^2*exp(2*x) + 4*x^7))/4 + log(x)^2/4 + (x*log(x^2 
)*exp(x)*log(x))/2)/(x^6*log(x^2)^2))*((log(x^2)^3*(exp(2*x)*(2*x^2 - x^3) 
 + 2*x^6 - 2*x^7))/2 + log(x)^2 - (log(x^2)^2*(x*exp(x) - exp(x)*log(x)*(5 
*x - x^2)))/2 + (log(x^2)*(3*log(x)^2 + log(x)*(2*x*exp(x) - 1)))/2))/(x^6 
*log(x^2)^3),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.82 \[ \int \frac {e^{-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}} \left (2 \log ^2(x)+\left (\left (-1+2 e^x x\right ) \log (x)+3 \log ^2(x)\right ) \log \left (x^2\right )+\left (-e^x x+e^x \left (5 x-x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (2 x^6-2 x^7+e^{2 x} \left (2 x^2-x^3\right )\right ) \log ^3\left (x^2\right )\right )}{2 x^6 \log ^3\left (x^2\right )} \, dx=\frac {x}{e^{\frac {e^{2 x} \mathrm {log}\left (x^{2}\right )^{2} x^{2}+2 e^{x} \mathrm {log}\left (x^{2}\right ) \mathrm {log}\left (x \right ) x +4 \mathrm {log}\left (x^{2}\right )^{2} x^{7}+\mathrm {log}\left (x \right )^{2}}{4 \mathrm {log}\left (x^{2}\right )^{2} x^{6}}}} \] Input:

int(1/2*(((-x^3+2*x^2)*exp(x)^2-2*x^7+2*x^6)*log(x^2)^3+((-x^2+5*x)*exp(x) 
*log(x)-exp(x)*x)*log(x^2)^2+(3*log(x)^2+(2*exp(x)*x-1)*log(x))*log(x^2)+2 
*log(x)^2)/x^6/log(x^2)^3/exp(1/4*((exp(x)^2*x^2+4*x^7)*log(x^2)^2+2*x*exp 
(x)*log(x)*log(x^2)+log(x)^2)/x^6/log(x^2)^2),x)
 

Output:

x/e**((e**(2*x)*log(x**2)**2*x**2 + 2*e**x*log(x**2)*log(x)*x + 4*log(x**2 
)**2*x**7 + log(x)**2)/(4*log(x**2)**2*x**6))