\(\int \frac {e^{\frac {x^2+(2+x^2) \log ^2(x^2)+x \log (e^{4/5} x) \log ^2(x^2)}{x}} (x^2+(8+4 x^2) \log (x^2)+4 x \log (e^{4/5} x) \log (x^2)+(-2+x+x^2) \log ^2(x^2))}{x^2} \, dx\) [649]

Optimal result

Integrand size = 86, antiderivative size = 26 \[ \int \frac {e^{\frac {x^2+\left (2+x^2\right ) \log ^2\left (x^2\right )+x \log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )}{x}} \left (x^2+\left (8+4 x^2\right ) \log \left (x^2\right )+4 x \log \left (e^{4/5} x\right ) \log \left (x^2\right )+\left (-2+x+x^2\right ) \log ^2\left (x^2\right )\right )}{x^2} \, dx=e^{x+\left (\frac {2}{x}+x+\log \left (e^{4/5} x\right )\right ) \log ^2\left (x^2\right )} \] Output:

exp(x+ln(x^2)^2*(2/x+x+ln(x*exp(4/5))))
 

Mathematica [A] (verified)

Time = 5.08 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {e^{\frac {x^2+\left (2+x^2\right ) \log ^2\left (x^2\right )+x \log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )}{x}} \left (x^2+\left (8+4 x^2\right ) \log \left (x^2\right )+4 x \log \left (e^{4/5} x\right ) \log \left (x^2\right )+\left (-2+x+x^2\right ) \log ^2\left (x^2\right )\right )}{x^2} \, dx=e^{x+\left (\frac {4}{5}+\frac {2}{x}+x\right ) \log ^2\left (x^2\right )} x^{\log ^2\left (x^2\right )} \] Input:

Integrate[(E^((x^2 + (2 + x^2)*Log[x^2]^2 + x*Log[E^(4/5)*x]*Log[x^2]^2)/x 
)*(x^2 + (8 + 4*x^2)*Log[x^2] + 4*x*Log[E^(4/5)*x]*Log[x^2] + (-2 + x + x^ 
2)*Log[x^2]^2))/x^2,x]
 

Output:

E^(x + (4/5 + 2/x + x)*Log[x^2]^2)*x^Log[x^2]^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.

\(\displaystyle \int \frac {\left (x^2+\left (x^2+x-2\right ) \log ^2\left (x^2\right )+4 x \log \left (e^{4/5} x\right ) \log \left (x^2\right )+\left (4 x^2+8\right ) \log \left (x^2\right )\right ) \exp \left (\frac {x^2+x \log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )+\left (x^2+2\right ) \log ^2\left (x^2\right )}{x}\right )}{x^2} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (x^2+\left (x^2+x-2\right ) \log ^2\left (x^2\right )+4 x \log \left (e^{4/5} x\right ) \log \left (x^2\right )+\left (4 x^2+8\right ) \log \left (x^2\right )\right ) \exp \left (\frac {\left (x^2+2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )+x\right )}{x^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {(x-1) (x+2) \log ^2\left (x^2\right ) \exp \left (\frac {\left (x^2+2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )+x\right )}{x^2}+\frac {4 \left (5 x^2+4 x+5 x \log (x)+10\right ) \log \left (x^2\right ) \exp \left (\frac {\left (x^2+2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )+x\right )}{5 x^2}+\exp \left (\frac {\left (x^2+2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )+x\right )\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \int \exp \left (\frac {\left (x^2+2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )+x\right )dx+4 \int \exp \left (\frac {\left (x^2+2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )+x\right ) \log \left (x^2\right )dx+8 \int \frac {\exp \left (\frac {\left (x^2+2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )+x\right ) \log \left (x^2\right )}{x^2}dx+\frac {16}{5} \int \frac {\exp \left (\frac {\left (x^2+2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )+x\right ) \log \left (x^2\right )}{x}dx+4 \int \frac {\exp \left (\frac {\left (x^2+2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )+x\right ) \log (x) \log \left (x^2\right )}{x}dx+\int \exp \left (\frac {\left (x^2+2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )+x\right ) \log ^2\left (x^2\right )dx-2 \int \frac {\exp \left (\frac {\left (x^2+2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )+x\right ) \log ^2\left (x^2\right )}{x^2}dx+\int \frac {\exp \left (\frac {\left (x^2+2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )+x\right ) \log ^2\left (x^2\right )}{x}dx\)

Input:

Int[(E^((x^2 + (2 + x^2)*Log[x^2]^2 + x*Log[E^(4/5)*x]*Log[x^2]^2)/x)*(x^2 
 + (8 + 4*x^2)*Log[x^2] + 4*x*Log[E^(4/5)*x]*Log[x^2] + (-2 + x + x^2)*Log 
[x^2]^2))/x^2,x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.68 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.35

method	result	size
parallelrisch	\({\mathrm e}^{\frac {x \ln \left (x^{2}\right )^{2} \ln \left (x \,{\mathrm e}^{\frac {4}{5}}\right )+\left (x^{2}+2\right ) \ln \left (x^{2}\right )^{2}+x^{2}}{x}}\)	\(35\)
risch	\(x^{\frac {8 i \pi \,\operatorname {csgn}\left (i x \right )}{x}} x^{-\frac {8 i \pi \,\operatorname {csgn}\left (i x^{2}\right )}{x}} x^{4 i x \pi \,\operatorname {csgn}\left (i x \right )} x^{-4 i x \pi \,\operatorname {csgn}\left (i x^{2}\right )} x^{2 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )} x^{-\frac {3 \pi ^{2}}{2}} x^{-\frac {\pi ^{2}}{2}} x^{-\frac {16 i \pi \,\operatorname {csgn}\left (i x^{2}\right )}{5}} x^{\frac {16 i \pi \,\operatorname {csgn}\left (i x \right )}{5}} {\mathrm e}^{\frac {-5 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{6} x^{2}+20 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{5} \operatorname {csgn}\left (i x \right ) x^{2}-30 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{4} \operatorname {csgn}\left (i x \right )^{2} x^{2}+20 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{3} \operatorname {csgn}\left (i x \right )^{3} x^{2}-5 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )^{4} x^{2}-4 x \,\pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{6}+16 x \,\pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{5} \operatorname {csgn}\left (i x \right )-24 x \,\pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{4} \operatorname {csgn}\left (i x \right )^{2}+16 x \,\pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{3} \operatorname {csgn}\left (i x \right )^{3}-4 x \,\pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )^{4}+80 i x \ln \left (x \right )^{2} \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )-40 i x \ln \left (x \right )^{2} \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}-40 i x \ln \left (x \right )^{2} \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-10 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{6}+40 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{5} \operatorname {csgn}\left (i x \right )-60 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{4} \operatorname {csgn}\left (i x \right )^{2}+40 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{3} \operatorname {csgn}\left (i x \right )^{3}-10 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )^{4}+80 x \ln \left (x \right )^{3}+80 x^{2} \ln \left (x \right )^{2}+64 x \ln \left (x \right )^{2}+160 \ln \left (x \right )^{2}+20 x^{2}}{20 x}}\)	\(537\)

Input:

int((4*x*ln(x^2)*ln(x*exp(4/5))+(x^2+x-2)*ln(x^2)^2+(4*x^2+8)*ln(x^2)+x^2) 
*exp((x*ln(x^2)^2*ln(x*exp(4/5))+(x^2+2)*ln(x^2)^2+x^2)/x)/x^2,x,method=_R 
ETURNVERBOSE)
 

Output:

exp((x*ln(x^2)^2*ln(x*exp(4/5))+(x^2+2)*ln(x^2)^2+x^2)/x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int \frac {e^{\frac {x^2+\left (2+x^2\right ) \log ^2\left (x^2\right )+x \log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )}{x}} \left (x^2+\left (8+4 x^2\right ) \log \left (x^2\right )+4 x \log \left (e^{4/5} x\right ) \log \left (x^2\right )+\left (-2+x+x^2\right ) \log ^2\left (x^2\right )\right )}{x^2} \, dx=e^{\left (\frac {5 \, x \log \left (x^{2}\right )^{3} + 2 \, {\left (5 \, x^{2} + 4 \, x + 10\right )} \log \left (x^{2}\right )^{2} + 10 \, x^{2}}{10 \, x}\right )} \] Input:

integrate((4*x*log(x^2)*log(x*exp(4/5))+(x^2+x-2)*log(x^2)^2+(4*x^2+8)*log 
(x^2)+x^2)*exp((x*log(x^2)^2*log(x*exp(4/5))+(x^2+2)*log(x^2)^2+x^2)/x)/x^ 
2,x, algorithm="fricas")
 

Output:

e^(1/10*(5*x*log(x^2)^3 + 2*(5*x^2 + 4*x + 10)*log(x^2)^2 + 10*x^2)/x)
 

Sympy [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \frac {e^{\frac {x^2+\left (2+x^2\right ) \log ^2\left (x^2\right )+x \log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )}{x}} \left (x^2+\left (8+4 x^2\right ) \log \left (x^2\right )+4 x \log \left (e^{4/5} x\right ) \log \left (x^2\right )+\left (-2+x+x^2\right ) \log ^2\left (x^2\right )\right )}{x^2} \, dx=e^{\frac {x^{2} + x \left (\frac {\log {\left (x^{2} \right )}}{2} + \frac {4}{5}\right ) \log {\left (x^{2} \right )}^{2} + \left (x^{2} + 2\right ) \log {\left (x^{2} \right )}^{2}}{x}} \] Input:

integrate((4*x*ln(x**2)*ln(x*exp(4/5))+(x**2+x-2)*ln(x**2)**2+(4*x**2+8)*l 
n(x**2)+x**2)*exp((x*ln(x**2)**2*ln(x*exp(4/5))+(x**2+2)*ln(x**2)**2+x**2) 
/x)/x**2,x)
 

Output:

exp((x**2 + x*(log(x**2)/2 + 4/5)*log(x**2)**2 + (x**2 + 2)*log(x**2)**2)/ 
x)
 

Maxima [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {e^{\frac {x^2+\left (2+x^2\right ) \log ^2\left (x^2\right )+x \log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )}{x}} \left (x^2+\left (8+4 x^2\right ) \log \left (x^2\right )+4 x \log \left (e^{4/5} x\right ) \log \left (x^2\right )+\left (-2+x+x^2\right ) \log ^2\left (x^2\right )\right )}{x^2} \, dx=e^{\left (4 \, x \log \left (x\right )^{2} + 4 \, \log \left (x\right )^{3} + \frac {16}{5} \, \log \left (x\right )^{2} + x + \frac {8 \, \log \left (x\right )^{2}}{x}\right )} \] Input:

integrate((4*x*log(x^2)*log(x*exp(4/5))+(x^2+x-2)*log(x^2)^2+(4*x^2+8)*log 
(x^2)+x^2)*exp((x*log(x^2)^2*log(x*exp(4/5))+(x^2+2)*log(x^2)^2+x^2)/x)/x^ 
2,x, algorithm="maxima")
 

Output:

e^(4*x*log(x)^2 + 4*log(x)^3 + 16/5*log(x)^2 + x + 8*log(x)^2/x)
 

Giac [F]

\[ \int \frac {e^{\frac {x^2+\left (2+x^2\right ) \log ^2\left (x^2\right )+x \log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )}{x}} \left (x^2+\left (8+4 x^2\right ) \log \left (x^2\right )+4 x \log \left (e^{4/5} x\right ) \log \left (x^2\right )+\left (-2+x+x^2\right ) \log ^2\left (x^2\right )\right )}{x^2} \, dx=\int { \frac {{\left ({\left (x^{2} + x - 2\right )} \log \left (x^{2}\right )^{2} + 4 \, x \log \left (x^{2}\right ) \log \left (x e^{\frac {4}{5}}\right ) + x^{2} + 4 \, {\left (x^{2} + 2\right )} \log \left (x^{2}\right )\right )} e^{\left (\frac {x \log \left (x^{2}\right )^{2} \log \left (x e^{\frac {4}{5}}\right ) + {\left (x^{2} + 2\right )} \log \left (x^{2}\right )^{2} + x^{2}}{x}\right )}}{x^{2}} \,d x } \] Input:

integrate((4*x*log(x^2)*log(x*exp(4/5))+(x^2+x-2)*log(x^2)^2+(4*x^2+8)*log 
(x^2)+x^2)*exp((x*log(x^2)^2*log(x*exp(4/5))+(x^2+2)*log(x^2)^2+x^2)/x)/x^ 
2,x, algorithm="giac")
 

Output:

integrate(((x^2 + x - 2)*log(x^2)^2 + 4*x*log(x^2)*log(x*e^(4/5)) + x^2 + 
4*(x^2 + 2)*log(x^2))*e^((x*log(x^2)^2*log(x*e^(4/5)) + (x^2 + 2)*log(x^2) 
^2 + x^2)/x)/x^2, x)
 

Mupad [B] (verification not implemented)

Time = 4.44 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58 \[ \int \frac {e^{\frac {x^2+\left (2+x^2\right ) \log ^2\left (x^2\right )+x \log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )}{x}} \left (x^2+\left (8+4 x^2\right ) \log \left (x^2\right )+4 x \log \left (e^{4/5} x\right ) \log \left (x^2\right )+\left (-2+x+x^2\right ) \log ^2\left (x^2\right )\right )}{x^2} \, dx=x^{{\ln \left (x^2\right )}^2}\,{\mathrm {e}}^{\frac {2\,{\ln \left (x^2\right )}^2}{x}}\,{\mathrm {e}}^{\frac {4\,{\ln \left (x^2\right )}^2}{5}}\,{\mathrm {e}}^x\,{\mathrm {e}}^{x\,{\ln \left (x^2\right )}^2} \] Input:

int((exp((log(x^2)^2*(x^2 + 2) + x^2 + x*log(x^2)^2*log(x*exp(4/5)))/x)*(l 
og(x^2)*(4*x^2 + 8) + log(x^2)^2*(x + x^2 - 2) + x^2 + 4*x*log(x^2)*log(x* 
exp(4/5))))/x^2,x)
 

Output:

x^(log(x^2)^2)*exp((2*log(x^2)^2)/x)*exp((4*log(x^2)^2)/5)*exp(x)*exp(x*lo 
g(x^2)^2)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62 \[ \int \frac {e^{\frac {x^2+\left (2+x^2\right ) \log ^2\left (x^2\right )+x \log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )}{x}} \left (x^2+\left (8+4 x^2\right ) \log \left (x^2\right )+4 x \log \left (e^{4/5} x\right ) \log \left (x^2\right )+\left (-2+x+x^2\right ) \log ^2\left (x^2\right )\right )}{x^2} \, dx=e^{\frac {\mathrm {log}\left (x^{2}\right )^{2} \mathrm {log}\left (e^{\frac {4}{5}} x \right ) x +\mathrm {log}\left (x^{2}\right )^{2} x^{2}+2 \mathrm {log}\left (x^{2}\right )^{2}+x^{2}}{x}} \] Input:

int((4*x*log(x^2)*log(x*exp(4/5))+(x^2+x-2)*log(x^2)^2+(4*x^2+8)*log(x^2)+ 
x^2)*exp((x*log(x^2)^2*log(x*exp(4/5))+(x^2+2)*log(x^2)^2+x^2)/x)/x^2,x)
 

Output:

e**((log(x**2)**2*log(e**(4/5)*x)*x + log(x**2)**2*x**2 + 2*log(x**2)**2 + 
 x**2)/x)