\(\int \frac {e^{\frac {-3-4 x+\log (\frac {x+x^4-2 x^2 \log (x^2)+\log ^2(x^2)}{x})}{x}} (3 x-4 x^2+6 x^4+(4-8 x^2) \log (x^2)+2 \log ^2(x^2)+(-x-x^4+2 x^2 \log (x^2)-\log ^2(x^2)) \log (\frac {x+x^4-2 x^2 \log (x^2)+\log ^2(x^2)}{x}))}{x^3+x^6-2 x^4 \log (x^2)+x^2 \log ^2(x^2)} \, dx\) [490]

Optimal result

Integrand size = 151, antiderivative size = 30 \[ \int \frac {e^{\frac {-3-4 x+\log \left (\frac {x+x^4-2 x^2 \log \left (x^2\right )+\log ^2\left (x^2\right )}{x}\right )}{x}} \left (3 x-4 x^2+6 x^4+\left (4-8 x^2\right ) \log \left (x^2\right )+2 \log ^2\left (x^2\right )+\left (-x-x^4+2 x^2 \log \left (x^2\right )-\log ^2\left (x^2\right )\right ) \log \left (\frac {x+x^4-2 x^2 \log \left (x^2\right )+\log ^2\left (x^2\right )}{x}\right )\right )}{x^3+x^6-2 x^4 \log \left (x^2\right )+x^2 \log ^2\left (x^2\right )} \, dx=e^{-5+\frac {-3+x+\log \left (\frac {x+\left (-x^2+\log \left (x^2\right )\right )^2}{x}\right )}{x}} \] Output:

exp((ln(((ln(x^2)-x^2)^2+x)/x)+x-3)/x-5)
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \frac {e^{\frac {-3-4 x+\log \left (\frac {x+x^4-2 x^2 \log \left (x^2\right )+\log ^2\left (x^2\right )}{x}\right )}{x}} \left (3 x-4 x^2+6 x^4+\left (4-8 x^2\right ) \log \left (x^2\right )+2 \log ^2\left (x^2\right )+\left (-x-x^4+2 x^2 \log \left (x^2\right )-\log ^2\left (x^2\right )\right ) \log \left (\frac {x+x^4-2 x^2 \log \left (x^2\right )+\log ^2\left (x^2\right )}{x}\right )\right )}{x^3+x^6-2 x^4 \log \left (x^2\right )+x^2 \log ^2\left (x^2\right )} \, dx=e^{-4-\frac {3}{x}} \left (1+x^3-2 x \log \left (x^2\right )+\frac {\log ^2\left (x^2\right )}{x}\right )^{\frac {1}{x}} \] Input:

Integrate[(E^((-3 - 4*x + Log[(x + x^4 - 2*x^2*Log[x^2] + Log[x^2]^2)/x])/ 
x)*(3*x - 4*x^2 + 6*x^4 + (4 - 8*x^2)*Log[x^2] + 2*Log[x^2]^2 + (-x - x^4 
+ 2*x^2*Log[x^2] - Log[x^2]^2)*Log[(x + x^4 - 2*x^2*Log[x^2] + Log[x^2]^2) 
/x]))/(x^3 + x^6 - 2*x^4*Log[x^2] + x^2*Log[x^2]^2),x]
 

Output:

E^(-4 - 3/x)*(1 + x^3 - 2*x*Log[x^2] + Log[x^2]^2/x)^x^(-1)
 

Rubi [A] (verified)

Time = 1.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.007, Rules used = {7257}

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.

\(\displaystyle \int \frac {\left (6 x^4-4 x^2+2 \log ^2\left (x^2\right )+\left (4-8 x^2\right ) \log \left (x^2\right )+\left (-x^4-\log ^2\left (x^2\right )+2 x^2 \log \left (x^2\right )-x\right ) \log \left (\frac {x^4+\log ^2\left (x^2\right )-2 x^2 \log \left (x^2\right )+x}{x}\right )+3 x\right ) \exp \left (\frac {\log \left (\frac {x^4+\log ^2\left (x^2\right )-2 x^2 \log \left (x^2\right )+x}{x}\right )-4 x-3}{x}\right )}{x^6+x^3+x^2 \log ^2\left (x^2\right )-2 x^4 \log \left (x^2\right )} \, dx\)

\(\Big \downarrow \) 7257

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

Input:

Int[(E^((-3 - 4*x + Log[(x + x^4 - 2*x^2*Log[x^2] + Log[x^2]^2)/x])/x)*(3* 
x - 4*x^2 + 6*x^4 + (4 - 8*x^2)*Log[x^2] + 2*Log[x^2]^2 + (-x - x^4 + 2*x^ 
2*Log[x^2] - Log[x^2]^2)*Log[(x + x^4 - 2*x^2*Log[x^2] + Log[x^2]^2)/x]))/ 
(x^3 + x^6 - 2*x^4*Log[x^2] + x^2*Log[x^2]^2),x]
 

Output:

((x + x^4 - 2*x^2*Log[x^2] + Log[x^2]^2)/x)^x^(-1)/E^((3 + 4*x)/x)
 

Defintions of rubi rules used

rule 7257

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Sim 
p[q*(F^v/Log[F]), x] /;  !FalseQ[q]] /; FreeQ[F, x]
 

Maple [A] (verified)

Time = 31.20 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20

method	result	size
parallelrisch	\({\mathrm e}^{\frac {\ln \left (\frac {\ln \left (x^{2}\right )^{2}-2 x^{2} \ln \left (x^{2}\right )+x^{4}+x}{x}\right )-4 x -3}{x}}\)	\(36\)

Input:

int(((-ln(x^2)^2+2*x^2*ln(x^2)-x^4-x)*ln((ln(x^2)^2-2*x^2*ln(x^2)+x^4+x)/x 
)+2*ln(x^2)^2+(-8*x^2+4)*ln(x^2)+6*x^4-4*x^2+3*x)*exp((ln((ln(x^2)^2-2*x^2 
*ln(x^2)+x^4+x)/x)-4*x-3)/x)/(x^2*ln(x^2)^2-2*x^4*ln(x^2)+x^6+x^3),x,metho 
d=_RETURNVERBOSE)
 

Output:

exp((ln((ln(x^2)^2-2*x^2*ln(x^2)+x^4+x)/x)-4*x-3)/x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.27 \[ \int \frac {e^{\frac {-3-4 x+\log \left (\frac {x+x^4-2 x^2 \log \left (x^2\right )+\log ^2\left (x^2\right )}{x}\right )}{x}} \left (3 x-4 x^2+6 x^4+\left (4-8 x^2\right ) \log \left (x^2\right )+2 \log ^2\left (x^2\right )+\left (-x-x^4+2 x^2 \log \left (x^2\right )-\log ^2\left (x^2\right )\right ) \log \left (\frac {x+x^4-2 x^2 \log \left (x^2\right )+\log ^2\left (x^2\right )}{x}\right )\right )}{x^3+x^6-2 x^4 \log \left (x^2\right )+x^2 \log ^2\left (x^2\right )} \, dx=e^{\left (-\frac {4 \, x - \log \left (\frac {x^{4} - 2 \, x^{2} \log \left (x^{2}\right ) + \log \left (x^{2}\right )^{2} + x}{x}\right ) + 3}{x}\right )} \] Input:

integrate(((-log(x^2)^2+2*x^2*log(x^2)-x^4-x)*log((log(x^2)^2-2*x^2*log(x^ 
2)+x^4+x)/x)+2*log(x^2)^2+(-8*x^2+4)*log(x^2)+6*x^4-4*x^2+3*x)*exp((log((l 
og(x^2)^2-2*x^2*log(x^2)+x^4+x)/x)-4*x-3)/x)/(x^2*log(x^2)^2-2*x^4*log(x^2 
)+x^6+x^3),x, algorithm="fricas")
 

Output:

e^(-(4*x - log((x^4 - 2*x^2*log(x^2) + log(x^2)^2 + x)/x) + 3)/x)
 

Sympy [A] (verification not implemented)

Time = 1.51 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {e^{\frac {-3-4 x+\log \left (\frac {x+x^4-2 x^2 \log \left (x^2\right )+\log ^2\left (x^2\right )}{x}\right )}{x}} \left (3 x-4 x^2+6 x^4+\left (4-8 x^2\right ) \log \left (x^2\right )+2 \log ^2\left (x^2\right )+\left (-x-x^4+2 x^2 \log \left (x^2\right )-\log ^2\left (x^2\right )\right ) \log \left (\frac {x+x^4-2 x^2 \log \left (x^2\right )+\log ^2\left (x^2\right )}{x}\right )\right )}{x^3+x^6-2 x^4 \log \left (x^2\right )+x^2 \log ^2\left (x^2\right )} \, dx=e^{\frac {- 4 x + \log {\left (\frac {x^{4} - 2 x^{2} \log {\left (x^{2} \right )} + x + \log {\left (x^{2} \right )}^{2}}{x} \right )} - 3}{x}} \] Input:

integrate(((-ln(x**2)**2+2*x**2*ln(x**2)-x**4-x)*ln((ln(x**2)**2-2*x**2*ln 
(x**2)+x**4+x)/x)+2*ln(x**2)**2+(-8*x**2+4)*ln(x**2)+6*x**4-4*x**2+3*x)*ex 
p((ln((ln(x**2)**2-2*x**2*ln(x**2)+x**4+x)/x)-4*x-3)/x)/(x**2*ln(x**2)**2- 
2*x**4*ln(x**2)+x**6+x**3),x)
 

Output:

exp((-4*x + log((x**4 - 2*x**2*log(x**2) + x + log(x**2)**2)/x) - 3)/x)
                                                                                    
                                                                                    
 

Maxima [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.27 \[ \int \frac {e^{\frac {-3-4 x+\log \left (\frac {x+x^4-2 x^2 \log \left (x^2\right )+\log ^2\left (x^2\right )}{x}\right )}{x}} \left (3 x-4 x^2+6 x^4+\left (4-8 x^2\right ) \log \left (x^2\right )+2 \log ^2\left (x^2\right )+\left (-x-x^4+2 x^2 \log \left (x^2\right )-\log ^2\left (x^2\right )\right ) \log \left (\frac {x+x^4-2 x^2 \log \left (x^2\right )+\log ^2\left (x^2\right )}{x}\right )\right )}{x^3+x^6-2 x^4 \log \left (x^2\right )+x^2 \log ^2\left (x^2\right )} \, dx=e^{\left (\frac {\log \left (x^{4} - 4 \, x^{2} \log \left (x\right ) + 4 \, \log \left (x\right )^{2} + x\right )}{x} - \frac {\log \left (x\right )}{x} - \frac {3}{x} - 4\right )} \] Input:

integrate(((-log(x^2)^2+2*x^2*log(x^2)-x^4-x)*log((log(x^2)^2-2*x^2*log(x^ 
2)+x^4+x)/x)+2*log(x^2)^2+(-8*x^2+4)*log(x^2)+6*x^4-4*x^2+3*x)*exp((log((l 
og(x^2)^2-2*x^2*log(x^2)+x^4+x)/x)-4*x-3)/x)/(x^2*log(x^2)^2-2*x^4*log(x^2 
)+x^6+x^3),x, algorithm="maxima")
 

Output:

e^(log(x^4 - 4*x^2*log(x) + 4*log(x)^2 + x)/x - log(x)/x - 3/x - 4)
 

Giac [A] (verification not implemented)

Time = 0.56 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {e^{\frac {-3-4 x+\log \left (\frac {x+x^4-2 x^2 \log \left (x^2\right )+\log ^2\left (x^2\right )}{x}\right )}{x}} \left (3 x-4 x^2+6 x^4+\left (4-8 x^2\right ) \log \left (x^2\right )+2 \log ^2\left (x^2\right )+\left (-x-x^4+2 x^2 \log \left (x^2\right )-\log ^2\left (x^2\right )\right ) \log \left (\frac {x+x^4-2 x^2 \log \left (x^2\right )+\log ^2\left (x^2\right )}{x}\right )\right )}{x^3+x^6-2 x^4 \log \left (x^2\right )+x^2 \log ^2\left (x^2\right )} \, dx=e^{\left (\frac {\log \left (x^{3} - 2 \, x \log \left (x^{2}\right ) + \frac {\log \left (x^{2}\right )^{2}}{x} + 1\right )}{x} - \frac {3}{x} - 4\right )} \] Input:

integrate(((-log(x^2)^2+2*x^2*log(x^2)-x^4-x)*log((log(x^2)^2-2*x^2*log(x^ 
2)+x^4+x)/x)+2*log(x^2)^2+(-8*x^2+4)*log(x^2)+6*x^4-4*x^2+3*x)*exp((log((l 
og(x^2)^2-2*x^2*log(x^2)+x^4+x)/x)-4*x-3)/x)/(x^2*log(x^2)^2-2*x^4*log(x^2 
)+x^6+x^3),x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 3.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {e^{\frac {-3-4 x+\log \left (\frac {x+x^4-2 x^2 \log \left (x^2\right )+\log ^2\left (x^2\right )}{x}\right )}{x}} \left (3 x-4 x^2+6 x^4+\left (4-8 x^2\right ) \log \left (x^2\right )+2 \log ^2\left (x^2\right )+\left (-x-x^4+2 x^2 \log \left (x^2\right )-\log ^2\left (x^2\right )\right ) \log \left (\frac {x+x^4-2 x^2 \log \left (x^2\right )+\log ^2\left (x^2\right )}{x}\right )\right )}{x^3+x^6-2 x^4 \log \left (x^2\right )+x^2 \log ^2\left (x^2\right )} \, dx={\mathrm {e}}^{-4}\,{\mathrm {e}}^{-\frac {3}{x}}\,{\left (x^3-2\,x\,\ln \left (x^2\right )+\frac {{\ln \left (x^2\right )}^2}{x}+1\right )}^{1/x} \] Input:

int((exp(-(4*x - log((x - 2*x^2*log(x^2) + log(x^2)^2 + x^4)/x) + 3)/x)*(3 
*x - log(x^2)*(8*x^2 - 4) - log((x - 2*x^2*log(x^2) + log(x^2)^2 + x^4)/x) 
*(x - 2*x^2*log(x^2) + log(x^2)^2 + x^4) + 2*log(x^2)^2 - 4*x^2 + 6*x^4))/ 
(x^3 - 2*x^4*log(x^2) + x^6 + x^2*log(x^2)^2),x)
 

Output:

exp(-4)*exp(-3/x)*(x^3 - 2*x*log(x^2) + log(x^2)^2/x + 1)^(1/x)
 

Reduce [F]

\[ \int \frac {e^{\frac {-3-4 x+\log \left (\frac {x+x^4-2 x^2 \log \left (x^2\right )+\log ^2\left (x^2\right )}{x}\right )}{x}} \left (3 x-4 x^2+6 x^4+\left (4-8 x^2\right ) \log \left (x^2\right )+2 \log ^2\left (x^2\right )+\left (-x-x^4+2 x^2 \log \left (x^2\right )-\log ^2\left (x^2\right )\right ) \log \left (\frac {x+x^4-2 x^2 \log \left (x^2\right )+\log ^2\left (x^2\right )}{x}\right )\right )}{x^3+x^6-2 x^4 \log \left (x^2\right )+x^2 \log ^2\left (x^2\right )} \, dx =\text {Too large to display} \] Input:

int(((-log(x^2)^2+2*x^2*log(x^2)-x^4-x)*log((log(x^2)^2-2*x^2*log(x^2)+x^4 
+x)/x)+2*log(x^2)^2+(-8*x^2+4)*log(x^2)+6*x^4-4*x^2+3*x)*exp((log((log(x^2 
)^2-2*x^2*log(x^2)+x^4+x)/x)-4*x-3)/x)/(x^2*log(x^2)^2-2*x^4*log(x^2)+x^6+ 
x^3),x)
 

Output:

(3*int(e**(log((log(x**2)**2 - 2*log(x**2)*x**2 + x**4 + x)/x)/x)/(e**(3/x 
)*log(x**2)**2*x - 2*e**(3/x)*log(x**2)*x**3 + e**(3/x)*x**5 + e**(3/x)*x* 
*2),x) - 4*int(e**(log((log(x**2)**2 - 2*log(x**2)*x**2 + x**4 + x)/x)/x)/ 
(e**(3/x)*log(x**2)**2 - 2*e**(3/x)*log(x**2)*x**2 + e**(3/x)*x**4 + e**(3 
/x)*x),x) - int((e**(log((log(x**2)**2 - 2*log(x**2)*x**2 + x**4 + x)/x)/x 
)*log(x**2)**2*log((log(x**2)**2 - 2*log(x**2)*x**2 + x**4 + x)/x))/(e**(3 
/x)*log(x**2)**2*x**2 - 2*e**(3/x)*log(x**2)*x**4 + e**(3/x)*x**6 + e**(3/ 
x)*x**3),x) + 2*int((e**(log((log(x**2)**2 - 2*log(x**2)*x**2 + x**4 + x)/ 
x)/x)*log(x**2)**2)/(e**(3/x)*log(x**2)**2*x**2 - 2*e**(3/x)*log(x**2)*x** 
4 + e**(3/x)*x**6 + e**(3/x)*x**3),x) + 6*int((e**(log((log(x**2)**2 - 2*l 
og(x**2)*x**2 + x**4 + x)/x)/x)*x**2)/(e**(3/x)*log(x**2)**2 - 2*e**(3/x)* 
log(x**2)*x**2 + e**(3/x)*x**4 + e**(3/x)*x),x) + 2*int((e**(log((log(x**2 
)**2 - 2*log(x**2)*x**2 + x**4 + x)/x)/x)*log(x**2)*log((log(x**2)**2 - 2* 
log(x**2)*x**2 + x**4 + x)/x))/(e**(3/x)*log(x**2)**2 - 2*e**(3/x)*log(x** 
2)*x**2 + e**(3/x)*x**4 + e**(3/x)*x),x) + 4*int((e**(log((log(x**2)**2 - 
2*log(x**2)*x**2 + x**4 + x)/x)/x)*log(x**2))/(e**(3/x)*log(x**2)**2*x**2 
- 2*e**(3/x)*log(x**2)*x**4 + e**(3/x)*x**6 + e**(3/x)*x**3),x) - 8*int((e 
**(log((log(x**2)**2 - 2*log(x**2)*x**2 + x**4 + x)/x)/x)*log(x**2))/(e**( 
3/x)*log(x**2)**2 - 2*e**(3/x)*log(x**2)*x**2 + e**(3/x)*x**4 + e**(3/x)*x 
),x) - int((e**(log((log(x**2)**2 - 2*log(x**2)*x**2 + x**4 + x)/x)/x)*...