\(\int \frac {-4+256 x^3-160 x^4+24 x^5+e^{x^2} (1+8 x-64 x^3+40 x^4-38 x^5+16 x^6-2 x^7)+(8-256 x^3+160 x^4-24 x^5+e^{x^2} (-2+64 x^3-40 x^4+6 x^5)) \log (4-e^{x^2})+(-4+e^{x^2}) \log ^2(4-e^{x^2})}{-4+e^{x^2}+(8-2 e^{x^2}) \log (4-e^{x^2})+(-4+e^{x^2}) \log ^2(4-e^{x^2})} \, dx\) [1699]

Optimal result

Integrand size = 174, antiderivative size = 34 \[ \int \frac {-4+256 x^3-160 x^4+24 x^5+e^{x^2} \left (1+8 x-64 x^3+40 x^4-38 x^5+16 x^6-2 x^7\right )+\left (8-256 x^3+160 x^4-24 x^5+e^{x^2} \left (-2+64 x^3-40 x^4+6 x^5\right )\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )}{-4+e^{x^2}+\left (8-2 e^{x^2}\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )} \, dx=-4+x-\frac {-4+(4-x)^2 x^4}{1-\log \left (4-e^{x^2}\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \frac {-4+256 x^3-160 x^4+24 x^5+e^{x^2} \left (1+8 x-64 x^3+40 x^4-38 x^5+16 x^6-2 x^7\right )+\left (8-256 x^3+160 x^4-24 x^5+e^{x^2} \left (-2+64 x^3-40 x^4+6 x^5\right )\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )}{-4+e^{x^2}+\left (8-2 e^{x^2}\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )} \, dx=x+\frac {-4+16 x^4-8 x^5+x^6}{-1+\log \left (4-e^{x^2}\right )} \] Input:

Integrate[(-4 + 256*x^3 - 160*x^4 + 24*x^5 + E^x^2*(1 + 8*x - 64*x^3 + 40* 
x^4 - 38*x^5 + 16*x^6 - 2*x^7) + (8 - 256*x^3 + 160*x^4 - 24*x^5 + E^x^2*( 
-2 + 64*x^3 - 40*x^4 + 6*x^5))*Log[4 - E^x^2] + (-4 + E^x^2)*Log[4 - E^x^2 
]^2)/(-4 + E^x^2 + (8 - 2*E^x^2)*Log[4 - E^x^2] + (-4 + E^x^2)*Log[4 - E^x 
^2]^2),x]
 

Output:

x + (-4 + 16*x^4 - 8*x^5 + x^6)/(-1 + Log[4 - E^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.

\(\displaystyle \int \frac {24 x^5-160 x^4+256 x^3+\left (e^{x^2}-4\right ) \log ^2\left (4-e^{x^2}\right )+\left (-24 x^5+160 x^4-256 x^3+e^{x^2} \left (6 x^5-40 x^4+64 x^3-2\right )+8\right ) \log \left (4-e^{x^2}\right )+e^{x^2} \left (-2 x^7+16 x^6-38 x^5+40 x^4-64 x^3+8 x+1\right )-4}{e^{x^2}+\left (e^{x^2}-4\right ) \log ^2\left (4-e^{x^2}\right )+\left (8-2 e^{x^2}\right ) \log \left (4-e^{x^2}\right )-4} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-24 x^5+160 x^4-256 x^3-\left (e^{x^2}-4\right ) \log ^2\left (4-e^{x^2}\right )-\left (-24 x^5+160 x^4-256 x^3+e^{x^2} \left (6 x^5-40 x^4+64 x^3-2\right )+8\right ) \log \left (4-e^{x^2}\right )-e^{x^2} \left (-2 x^7+16 x^6-38 x^5+40 x^4-64 x^3+8 x+1\right )+4}{\left (4-e^{x^2}\right ) \left (1-\log \left (4-e^{x^2}\right )\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {-2 x^7+16 x^6-38 x^5+40 x^4-64 x^3+\log ^2\left (4-e^{x^2}\right )-2 \log \left (4-e^{x^2}\right )+6 x^5 \log \left (4-e^{x^2}\right )-40 x^4 \log \left (4-e^{x^2}\right )+64 x^3 \log \left (4-e^{x^2}\right )+8 x+1}{\left (\log \left (4-e^{x^2}\right )-1\right )^2}-\frac {8 x \left (x^6-8 x^5+16 x^4-4\right )}{\left (e^{x^2}-4\right ) \left (\log \left (4-e^{x^2}\right )-1\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -16 \text {Subst}\left (\int \frac {x^2}{\left (\log \left (4-e^x\right )-1\right )^2}dx,x,x^2\right )-64 \text {Subst}\left (\int \frac {x^2}{\left (-4+e^x\right ) \left (\log \left (4-e^x\right )-1\right )^2}dx,x,x^2\right )+32 \text {Subst}\left (\int \frac {x}{\log \left (4-e^x\right )-1}dx,x,x^2\right )+3 \text {Subst}\left (\int \frac {x^2}{\log \left (4-e^x\right )-1}dx,x,x^2\right )-\text {Subst}\left (\int \frac {x^3}{\left (\log \left (4-e^x\right )-1\right )^2}dx,x,x^2\right )-4 \text {Subst}\left (\int \frac {x^3}{\left (-4+e^x\right ) \left (\log \left (4-e^x\right )-1\right )^2}dx,x,x^2\right )+16 \int \frac {x^6}{\left (\log \left (4-e^{x^2}\right )-1\right )^2}dx+64 \int \frac {x^6}{\left (-4+e^{x^2}\right ) \left (\log \left (4-e^{x^2}\right )-1\right )^2}dx-40 \int \frac {x^4}{\log \left (4-e^{x^2}\right )-1}dx+\frac {4}{1-\log \left (4-e^{x^2}\right )}+x\)

Input:

Int[(-4 + 256*x^3 - 160*x^4 + 24*x^5 + E^x^2*(1 + 8*x - 64*x^3 + 40*x^4 - 
38*x^5 + 16*x^6 - 2*x^7) + (8 - 256*x^3 + 160*x^4 - 24*x^5 + E^x^2*(-2 + 6 
4*x^3 - 40*x^4 + 6*x^5))*Log[4 - E^x^2] + (-4 + E^x^2)*Log[4 - E^x^2]^2)/( 
-4 + E^x^2 + (8 - 2*E^x^2)*Log[4 - E^x^2] + (-4 + E^x^2)*Log[4 - E^x^2]^2) 
,x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94

method	result	size
risch	\(x +\frac {x^{6}-8 x^{5}+16 x^{4}-4}{\ln \left (4-{\mathrm e}^{x^{2}}\right )-1}\)	\(32\)
parallelrisch	\(\frac {8 x^{6}-64 x^{5}+128 x^{4}+8 x \ln \left (4-{\mathrm e}^{x^{2}}\right )-8 x -32 \ln \left (4-{\mathrm e}^{x^{2}}\right )}{8 \ln \left (4-{\mathrm e}^{x^{2}}\right )-8}\)	\(58\)

Input:

int(((exp(x^2)-4)*ln(4-exp(x^2))^2+((6*x^5-40*x^4+64*x^3-2)*exp(x^2)-24*x^ 
5+160*x^4-256*x^3+8)*ln(4-exp(x^2))+(-2*x^7+16*x^6-38*x^5+40*x^4-64*x^3+8* 
x+1)*exp(x^2)+24*x^5-160*x^4+256*x^3-4)/((exp(x^2)-4)*ln(4-exp(x^2))^2+(-2 
*exp(x^2)+8)*ln(4-exp(x^2))+exp(x^2)-4),x,method=_RETURNVERBOSE)
 

Output:

x+(x^6-8*x^5+16*x^4-4)/(ln(4-exp(x^2))-1)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.26 \[ \int \frac {-4+256 x^3-160 x^4+24 x^5+e^{x^2} \left (1+8 x-64 x^3+40 x^4-38 x^5+16 x^6-2 x^7\right )+\left (8-256 x^3+160 x^4-24 x^5+e^{x^2} \left (-2+64 x^3-40 x^4+6 x^5\right )\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )}{-4+e^{x^2}+\left (8-2 e^{x^2}\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )} \, dx=\frac {x^{6} - 8 \, x^{5} + 16 \, x^{4} + x \log \left (-e^{\left (x^{2}\right )} + 4\right ) - x - 4}{\log \left (-e^{\left (x^{2}\right )} + 4\right ) - 1} \] Input:

integrate(((exp(x^2)-4)*log(4-exp(x^2))^2+((6*x^5-40*x^4+64*x^3-2)*exp(x^2 
)-24*x^5+160*x^4-256*x^3+8)*log(4-exp(x^2))+(-2*x^7+16*x^6-38*x^5+40*x^4-6 
4*x^3+8*x+1)*exp(x^2)+24*x^5-160*x^4+256*x^3-4)/((exp(x^2)-4)*log(4-exp(x^ 
2))^2+(-2*exp(x^2)+8)*log(4-exp(x^2))+exp(x^2)-4),x, algorithm="fricas")
 

Output:

(x^6 - 8*x^5 + 16*x^4 + x*log(-e^(x^2) + 4) - x - 4)/(log(-e^(x^2) + 4) - 
1)
 

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.76 \[ \int \frac {-4+256 x^3-160 x^4+24 x^5+e^{x^2} \left (1+8 x-64 x^3+40 x^4-38 x^5+16 x^6-2 x^7\right )+\left (8-256 x^3+160 x^4-24 x^5+e^{x^2} \left (-2+64 x^3-40 x^4+6 x^5\right )\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )}{-4+e^{x^2}+\left (8-2 e^{x^2}\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )} \, dx=x + \frac {x^{6} - 8 x^{5} + 16 x^{4} - 4}{\log {\left (4 - e^{x^{2}} \right )} - 1} \] Input:

integrate(((exp(x**2)-4)*ln(4-exp(x**2))**2+((6*x**5-40*x**4+64*x**3-2)*ex 
p(x**2)-24*x**5+160*x**4-256*x**3+8)*ln(4-exp(x**2))+(-2*x**7+16*x**6-38*x 
**5+40*x**4-64*x**3+8*x+1)*exp(x**2)+24*x**5-160*x**4+256*x**3-4)/((exp(x* 
*2)-4)*ln(4-exp(x**2))**2+(-2*exp(x**2)+8)*ln(4-exp(x**2))+exp(x**2)-4),x)
 

Output:

x + (x**6 - 8*x**5 + 16*x**4 - 4)/(log(4 - exp(x**2)) - 1)
 

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.26 \[ \int \frac {-4+256 x^3-160 x^4+24 x^5+e^{x^2} \left (1+8 x-64 x^3+40 x^4-38 x^5+16 x^6-2 x^7\right )+\left (8-256 x^3+160 x^4-24 x^5+e^{x^2} \left (-2+64 x^3-40 x^4+6 x^5\right )\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )}{-4+e^{x^2}+\left (8-2 e^{x^2}\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )} \, dx=\frac {x^{6} - 8 \, x^{5} + 16 \, x^{4} + x \log \left (-e^{\left (x^{2}\right )} + 4\right ) - x - 4}{\log \left (-e^{\left (x^{2}\right )} + 4\right ) - 1} \] Input:

integrate(((exp(x^2)-4)*log(4-exp(x^2))^2+((6*x^5-40*x^4+64*x^3-2)*exp(x^2 
)-24*x^5+160*x^4-256*x^3+8)*log(4-exp(x^2))+(-2*x^7+16*x^6-38*x^5+40*x^4-6 
4*x^3+8*x+1)*exp(x^2)+24*x^5-160*x^4+256*x^3-4)/((exp(x^2)-4)*log(4-exp(x^ 
2))^2+(-2*exp(x^2)+8)*log(4-exp(x^2))+exp(x^2)-4),x, algorithm="maxima")
 

Output:

(x^6 - 8*x^5 + 16*x^4 + x*log(-e^(x^2) + 4) - x - 4)/(log(-e^(x^2) + 4) - 
1)
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.26 \[ \int \frac {-4+256 x^3-160 x^4+24 x^5+e^{x^2} \left (1+8 x-64 x^3+40 x^4-38 x^5+16 x^6-2 x^7\right )+\left (8-256 x^3+160 x^4-24 x^5+e^{x^2} \left (-2+64 x^3-40 x^4+6 x^5\right )\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )}{-4+e^{x^2}+\left (8-2 e^{x^2}\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )} \, dx=\frac {x^{6} - 8 \, x^{5} + 16 \, x^{4} + x \log \left (-e^{\left (x^{2}\right )} + 4\right ) - x - 4}{\log \left (-e^{\left (x^{2}\right )} + 4\right ) - 1} \] Input:

integrate(((exp(x^2)-4)*log(4-exp(x^2))^2+((6*x^5-40*x^4+64*x^3-2)*exp(x^2 
)-24*x^5+160*x^4-256*x^3+8)*log(4-exp(x^2))+(-2*x^7+16*x^6-38*x^5+40*x^4-6 
4*x^3+8*x+1)*exp(x^2)+24*x^5-160*x^4+256*x^3-4)/((exp(x^2)-4)*log(4-exp(x^ 
2))^2+(-2*exp(x^2)+8)*log(4-exp(x^2))+exp(x^2)-4),x, algorithm="giac")
 

Output:

(x^6 - 8*x^5 + 16*x^4 + x*log(-e^(x^2) + 4) - x - 4)/(log(-e^(x^2) + 4) - 
1)
 

Mupad [B] (verification not implemented)

Time = 3.35 (sec) , antiderivative size = 169, normalized size of antiderivative = 4.97 \[ \int \frac {-4+256 x^3-160 x^4+24 x^5+e^{x^2} \left (1+8 x-64 x^3+40 x^4-38 x^5+16 x^6-2 x^7\right )+\left (8-256 x^3+160 x^4-24 x^5+e^{x^2} \left (-2+64 x^3-40 x^4+6 x^5\right )\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )}{-4+e^{x^2}+\left (8-2 e^{x^2}\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )} \, dx=x-\frac {{\mathrm {e}}^{-x^2}\,\left (4\,{\mathrm {e}}^{x^2}-32\,x^2\,{\mathrm {e}}^{x^2}+20\,x^3\,{\mathrm {e}}^{x^2}-19\,x^4\,{\mathrm {e}}^{x^2}+8\,x^5\,{\mathrm {e}}^{x^2}-x^6\,{\mathrm {e}}^{x^2}+128\,x^2-80\,x^3+12\,x^4\right )+{\mathrm {e}}^{-x^2}\,\ln \left (4-{\mathrm {e}}^{x^2}\right )\,\left ({\mathrm {e}}^{x^2}-4\right )\,\left (3\,x^4-20\,x^3+32\,x^2\right )}{\ln \left (4-{\mathrm {e}}^{x^2}\right )-1}+32\,x^2-20\,x^3+3\,x^4-{\mathrm {e}}^{-x^2}\,\left (12\,x^4-80\,x^3+128\,x^2\right ) \] Input:

int((log(4 - exp(x^2))*(exp(x^2)*(64*x^3 - 40*x^4 + 6*x^5 - 2) - 256*x^3 + 
 160*x^4 - 24*x^5 + 8) + log(4 - exp(x^2))^2*(exp(x^2) - 4) + 256*x^3 - 16 
0*x^4 + 24*x^5 + exp(x^2)*(8*x - 64*x^3 + 40*x^4 - 38*x^5 + 16*x^6 - 2*x^7 
 + 1) - 4)/(exp(x^2) - log(4 - exp(x^2))*(2*exp(x^2) - 8) + log(4 - exp(x^ 
2))^2*(exp(x^2) - 4) - 4),x)
 

Output:

x - (exp(-x^2)*(4*exp(x^2) - 32*x^2*exp(x^2) + 20*x^3*exp(x^2) - 19*x^4*ex 
p(x^2) + 8*x^5*exp(x^2) - x^6*exp(x^2) + 128*x^2 - 80*x^3 + 12*x^4) + exp( 
-x^2)*log(4 - exp(x^2))*(exp(x^2) - 4)*(32*x^2 - 20*x^3 + 3*x^4))/(log(4 - 
 exp(x^2)) - 1) + 32*x^2 - 20*x^3 + 3*x^4 - exp(-x^2)*(128*x^2 - 80*x^3 + 
12*x^4)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.65 \[ \int \frac {-4+256 x^3-160 x^4+24 x^5+e^{x^2} \left (1+8 x-64 x^3+40 x^4-38 x^5+16 x^6-2 x^7\right )+\left (8-256 x^3+160 x^4-24 x^5+e^{x^2} \left (-2+64 x^3-40 x^4+6 x^5\right )\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )}{-4+e^{x^2}+\left (8-2 e^{x^2}\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )} \, dx=\frac {\mathrm {log}\left (-e^{x^{2}}+4\right ) x -4 \,\mathrm {log}\left (-e^{x^{2}}+4\right )+x^{6}-8 x^{5}+16 x^{4}-x}{\mathrm {log}\left (-e^{x^{2}}+4\right )-1} \] Input:

int(((exp(x^2)-4)*log(4-exp(x^2))^2+((6*x^5-40*x^4+64*x^3-2)*exp(x^2)-24*x 
^5+160*x^4-256*x^3+8)*log(4-exp(x^2))+(-2*x^7+16*x^6-38*x^5+40*x^4-64*x^3+ 
8*x+1)*exp(x^2)+24*x^5-160*x^4+256*x^3-4)/((exp(x^2)-4)*log(4-exp(x^2))^2+ 
(-2*exp(x^2)+8)*log(4-exp(x^2))+exp(x^2)-4),x)
 

Output:

(log( - e**(x**2) + 4)*x - 4*log( - e**(x**2) + 4) + x**6 - 8*x**5 + 16*x* 
*4 - x)/(log( - e**(x**2) + 4) - 1)