\(\int \frac {e^{-x^2+x^4} (x+6 \log (4 x+5 x^2+x^3)) (24+64 x+23 x^2-7 x^3-10 x^4+14 x^5+20 x^6+4 x^7+(-48 x^2-60 x^3+84 x^4+120 x^5+24 x^6) \log (4 x+5 x^2+x^3))}{4 x^2+5 x^3+x^4+(24 x+30 x^2+6 x^3) \log (4 x+5 x^2+x^3)} \, dx\) [2033]

Optimal result

Integrand size = 148, antiderivative size = 26 \[ \int \frac {e^{-x^2+x^4} \left (x+6 \log \left (4 x+5 x^2+x^3\right )\right ) \left (24+64 x+23 x^2-7 x^3-10 x^4+14 x^5+20 x^6+4 x^7+\left (-48 x^2-60 x^3+84 x^4+120 x^5+24 x^6\right ) \log \left (4 x+5 x^2+x^3\right )\right )}{4 x^2+5 x^3+x^4+\left (24 x+30 x^2+6 x^3\right ) \log \left (4 x+5 x^2+x^3\right )} \, dx=e^{-x^2+x^4} \left (x+6 \log \left ((4+x) \left (x+x^2\right )\right )\right ) \] Output:

exp(x^4-x^2+ln(x+6*ln((4+x)*(x^2+x))))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {e^{-x^2+x^4} \left (x+6 \log \left (4 x+5 x^2+x^3\right )\right ) \left (24+64 x+23 x^2-7 x^3-10 x^4+14 x^5+20 x^6+4 x^7+\left (-48 x^2-60 x^3+84 x^4+120 x^5+24 x^6\right ) \log \left (4 x+5 x^2+x^3\right )\right )}{4 x^2+5 x^3+x^4+\left (24 x+30 x^2+6 x^3\right ) \log \left (4 x+5 x^2+x^3\right )} \, dx=e^{-x^2+x^4} \left (x+6 \log \left (x \left (4+5 x+x^2\right )\right )\right ) \] Input:

Integrate[(E^(-x^2 + x^4)*(x + 6*Log[4*x + 5*x^2 + x^3])*(24 + 64*x + 23*x 
^2 - 7*x^3 - 10*x^4 + 14*x^5 + 20*x^6 + 4*x^7 + (-48*x^2 - 60*x^3 + 84*x^4 
 + 120*x^5 + 24*x^6)*Log[4*x + 5*x^2 + x^3]))/(4*x^2 + 5*x^3 + x^4 + (24*x 
 + 30*x^2 + 6*x^3)*Log[4*x + 5*x^2 + x^3]),x]
 

Output:

E^(-x^2 + x^4)*(x + 6*Log[x*(4 + 5*x + 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.

\(\displaystyle \int \frac {e^{x^4-x^2} \left (6 \log \left (x^3+5 x^2+4 x\right )+x\right ) \left (4 x^7+20 x^6+14 x^5-10 x^4-7 x^3+23 x^2+\left (24 x^6+120 x^5+84 x^4-60 x^3-48 x^2\right ) \log \left (x^3+5 x^2+4 x\right )+64 x+24\right )}{x^4+5 x^3+4 x^2+\left (6 x^3+30 x^2+24 x\right ) \log \left (x^3+5 x^2+4 x\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {e^{x^4-x^2} \left (6 \log \left (x^3+5 x^2+4 x\right )+x\right ) \left (4 x^7+20 x^6+14 x^5-10 x^4-7 x^3+23 x^2+\left (24 x^6+120 x^5+84 x^4-60 x^3-48 x^2\right ) \log \left (x^3+5 x^2+4 x\right )+64 x+24\right )}{x \left (x^2+5 x+4\right ) \left (6 \log \left (x \left (x^2+5 x+4\right )\right )+x\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle \int \left (\frac {72 e^{x^4-x^2} \left (2 x^2-1\right ) x \log ^2\left (x \left (x^2+5 x+4\right )\right )}{6 \log \left (x \left (x^2+5 x+4\right )\right )+x}-\frac {10 e^{x^4-x^2} x^4}{(x+1) (x+4) \left (6 \log \left (x \left (x^2+5 x+4\right )\right )+x\right )}+\frac {23 e^{x^4-x^2} x^2}{(x+1) (x+4) \left (6 \log \left (x \left (x^2+5 x+4\right )\right )+x\right )}+\frac {64 e^{x^4-x^2} x}{(x+1) (x+4) \left (6 \log \left (x \left (x^2+5 x+4\right )\right )+x\right )}+\frac {24 e^{x^4-x^2}}{(x+1) (x+4) \left (6 \log \left (x \left (x^2+5 x+4\right )\right )+x\right )}+\frac {4 e^{x^4-x^2} x^7}{(x+1) (x+4) \left (6 \log \left (x \left (x^2+5 x+4\right )\right )+x\right )}+\frac {20 e^{x^4-x^2} x^6}{(x+1) (x+4) \left (6 \log \left (x \left (x^2+5 x+4\right )\right )+x\right )}+\frac {14 e^{x^4-x^2} x^5}{(x+1) (x+4) \left (6 \log \left (x \left (x^2+5 x+4\right )\right )+x\right )}-\frac {7 e^{x^4-x^2} x^3}{(x+1) (x+4) \left (6 \log \left (x \left (x^2+5 x+4\right )\right )+x\right )}+\frac {6 e^{x^4-x^2} \left (8 x^7+40 x^6+28 x^5-20 x^4-15 x^3+23 x^2+64 x+24\right ) \log \left (x \left (x^2+5 x+4\right )\right )}{(x+1) (x+4) x \left (6 \log \left (x \left (x^2+5 x+4\right )\right )+x\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {e^{x^4-x^2} \left (4 x^7+20 x^6+14 x^5-10 x^4-7 x^3+23 x^2+12 \left (2 x^4+10 x^3+7 x^2-5 x-4\right ) x^2 \log \left (x \left (x^2+5 x+4\right )\right )+64 x+24\right )}{x (x+1) (x+4)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {14 e^{x^4-x^2} x^4}{x^2+5 x+4}-\frac {7 e^{x^4-x^2} x^2}{x^2+5 x+4}+\frac {23 e^{x^4-x^2} x}{x^2+5 x+4}+\frac {64 e^{x^4-x^2}}{x^2+5 x+4}+\frac {24 e^{x^4-x^2}}{\left (x^2+5 x+4\right ) x}+12 e^{x^4-x^2} \left (2 x^2-1\right ) x \log \left (x \left (x^2+5 x+4\right )\right )+\frac {4 e^{x^4-x^2} x^6}{x^2+5 x+4}+\frac {20 e^{x^4-x^2} x^5}{x^2+5 x+4}-\frac {10 e^{x^4-x^2} x^3}{x^2+5 x+4}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \int e^{x^4-x^2}dx-\frac {128}{3} \int \frac {e^{x^4-x^2}}{-2 x-2}dx-2 \int e^{x^4-x^2} x^2dx+4 \int e^{x^4-x^2} x^4dx-14 \int \frac {e^{x^4-x^2}}{x+1}dx-4 \int \frac {e^{x^4-x^2}}{x+4}dx-\frac {44}{3} \int \frac {e^{x^4-x^2}}{2 x+2}dx+8 \int \frac {e^{x^4-x^2}}{2 x+8}dx+6 e^{x^4-x^2} \log \left (x \left (x^2+5 x+4\right )\right )\)

Input:

Int[(E^(-x^2 + x^4)*(x + 6*Log[4*x + 5*x^2 + x^3])*(24 + 64*x + 23*x^2 - 7 
*x^3 - 10*x^4 + 14*x^5 + 20*x^6 + 4*x^7 + (-48*x^2 - 60*x^3 + 84*x^4 + 120 
*x^5 + 24*x^6)*Log[4*x + 5*x^2 + x^3]))/(4*x^2 + 5*x^3 + x^4 + (24*x + 30* 
x^2 + 6*x^3)*Log[4*x + 5*x^2 + x^3]),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 127.74 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12

method	result	size
parallelrisch	\({\mathrm e}^{\ln \left (6 \ln \left (x^{3}+5 x^{2}+4 x \right )+x \right )+x^{4}-x^{2}}\)	\(29\)

Input:

int(((24*x^6+120*x^5+84*x^4-60*x^3-48*x^2)*ln(x^3+5*x^2+4*x)+4*x^7+20*x^6+ 
14*x^5-10*x^4-7*x^3+23*x^2+64*x+24)*exp(ln(6*ln(x^3+5*x^2+4*x)+x)+x^4-x^2) 
/((6*x^3+30*x^2+24*x)*ln(x^3+5*x^2+4*x)+x^4+5*x^3+4*x^2),x,method=_RETURNV 
ERBOSE)
 

Output:

exp(ln(6*ln(x^3+5*x^2+4*x)+x)+x^4-x^2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {e^{-x^2+x^4} \left (x+6 \log \left (4 x+5 x^2+x^3\right )\right ) \left (24+64 x+23 x^2-7 x^3-10 x^4+14 x^5+20 x^6+4 x^7+\left (-48 x^2-60 x^3+84 x^4+120 x^5+24 x^6\right ) \log \left (4 x+5 x^2+x^3\right )\right )}{4 x^2+5 x^3+x^4+\left (24 x+30 x^2+6 x^3\right ) \log \left (4 x+5 x^2+x^3\right )} \, dx=e^{\left (x^{4} - x^{2} + \log \left (x + 6 \, \log \left (x^{3} + 5 \, x^{2} + 4 \, x\right )\right )\right )} \] Input:

integrate(((24*x^6+120*x^5+84*x^4-60*x^3-48*x^2)*log(x^3+5*x^2+4*x)+4*x^7+ 
20*x^6+14*x^5-10*x^4-7*x^3+23*x^2+64*x+24)*exp(log(6*log(x^3+5*x^2+4*x)+x) 
+x^4-x^2)/((6*x^3+30*x^2+24*x)*log(x^3+5*x^2+4*x)+x^4+5*x^3+4*x^2),x, algo 
rithm="fricas")
 

Output:

e^(x^4 - x^2 + log(x + 6*log(x^3 + 5*x^2 + 4*x)))
 

Sympy [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {e^{-x^2+x^4} \left (x+6 \log \left (4 x+5 x^2+x^3\right )\right ) \left (24+64 x+23 x^2-7 x^3-10 x^4+14 x^5+20 x^6+4 x^7+\left (-48 x^2-60 x^3+84 x^4+120 x^5+24 x^6\right ) \log \left (4 x+5 x^2+x^3\right )\right )}{4 x^2+5 x^3+x^4+\left (24 x+30 x^2+6 x^3\right ) \log \left (4 x+5 x^2+x^3\right )} \, dx=\left (x + 6 \log {\left (x^{3} + 5 x^{2} + 4 x \right )}\right ) e^{x^{4} - x^{2}} \] Input:

integrate(((24*x**6+120*x**5+84*x**4-60*x**3-48*x**2)*ln(x**3+5*x**2+4*x)+ 
4*x**7+20*x**6+14*x**5-10*x**4-7*x**3+23*x**2+64*x+24)*exp(ln(6*ln(x**3+5* 
x**2+4*x)+x)+x**4-x**2)/((6*x**3+30*x**2+24*x)*ln(x**3+5*x**2+4*x)+x**4+5* 
x**3+4*x**2),x)
 

Output:

(x + 6*log(x**3 + 5*x**2 + 4*x))*exp(x**4 - x**2)
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {e^{-x^2+x^4} \left (x+6 \log \left (4 x+5 x^2+x^3\right )\right ) \left (24+64 x+23 x^2-7 x^3-10 x^4+14 x^5+20 x^6+4 x^7+\left (-48 x^2-60 x^3+84 x^4+120 x^5+24 x^6\right ) \log \left (4 x+5 x^2+x^3\right )\right )}{4 x^2+5 x^3+x^4+\left (24 x+30 x^2+6 x^3\right ) \log \left (4 x+5 x^2+x^3\right )} \, dx={\left (x + 6 \, \log \left (x + 4\right ) + 6 \, \log \left (x + 1\right ) + 6 \, \log \left (x\right )\right )} e^{\left (x^{4} - x^{2}\right )} \] Input:

integrate(((24*x^6+120*x^5+84*x^4-60*x^3-48*x^2)*log(x^3+5*x^2+4*x)+4*x^7+ 
20*x^6+14*x^5-10*x^4-7*x^3+23*x^2+64*x+24)*exp(log(6*log(x^3+5*x^2+4*x)+x) 
+x^4-x^2)/((6*x^3+30*x^2+24*x)*log(x^3+5*x^2+4*x)+x^4+5*x^3+4*x^2),x, algo 
rithm="maxima")
 

Output:

(x + 6*log(x + 4) + 6*log(x + 1) + 6*log(x))*e^(x^4 - x^2)
 

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {e^{-x^2+x^4} \left (x+6 \log \left (4 x+5 x^2+x^3\right )\right ) \left (24+64 x+23 x^2-7 x^3-10 x^4+14 x^5+20 x^6+4 x^7+\left (-48 x^2-60 x^3+84 x^4+120 x^5+24 x^6\right ) \log \left (4 x+5 x^2+x^3\right )\right )}{4 x^2+5 x^3+x^4+\left (24 x+30 x^2+6 x^3\right ) \log \left (4 x+5 x^2+x^3\right )} \, dx=e^{\left (x^{4} - x^{2} + \log \left (x + 6 \, \log \left (x^{3} + 5 \, x^{2} + 4 \, x\right )\right )\right )} \] Input:

integrate(((24*x^6+120*x^5+84*x^4-60*x^3-48*x^2)*log(x^3+5*x^2+4*x)+4*x^7+ 
20*x^6+14*x^5-10*x^4-7*x^3+23*x^2+64*x+24)*exp(log(6*log(x^3+5*x^2+4*x)+x) 
+x^4-x^2)/((6*x^3+30*x^2+24*x)*log(x^3+5*x^2+4*x)+x^4+5*x^3+4*x^2),x, algo 
rithm="giac")
 

Output:

e^(x^4 - x^2 + log(x + 6*log(x^3 + 5*x^2 + 4*x)))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{-x^2+x^4} \left (x+6 \log \left (4 x+5 x^2+x^3\right )\right ) \left (24+64 x+23 x^2-7 x^3-10 x^4+14 x^5+20 x^6+4 x^7+\left (-48 x^2-60 x^3+84 x^4+120 x^5+24 x^6\right ) \log \left (4 x+5 x^2+x^3\right )\right )}{4 x^2+5 x^3+x^4+\left (24 x+30 x^2+6 x^3\right ) \log \left (4 x+5 x^2+x^3\right )} \, dx=\int \frac {{\mathrm {e}}^{\ln \left (x+6\,\ln \left (x^3+5\,x^2+4\,x\right )\right )-x^2+x^4}\,\left (64\,x+\ln \left (x^3+5\,x^2+4\,x\right )\,\left (24\,x^6+120\,x^5+84\,x^4-60\,x^3-48\,x^2\right )+23\,x^2-7\,x^3-10\,x^4+14\,x^5+20\,x^6+4\,x^7+24\right )}{\ln \left (x^3+5\,x^2+4\,x\right )\,\left (6\,x^3+30\,x^2+24\,x\right )+4\,x^2+5\,x^3+x^4} \,d x \] Input:

int((exp(log(x + 6*log(4*x + 5*x^2 + x^3)) - x^2 + x^4)*(64*x + log(4*x + 
5*x^2 + x^3)*(84*x^4 - 60*x^3 - 48*x^2 + 120*x^5 + 24*x^6) + 23*x^2 - 7*x^ 
3 - 10*x^4 + 14*x^5 + 20*x^6 + 4*x^7 + 24))/(log(4*x + 5*x^2 + x^3)*(24*x 
+ 30*x^2 + 6*x^3) + 4*x^2 + 5*x^3 + x^4),x)
 

Output:

int((exp(log(x + 6*log(4*x + 5*x^2 + x^3)) - x^2 + x^4)*(64*x + log(4*x + 
5*x^2 + x^3)*(84*x^4 - 60*x^3 - 48*x^2 + 120*x^5 + 24*x^6) + 23*x^2 - 7*x^ 
3 - 10*x^4 + 14*x^5 + 20*x^6 + 4*x^7 + 24))/(log(4*x + 5*x^2 + x^3)*(24*x 
+ 30*x^2 + 6*x^3) + 4*x^2 + 5*x^3 + x^4), x)
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {e^{-x^2+x^4} \left (x+6 \log \left (4 x+5 x^2+x^3\right )\right ) \left (24+64 x+23 x^2-7 x^3-10 x^4+14 x^5+20 x^6+4 x^7+\left (-48 x^2-60 x^3+84 x^4+120 x^5+24 x^6\right ) \log \left (4 x+5 x^2+x^3\right )\right )}{4 x^2+5 x^3+x^4+\left (24 x+30 x^2+6 x^3\right ) \log \left (4 x+5 x^2+x^3\right )} \, dx=\frac {e^{x^{4}} \left (6 \,\mathrm {log}\left (x^{3}+5 x^{2}+4 x \right )+x \right )}{e^{x^{2}}} \] Input:

int(((24*x^6+120*x^5+84*x^4-60*x^3-48*x^2)*log(x^3+5*x^2+4*x)+4*x^7+20*x^6 
+14*x^5-10*x^4-7*x^3+23*x^2+64*x+24)*exp(log(6*log(x^3+5*x^2+4*x)+x)+x^4-x 
^2)/((6*x^3+30*x^2+24*x)*log(x^3+5*x^2+4*x)+x^4+5*x^3+4*x^2),x)
 

Output:

(e**(x**4)*(6*log(x**3 + 5*x**2 + 4*x) + x))/e**(x**2)