\(\int \frac {-80 x-80 x^3+e^x (-16 x-16 x^2-16 x^3)-32 \log (x)-16 x^2 \log ^2(x)}{e^{2 x^2} (3125 x^6+3125 e^x x^6+1250 e^{2 x} x^6+250 e^{3 x} x^6+25 e^{4 x} x^6+e^{5 x} x^6)+e^{2 x^2} (3125 x^5+2500 e^x x^5+750 e^{2 x} x^5+100 e^{3 x} x^5+5 e^{4 x} x^5) \log ^2(x)+e^{2 x^2} (1250 x^4+750 e^x x^4+150 e^{2 x} x^4+10 e^{3 x} x^4) \log ^4(x)+e^{2 x^2} (250 x^3+100 e^x x^3+10 e^{2 x} x^3) \log ^6(x)+e^{2 x^2} (25 x^2+5 e^x x^2) \log ^8(x)+e^{2 x^2} x \log ^{10}(x)} \, dx\) [836]

Optimal result

Integrand size = 282, antiderivative size = 24 \[ \int \frac {-80 x-80 x^3+e^x \left (-16 x-16 x^2-16 x^3\right )-32 \log (x)-16 x^2 \log ^2(x)}{e^{2 x^2} \left (3125 x^6+3125 e^x x^6+1250 e^{2 x} x^6+250 e^{3 x} x^6+25 e^{4 x} x^6+e^{5 x} x^6\right )+e^{2 x^2} \left (3125 x^5+2500 e^x x^5+750 e^{2 x} x^5+100 e^{3 x} x^5+5 e^{4 x} x^5\right ) \log ^2(x)+e^{2 x^2} \left (1250 x^4+750 e^x x^4+150 e^{2 x} x^4+10 e^{3 x} x^4\right ) \log ^4(x)+e^{2 x^2} \left (250 x^3+100 e^x x^3+10 e^{2 x} x^3\right ) \log ^6(x)+e^{2 x^2} \left (25 x^2+5 e^x x^2\right ) \log ^8(x)+e^{2 x^2} x \log ^{10}(x)} \, dx=\frac {4 e^{-2 x^2}}{\left (5 x+e^x x+\log ^2(x)\right )^4} \] Output:

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

Mathematica [A] (verified)

Time = 1.35 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {-80 x-80 x^3+e^x \left (-16 x-16 x^2-16 x^3\right )-32 \log (x)-16 x^2 \log ^2(x)}{e^{2 x^2} \left (3125 x^6+3125 e^x x^6+1250 e^{2 x} x^6+250 e^{3 x} x^6+25 e^{4 x} x^6+e^{5 x} x^6\right )+e^{2 x^2} \left (3125 x^5+2500 e^x x^5+750 e^{2 x} x^5+100 e^{3 x} x^5+5 e^{4 x} x^5\right ) \log ^2(x)+e^{2 x^2} \left (1250 x^4+750 e^x x^4+150 e^{2 x} x^4+10 e^{3 x} x^4\right ) \log ^4(x)+e^{2 x^2} \left (250 x^3+100 e^x x^3+10 e^{2 x} x^3\right ) \log ^6(x)+e^{2 x^2} \left (25 x^2+5 e^x x^2\right ) \log ^8(x)+e^{2 x^2} x \log ^{10}(x)} \, dx=\frac {4 e^{-2 x^2}}{\left (\left (5+e^x\right ) x+\log ^2(x)\right )^4} \] Input:

Integrate[(-80*x - 80*x^3 + E^x*(-16*x - 16*x^2 - 16*x^3) - 32*Log[x] - 16 
*x^2*Log[x]^2)/(E^(2*x^2)*(3125*x^6 + 3125*E^x*x^6 + 1250*E^(2*x)*x^6 + 25 
0*E^(3*x)*x^6 + 25*E^(4*x)*x^6 + E^(5*x)*x^6) + E^(2*x^2)*(3125*x^5 + 2500 
*E^x*x^5 + 750*E^(2*x)*x^5 + 100*E^(3*x)*x^5 + 5*E^(4*x)*x^5)*Log[x]^2 + E 
^(2*x^2)*(1250*x^4 + 750*E^x*x^4 + 150*E^(2*x)*x^4 + 10*E^(3*x)*x^4)*Log[x 
]^4 + E^(2*x^2)*(250*x^3 + 100*E^x*x^3 + 10*E^(2*x)*x^3)*Log[x]^6 + E^(2*x 
^2)*(25*x^2 + 5*E^x*x^2)*Log[x]^8 + E^(2*x^2)*x*Log[x]^10),x]
 

Output:

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

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.

\(\displaystyle \int \frac {-80 x^3-16 x^2 \log ^2(x)+e^x \left (-16 x^3-16 x^2-16 x\right )-80 x-32 \log (x)}{e^{2 x^2} x \log ^{10}(x)+e^{2 x^2} \left (5 e^x x^2+25 x^2\right ) \log ^8(x)+e^{2 x^2} \left (3125 e^x x^6+1250 e^{2 x} x^6+250 e^{3 x} x^6+25 e^{4 x} x^6+e^{5 x} x^6+3125 x^6\right )+e^{2 x^2} \left (2500 e^x x^5+750 e^{2 x} x^5+100 e^{3 x} x^5+5 e^{4 x} x^5+3125 x^5\right ) \log ^2(x)+e^{2 x^2} \left (750 e^x x^4+150 e^{2 x} x^4+10 e^{3 x} x^4+1250 x^4\right ) \log ^4(x)+e^{2 x^2} \left (100 e^x x^3+10 e^{2 x} x^3+250 x^3\right ) \log ^6(x)} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {16 e^{-2 x^2} \left (-5 x \left (x^2+1\right )-e^x x \left (x^2+x+1\right )-x^2 \log ^2(x)-2 \log (x)\right )}{x \left (\left (e^x+5\right ) x+\log ^2(x)\right )^5}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 16 \int -\frac {e^{-2 x^2} \left (x^2 \log ^2(x)+2 \log (x)+5 x \left (x^2+1\right )+e^x x \left (x^2+x+1\right )\right )}{x \left (\log ^2(x)+\left (5+e^x\right ) x\right )^5}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -16 \int \frac {e^{-2 x^2} \left (x^2 \log ^2(x)+2 \log (x)+5 x \left (x^2+1\right )+e^x x \left (x^2+x+1\right )\right )}{x \left (\log ^2(x)+\left (5+e^x\right ) x\right )^5}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -16 \int \left (\frac {e^{-2 x^2} \left (x^2+x+1\right )}{x \left (\log ^2(x)+e^x x+5 x\right )^4}-\frac {e^{-2 x^2} \left (5 x^2+\log ^2(x) x+\log ^2(x)-2 \log (x)\right )}{x \left (\log ^2(x)+e^x x+5 x\right )^5}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

Int[(-80*x - 80*x^3 + E^x*(-16*x - 16*x^2 - 16*x^3) - 32*Log[x] - 16*x^2*L 
og[x]^2)/(E^(2*x^2)*(3125*x^6 + 3125*E^x*x^6 + 1250*E^(2*x)*x^6 + 250*E^(3 
*x)*x^6 + 25*E^(4*x)*x^6 + E^(5*x)*x^6) + E^(2*x^2)*(3125*x^5 + 2500*E^x*x 
^5 + 750*E^(2*x)*x^5 + 100*E^(3*x)*x^5 + 5*E^(4*x)*x^5)*Log[x]^2 + E^(2*x^ 
2)*(1250*x^4 + 750*E^x*x^4 + 150*E^(2*x)*x^4 + 10*E^(3*x)*x^4)*Log[x]^4 + 
E^(2*x^2)*(250*x^3 + 100*E^x*x^3 + 10*E^(2*x)*x^3)*Log[x]^6 + E^(2*x^2)*(2 
5*x^2 + 5*E^x*x^2)*Log[x]^8 + E^(2*x^2)*x*Log[x]^10),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96

\[\frac {4 \,{\mathrm e}^{-2 x^{2}}}{\left (\ln \left (x \right )^{2}+5 x +{\mathrm e}^{x} x \right )^{4}}\]

Input:

int((-16*x^2*ln(x)^2-32*ln(x)+(-16*x^3-16*x^2-16*x)*exp(x)-80*x^3-80*x)/(x 
*exp(x^2)^2*ln(x)^10+(5*exp(x)*x^2+25*x^2)*exp(x^2)^2*ln(x)^8+(10*exp(x)^2 
*x^3+100*exp(x)*x^3+250*x^3)*exp(x^2)^2*ln(x)^6+(10*x^4*exp(x)^3+150*exp(x 
)^2*x^4+750*exp(x)*x^4+1250*x^4)*exp(x^2)^2*ln(x)^4+(5*x^5*exp(x)^4+100*x^ 
5*exp(x)^3+750*x^5*exp(x)^2+2500*x^5*exp(x)+3125*x^5)*exp(x^2)^2*ln(x)^2+( 
x^6*exp(x)^5+25*x^6*exp(x)^4+250*x^6*exp(x)^3+1250*x^6*exp(x)^2+3125*x^6*e 
xp(x)+3125*x^6)*exp(x^2)^2),x)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (22) = 44\).

Time = 0.10 (sec) , antiderivative size = 157, normalized size of antiderivative = 6.54 \[ \int \frac {-80 x-80 x^3+e^x \left (-16 x-16 x^2-16 x^3\right )-32 \log (x)-16 x^2 \log ^2(x)}{e^{2 x^2} \left (3125 x^6+3125 e^x x^6+1250 e^{2 x} x^6+250 e^{3 x} x^6+25 e^{4 x} x^6+e^{5 x} x^6\right )+e^{2 x^2} \left (3125 x^5+2500 e^x x^5+750 e^{2 x} x^5+100 e^{3 x} x^5+5 e^{4 x} x^5\right ) \log ^2(x)+e^{2 x^2} \left (1250 x^4+750 e^x x^4+150 e^{2 x} x^4+10 e^{3 x} x^4\right ) \log ^4(x)+e^{2 x^2} \left (250 x^3+100 e^x x^3+10 e^{2 x} x^3\right ) \log ^6(x)+e^{2 x^2} \left (25 x^2+5 e^x x^2\right ) \log ^8(x)+e^{2 x^2} x \log ^{10}(x)} \, dx=\frac {4}{e^{\left (2 \, x^{2}\right )} \log \left (x\right )^{8} + 4 \, {\left (x e^{x} + 5 \, x\right )} e^{\left (2 \, x^{2}\right )} \log \left (x\right )^{6} + 6 \, {\left (x^{2} e^{\left (2 \, x\right )} + 10 \, x^{2} e^{x} + 25 \, x^{2}\right )} e^{\left (2 \, x^{2}\right )} \log \left (x\right )^{4} + 4 \, {\left (x^{3} e^{\left (3 \, x\right )} + 15 \, x^{3} e^{\left (2 \, x\right )} + 75 \, x^{3} e^{x} + 125 \, x^{3}\right )} e^{\left (2 \, x^{2}\right )} \log \left (x\right )^{2} + {\left (x^{4} e^{\left (4 \, x\right )} + 20 \, x^{4} e^{\left (3 \, x\right )} + 150 \, x^{4} e^{\left (2 \, x\right )} + 500 \, x^{4} e^{x} + 625 \, x^{4}\right )} e^{\left (2 \, x^{2}\right )}} \] Input:

integrate((-16*x^2*log(x)^2-32*log(x)+(-16*x^3-16*x^2-16*x)*exp(x)-80*x^3- 
80*x)/(x*exp(x^2)^2*log(x)^10+(5*exp(x)*x^2+25*x^2)*exp(x^2)^2*log(x)^8+(1 
0*exp(x)^2*x^3+100*exp(x)*x^3+250*x^3)*exp(x^2)^2*log(x)^6+(10*x^4*exp(x)^ 
3+150*exp(x)^2*x^4+750*exp(x)*x^4+1250*x^4)*exp(x^2)^2*log(x)^4+(5*x^5*exp 
(x)^4+100*x^5*exp(x)^3+750*x^5*exp(x)^2+2500*x^5*exp(x)+3125*x^5)*exp(x^2) 
^2*log(x)^2+(x^6*exp(x)^5+25*x^6*exp(x)^4+250*x^6*exp(x)^3+1250*x^6*exp(x) 
^2+3125*x^6*exp(x)+3125*x^6)*exp(x^2)^2),x, algorithm="fricas")
 

Output:

4/(e^(2*x^2)*log(x)^8 + 4*(x*e^x + 5*x)*e^(2*x^2)*log(x)^6 + 6*(x^2*e^(2*x 
) + 10*x^2*e^x + 25*x^2)*e^(2*x^2)*log(x)^4 + 4*(x^3*e^(3*x) + 15*x^3*e^(2 
*x) + 75*x^3*e^x + 125*x^3)*e^(2*x^2)*log(x)^2 + (x^4*e^(4*x) + 20*x^4*e^( 
3*x) + 150*x^4*e^(2*x) + 500*x^4*e^x + 625*x^4)*e^(2*x^2))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (22) = 44\).

Time = 0.44 (sec) , antiderivative size = 168, normalized size of antiderivative = 7.00 \[ \int \frac {-80 x-80 x^3+e^x \left (-16 x-16 x^2-16 x^3\right )-32 \log (x)-16 x^2 \log ^2(x)}{e^{2 x^2} \left (3125 x^6+3125 e^x x^6+1250 e^{2 x} x^6+250 e^{3 x} x^6+25 e^{4 x} x^6+e^{5 x} x^6\right )+e^{2 x^2} \left (3125 x^5+2500 e^x x^5+750 e^{2 x} x^5+100 e^{3 x} x^5+5 e^{4 x} x^5\right ) \log ^2(x)+e^{2 x^2} \left (1250 x^4+750 e^x x^4+150 e^{2 x} x^4+10 e^{3 x} x^4\right ) \log ^4(x)+e^{2 x^2} \left (250 x^3+100 e^x x^3+10 e^{2 x} x^3\right ) \log ^6(x)+e^{2 x^2} \left (25 x^2+5 e^x x^2\right ) \log ^8(x)+e^{2 x^2} x \log ^{10}(x)} \, dx=\frac {4 e^{- 2 x^{2}}}{x^{4} e^{4 x} + 20 x^{4} e^{3 x} + 150 x^{4} e^{2 x} + 500 x^{4} e^{x} + 625 x^{4} + 4 x^{3} e^{3 x} \log {\left (x \right )}^{2} + 60 x^{3} e^{2 x} \log {\left (x \right )}^{2} + 300 x^{3} e^{x} \log {\left (x \right )}^{2} + 500 x^{3} \log {\left (x \right )}^{2} + 6 x^{2} e^{2 x} \log {\left (x \right )}^{4} + 60 x^{2} e^{x} \log {\left (x \right )}^{4} + 150 x^{2} \log {\left (x \right )}^{4} + 4 x e^{x} \log {\left (x \right )}^{6} + 20 x \log {\left (x \right )}^{6} + \log {\left (x \right )}^{8}} \] Input:

integrate((-16*x**2*ln(x)**2-32*ln(x)+(-16*x**3-16*x**2-16*x)*exp(x)-80*x* 
*3-80*x)/(x*exp(x**2)**2*ln(x)**10+(5*exp(x)*x**2+25*x**2)*exp(x**2)**2*ln 
(x)**8+(10*exp(x)**2*x**3+100*exp(x)*x**3+250*x**3)*exp(x**2)**2*ln(x)**6+ 
(10*x**4*exp(x)**3+150*exp(x)**2*x**4+750*exp(x)*x**4+1250*x**4)*exp(x**2) 
**2*ln(x)**4+(5*x**5*exp(x)**4+100*x**5*exp(x)**3+750*x**5*exp(x)**2+2500* 
x**5*exp(x)+3125*x**5)*exp(x**2)**2*ln(x)**2+(x**6*exp(x)**5+25*x**6*exp(x 
)**4+250*x**6*exp(x)**3+1250*x**6*exp(x)**2+3125*x**6*exp(x)+3125*x**6)*ex 
p(x**2)**2),x)
 

Output:

4*exp(-2*x**2)/(x**4*exp(4*x) + 20*x**4*exp(3*x) + 150*x**4*exp(2*x) + 500 
*x**4*exp(x) + 625*x**4 + 4*x**3*exp(3*x)*log(x)**2 + 60*x**3*exp(2*x)*log 
(x)**2 + 300*x**3*exp(x)*log(x)**2 + 500*x**3*log(x)**2 + 6*x**2*exp(2*x)* 
log(x)**4 + 60*x**2*exp(x)*log(x)**4 + 150*x**2*log(x)**4 + 4*x*exp(x)*log 
(x)**6 + 20*x*log(x)**6 + log(x)**8)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (22) = 44\).

Time = 0.13 (sec) , antiderivative size = 136, normalized size of antiderivative = 5.67 \[ \int \frac {-80 x-80 x^3+e^x \left (-16 x-16 x^2-16 x^3\right )-32 \log (x)-16 x^2 \log ^2(x)}{e^{2 x^2} \left (3125 x^6+3125 e^x x^6+1250 e^{2 x} x^6+250 e^{3 x} x^6+25 e^{4 x} x^6+e^{5 x} x^6\right )+e^{2 x^2} \left (3125 x^5+2500 e^x x^5+750 e^{2 x} x^5+100 e^{3 x} x^5+5 e^{4 x} x^5\right ) \log ^2(x)+e^{2 x^2} \left (1250 x^4+750 e^x x^4+150 e^{2 x} x^4+10 e^{3 x} x^4\right ) \log ^4(x)+e^{2 x^2} \left (250 x^3+100 e^x x^3+10 e^{2 x} x^3\right ) \log ^6(x)+e^{2 x^2} \left (25 x^2+5 e^x x^2\right ) \log ^8(x)+e^{2 x^2} x \log ^{10}(x)} \, dx=\frac {4 \, e^{\left (-2 \, x^{2}\right )}}{\log \left (x\right )^{8} + 20 \, x \log \left (x\right )^{6} + 150 \, x^{2} \log \left (x\right )^{4} + x^{4} e^{\left (4 \, x\right )} + 500 \, x^{3} \log \left (x\right )^{2} + 625 \, x^{4} + 4 \, {\left (x^{3} \log \left (x\right )^{2} + 5 \, x^{4}\right )} e^{\left (3 \, x\right )} + 6 \, {\left (x^{2} \log \left (x\right )^{4} + 10 \, x^{3} \log \left (x\right )^{2} + 25 \, x^{4}\right )} e^{\left (2 \, x\right )} + 4 \, {\left (x \log \left (x\right )^{6} + 15 \, x^{2} \log \left (x\right )^{4} + 75 \, x^{3} \log \left (x\right )^{2} + 125 \, x^{4}\right )} e^{x}} \] Input:

integrate((-16*x^2*log(x)^2-32*log(x)+(-16*x^3-16*x^2-16*x)*exp(x)-80*x^3- 
80*x)/(x*exp(x^2)^2*log(x)^10+(5*exp(x)*x^2+25*x^2)*exp(x^2)^2*log(x)^8+(1 
0*exp(x)^2*x^3+100*exp(x)*x^3+250*x^3)*exp(x^2)^2*log(x)^6+(10*x^4*exp(x)^ 
3+150*exp(x)^2*x^4+750*exp(x)*x^4+1250*x^4)*exp(x^2)^2*log(x)^4+(5*x^5*exp 
(x)^4+100*x^5*exp(x)^3+750*x^5*exp(x)^2+2500*x^5*exp(x)+3125*x^5)*exp(x^2) 
^2*log(x)^2+(x^6*exp(x)^5+25*x^6*exp(x)^4+250*x^6*exp(x)^3+1250*x^6*exp(x) 
^2+3125*x^6*exp(x)+3125*x^6)*exp(x^2)^2),x, algorithm="maxima")
 

Output:

4*e^(-2*x^2)/(log(x)^8 + 20*x*log(x)^6 + 150*x^2*log(x)^4 + x^4*e^(4*x) + 
500*x^3*log(x)^2 + 625*x^4 + 4*(x^3*log(x)^2 + 5*x^4)*e^(3*x) + 6*(x^2*log 
(x)^4 + 10*x^3*log(x)^2 + 25*x^4)*e^(2*x) + 4*(x*log(x)^6 + 15*x^2*log(x)^ 
4 + 75*x^3*log(x)^2 + 125*x^4)*e^x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (22) = 44\).

Time = 1.13 (sec) , antiderivative size = 239, normalized size of antiderivative = 9.96 \[ \int \frac {-80 x-80 x^3+e^x \left (-16 x-16 x^2-16 x^3\right )-32 \log (x)-16 x^2 \log ^2(x)}{e^{2 x^2} \left (3125 x^6+3125 e^x x^6+1250 e^{2 x} x^6+250 e^{3 x} x^6+25 e^{4 x} x^6+e^{5 x} x^6\right )+e^{2 x^2} \left (3125 x^5+2500 e^x x^5+750 e^{2 x} x^5+100 e^{3 x} x^5+5 e^{4 x} x^5\right ) \log ^2(x)+e^{2 x^2} \left (1250 x^4+750 e^x x^4+150 e^{2 x} x^4+10 e^{3 x} x^4\right ) \log ^4(x)+e^{2 x^2} \left (250 x^3+100 e^x x^3+10 e^{2 x} x^3\right ) \log ^6(x)+e^{2 x^2} \left (25 x^2+5 e^x x^2\right ) \log ^8(x)+e^{2 x^2} x \log ^{10}(x)} \, dx=\frac {4 \, e^{\left (6 \, x^{2}\right )}}{e^{\left (8 \, x^{2}\right )} \log \left (x\right )^{8} + 20 \, x e^{\left (8 \, x^{2}\right )} \log \left (x\right )^{6} + 4 \, x e^{\left (8 \, x^{2} + x\right )} \log \left (x\right )^{6} + 150 \, x^{2} e^{\left (8 \, x^{2}\right )} \log \left (x\right )^{4} + 6 \, x^{2} e^{\left (8 \, x^{2} + 2 \, x\right )} \log \left (x\right )^{4} + 60 \, x^{2} e^{\left (8 \, x^{2} + x\right )} \log \left (x\right )^{4} + 500 \, x^{3} e^{\left (8 \, x^{2}\right )} \log \left (x\right )^{2} + 4 \, x^{3} e^{\left (8 \, x^{2} + 3 \, x\right )} \log \left (x\right )^{2} + 60 \, x^{3} e^{\left (8 \, x^{2} + 2 \, x\right )} \log \left (x\right )^{2} + 300 \, x^{3} e^{\left (8 \, x^{2} + x\right )} \log \left (x\right )^{2} + 625 \, x^{4} e^{\left (8 \, x^{2}\right )} + x^{4} e^{\left (8 \, x^{2} + 4 \, x\right )} + 20 \, x^{4} e^{\left (8 \, x^{2} + 3 \, x\right )} + 150 \, x^{4} e^{\left (8 \, x^{2} + 2 \, x\right )} + 500 \, x^{4} e^{\left (8 \, x^{2} + x\right )}} \] Input:

integrate((-16*x^2*log(x)^2-32*log(x)+(-16*x^3-16*x^2-16*x)*exp(x)-80*x^3- 
80*x)/(x*exp(x^2)^2*log(x)^10+(5*exp(x)*x^2+25*x^2)*exp(x^2)^2*log(x)^8+(1 
0*exp(x)^2*x^3+100*exp(x)*x^3+250*x^3)*exp(x^2)^2*log(x)^6+(10*x^4*exp(x)^ 
3+150*exp(x)^2*x^4+750*exp(x)*x^4+1250*x^4)*exp(x^2)^2*log(x)^4+(5*x^5*exp 
(x)^4+100*x^5*exp(x)^3+750*x^5*exp(x)^2+2500*x^5*exp(x)+3125*x^5)*exp(x^2) 
^2*log(x)^2+(x^6*exp(x)^5+25*x^6*exp(x)^4+250*x^6*exp(x)^3+1250*x^6*exp(x) 
^2+3125*x^6*exp(x)+3125*x^6)*exp(x^2)^2),x, algorithm="giac")
 

Output:

4*e^(6*x^2)/(e^(8*x^2)*log(x)^8 + 20*x*e^(8*x^2)*log(x)^6 + 4*x*e^(8*x^2 + 
 x)*log(x)^6 + 150*x^2*e^(8*x^2)*log(x)^4 + 6*x^2*e^(8*x^2 + 2*x)*log(x)^4 
 + 60*x^2*e^(8*x^2 + x)*log(x)^4 + 500*x^3*e^(8*x^2)*log(x)^2 + 4*x^3*e^(8 
*x^2 + 3*x)*log(x)^2 + 60*x^3*e^(8*x^2 + 2*x)*log(x)^2 + 300*x^3*e^(8*x^2 
+ x)*log(x)^2 + 625*x^4*e^(8*x^2) + x^4*e^(8*x^2 + 4*x) + 20*x^4*e^(8*x^2 
+ 3*x) + 150*x^4*e^(8*x^2 + 2*x) + 500*x^4*e^(8*x^2 + x))
 

Mupad [B] (verification not implemented)

Time = 3.53 (sec) , antiderivative size = 148, normalized size of antiderivative = 6.17 \[ \int \frac {-80 x-80 x^3+e^x \left (-16 x-16 x^2-16 x^3\right )-32 \log (x)-16 x^2 \log ^2(x)}{e^{2 x^2} \left (3125 x^6+3125 e^x x^6+1250 e^{2 x} x^6+250 e^{3 x} x^6+25 e^{4 x} x^6+e^{5 x} x^6\right )+e^{2 x^2} \left (3125 x^5+2500 e^x x^5+750 e^{2 x} x^5+100 e^{3 x} x^5+5 e^{4 x} x^5\right ) \log ^2(x)+e^{2 x^2} \left (1250 x^4+750 e^x x^4+150 e^{2 x} x^4+10 e^{3 x} x^4\right ) \log ^4(x)+e^{2 x^2} \left (250 x^3+100 e^x x^3+10 e^{2 x} x^3\right ) \log ^6(x)+e^{2 x^2} \left (25 x^2+5 e^x x^2\right ) \log ^8(x)+e^{2 x^2} x \log ^{10}(x)} \, dx=\frac {4\,{\mathrm {e}}^{-2\,x^2}}{500\,x^4\,{\mathrm {e}}^x+20\,x\,{\ln \left (x\right )}^6+{\ln \left (x\right )}^8+150\,x^4\,{\mathrm {e}}^{2\,x}+20\,x^4\,{\mathrm {e}}^{3\,x}+x^4\,{\mathrm {e}}^{4\,x}+500\,x^3\,{\ln \left (x\right )}^2+150\,x^2\,{\ln \left (x\right )}^4+625\,x^4+300\,x^3\,{\mathrm {e}}^x\,{\ln \left (x\right )}^2+60\,x^2\,{\mathrm {e}}^x\,{\ln \left (x\right )}^4+60\,x^3\,{\mathrm {e}}^{2\,x}\,{\ln \left (x\right )}^2+6\,x^2\,{\mathrm {e}}^{2\,x}\,{\ln \left (x\right )}^4+4\,x^3\,{\mathrm {e}}^{3\,x}\,{\ln \left (x\right )}^2+4\,x\,{\mathrm {e}}^x\,{\ln \left (x\right )}^6} \] Input:

int(-(80*x + 32*log(x) + 16*x^2*log(x)^2 + 80*x^3 + exp(x)*(16*x + 16*x^2 
+ 16*x^3))/(exp(2*x^2)*(3125*x^6*exp(x) + 1250*x^6*exp(2*x) + 250*x^6*exp( 
3*x) + 25*x^6*exp(4*x) + x^6*exp(5*x) + 3125*x^6) + exp(2*x^2)*log(x)^2*(2 
500*x^5*exp(x) + 750*x^5*exp(2*x) + 100*x^5*exp(3*x) + 5*x^5*exp(4*x) + 31 
25*x^5) + exp(2*x^2)*log(x)^6*(100*x^3*exp(x) + 10*x^3*exp(2*x) + 250*x^3) 
 + x*exp(2*x^2)*log(x)^10 + exp(2*x^2)*log(x)^4*(750*x^4*exp(x) + 150*x^4* 
exp(2*x) + 10*x^4*exp(3*x) + 1250*x^4) + exp(2*x^2)*log(x)^8*(5*x^2*exp(x) 
 + 25*x^2)),x)
 

Output:

(4*exp(-2*x^2))/(500*x^4*exp(x) + 20*x*log(x)^6 + log(x)^8 + 150*x^4*exp(2 
*x) + 20*x^4*exp(3*x) + x^4*exp(4*x) + 500*x^3*log(x)^2 + 150*x^2*log(x)^4 
 + 625*x^4 + 300*x^3*exp(x)*log(x)^2 + 60*x^2*exp(x)*log(x)^4 + 60*x^3*exp 
(2*x)*log(x)^2 + 6*x^2*exp(2*x)*log(x)^4 + 4*x^3*exp(3*x)*log(x)^2 + 4*x*e 
xp(x)*log(x)^6)
 

Reduce [F]

\[ \int \frac {-80 x-80 x^3+e^x \left (-16 x-16 x^2-16 x^3\right )-32 \log (x)-16 x^2 \log ^2(x)}{e^{2 x^2} \left (3125 x^6+3125 e^x x^6+1250 e^{2 x} x^6+250 e^{3 x} x^6+25 e^{4 x} x^6+e^{5 x} x^6\right )+e^{2 x^2} \left (3125 x^5+2500 e^x x^5+750 e^{2 x} x^5+100 e^{3 x} x^5+5 e^{4 x} x^5\right ) \log ^2(x)+e^{2 x^2} \left (1250 x^4+750 e^x x^4+150 e^{2 x} x^4+10 e^{3 x} x^4\right ) \log ^4(x)+e^{2 x^2} \left (250 x^3+100 e^x x^3+10 e^{2 x} x^3\right ) \log ^6(x)+e^{2 x^2} \left (25 x^2+5 e^x x^2\right ) \log ^8(x)+e^{2 x^2} x \log ^{10}(x)} \, dx=\int \frac {-16 \mathrm {log}\left (x \right )^{2} x^{2}-32 \,\mathrm {log}\left (x \right )+\left (-16 x^{3}-16 x^{2}-16 x \right ) {\mathrm e}^{x}-80 x^{3}-80 x}{x \left ({\mathrm e}^{x^{2}}\right )^{2} \mathrm {log}\left (x \right )^{10}+\left (5 \,{\mathrm e}^{x} x^{2}+25 x^{2}\right ) \left ({\mathrm e}^{x^{2}}\right )^{2} \mathrm {log}\left (x \right )^{8}+\left (10 \left ({\mathrm e}^{x}\right )^{2} x^{3}+100 \,{\mathrm e}^{x} x^{3}+250 x^{3}\right ) \left ({\mathrm e}^{x^{2}}\right )^{2} \mathrm {log}\left (x \right )^{6}+\left (10 x^{4} \left ({\mathrm e}^{x}\right )^{3}+150 \left ({\mathrm e}^{x}\right )^{2} x^{4}+750 \,{\mathrm e}^{x} x^{4}+1250 x^{4}\right ) \left ({\mathrm e}^{x^{2}}\right )^{2} \mathrm {log}\left (x \right )^{4}+\left (5 x^{5} \left ({\mathrm e}^{x}\right )^{4}+100 x^{5} \left ({\mathrm e}^{x}\right )^{3}+750 x^{5} \left ({\mathrm e}^{x}\right )^{2}+2500 x^{5} {\mathrm e}^{x}+3125 x^{5}\right ) \left ({\mathrm e}^{x^{2}}\right )^{2} \mathrm {log}\left (x \right )^{2}+\left (x^{6} \left ({\mathrm e}^{x}\right )^{5}+25 x^{6} \left ({\mathrm e}^{x}\right )^{4}+250 x^{6} \left ({\mathrm e}^{x}\right )^{3}+1250 x^{6} \left ({\mathrm e}^{x}\right )^{2}+3125 x^{6} {\mathrm e}^{x}+3125 x^{6}\right ) \left ({\mathrm e}^{x^{2}}\right )^{2}}d x \] Input:

int((-16*x^2*log(x)^2-32*log(x)+(-16*x^3-16*x^2-16*x)*exp(x)-80*x^3-80*x)/ 
(x*exp(x^2)^2*log(x)^10+(5*exp(x)*x^2+25*x^2)*exp(x^2)^2*log(x)^8+(10*exp( 
x)^2*x^3+100*exp(x)*x^3+250*x^3)*exp(x^2)^2*log(x)^6+(10*x^4*exp(x)^3+150* 
exp(x)^2*x^4+750*exp(x)*x^4+1250*x^4)*exp(x^2)^2*log(x)^4+(5*x^5*exp(x)^4+ 
100*x^5*exp(x)^3+750*x^5*exp(x)^2+2500*x^5*exp(x)+3125*x^5)*exp(x^2)^2*log 
(x)^2+(x^6*exp(x)^5+25*x^6*exp(x)^4+250*x^6*exp(x)^3+1250*x^6*exp(x)^2+312 
5*x^6*exp(x)+3125*x^6)*exp(x^2)^2),x)
 

Output:

int((-16*x^2*log(x)^2-32*log(x)+(-16*x^3-16*x^2-16*x)*exp(x)-80*x^3-80*x)/ 
(x*exp(x^2)^2*log(x)^10+(5*exp(x)*x^2+25*x^2)*exp(x^2)^2*log(x)^8+(10*exp( 
x)^2*x^3+100*exp(x)*x^3+250*x^3)*exp(x^2)^2*log(x)^6+(10*x^4*exp(x)^3+150* 
exp(x)^2*x^4+750*exp(x)*x^4+1250*x^4)*exp(x^2)^2*log(x)^4+(5*x^5*exp(x)^4+ 
100*x^5*exp(x)^3+750*x^5*exp(x)^2+2500*x^5*exp(x)+3125*x^5)*exp(x^2)^2*log 
(x)^2+(x^6*exp(x)^5+25*x^6*exp(x)^4+250*x^6*exp(x)^3+1250*x^6*exp(x)^2+312 
5*x^6*exp(x)+3125*x^6)*exp(x^2)^2),x)