\(\int \frac {-1000-1500 x-1500 x^2+e^{x^2} (200+340 x+300 x^2-60 x^3)+(400+300 x+600 x^2+e^{x^2} (-80-60 x-120 x^2)) \log (5-e^{x^2})+(-40-60 x^2+e^{x^2} (8+12 x^2)) \log ^2(5-e^{x^2})}{-500+1500 x^2-1125 x^4+e^{x^2} (100-300 x^2+225 x^4)+(200-600 x^2+450 x^4+e^{x^2} (-40+120 x^2-90 x^4)) \log (5-e^{x^2})+(-20+60 x^2-45 x^4+e^{x^2} (4-12 x^2+9 x^4)) \log ^2(5-e^{x^2})} \, dx\) [966]

Optimal result

Integrand size = 218, antiderivative size = 33 \[ \int \frac {-1000-1500 x-1500 x^2+e^{x^2} \left (200+340 x+300 x^2-60 x^3\right )+\left (400+300 x+600 x^2+e^{x^2} \left (-80-60 x-120 x^2\right )\right ) \log \left (5-e^{x^2}\right )+\left (-40-60 x^2+e^{x^2} \left (8+12 x^2\right )\right ) \log ^2\left (5-e^{x^2}\right )}{-500+1500 x^2-1125 x^4+e^{x^2} \left (100-300 x^2+225 x^4\right )+\left (200-600 x^2+450 x^4+e^{x^2} \left (-40+120 x^2-90 x^4\right )\right ) \log \left (5-e^{x^2}\right )+\left (-20+60 x^2-45 x^4+e^{x^2} \left (4-12 x^2+9 x^4\right )\right ) \log ^2\left (5-e^{x^2}\right )} \, dx=\frac {2 \left (2 x+\frac {5}{5-\log \left (5-e^{x^2}\right )}\right )}{2-3 x^2} \] Output:

(2*x+5/(5-ln(5-exp(x^2))))/(-3/2*x^2+1)
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {-1000-1500 x-1500 x^2+e^{x^2} \left (200+340 x+300 x^2-60 x^3\right )+\left (400+300 x+600 x^2+e^{x^2} \left (-80-60 x-120 x^2\right )\right ) \log \left (5-e^{x^2}\right )+\left (-40-60 x^2+e^{x^2} \left (8+12 x^2\right )\right ) \log ^2\left (5-e^{x^2}\right )}{-500+1500 x^2-1125 x^4+e^{x^2} \left (100-300 x^2+225 x^4\right )+\left (200-600 x^2+450 x^4+e^{x^2} \left (-40+120 x^2-90 x^4\right )\right ) \log \left (5-e^{x^2}\right )+\left (-20+60 x^2-45 x^4+e^{x^2} \left (4-12 x^2+9 x^4\right )\right ) \log ^2\left (5-e^{x^2}\right )} \, dx=-\frac {2 \left (2 x-\frac {5}{-5+\log \left (5-e^{x^2}\right )}\right )}{-2+3 x^2} \] Input:

Integrate[(-1000 - 1500*x - 1500*x^2 + E^x^2*(200 + 340*x + 300*x^2 - 60*x 
^3) + (400 + 300*x + 600*x^2 + E^x^2*(-80 - 60*x - 120*x^2))*Log[5 - E^x^2 
] + (-40 - 60*x^2 + E^x^2*(8 + 12*x^2))*Log[5 - E^x^2]^2)/(-500 + 1500*x^2 
 - 1125*x^4 + E^x^2*(100 - 300*x^2 + 225*x^4) + (200 - 600*x^2 + 450*x^4 + 
 E^x^2*(-40 + 120*x^2 - 90*x^4))*Log[5 - E^x^2] + (-20 + 60*x^2 - 45*x^4 + 
 E^x^2*(4 - 12*x^2 + 9*x^4))*Log[5 - E^x^2]^2),x]
 

Output:

(-2*(2*x - 5/(-5 + Log[5 - E^x^2])))/(-2 + 3*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.

\(\displaystyle \int \frac {-1500 x^2+\left (-60 x^2+e^{x^2} \left (12 x^2+8\right )-40\right ) \log ^2\left (5-e^{x^2}\right )+\left (600 x^2+e^{x^2} \left (-120 x^2-60 x-80\right )+300 x+400\right ) \log \left (5-e^{x^2}\right )+e^{x^2} \left (-60 x^3+300 x^2+340 x+200\right )-1500 x-1000}{-1125 x^4+1500 x^2+e^{x^2} \left (225 x^4-300 x^2+100\right )+\left (-45 x^4+60 x^2+e^{x^2} \left (9 x^4-12 x^2+4\right )-20\right ) \log ^2\left (5-e^{x^2}\right )+\left (450 x^4-600 x^2+e^{x^2} \left (-90 x^4+120 x^2-40\right )+200\right ) \log \left (5-e^{x^2}\right )-500} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {4 \left (125 \left (3 x^2+3 x+2\right )-\left (\left (e^{x^2}-5\right ) \left (3 x^2+2\right ) \log ^2\left (5-e^{x^2}\right )\right )+5 \left (e^{x^2}-5\right ) \left (6 x^2+3 x+4\right ) \log \left (5-e^{x^2}\right )+5 e^{x^2} \left (3 x^3-15 x^2-17 x-10\right )\right )}{\left (5-e^{x^2}\right ) \left (2-3 x^2\right )^2 \left (5-\log \left (5-e^{x^2}\right )\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 4 \int \frac {\left (5-e^{x^2}\right ) \left (3 x^2+2\right ) \log ^2\left (5-e^{x^2}\right )-5 \left (5-e^{x^2}\right ) \left (6 x^2+3 x+4\right ) \log \left (5-e^{x^2}\right )+125 \left (3 x^2+3 x+2\right )-5 e^{x^2} \left (-3 x^3+15 x^2+17 x+10\right )}{\left (5-e^{x^2}\right ) \left (2-3 x^2\right )^2 \left (5-\log \left (5-e^{x^2}\right )\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 4 \int \left (\frac {-15 x^3+3 \log ^2\left (5-e^{x^2}\right ) x^2-30 \log \left (5-e^{x^2}\right ) x^2+75 x^2-15 \log \left (5-e^{x^2}\right ) x+85 x+2 \log ^2\left (5-e^{x^2}\right )-20 \log \left (5-e^{x^2}\right )+50}{\left (3 x^2-2\right )^2 \left (\log \left (5-e^{x^2}\right )-5\right )^2}-\frac {25 x}{\left (-5+e^{x^2}\right ) \left (3 x^2-2\right ) \left (\log \left (5-e^{x^2}\right )-5\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 4 \left (-\frac {5}{2} \text {Subst}\left (\int \frac {1}{(3 x-2) \left (\log \left (5-e^x\right )-5\right )^2}dx,x,x^2\right )-\frac {25}{2} \text {Subst}\left (\int \frac {1}{\left (-5+e^x\right ) (3 x-2) \left (\log \left (5-e^x\right )-5\right )^2}dx,x,x^2\right )-\frac {15}{2} \text {Subst}\left (\int \frac {1}{(3 x-2)^2 \left (\log \left (5-e^x\right )-5\right )}dx,x,x^2\right )+\frac {x}{2-3 x^2}\right )\)

Input:

Int[(-1000 - 1500*x - 1500*x^2 + E^x^2*(200 + 340*x + 300*x^2 - 60*x^3) + 
(400 + 300*x + 600*x^2 + E^x^2*(-80 - 60*x - 120*x^2))*Log[5 - E^x^2] + (- 
40 - 60*x^2 + E^x^2*(8 + 12*x^2))*Log[5 - E^x^2]^2)/(-500 + 1500*x^2 - 112 
5*x^4 + E^x^2*(100 - 300*x^2 + 225*x^4) + (200 - 600*x^2 + 450*x^4 + E^x^2 
*(-40 + 120*x^2 - 90*x^4))*Log[5 - E^x^2] + (-20 + 60*x^2 - 45*x^4 + E^x^2 
*(4 - 12*x^2 + 9*x^4))*Log[5 - E^x^2]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.68 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.15

method	result	size
risch	\(-\frac {4 x}{3 x^{2}-2}+\frac {10}{\left (3 x^{2}-2\right ) \left (\ln \left (5-{\mathrm e}^{x^{2}}\right )-5\right )}\)	\(38\)
parallelrisch	\(\frac {30-12 \ln \left (5-{\mathrm e}^{x^{2}}\right ) x +60 x}{3 \left (\ln \left (5-{\mathrm e}^{x^{2}}\right )-5\right ) \left (3 x^{2}-2\right )}\)	\(42\)

Input:

int((((12*x^2+8)*exp(x^2)-60*x^2-40)*ln(5-exp(x^2))^2+((-120*x^2-60*x-80)* 
exp(x^2)+600*x^2+300*x+400)*ln(5-exp(x^2))+(-60*x^3+300*x^2+340*x+200)*exp 
(x^2)-1500*x^2-1500*x-1000)/(((9*x^4-12*x^2+4)*exp(x^2)-45*x^4+60*x^2-20)* 
ln(5-exp(x^2))^2+((-90*x^4+120*x^2-40)*exp(x^2)+450*x^4-600*x^2+200)*ln(5- 
exp(x^2))+(225*x^4-300*x^2+100)*exp(x^2)-1125*x^4+1500*x^2-500),x,method=_ 
RETURNVERBOSE)
 

Output:

-4*x/(3*x^2-2)+10/(3*x^2-2)/(ln(5-exp(x^2))-5)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.39 \[ \int \frac {-1000-1500 x-1500 x^2+e^{x^2} \left (200+340 x+300 x^2-60 x^3\right )+\left (400+300 x+600 x^2+e^{x^2} \left (-80-60 x-120 x^2\right )\right ) \log \left (5-e^{x^2}\right )+\left (-40-60 x^2+e^{x^2} \left (8+12 x^2\right )\right ) \log ^2\left (5-e^{x^2}\right )}{-500+1500 x^2-1125 x^4+e^{x^2} \left (100-300 x^2+225 x^4\right )+\left (200-600 x^2+450 x^4+e^{x^2} \left (-40+120 x^2-90 x^4\right )\right ) \log \left (5-e^{x^2}\right )+\left (-20+60 x^2-45 x^4+e^{x^2} \left (4-12 x^2+9 x^4\right )\right ) \log ^2\left (5-e^{x^2}\right )} \, dx=\frac {2 \, {\left (2 \, x \log \left (-e^{\left (x^{2}\right )} + 5\right ) - 10 \, x - 5\right )}}{15 \, x^{2} - {\left (3 \, x^{2} - 2\right )} \log \left (-e^{\left (x^{2}\right )} + 5\right ) - 10} \] Input:

integrate((((12*x^2+8)*exp(x^2)-60*x^2-40)*log(5-exp(x^2))^2+((-120*x^2-60 
*x-80)*exp(x^2)+600*x^2+300*x+400)*log(5-exp(x^2))+(-60*x^3+300*x^2+340*x+ 
200)*exp(x^2)-1500*x^2-1500*x-1000)/(((9*x^4-12*x^2+4)*exp(x^2)-45*x^4+60* 
x^2-20)*log(5-exp(x^2))^2+((-90*x^4+120*x^2-40)*exp(x^2)+450*x^4-600*x^2+2 
00)*log(5-exp(x^2))+(225*x^4-300*x^2+100)*exp(x^2)-1125*x^4+1500*x^2-500), 
x, algorithm="fricas")
 

Output:

2*(2*x*log(-e^(x^2) + 5) - 10*x - 5)/(15*x^2 - (3*x^2 - 2)*log(-e^(x^2) + 
5) - 10)
 

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \frac {-1000-1500 x-1500 x^2+e^{x^2} \left (200+340 x+300 x^2-60 x^3\right )+\left (400+300 x+600 x^2+e^{x^2} \left (-80-60 x-120 x^2\right )\right ) \log \left (5-e^{x^2}\right )+\left (-40-60 x^2+e^{x^2} \left (8+12 x^2\right )\right ) \log ^2\left (5-e^{x^2}\right )}{-500+1500 x^2-1125 x^4+e^{x^2} \left (100-300 x^2+225 x^4\right )+\left (200-600 x^2+450 x^4+e^{x^2} \left (-40+120 x^2-90 x^4\right )\right ) \log \left (5-e^{x^2}\right )+\left (-20+60 x^2-45 x^4+e^{x^2} \left (4-12 x^2+9 x^4\right )\right ) \log ^2\left (5-e^{x^2}\right )} \, dx=- \frac {4 x}{3 x^{2} - 2} + \frac {10}{- 15 x^{2} + \left (3 x^{2} - 2\right ) \log {\left (5 - e^{x^{2}} \right )} + 10} \] Input:

integrate((((12*x**2+8)*exp(x**2)-60*x**2-40)*ln(5-exp(x**2))**2+((-120*x* 
*2-60*x-80)*exp(x**2)+600*x**2+300*x+400)*ln(5-exp(x**2))+(-60*x**3+300*x* 
*2+340*x+200)*exp(x**2)-1500*x**2-1500*x-1000)/(((9*x**4-12*x**2+4)*exp(x* 
*2)-45*x**4+60*x**2-20)*ln(5-exp(x**2))**2+((-90*x**4+120*x**2-40)*exp(x** 
2)+450*x**4-600*x**2+200)*ln(5-exp(x**2))+(225*x**4-300*x**2+100)*exp(x**2 
)-1125*x**4+1500*x**2-500),x)
 

Output:

-4*x/(3*x**2 - 2) + 10/(-15*x**2 + (3*x**2 - 2)*log(5 - exp(x**2)) + 10)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.39 \[ \int \frac {-1000-1500 x-1500 x^2+e^{x^2} \left (200+340 x+300 x^2-60 x^3\right )+\left (400+300 x+600 x^2+e^{x^2} \left (-80-60 x-120 x^2\right )\right ) \log \left (5-e^{x^2}\right )+\left (-40-60 x^2+e^{x^2} \left (8+12 x^2\right )\right ) \log ^2\left (5-e^{x^2}\right )}{-500+1500 x^2-1125 x^4+e^{x^2} \left (100-300 x^2+225 x^4\right )+\left (200-600 x^2+450 x^4+e^{x^2} \left (-40+120 x^2-90 x^4\right )\right ) \log \left (5-e^{x^2}\right )+\left (-20+60 x^2-45 x^4+e^{x^2} \left (4-12 x^2+9 x^4\right )\right ) \log ^2\left (5-e^{x^2}\right )} \, dx=\frac {2 \, {\left (2 \, x \log \left (-e^{\left (x^{2}\right )} + 5\right ) - 10 \, x - 5\right )}}{15 \, x^{2} - {\left (3 \, x^{2} - 2\right )} \log \left (-e^{\left (x^{2}\right )} + 5\right ) - 10} \] Input:

integrate((((12*x^2+8)*exp(x^2)-60*x^2-40)*log(5-exp(x^2))^2+((-120*x^2-60 
*x-80)*exp(x^2)+600*x^2+300*x+400)*log(5-exp(x^2))+(-60*x^3+300*x^2+340*x+ 
200)*exp(x^2)-1500*x^2-1500*x-1000)/(((9*x^4-12*x^2+4)*exp(x^2)-45*x^4+60* 
x^2-20)*log(5-exp(x^2))^2+((-90*x^4+120*x^2-40)*exp(x^2)+450*x^4-600*x^2+2 
00)*log(5-exp(x^2))+(225*x^4-300*x^2+100)*exp(x^2)-1125*x^4+1500*x^2-500), 
x, algorithm="maxima")
 

Output:

2*(2*x*log(-e^(x^2) + 5) - 10*x - 5)/(15*x^2 - (3*x^2 - 2)*log(-e^(x^2) + 
5) - 10)
 

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.61 \[ \int \frac {-1000-1500 x-1500 x^2+e^{x^2} \left (200+340 x+300 x^2-60 x^3\right )+\left (400+300 x+600 x^2+e^{x^2} \left (-80-60 x-120 x^2\right )\right ) \log \left (5-e^{x^2}\right )+\left (-40-60 x^2+e^{x^2} \left (8+12 x^2\right )\right ) \log ^2\left (5-e^{x^2}\right )}{-500+1500 x^2-1125 x^4+e^{x^2} \left (100-300 x^2+225 x^4\right )+\left (200-600 x^2+450 x^4+e^{x^2} \left (-40+120 x^2-90 x^4\right )\right ) \log \left (5-e^{x^2}\right )+\left (-20+60 x^2-45 x^4+e^{x^2} \left (4-12 x^2+9 x^4\right )\right ) \log ^2\left (5-e^{x^2}\right )} \, dx=-\frac {2 \, {\left (2 \, x \log \left (-e^{\left (x^{2}\right )} + 5\right ) - 10 \, x - 5\right )}}{3 \, x^{2} \log \left (-e^{\left (x^{2}\right )} + 5\right ) - 15 \, x^{2} - 2 \, \log \left (-e^{\left (x^{2}\right )} + 5\right ) + 10} \] Input:

integrate((((12*x^2+8)*exp(x^2)-60*x^2-40)*log(5-exp(x^2))^2+((-120*x^2-60 
*x-80)*exp(x^2)+600*x^2+300*x+400)*log(5-exp(x^2))+(-60*x^3+300*x^2+340*x+ 
200)*exp(x^2)-1500*x^2-1500*x-1000)/(((9*x^4-12*x^2+4)*exp(x^2)-45*x^4+60* 
x^2-20)*log(5-exp(x^2))^2+((-90*x^4+120*x^2-40)*exp(x^2)+450*x^4-600*x^2+2 
00)*log(5-exp(x^2))+(225*x^4-300*x^2+100)*exp(x^2)-1125*x^4+1500*x^2-500), 
x, algorithm="giac")
 

Output:

-2*(2*x*log(-e^(x^2) + 5) - 10*x - 5)/(3*x^2*log(-e^(x^2) + 5) - 15*x^2 - 
2*log(-e^(x^2) + 5) + 10)
 

Mupad [B] (verification not implemented)

Time = 3.19 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.24 \[ \int \frac {-1000-1500 x-1500 x^2+e^{x^2} \left (200+340 x+300 x^2-60 x^3\right )+\left (400+300 x+600 x^2+e^{x^2} \left (-80-60 x-120 x^2\right )\right ) \log \left (5-e^{x^2}\right )+\left (-40-60 x^2+e^{x^2} \left (8+12 x^2\right )\right ) \log ^2\left (5-e^{x^2}\right )}{-500+1500 x^2-1125 x^4+e^{x^2} \left (100-300 x^2+225 x^4\right )+\left (200-600 x^2+450 x^4+e^{x^2} \left (-40+120 x^2-90 x^4\right )\right ) \log \left (5-e^{x^2}\right )+\left (-20+60 x^2-45 x^4+e^{x^2} \left (4-12 x^2+9 x^4\right )\right ) \log ^2\left (5-e^{x^2}\right )} \, dx=\frac {2\,\left (10\,x-2\,x\,\ln \left (5-{\mathrm {e}}^{x^2}\right )+5\right )}{\left (3\,x^2-2\right )\,\left (\ln \left (5-{\mathrm {e}}^{x^2}\right )-5\right )} \] Input:

int(-(1500*x - log(5 - exp(x^2))*(300*x - exp(x^2)*(60*x + 120*x^2 + 80) + 
 600*x^2 + 400) - exp(x^2)*(340*x + 300*x^2 - 60*x^3 + 200) + 1500*x^2 + l 
og(5 - exp(x^2))^2*(60*x^2 - exp(x^2)*(12*x^2 + 8) + 40) + 1000)/(log(5 - 
exp(x^2))^2*(exp(x^2)*(9*x^4 - 12*x^2 + 4) + 60*x^2 - 45*x^4 - 20) + exp(x 
^2)*(225*x^4 - 300*x^2 + 100) + 1500*x^2 - 1125*x^4 - log(5 - exp(x^2))*(e 
xp(x^2)*(90*x^4 - 120*x^2 + 40) + 600*x^2 - 450*x^4 - 200) - 500),x)
 

Output:

(2*(10*x - 2*x*log(5 - exp(x^2)) + 5))/((3*x^2 - 2)*(log(5 - exp(x^2)) - 5 
))
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 156, normalized size of antiderivative = 4.73 \[ \int \frac {-1000-1500 x-1500 x^2+e^{x^2} \left (200+340 x+300 x^2-60 x^3\right )+\left (400+300 x+600 x^2+e^{x^2} \left (-80-60 x-120 x^2\right )\right ) \log \left (5-e^{x^2}\right )+\left (-40-60 x^2+e^{x^2} \left (8+12 x^2\right )\right ) \log ^2\left (5-e^{x^2}\right )}{-500+1500 x^2-1125 x^4+e^{x^2} \left (100-300 x^2+225 x^4\right )+\left (200-600 x^2+450 x^4+e^{x^2} \left (-40+120 x^2-90 x^4\right )\right ) \log \left (5-e^{x^2}\right )+\left (-20+60 x^2-45 x^4+e^{x^2} \left (4-12 x^2+9 x^4\right )\right ) \log ^2\left (5-e^{x^2}\right )} \, dx=\frac {3 \,\mathrm {log}\left (e^{x^{2}}-5\right ) \mathrm {log}\left (-e^{x^{2}}+5\right ) x^{2}-2 \,\mathrm {log}\left (e^{x^{2}}-5\right ) \mathrm {log}\left (-e^{x^{2}}+5\right )-15 \,\mathrm {log}\left (e^{x^{2}}-5\right ) x^{2}+10 \,\mathrm {log}\left (e^{x^{2}}-5\right )-3 \mathrm {log}\left (-e^{x^{2}}+5\right )^{2} x^{2}+2 \mathrm {log}\left (-e^{x^{2}}+5\right )^{2}-20 \,\mathrm {log}\left (-e^{x^{2}}+5\right ) x +75 x^{2}+100 x}{15 \,\mathrm {log}\left (-e^{x^{2}}+5\right ) x^{2}-10 \,\mathrm {log}\left (-e^{x^{2}}+5\right )-75 x^{2}+50} \] Input:

int((((12*x^2+8)*exp(x^2)-60*x^2-40)*log(5-exp(x^2))^2+((-120*x^2-60*x-80) 
*exp(x^2)+600*x^2+300*x+400)*log(5-exp(x^2))+(-60*x^3+300*x^2+340*x+200)*e 
xp(x^2)-1500*x^2-1500*x-1000)/(((9*x^4-12*x^2+4)*exp(x^2)-45*x^4+60*x^2-20 
)*log(5-exp(x^2))^2+((-90*x^4+120*x^2-40)*exp(x^2)+450*x^4-600*x^2+200)*lo 
g(5-exp(x^2))+(225*x^4-300*x^2+100)*exp(x^2)-1125*x^4+1500*x^2-500),x)
 

Output:

(3*log(e**(x**2) - 5)*log( - e**(x**2) + 5)*x**2 - 2*log(e**(x**2) - 5)*lo 
g( - e**(x**2) + 5) - 15*log(e**(x**2) - 5)*x**2 + 10*log(e**(x**2) - 5) - 
 3*log( - e**(x**2) + 5)**2*x**2 + 2*log( - e**(x**2) + 5)**2 - 20*log( - 
e**(x**2) + 5)*x + 75*x**2 + 100*x)/(5*(3*log( - e**(x**2) + 5)*x**2 - 2*l 
og( - e**(x**2) + 5) - 15*x**2 + 10))