\(\int \frac {-1000-1500 x-1500 x^2+e^{x^2} (200+340 x+300 x^2-60 x^3)+(400+300 x+600 x^2+e^{x^2} (-80-60 x-120 x^2)) \log (5-e^{x^2})+(-40-60 x^2+e^{x^2} (8+12 x^2)) \log ^2(5-e^{x^2})}{-500+1500 x^2-1125 x^4+e^{x^2} (100-300 x^2+225 x^4)+(200-600 x^2+450 x^4+e^{x^2} (-40+120 x^2-90 x^4)) \log (5-e^{x^2})+(-20+60 x^2-45 x^4+e^{x^2} (4-12 x^2+9 x^4)) \log ^2(5-e^{x^2})} \, dx\) [3966]

Optimal result

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

[Out]

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

Rubi [F]

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

[In]

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 - 1
125*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]

[Out]

(4*x)/(2 - 3*x^2) - 10*Defer[Subst][Defer[Int][1/((-2 + 3*x)*(-5 + Log[5 - E^x])^2), x], x, x^2] - 50*Defer[Su
bst][Defer[Int][1/((-5 + E^x)*(-2 + 3*x)*(-5 + Log[5 - E^x])^2), x], x, x^2] - 30*Defer[Subst][Defer[Int][1/((
-2 + 3*x)^2*(-5 + Log[5 - E^x])), x], x, x^2]

Rubi steps \begin{align*} \text {integral}& = \int \frac {4 \left (125 \left (2+3 x+3 x^2\right )+5 e^{x^2} \left (-10-17 x-15 x^2+3 x^3\right )+5 \left (-5+e^{x^2}\right ) \left (4+3 x+6 x^2\right ) \log \left (5-e^{x^2}\right )-\left (-5+e^{x^2}\right ) \left (2+3 x^2\right ) \log ^2\left (5-e^{x^2}\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 \\ & = 4 \int \frac {125 \left (2+3 x+3 x^2\right )+5 e^{x^2} \left (-10-17 x-15 x^2+3 x^3\right )+5 \left (-5+e^{x^2}\right ) \left (4+3 x+6 x^2\right ) \log \left (5-e^{x^2}\right )-\left (-5+e^{x^2}\right ) \left (2+3 x^2\right ) \log ^2\left (5-e^{x^2}\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 \\ & = 4 \int \left (-\frac {25 x}{\left (-5+e^{x^2}\right ) \left (-2+3 x^2\right ) \left (-5+\log \left (5-e^{x^2}\right )\right )^2}+\frac {50+85 x+75 x^2-15 x^3-20 \log \left (5-e^{x^2}\right )-15 x \log \left (5-e^{x^2}\right )-30 x^2 \log \left (5-e^{x^2}\right )+2 \log ^2\left (5-e^{x^2}\right )+3 x^2 \log ^2\left (5-e^{x^2}\right )}{\left (-2+3 x^2\right )^2 \left (-5+\log \left (5-e^{x^2}\right )\right )^2}\right ) \, dx \\ & = 4 \int \frac {50+85 x+75 x^2-15 x^3-20 \log \left (5-e^{x^2}\right )-15 x \log \left (5-e^{x^2}\right )-30 x^2 \log \left (5-e^{x^2}\right )+2 \log ^2\left (5-e^{x^2}\right )+3 x^2 \log ^2\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-100 \int \frac {x}{\left (-5+e^{x^2}\right ) \left (-2+3 x^2\right ) \left (-5+\log \left (5-e^{x^2}\right )\right )^2} \, dx \\ & = 4 \int \left (\frac {2+3 x^2}{\left (-2+3 x^2\right )^2}-\frac {5 x}{\left (-2+3 x^2\right ) \left (-5+\log \left (5-e^{x^2}\right )\right )^2}-\frac {15 x}{\left (-2+3 x^2\right )^2 \left (-5+\log \left (5-e^{x^2}\right )\right )}\right ) \, dx-50 \text {Subst}\left (\int \frac {1}{\left (-5+e^x\right ) (-2+3 x) \left (-5+\log \left (5-e^x\right )\right )^2} \, dx,x,x^2\right ) \\ & = 4 \int \frac {2+3 x^2}{\left (-2+3 x^2\right )^2} \, dx-20 \int \frac {x}{\left (-2+3 x^2\right ) \left (-5+\log \left (5-e^{x^2}\right )\right )^2} \, dx-50 \text {Subst}\left (\int \frac {1}{\left (-5+e^x\right ) (-2+3 x) \left (-5+\log \left (5-e^x\right )\right )^2} \, dx,x,x^2\right )-60 \int \frac {x}{\left (-2+3 x^2\right )^2 \left (-5+\log \left (5-e^{x^2}\right )\right )} \, dx \\ & = \frac {4 x}{2-3 x^2}-10 \text {Subst}\left (\int \frac {1}{(-2+3 x) \left (-5+\log \left (5-e^x\right )\right )^2} \, dx,x,x^2\right )-30 \text {Subst}\left (\int \frac {1}{(-2+3 x)^2 \left (-5+\log \left (5-e^x\right )\right )} \, dx,x,x^2\right )-50 \text {Subst}\left (\int \frac {1}{\left (-5+e^x\right ) (-2+3 x) \left (-5+\log \left (5-e^x\right )\right )^2} \, dx,x,x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {-1000-1500 x-1500 x^2+e^{x^2} \left (200+340 x+300 x^2-60 x^3\right )+\left (400+300 x+600 x^2+e^{x^2} \left (-80-60 x-120 x^2\right )\right ) \log \left (5-e^{x^2}\right )+\left (-40-60 x^2+e^{x^2} \left (8+12 x^2\right )\right ) \log ^2\left (5-e^{x^2}\right )}{-500+1500 x^2-1125 x^4+e^{x^2} \left (100-300 x^2+225 x^4\right )+\left (200-600 x^2+450 x^4+e^{x^2} \left (-40+120 x^2-90 x^4\right )\right ) \log \left (5-e^{x^2}\right )+\left (-20+60 x^2-45 x^4+e^{x^2} \left (4-12 x^2+9 x^4\right )\right ) \log ^2\left (5-e^{x^2}\right )} \, dx=-\frac {2 \left (2 x-\frac {5}{-5+\log \left (5-e^{x^2}\right )}\right )}{-2+3 x^2} \]

[In]

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))*L
og[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]

[Out]

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

Maple [A] (verified)

Time = 1.53 (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	\(\frac {30-12 \ln \left (5-{\mathrm e}^{x^{2}}\right ) x +60 x}{3 \left (\ln \left (5-{\mathrm e}^{x^{2}}\right )-5\right ) \left (3 x^{2}-2\right )}\)	\(42\)

[In]

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)*ex
p(x^2)-1125*x^4+1500*x^2-500),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.39 \[ \int \frac {-1000-1500 x-1500 x^2+e^{x^2} \left (200+340 x+300 x^2-60 x^3\right )+\left (400+300 x+600 x^2+e^{x^2} \left (-80-60 x-120 x^2\right )\right ) \log \left (5-e^{x^2}\right )+\left (-40-60 x^2+e^{x^2} \left (8+12 x^2\right )\right ) \log ^2\left (5-e^{x^2}\right )}{-500+1500 x^2-1125 x^4+e^{x^2} \left (100-300 x^2+225 x^4\right )+\left (200-600 x^2+450 x^4+e^{x^2} \left (-40+120 x^2-90 x^4\right )\right ) \log \left (5-e^{x^2}\right )+\left (-20+60 x^2-45 x^4+e^{x^2} \left (4-12 x^2+9 x^4\right )\right ) \log ^2\left (5-e^{x^2}\right )} \, dx=\frac {2 \, {\left (2 \, x \log \left (-e^{\left (x^{2}\right )} + 5\right ) - 10 \, x - 5\right )}}{15 \, x^{2} - {\left (3 \, x^{2} - 2\right )} \log \left (-e^{\left (x^{2}\right )} + 5\right ) - 10} \]

[In]

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)*l
og(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+6
0*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)*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")

[Out]

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

[In]

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)

[Out]

-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)

none

Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.39 \[ \int \frac {-1000-1500 x-1500 x^2+e^{x^2} \left (200+340 x+300 x^2-60 x^3\right )+\left (400+300 x+600 x^2+e^{x^2} \left (-80-60 x-120 x^2\right )\right ) \log \left (5-e^{x^2}\right )+\left (-40-60 x^2+e^{x^2} \left (8+12 x^2\right )\right ) \log ^2\left (5-e^{x^2}\right )}{-500+1500 x^2-1125 x^4+e^{x^2} \left (100-300 x^2+225 x^4\right )+\left (200-600 x^2+450 x^4+e^{x^2} \left (-40+120 x^2-90 x^4\right )\right ) \log \left (5-e^{x^2}\right )+\left (-20+60 x^2-45 x^4+e^{x^2} \left (4-12 x^2+9 x^4\right )\right ) \log ^2\left (5-e^{x^2}\right )} \, dx=\frac {2 \, {\left (2 \, x \log \left (-e^{\left (x^{2}\right )} + 5\right ) - 10 \, x - 5\right )}}{15 \, x^{2} - {\left (3 \, x^{2} - 2\right )} \log \left (-e^{\left (x^{2}\right )} + 5\right ) - 10} \]

[In]

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)*l
og(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+6
0*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)*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")

[Out]

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)

none

Time = 0.37 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.61 \[ \int \frac {-1000-1500 x-1500 x^2+e^{x^2} \left (200+340 x+300 x^2-60 x^3\right )+\left (400+300 x+600 x^2+e^{x^2} \left (-80-60 x-120 x^2\right )\right ) \log \left (5-e^{x^2}\right )+\left (-40-60 x^2+e^{x^2} \left (8+12 x^2\right )\right ) \log ^2\left (5-e^{x^2}\right )}{-500+1500 x^2-1125 x^4+e^{x^2} \left (100-300 x^2+225 x^4\right )+\left (200-600 x^2+450 x^4+e^{x^2} \left (-40+120 x^2-90 x^4\right )\right ) \log \left (5-e^{x^2}\right )+\left (-20+60 x^2-45 x^4+e^{x^2} \left (4-12 x^2+9 x^4\right )\right ) \log ^2\left (5-e^{x^2}\right )} \, dx=-\frac {2 \, {\left (2 \, x \log \left (-e^{\left (x^{2}\right )} + 5\right ) - 10 \, x - 5\right )}}{3 \, x^{2} \log \left (-e^{\left (x^{2}\right )} + 5\right ) - 15 \, x^{2} - 2 \, \log \left (-e^{\left (x^{2}\right )} + 5\right ) + 10} \]

[In]

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)*l
og(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+6
0*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)*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")

[Out]

-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 = 9.36 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.24 \[ \int \frac {-1000-1500 x-1500 x^2+e^{x^2} \left (200+340 x+300 x^2-60 x^3\right )+\left (400+300 x+600 x^2+e^{x^2} \left (-80-60 x-120 x^2\right )\right ) \log \left (5-e^{x^2}\right )+\left (-40-60 x^2+e^{x^2} \left (8+12 x^2\right )\right ) \log ^2\left (5-e^{x^2}\right )}{-500+1500 x^2-1125 x^4+e^{x^2} \left (100-300 x^2+225 x^4\right )+\left (200-600 x^2+450 x^4+e^{x^2} \left (-40+120 x^2-90 x^4\right )\right ) \log \left (5-e^{x^2}\right )+\left (-20+60 x^2-45 x^4+e^{x^2} \left (4-12 x^2+9 x^4\right )\right ) \log ^2\left (5-e^{x^2}\right )} \, dx=\frac {2\,\left (10\,x-2\,x\,\ln \left (5-{\mathrm {e}}^{x^2}\right )+5\right )}{\left (3\,x^2-2\right )\,\left (\ln \left (5-{\mathrm {e}}^{x^2}\right )-5\right )} \]

[In]

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 + log(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) + 15
00*x^2 - 1125*x^4 - log(5 - exp(x^2))*(exp(x^2)*(90*x^4 - 120*x^2 + 40) + 600*x^2 - 450*x^4 - 200) - 500),x)

[Out]

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