\(\int \frac {(20 x-996 x^2+11900 x^3+12380 x^4+4992 x^5+1000 x^6+100 x^7+4 x^8+e^{x^2} (-40+1992 x-23800 x^2-24760 x^3-9984 x^4-2000 x^5-200 x^6-8 x^7)) \log (2 x^2)+(-5 x-5 x^2+3225 x^3+3165 x^4+1254 x^5+250 x^6+25 x^7+x^8+e^{x^2} (508 x-12380 x^2-13516 x^3+6896 x^4+11380 x^5+4892 x^6+996 x^7+100 x^8+4 x^9)) \log ^2(2 x^2)}{3125 x^3+3125 x^4+1250 x^5+250 x^6+25 x^7+x^8+e^{2 x^2} (12500 x+12500 x^2+5000 x^3+1000 x^4+100 x^5+4 x^6)+e^{x^2} (-12500 x^2-12500 x^3-5000 x^4-1000 x^5-100 x^6-4 x^7)} \, dx\) [2822]

Optimal result

Integrand size = 290, antiderivative size = 31 \[ \int \frac {\left (20 x-996 x^2+11900 x^3+12380 x^4+4992 x^5+1000 x^6+100 x^7+4 x^8+e^{x^2} \left (-40+1992 x-23800 x^2-24760 x^3-9984 x^4-2000 x^5-200 x^6-8 x^7\right )\right ) \log \left (2 x^2\right )+\left (-5 x-5 x^2+3225 x^3+3165 x^4+1254 x^5+250 x^6+25 x^7+x^8+e^{x^2} \left (508 x-12380 x^2-13516 x^3+6896 x^4+11380 x^5+4892 x^6+996 x^7+100 x^8+4 x^9\right )\right ) \log ^2\left (2 x^2\right )}{3125 x^3+3125 x^4+1250 x^5+250 x^6+25 x^7+x^8+e^{2 x^2} \left (12500 x+12500 x^2+5000 x^3+1000 x^4+100 x^5+4 x^6\right )+e^{x^2} \left (-12500 x^2-12500 x^3-5000 x^4-1000 x^5-100 x^6-4 x^7\right )} \, dx=\frac {\left (-x+\frac {1}{(5+x)^2}\right )^2 \log ^2\left (2 x^2\right )}{-2 e^{x^2}+x} \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.39 \[ \int \frac {\left (20 x-996 x^2+11900 x^3+12380 x^4+4992 x^5+1000 x^6+100 x^7+4 x^8+e^{x^2} \left (-40+1992 x-23800 x^2-24760 x^3-9984 x^4-2000 x^5-200 x^6-8 x^7\right )\right ) \log \left (2 x^2\right )+\left (-5 x-5 x^2+3225 x^3+3165 x^4+1254 x^5+250 x^6+25 x^7+x^8+e^{x^2} \left (508 x-12380 x^2-13516 x^3+6896 x^4+11380 x^5+4892 x^6+996 x^7+100 x^8+4 x^9\right )\right ) \log ^2\left (2 x^2\right )}{3125 x^3+3125 x^4+1250 x^5+250 x^6+25 x^7+x^8+e^{2 x^2} \left (12500 x+12500 x^2+5000 x^3+1000 x^4+100 x^5+4 x^6\right )+e^{x^2} \left (-12500 x^2-12500 x^3-5000 x^4-1000 x^5-100 x^6-4 x^7\right )} \, dx=-\frac {\left (-1+25 x+10 x^2+x^3\right )^2 \log ^2\left (2 x^2\right )}{\left (2 e^{x^2}-x\right ) (5+x)^4} \] Input:

Integrate[((20*x - 996*x^2 + 11900*x^3 + 12380*x^4 + 4992*x^5 + 1000*x^6 + 
 100*x^7 + 4*x^8 + E^x^2*(-40 + 1992*x - 23800*x^2 - 24760*x^3 - 9984*x^4 
- 2000*x^5 - 200*x^6 - 8*x^7))*Log[2*x^2] + (-5*x - 5*x^2 + 3225*x^3 + 316 
5*x^4 + 1254*x^5 + 250*x^6 + 25*x^7 + x^8 + E^x^2*(508*x - 12380*x^2 - 135 
16*x^3 + 6896*x^4 + 11380*x^5 + 4892*x^6 + 996*x^7 + 100*x^8 + 4*x^9))*Log 
[2*x^2]^2)/(3125*x^3 + 3125*x^4 + 1250*x^5 + 250*x^6 + 25*x^7 + x^8 + E^(2 
*x^2)*(12500*x + 12500*x^2 + 5000*x^3 + 1000*x^4 + 100*x^5 + 4*x^6) + E^x^ 
2*(-12500*x^2 - 12500*x^3 - 5000*x^4 - 1000*x^5 - 100*x^6 - 4*x^7)),x]
 

Output:

-(((-1 + 25*x + 10*x^2 + x^3)^2*Log[2*x^2]^2)/((2*E^x^2 - x)*(5 + x)^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.

\(\displaystyle \int \frac {\left (4 x^8+100 x^7+1000 x^6+4992 x^5+12380 x^4+11900 x^3-996 x^2+e^{x^2} \left (-8 x^7-200 x^6-2000 x^5-9984 x^4-24760 x^3-23800 x^2+1992 x-40\right )+20 x\right ) \log \left (2 x^2\right )+\left (x^8+25 x^7+250 x^6+1254 x^5+3165 x^4+3225 x^3-5 x^2+e^{x^2} \left (4 x^9+100 x^8+996 x^7+4892 x^6+11380 x^5+6896 x^4-13516 x^3-12380 x^2+508 x\right )-5 x\right ) \log ^2\left (2 x^2\right )}{x^8+25 x^7+250 x^6+1250 x^5+3125 x^4+3125 x^3+e^{2 x^2} \left (4 x^6+100 x^5+1000 x^4+5000 x^3+12500 x^2+12500 x\right )+e^{x^2} \left (-4 x^7-100 x^6-1000 x^5-5000 x^4-12500 x^3-12500 x^2\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (-x^3-10 x^2-25 x+1\right ) \log \left (2 x^2\right ) \left (4 \left (2 e^{x^2}-x\right ) \left (x^4+15 x^3+75 x^2+124 x-5\right )-x \left (x^4+15 x^3+75 x^2+4 e^{x^2} \left (x^5+15 x^4+74 x^3+109 x^2-80 x-127\right )+130 x+5\right ) \log \left (2 x^2\right )\right )}{\left (2 e^{x^2}-x\right )^2 x (x+5)^5}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\left (2 x^2-1\right ) \left (x^3+10 x^2+25 x-1\right )^2 \log ^2\left (2 x^2\right )}{(x+5)^4 \left (x-2 e^{x^2}\right )^2}+\frac {2 \left (x^3+10 x^2+25 x-1\right ) \left (-2 x^4-30 x^3-150 x^2-80 x^2 \log \left (2 x^2\right )-127 x \log \left (2 x^2\right )+x^6 \log \left (2 x^2\right )+15 x^5 \log \left (2 x^2\right )+74 x^4 \log \left (2 x^2\right )+109 x^3 \log \left (2 x^2\right )-248 x+10\right ) \log \left (2 x^2\right )}{\left (2 e^{x^2}-x\right ) x (x+5)^5}\right )dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \left (\frac {\left (2 x^2-1\right ) \left (x^3+10 x^2+25 x-1\right )^2 \log ^2\left (2 x^2\right )}{(x+5)^4 \left (x-2 e^{x^2}\right )^2}+\frac {2 \left (x^3+10 x^2+25 x-1\right ) \left (-2 x^4-30 x^3-150 x^2-80 x^2 \log \left (2 x^2\right )-127 x \log \left (2 x^2\right )+x^6 \log \left (2 x^2\right )+15 x^5 \log \left (2 x^2\right )+74 x^4 \log \left (2 x^2\right )+109 x^3 \log \left (2 x^2\right )-248 x+10\right ) \log \left (2 x^2\right )}{\left (2 e^{x^2}-x\right ) x (x+5)^5}\right )dx\)

Input:

Int[((20*x - 996*x^2 + 11900*x^3 + 12380*x^4 + 4992*x^5 + 1000*x^6 + 100*x 
^7 + 4*x^8 + E^x^2*(-40 + 1992*x - 23800*x^2 - 24760*x^3 - 9984*x^4 - 2000 
*x^5 - 200*x^6 - 8*x^7))*Log[2*x^2] + (-5*x - 5*x^2 + 3225*x^3 + 3165*x^4 
+ 1254*x^5 + 250*x^6 + 25*x^7 + x^8 + E^x^2*(508*x - 12380*x^2 - 13516*x^3 
 + 6896*x^4 + 11380*x^5 + 4892*x^6 + 996*x^7 + 100*x^8 + 4*x^9))*Log[2*x^2 
]^2)/(3125*x^3 + 3125*x^4 + 1250*x^5 + 250*x^6 + 25*x^7 + x^8 + E^(2*x^2)* 
(12500*x + 12500*x^2 + 5000*x^3 + 1000*x^4 + 100*x^5 + 4*x^6) + E^x^2*(-12 
500*x^2 - 12500*x^3 - 5000*x^4 - 1000*x^5 - 100*x^6 - 4*x^7)),x]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.21 (sec) , antiderivative size = 1812, normalized size of antiderivative = 58.45

\[\text {Expression too large to display}\]

Input:

int((((4*x^9+100*x^8+996*x^7+4892*x^6+11380*x^5+6896*x^4-13516*x^3-12380*x 
^2+508*x)*exp(x^2)+x^8+25*x^7+250*x^6+1254*x^5+3165*x^4+3225*x^3-5*x^2-5*x 
)*ln(2*x^2)^2+((-8*x^7-200*x^6-2000*x^5-9984*x^4-24760*x^3-23800*x^2+1992* 
x-40)*exp(x^2)+4*x^8+100*x^7+1000*x^6+4992*x^5+12380*x^4+11900*x^3-996*x^2 
+20*x)*ln(2*x^2))/((4*x^6+100*x^5+1000*x^4+5000*x^3+12500*x^2+12500*x)*exp 
(x^2)^2+(-4*x^7-100*x^6-1000*x^5-5000*x^4-12500*x^3-12500*x^2)*exp(x^2)+x^ 
8+25*x^7+250*x^6+1250*x^5+3125*x^4+3125*x^3),x)
 

Output:

4*(x^6+20*x^5+150*x^4+498*x^3+605*x^2-50*x+1)/(x^4+20*x^3+150*x^2+500*x+62 
5)/(x-2*exp(x^2))*ln(x)^2+2*(-I*Pi*csgn(I*x)^2*csgn(I*x^2)+2*I*Pi*csgn(I*x 
)*csgn(I*x^2)^2+2*x^6*ln(2)+300*x^4*ln(2)+1210*x^2*ln(2)-100*x*ln(2)+40*x^ 
5*ln(2)+996*x^3*ln(2)+2*ln(2)-I*Pi*csgn(I*x^2)^3+2*I*Pi*x^6*csgn(I*x)*csgn 
(I*x^2)^2-20*I*Pi*x^5*csgn(I*x)^2*csgn(I*x^2)+40*I*Pi*x^5*csgn(I*x)*csgn(I 
*x^2)^2-150*I*Pi*x^4*csgn(I*x)^2*csgn(I*x^2)+300*I*Pi*x^4*csgn(I*x)*csgn(I 
*x^2)^2-498*I*Pi*x^3*csgn(I*x)^2*csgn(I*x^2)+996*I*Pi*x^3*csgn(I*x)*csgn(I 
*x^2)^2-605*I*Pi*x^2*csgn(I*x)^2*csgn(I*x^2)+1210*I*Pi*x^2*csgn(I*x)*csgn( 
I*x^2)^2+50*I*Pi*x*csgn(I*x)^2*csgn(I*x^2)-100*I*Pi*x*csgn(I*x)*csgn(I*x^2 
)^2-I*Pi*x^6*csgn(I*x)^2*csgn(I*x^2)-I*Pi*x^6*csgn(I*x^2)^3-20*I*Pi*x^5*cs 
gn(I*x^2)^3-150*I*Pi*x^4*csgn(I*x^2)^3-498*I*Pi*x^3*csgn(I*x^2)^3-605*I*Pi 
*x^2*csgn(I*x^2)^3+50*I*Pi*x*csgn(I*x^2)^3)/(5+x)/(x^3+15*x^2+75*x+125)/(x 
-2*exp(x^2))*ln(x)+1/4*(-605*x^2*Pi^2*csgn(I*x)^4*csgn(I*x^2)^2+2420*x^2*P 
i^2*csgn(I*x)^3*csgn(I*x^2)^3-3630*x^2*Pi^2*csgn(I*x)^2*csgn(I*x^2)^4+2420 
*x^2*Pi^2*csgn(I*x)*csgn(I*x^2)^5+4*x^6*ln(2)^2+600*x^4*ln(2)^2+1992*x^3*l 
n(2)^2+2420*x^2*ln(2)^2+80*x^5*ln(2)^2-200*x*ln(2)^2+4*ln(2)^2-120*Pi^2*x^ 
5*csgn(I*x^2)^4*csgn(I*x)^2+80*Pi^2*x^5*csgn(I*x^2)^5*csgn(I*x)-900*Pi^2*x 
^4*csgn(I*x^2)^4*csgn(I*x)^2+600*Pi^2*x^4*csgn(I*x^2)^5*csgn(I*x)-2988*Pi^ 
2*x^3*csgn(I*x^2)^4*csgn(I*x)^2+1992*Pi^2*x^3*csgn(I*x^2)^5*csgn(I*x)-Pi^2 
*csgn(I*x)^4*csgn(I*x^2)^2+4*Pi^2*csgn(I*x)^3*csgn(I*x^2)^3-6*Pi^2*csgn...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (30) = 60\).

Time = 0.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.74 \[ \int \frac {\left (20 x-996 x^2+11900 x^3+12380 x^4+4992 x^5+1000 x^6+100 x^7+4 x^8+e^{x^2} \left (-40+1992 x-23800 x^2-24760 x^3-9984 x^4-2000 x^5-200 x^6-8 x^7\right )\right ) \log \left (2 x^2\right )+\left (-5 x-5 x^2+3225 x^3+3165 x^4+1254 x^5+250 x^6+25 x^7+x^8+e^{x^2} \left (508 x-12380 x^2-13516 x^3+6896 x^4+11380 x^5+4892 x^6+996 x^7+100 x^8+4 x^9\right )\right ) \log ^2\left (2 x^2\right )}{3125 x^3+3125 x^4+1250 x^5+250 x^6+25 x^7+x^8+e^{2 x^2} \left (12500 x+12500 x^2+5000 x^3+1000 x^4+100 x^5+4 x^6\right )+e^{x^2} \left (-12500 x^2-12500 x^3-5000 x^4-1000 x^5-100 x^6-4 x^7\right )} \, dx=\frac {{\left (x^{6} + 20 \, x^{5} + 150 \, x^{4} + 498 \, x^{3} + 605 \, x^{2} - 50 \, x + 1\right )} \log \left (2 \, x^{2}\right )^{2}}{x^{5} + 20 \, x^{4} + 150 \, x^{3} + 500 \, x^{2} - 2 \, {\left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + 625\right )} e^{\left (x^{2}\right )} + 625 \, x} \] Input:

integrate((((4*x^9+100*x^8+996*x^7+4892*x^6+11380*x^5+6896*x^4-13516*x^3-1 
2380*x^2+508*x)*exp(x^2)+x^8+25*x^7+250*x^6+1254*x^5+3165*x^4+3225*x^3-5*x 
^2-5*x)*log(2*x^2)^2+((-8*x^7-200*x^6-2000*x^5-9984*x^4-24760*x^3-23800*x^ 
2+1992*x-40)*exp(x^2)+4*x^8+100*x^7+1000*x^6+4992*x^5+12380*x^4+11900*x^3- 
996*x^2+20*x)*log(2*x^2))/((4*x^6+100*x^5+1000*x^4+5000*x^3+12500*x^2+1250 
0*x)*exp(x^2)^2+(-4*x^7-100*x^6-1000*x^5-5000*x^4-12500*x^3-12500*x^2)*exp 
(x^2)+x^8+25*x^7+250*x^6+1250*x^5+3125*x^4+3125*x^3),x, algorithm="fricas" 
)
 

Output:

(x^6 + 20*x^5 + 150*x^4 + 498*x^3 + 605*x^2 - 50*x + 1)*log(2*x^2)^2/(x^5 
+ 20*x^4 + 150*x^3 + 500*x^2 - 2*(x^4 + 20*x^3 + 150*x^2 + 500*x + 625)*e^ 
(x^2) + 625*x)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (29) = 58\).

Time = 0.31 (sec) , antiderivative size = 133, normalized size of antiderivative = 4.29 \[ \int \frac {\left (20 x-996 x^2+11900 x^3+12380 x^4+4992 x^5+1000 x^6+100 x^7+4 x^8+e^{x^2} \left (-40+1992 x-23800 x^2-24760 x^3-9984 x^4-2000 x^5-200 x^6-8 x^7\right )\right ) \log \left (2 x^2\right )+\left (-5 x-5 x^2+3225 x^3+3165 x^4+1254 x^5+250 x^6+25 x^7+x^8+e^{x^2} \left (508 x-12380 x^2-13516 x^3+6896 x^4+11380 x^5+4892 x^6+996 x^7+100 x^8+4 x^9\right )\right ) \log ^2\left (2 x^2\right )}{3125 x^3+3125 x^4+1250 x^5+250 x^6+25 x^7+x^8+e^{2 x^2} \left (12500 x+12500 x^2+5000 x^3+1000 x^4+100 x^5+4 x^6\right )+e^{x^2} \left (-12500 x^2-12500 x^3-5000 x^4-1000 x^5-100 x^6-4 x^7\right )} \, dx=\frac {- x^{6} \log {\left (2 x^{2} \right )}^{2} - 20 x^{5} \log {\left (2 x^{2} \right )}^{2} - 150 x^{4} \log {\left (2 x^{2} \right )}^{2} - 498 x^{3} \log {\left (2 x^{2} \right )}^{2} - 605 x^{2} \log {\left (2 x^{2} \right )}^{2} + 50 x \log {\left (2 x^{2} \right )}^{2} - \log {\left (2 x^{2} \right )}^{2}}{- x^{5} - 20 x^{4} - 150 x^{3} - 500 x^{2} - 625 x + \left (2 x^{4} + 40 x^{3} + 300 x^{2} + 1000 x + 1250\right ) e^{x^{2}}} \] Input:

integrate((((4*x**9+100*x**8+996*x**7+4892*x**6+11380*x**5+6896*x**4-13516 
*x**3-12380*x**2+508*x)*exp(x**2)+x**8+25*x**7+250*x**6+1254*x**5+3165*x** 
4+3225*x**3-5*x**2-5*x)*ln(2*x**2)**2+((-8*x**7-200*x**6-2000*x**5-9984*x* 
*4-24760*x**3-23800*x**2+1992*x-40)*exp(x**2)+4*x**8+100*x**7+1000*x**6+49 
92*x**5+12380*x**4+11900*x**3-996*x**2+20*x)*ln(2*x**2))/((4*x**6+100*x**5 
+1000*x**4+5000*x**3+12500*x**2+12500*x)*exp(x**2)**2+(-4*x**7-100*x**6-10 
00*x**5-5000*x**4-12500*x**3-12500*x**2)*exp(x**2)+x**8+25*x**7+250*x**6+1 
250*x**5+3125*x**4+3125*x**3),x)
 

Output:

(-x**6*log(2*x**2)**2 - 20*x**5*log(2*x**2)**2 - 150*x**4*log(2*x**2)**2 - 
 498*x**3*log(2*x**2)**2 - 605*x**2*log(2*x**2)**2 + 50*x*log(2*x**2)**2 - 
 log(2*x**2)**2)/(-x**5 - 20*x**4 - 150*x**3 - 500*x**2 - 625*x + (2*x**4 
+ 40*x**3 + 300*x**2 + 1000*x + 1250)*exp(x**2))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (30) = 60\).

Time = 0.20 (sec) , antiderivative size = 185, normalized size of antiderivative = 5.97 \[ \int \frac {\left (20 x-996 x^2+11900 x^3+12380 x^4+4992 x^5+1000 x^6+100 x^7+4 x^8+e^{x^2} \left (-40+1992 x-23800 x^2-24760 x^3-9984 x^4-2000 x^5-200 x^6-8 x^7\right )\right ) \log \left (2 x^2\right )+\left (-5 x-5 x^2+3225 x^3+3165 x^4+1254 x^5+250 x^6+25 x^7+x^8+e^{x^2} \left (508 x-12380 x^2-13516 x^3+6896 x^4+11380 x^5+4892 x^6+996 x^7+100 x^8+4 x^9\right )\right ) \log ^2\left (2 x^2\right )}{3125 x^3+3125 x^4+1250 x^5+250 x^6+25 x^7+x^8+e^{2 x^2} \left (12500 x+12500 x^2+5000 x^3+1000 x^4+100 x^5+4 x^6\right )+e^{x^2} \left (-12500 x^2-12500 x^3-5000 x^4-1000 x^5-100 x^6-4 x^7\right )} \, dx=\frac {x^{6} \log \left (2\right )^{2} + 20 \, x^{5} \log \left (2\right )^{2} + 150 \, x^{4} \log \left (2\right )^{2} + 498 \, x^{3} \log \left (2\right )^{2} + 605 \, x^{2} \log \left (2\right )^{2} - 50 \, x \log \left (2\right )^{2} + 4 \, {\left (x^{6} + 20 \, x^{5} + 150 \, x^{4} + 498 \, x^{3} + 605 \, x^{2} - 50 \, x + 1\right )} \log \left (x\right )^{2} + \log \left (2\right )^{2} + 4 \, {\left (x^{6} \log \left (2\right ) + 20 \, x^{5} \log \left (2\right ) + 150 \, x^{4} \log \left (2\right ) + 498 \, x^{3} \log \left (2\right ) + 605 \, x^{2} \log \left (2\right ) - 50 \, x \log \left (2\right ) + \log \left (2\right )\right )} \log \left (x\right )}{x^{5} + 20 \, x^{4} + 150 \, x^{3} + 500 \, x^{2} - 2 \, {\left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + 625\right )} e^{\left (x^{2}\right )} + 625 \, x} \] Input:

integrate((((4*x^9+100*x^8+996*x^7+4892*x^6+11380*x^5+6896*x^4-13516*x^3-1 
2380*x^2+508*x)*exp(x^2)+x^8+25*x^7+250*x^6+1254*x^5+3165*x^4+3225*x^3-5*x 
^2-5*x)*log(2*x^2)^2+((-8*x^7-200*x^6-2000*x^5-9984*x^4-24760*x^3-23800*x^ 
2+1992*x-40)*exp(x^2)+4*x^8+100*x^7+1000*x^6+4992*x^5+12380*x^4+11900*x^3- 
996*x^2+20*x)*log(2*x^2))/((4*x^6+100*x^5+1000*x^4+5000*x^3+12500*x^2+1250 
0*x)*exp(x^2)^2+(-4*x^7-100*x^6-1000*x^5-5000*x^4-12500*x^3-12500*x^2)*exp 
(x^2)+x^8+25*x^7+250*x^6+1250*x^5+3125*x^4+3125*x^3),x, algorithm="maxima" 
)
 

Output:

(x^6*log(2)^2 + 20*x^5*log(2)^2 + 150*x^4*log(2)^2 + 498*x^3*log(2)^2 + 60 
5*x^2*log(2)^2 - 50*x*log(2)^2 + 4*(x^6 + 20*x^5 + 150*x^4 + 498*x^3 + 605 
*x^2 - 50*x + 1)*log(x)^2 + log(2)^2 + 4*(x^6*log(2) + 20*x^5*log(2) + 150 
*x^4*log(2) + 498*x^3*log(2) + 605*x^2*log(2) - 50*x*log(2) + log(2))*log( 
x))/(x^5 + 20*x^4 + 150*x^3 + 500*x^2 - 2*(x^4 + 20*x^3 + 150*x^2 + 500*x 
+ 625)*e^(x^2) + 625*x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (30) = 60\).

Time = 0.26 (sec) , antiderivative size = 149, normalized size of antiderivative = 4.81 \[ \int \frac {\left (20 x-996 x^2+11900 x^3+12380 x^4+4992 x^5+1000 x^6+100 x^7+4 x^8+e^{x^2} \left (-40+1992 x-23800 x^2-24760 x^3-9984 x^4-2000 x^5-200 x^6-8 x^7\right )\right ) \log \left (2 x^2\right )+\left (-5 x-5 x^2+3225 x^3+3165 x^4+1254 x^5+250 x^6+25 x^7+x^8+e^{x^2} \left (508 x-12380 x^2-13516 x^3+6896 x^4+11380 x^5+4892 x^6+996 x^7+100 x^8+4 x^9\right )\right ) \log ^2\left (2 x^2\right )}{3125 x^3+3125 x^4+1250 x^5+250 x^6+25 x^7+x^8+e^{2 x^2} \left (12500 x+12500 x^2+5000 x^3+1000 x^4+100 x^5+4 x^6\right )+e^{x^2} \left (-12500 x^2-12500 x^3-5000 x^4-1000 x^5-100 x^6-4 x^7\right )} \, dx=\frac {x^{6} \log \left (2 \, x^{2}\right )^{2} + 20 \, x^{5} \log \left (2 \, x^{2}\right )^{2} + 150 \, x^{4} \log \left (2 \, x^{2}\right )^{2} + 498 \, x^{3} \log \left (2 \, x^{2}\right )^{2} + 605 \, x^{2} \log \left (2 \, x^{2}\right )^{2} - 50 \, x \log \left (2 \, x^{2}\right )^{2} + \log \left (2 \, x^{2}\right )^{2}}{x^{5} - 2 \, x^{4} e^{\left (x^{2}\right )} + 20 \, x^{4} - 40 \, x^{3} e^{\left (x^{2}\right )} + 150 \, x^{3} - 300 \, x^{2} e^{\left (x^{2}\right )} + 500 \, x^{2} - 1000 \, x e^{\left (x^{2}\right )} + 625 \, x - 1250 \, e^{\left (x^{2}\right )}} \] Input:

integrate((((4*x^9+100*x^8+996*x^7+4892*x^6+11380*x^5+6896*x^4-13516*x^3-1 
2380*x^2+508*x)*exp(x^2)+x^8+25*x^7+250*x^6+1254*x^5+3165*x^4+3225*x^3-5*x 
^2-5*x)*log(2*x^2)^2+((-8*x^7-200*x^6-2000*x^5-9984*x^4-24760*x^3-23800*x^ 
2+1992*x-40)*exp(x^2)+4*x^8+100*x^7+1000*x^6+4992*x^5+12380*x^4+11900*x^3- 
996*x^2+20*x)*log(2*x^2))/((4*x^6+100*x^5+1000*x^4+5000*x^3+12500*x^2+1250 
0*x)*exp(x^2)^2+(-4*x^7-100*x^6-1000*x^5-5000*x^4-12500*x^3-12500*x^2)*exp 
(x^2)+x^8+25*x^7+250*x^6+1250*x^5+3125*x^4+3125*x^3),x, algorithm="giac")
 

Output:

(x^6*log(2*x^2)^2 + 20*x^5*log(2*x^2)^2 + 150*x^4*log(2*x^2)^2 + 498*x^3*l 
og(2*x^2)^2 + 605*x^2*log(2*x^2)^2 - 50*x*log(2*x^2)^2 + log(2*x^2)^2)/(x^ 
5 - 2*x^4*e^(x^2) + 20*x^4 - 40*x^3*e^(x^2) + 150*x^3 - 300*x^2*e^(x^2) + 
500*x^2 - 1000*x*e^(x^2) + 625*x - 1250*e^(x^2))
 

Mupad [B] (verification not implemented)

Time = 3.48 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.26 \[ \int \frac {\left (20 x-996 x^2+11900 x^3+12380 x^4+4992 x^5+1000 x^6+100 x^7+4 x^8+e^{x^2} \left (-40+1992 x-23800 x^2-24760 x^3-9984 x^4-2000 x^5-200 x^6-8 x^7\right )\right ) \log \left (2 x^2\right )+\left (-5 x-5 x^2+3225 x^3+3165 x^4+1254 x^5+250 x^6+25 x^7+x^8+e^{x^2} \left (508 x-12380 x^2-13516 x^3+6896 x^4+11380 x^5+4892 x^6+996 x^7+100 x^8+4 x^9\right )\right ) \log ^2\left (2 x^2\right )}{3125 x^3+3125 x^4+1250 x^5+250 x^6+25 x^7+x^8+e^{2 x^2} \left (12500 x+12500 x^2+5000 x^3+1000 x^4+100 x^5+4 x^6\right )+e^{x^2} \left (-12500 x^2-12500 x^3-5000 x^4-1000 x^5-100 x^6-4 x^7\right )} \, dx=\frac {{\ln \left (2\,x^2\right )}^2\,{\left (x^3+10\,x^2+25\,x-1\right )}^2}{\left (x-2\,{\mathrm {e}}^{x^2}\right )\,{\left (x+5\right )}^4} \] Input:

int((log(2*x^2)^2*(exp(x^2)*(508*x - 12380*x^2 - 13516*x^3 + 6896*x^4 + 11 
380*x^5 + 4892*x^6 + 996*x^7 + 100*x^8 + 4*x^9) - 5*x - 5*x^2 + 3225*x^3 + 
 3165*x^4 + 1254*x^5 + 250*x^6 + 25*x^7 + x^8) + log(2*x^2)*(20*x - exp(x^ 
2)*(23800*x^2 - 1992*x + 24760*x^3 + 9984*x^4 + 2000*x^5 + 200*x^6 + 8*x^7 
 + 40) - 996*x^2 + 11900*x^3 + 12380*x^4 + 4992*x^5 + 1000*x^6 + 100*x^7 + 
 4*x^8))/(exp(2*x^2)*(12500*x + 12500*x^2 + 5000*x^3 + 1000*x^4 + 100*x^5 
+ 4*x^6) - exp(x^2)*(12500*x^2 + 12500*x^3 + 5000*x^4 + 1000*x^5 + 100*x^6 
 + 4*x^7) + 3125*x^3 + 3125*x^4 + 1250*x^5 + 250*x^6 + 25*x^7 + x^8),x)
 

Output:

(log(2*x^2)^2*(25*x + 10*x^2 + x^3 - 1)^2)/((x - 2*exp(x^2))*(x + 5)^4)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 110, normalized size of antiderivative = 3.55 \[ \int \frac {\left (20 x-996 x^2+11900 x^3+12380 x^4+4992 x^5+1000 x^6+100 x^7+4 x^8+e^{x^2} \left (-40+1992 x-23800 x^2-24760 x^3-9984 x^4-2000 x^5-200 x^6-8 x^7\right )\right ) \log \left (2 x^2\right )+\left (-5 x-5 x^2+3225 x^3+3165 x^4+1254 x^5+250 x^6+25 x^7+x^8+e^{x^2} \left (508 x-12380 x^2-13516 x^3+6896 x^4+11380 x^5+4892 x^6+996 x^7+100 x^8+4 x^9\right )\right ) \log ^2\left (2 x^2\right )}{3125 x^3+3125 x^4+1250 x^5+250 x^6+25 x^7+x^8+e^{2 x^2} \left (12500 x+12500 x^2+5000 x^3+1000 x^4+100 x^5+4 x^6\right )+e^{x^2} \left (-12500 x^2-12500 x^3-5000 x^4-1000 x^5-100 x^6-4 x^7\right )} \, dx=\frac {\mathrm {log}\left (2 x^{2}\right )^{2} \left (-x^{6}-20 x^{5}-150 x^{4}-498 x^{3}-605 x^{2}+50 x -1\right )}{2 e^{x^{2}} x^{4}+40 e^{x^{2}} x^{3}+300 e^{x^{2}} x^{2}+1000 e^{x^{2}} x +1250 e^{x^{2}}-x^{5}-20 x^{4}-150 x^{3}-500 x^{2}-625 x} \] Input:

int((((4*x^9+100*x^8+996*x^7+4892*x^6+11380*x^5+6896*x^4-13516*x^3-12380*x 
^2+508*x)*exp(x^2)+x^8+25*x^7+250*x^6+1254*x^5+3165*x^4+3225*x^3-5*x^2-5*x 
)*log(2*x^2)^2+((-8*x^7-200*x^6-2000*x^5-9984*x^4-24760*x^3-23800*x^2+1992 
*x-40)*exp(x^2)+4*x^8+100*x^7+1000*x^6+4992*x^5+12380*x^4+11900*x^3-996*x^ 
2+20*x)*log(2*x^2))/((4*x^6+100*x^5+1000*x^4+5000*x^3+12500*x^2+12500*x)*e 
xp(x^2)^2+(-4*x^7-100*x^6-1000*x^5-5000*x^4-12500*x^3-12500*x^2)*exp(x^2)+ 
x^8+25*x^7+250*x^6+1250*x^5+3125*x^4+3125*x^3),x)
 

Output:

(log(2*x**2)**2*( - x**6 - 20*x**5 - 150*x**4 - 498*x**3 - 605*x**2 + 50*x 
 - 1))/(2*e**(x**2)*x**4 + 40*e**(x**2)*x**3 + 300*e**(x**2)*x**2 + 1000*e 
**(x**2)*x + 1250*e**(x**2) - x**5 - 20*x**4 - 150*x**3 - 500*x**2 - 625*x 
)