\(\int \frac {3125-9395 x+1867 x^2+8632 x^3-6351 x^4+566 x^5+832 x^6-304 x^7+32 x^8+(6250-3125 x-5625 x^2+4750 x^3-200 x^4-864 x^5+304 x^6-32 x^7) \log (2+x)+(6250 x-9385 x^2+3749 x^3+1002 x^4-1200 x^5+336 x^6-32 x^7+(-6250+9375 x-3750 x^2-1000 x^3+1200 x^4-336 x^5+32 x^6) \log (2+x)) \log (\frac {625 x-1001 x^2+600 x^3-160 x^4+16 x^5+(-625+1000 x-600 x^2+160 x^3-16 x^4) \log (2+x)}{625-1000 x+600 x^2-160 x^3+16 x^4})}{-6250 x^2+9385 x^3-3749 x^4-1002 x^5+1200 x^6-336 x^7+32 x^8+(6250 x-9375 x^2+3750 x^3+1000 x^4-1200 x^5+336 x^6-32 x^7) \log (2+x)+(6250 x-9385 x^2+3749 x^3+1002 x^4-1200 x^5+336 x^6-32 x^7+(-6250+9375 x-3750 x^2-1000 x^3+1200 x^4-336 x^5+32 x^6) \log (2+x)) \log (\frac {625 x-1001 x^2+600 x^3-160 x^4+16 x^5+(-625+1000 x-600 x^2+160 x^3-16 x^4) \log (2+x)}{625-1000 x+600 x^2-160 x^3+16 x^4})} \, dx\) [1202]

Optimal result

Integrand size = 444, antiderivative size = 29 \[ \int \frac {3125-9395 x+1867 x^2+8632 x^3-6351 x^4+566 x^5+832 x^6-304 x^7+32 x^8+\left (6250-3125 x-5625 x^2+4750 x^3-200 x^4-864 x^5+304 x^6-32 x^7\right ) \log (2+x)+\left (6250 x-9385 x^2+3749 x^3+1002 x^4-1200 x^5+336 x^6-32 x^7+\left (-6250+9375 x-3750 x^2-1000 x^3+1200 x^4-336 x^5+32 x^6\right ) \log (2+x)\right ) \log \left (\frac {625 x-1001 x^2+600 x^3-160 x^4+16 x^5+\left (-625+1000 x-600 x^2+160 x^3-16 x^4\right ) \log (2+x)}{625-1000 x+600 x^2-160 x^3+16 x^4}\right )}{-6250 x^2+9385 x^3-3749 x^4-1002 x^5+1200 x^6-336 x^7+32 x^8+\left (6250 x-9375 x^2+3750 x^3+1000 x^4-1200 x^5+336 x^6-32 x^7\right ) \log (2+x)+\left (6250 x-9385 x^2+3749 x^3+1002 x^4-1200 x^5+336 x^6-32 x^7+\left (-6250+9375 x-3750 x^2-1000 x^3+1200 x^4-336 x^5+32 x^6\right ) \log (2+x)\right ) \log \left (\frac {625 x-1001 x^2+600 x^3-160 x^4+16 x^5+\left (-625+1000 x-600 x^2+160 x^3-16 x^4\right ) \log (2+x)}{625-1000 x+600 x^2-160 x^3+16 x^4}\right )} \, dx=1+x+\log \left (-x+\log \left (x-\frac {x^2}{(5-2 x)^4}-\log (2+x)\right )\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.52 \[ \int \frac {3125-9395 x+1867 x^2+8632 x^3-6351 x^4+566 x^5+832 x^6-304 x^7+32 x^8+\left (6250-3125 x-5625 x^2+4750 x^3-200 x^4-864 x^5+304 x^6-32 x^7\right ) \log (2+x)+\left (6250 x-9385 x^2+3749 x^3+1002 x^4-1200 x^5+336 x^6-32 x^7+\left (-6250+9375 x-3750 x^2-1000 x^3+1200 x^4-336 x^5+32 x^6\right ) \log (2+x)\right ) \log \left (\frac {625 x-1001 x^2+600 x^3-160 x^4+16 x^5+\left (-625+1000 x-600 x^2+160 x^3-16 x^4\right ) \log (2+x)}{625-1000 x+600 x^2-160 x^3+16 x^4}\right )}{-6250 x^2+9385 x^3-3749 x^4-1002 x^5+1200 x^6-336 x^7+32 x^8+\left (6250 x-9375 x^2+3750 x^3+1000 x^4-1200 x^5+336 x^6-32 x^7\right ) \log (2+x)+\left (6250 x-9385 x^2+3749 x^3+1002 x^4-1200 x^5+336 x^6-32 x^7+\left (-6250+9375 x-3750 x^2-1000 x^3+1200 x^4-336 x^5+32 x^6\right ) \log (2+x)\right ) \log \left (\frac {625 x-1001 x^2+600 x^3-160 x^4+16 x^5+\left (-625+1000 x-600 x^2+160 x^3-16 x^4\right ) \log (2+x)}{625-1000 x+600 x^2-160 x^3+16 x^4}\right )} \, dx=x+\log \left (x-\log \left (\frac {x \left (625-1001 x+600 x^2-160 x^3+16 x^4\right )}{(5-2 x)^4}-\log (2+x)\right )\right ) \] Input:

Integrate[(3125 - 9395*x + 1867*x^2 + 8632*x^3 - 6351*x^4 + 566*x^5 + 832* 
x^6 - 304*x^7 + 32*x^8 + (6250 - 3125*x - 5625*x^2 + 4750*x^3 - 200*x^4 - 
864*x^5 + 304*x^6 - 32*x^7)*Log[2 + x] + (6250*x - 9385*x^2 + 3749*x^3 + 1 
002*x^4 - 1200*x^5 + 336*x^6 - 32*x^7 + (-6250 + 9375*x - 3750*x^2 - 1000* 
x^3 + 1200*x^4 - 336*x^5 + 32*x^6)*Log[2 + x])*Log[(625*x - 1001*x^2 + 600 
*x^3 - 160*x^4 + 16*x^5 + (-625 + 1000*x - 600*x^2 + 160*x^3 - 16*x^4)*Log 
[2 + x])/(625 - 1000*x + 600*x^2 - 160*x^3 + 16*x^4)])/(-6250*x^2 + 9385*x 
^3 - 3749*x^4 - 1002*x^5 + 1200*x^6 - 336*x^7 + 32*x^8 + (6250*x - 9375*x^ 
2 + 3750*x^3 + 1000*x^4 - 1200*x^5 + 336*x^6 - 32*x^7)*Log[2 + x] + (6250* 
x - 9385*x^2 + 3749*x^3 + 1002*x^4 - 1200*x^5 + 336*x^6 - 32*x^7 + (-6250 
+ 9375*x - 3750*x^2 - 1000*x^3 + 1200*x^4 - 336*x^5 + 32*x^6)*Log[2 + x])* 
Log[(625*x - 1001*x^2 + 600*x^3 - 160*x^4 + 16*x^5 + (-625 + 1000*x - 600* 
x^2 + 160*x^3 - 16*x^4)*Log[2 + x])/(625 - 1000*x + 600*x^2 - 160*x^3 + 16 
*x^4)]),x]
 

Output:

x + Log[x - Log[(x*(625 - 1001*x + 600*x^2 - 160*x^3 + 16*x^4))/(5 - 2*x)^ 
4 - Log[2 + x]]]
 

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.

\(\displaystyle \int \frac {32 x^8-304 x^7+832 x^6+566 x^5-6351 x^4+8632 x^3+1867 x^2+\left (-32 x^7+304 x^6-864 x^5-200 x^4+4750 x^3-5625 x^2-3125 x+6250\right ) \log (x+2)+\left (-32 x^7+336 x^6-1200 x^5+1002 x^4+3749 x^3-9385 x^2+\left (32 x^6-336 x^5+1200 x^4-1000 x^3-3750 x^2+9375 x-6250\right ) \log (x+2)+6250 x\right ) \log \left (\frac {16 x^5-160 x^4+600 x^3-1001 x^2+\left (-16 x^4+160 x^3-600 x^2+1000 x-625\right ) \log (x+2)+625 x}{16 x^4-160 x^3+600 x^2-1000 x+625}\right )-9395 x+3125}{32 x^8-336 x^7+1200 x^6-1002 x^5-3749 x^4+9385 x^3-6250 x^2+\left (-32 x^7+336 x^6-1200 x^5+1000 x^4+3750 x^3-9375 x^2+6250 x\right ) \log (x+2)+\left (-32 x^7+336 x^6-1200 x^5+1002 x^4+3749 x^3-9385 x^2+\left (32 x^6-336 x^5+1200 x^4-1000 x^3-3750 x^2+9375 x-6250\right ) \log (x+2)+6250 x\right ) \log \left (\frac {16 x^5-160 x^4+600 x^3-1001 x^2+\left (-16 x^4+160 x^3-600 x^2+1000 x-625\right ) \log (x+2)+625 x}{16 x^4-160 x^3+600 x^2-1000 x+625}\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-32 x^8+304 x^7-832 x^6-566 x^5+6351 x^4-8632 x^3-1867 x^2+(2 x-5)^5 \left (x^2+3 x+2\right ) \log (x+2)+\left (2 x^2-x-10\right ) \left (x \left (16 x^4-160 x^3+600 x^2-1001 x+625\right )-(5-2 x)^4 \log (x+2)\right ) \log \left (\frac {x \left (16 x^4-160 x^3+600 x^2-1001 x+625\right )}{(5-2 x)^4}-\log (x+2)\right )+9395 x-3125}{\left (-2 x^2+x+10\right ) \left (x \left (16 x^4-160 x^3+600 x^2-1001 x+625\right )-(5-2 x)^4 \log (x+2)\right ) \left (x-\log \left (\frac {x \left (16 x^4-160 x^3+600 x^2-1001 x+625\right )}{(5-2 x)^4}-\log (x+2)\right )\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle \int \left (\frac {32 x^7-368 x^6-32 x^6 \log (x+2)+1568 x^5+336 x^5 \log (x+2)-2602 x^4-1200 x^4 \log (x+2)-753 x^3+1000 x^3 \log (x+2)+8117 x^2+3750 x^2 \log (x+2)-9395 x-9375 x \log (x+2)+6250 \log (x+2)+3125}{(x+2) (2 x-5) \left (16 x^5-160 x^4-16 x^4 \log (x+2)+600 x^3+160 x^3 \log (x+2)-1001 x^2-600 x^2 \log (x+2)+625 x+1000 x \log (x+2)-625 \log (x+2)\right ) \left (x-\log \left (\frac {x \left (16 x^4-160 x^3+600 x^2-1001 x+625\right )}{(5-2 x)^4}-\log (x+2)\right )\right )}+1\right )dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \left (\frac {32 x^7-368 x^6-32 x^6 \log (x+2)+1568 x^5+336 x^5 \log (x+2)-2602 x^4-1200 x^4 \log (x+2)-753 x^3+1000 x^3 \log (x+2)+8117 x^2+3750 x^2 \log (x+2)-9395 x-9375 x \log (x+2)+6250 \log (x+2)+3125}{(x+2) (2 x-5) \left (16 x^5-160 x^4-16 x^4 \log (x+2)+600 x^3+160 x^3 \log (x+2)-1001 x^2-600 x^2 \log (x+2)+625 x+1000 x \log (x+2)-625 \log (x+2)\right ) \left (x-\log \left (\frac {x \left (16 x^4-160 x^3+600 x^2-1001 x+625\right )}{(5-2 x)^4}-\log (x+2)\right )\right )}+1\right )dx\)

Input:

Int[(3125 - 9395*x + 1867*x^2 + 8632*x^3 - 6351*x^4 + 566*x^5 + 832*x^6 - 
304*x^7 + 32*x^8 + (6250 - 3125*x - 5625*x^2 + 4750*x^3 - 200*x^4 - 864*x^ 
5 + 304*x^6 - 32*x^7)*Log[2 + x] + (6250*x - 9385*x^2 + 3749*x^3 + 1002*x^ 
4 - 1200*x^5 + 336*x^6 - 32*x^7 + (-6250 + 9375*x - 3750*x^2 - 1000*x^3 + 
1200*x^4 - 336*x^5 + 32*x^6)*Log[2 + x])*Log[(625*x - 1001*x^2 + 600*x^3 - 
 160*x^4 + 16*x^5 + (-625 + 1000*x - 600*x^2 + 160*x^3 - 16*x^4)*Log[2 + x 
])/(625 - 1000*x + 600*x^2 - 160*x^3 + 16*x^4)])/(-6250*x^2 + 9385*x^3 - 3 
749*x^4 - 1002*x^5 + 1200*x^6 - 336*x^7 + 32*x^8 + (6250*x - 9375*x^2 + 37 
50*x^3 + 1000*x^4 - 1200*x^5 + 336*x^6 - 32*x^7)*Log[2 + x] + (6250*x - 93 
85*x^2 + 3749*x^3 + 1002*x^4 - 1200*x^5 + 336*x^6 - 32*x^7 + (-6250 + 9375 
*x - 3750*x^2 - 1000*x^3 + 1200*x^4 - 336*x^5 + 32*x^6)*Log[2 + x])*Log[(6 
25*x - 1001*x^2 + 600*x^3 - 160*x^4 + 16*x^5 + (-625 + 1000*x - 600*x^2 + 
160*x^3 - 16*x^4)*Log[2 + x])/(625 - 1000*x + 600*x^2 - 160*x^3 + 16*x^4)] 
),x]
 

Output:

$Aborted
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(80\) vs. \(2(29)=58\).

Time = 150.48 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.79

method	result	size
parallelrisch	\(\ln \left (x -\ln \left (\frac {\left (-16 x^{4}+160 x^{3}-600 x^{2}+1000 x -625\right ) \ln \left (2+x \right )+16 x^{5}-160 x^{4}+600 x^{3}-1001 x^{2}+625 x}{16 x^{4}-160 x^{3}+600 x^{2}-1000 x +625}\right )\right )+x\)	\(81\)
risch	\(\text {Expression too large to display}\)	\(727\)
default	\(\text {Expression too large to display}\)	\(838\)

Input:

int((((32*x^6-336*x^5+1200*x^4-1000*x^3-3750*x^2+9375*x-6250)*ln(2+x)-32*x 
^7+336*x^6-1200*x^5+1002*x^4+3749*x^3-9385*x^2+6250*x)*ln(((-16*x^4+160*x^ 
3-600*x^2+1000*x-625)*ln(2+x)+16*x^5-160*x^4+600*x^3-1001*x^2+625*x)/(16*x 
^4-160*x^3+600*x^2-1000*x+625))+(-32*x^7+304*x^6-864*x^5-200*x^4+4750*x^3- 
5625*x^2-3125*x+6250)*ln(2+x)+32*x^8-304*x^7+832*x^6+566*x^5-6351*x^4+8632 
*x^3+1867*x^2-9395*x+3125)/(((32*x^6-336*x^5+1200*x^4-1000*x^3-3750*x^2+93 
75*x-6250)*ln(2+x)-32*x^7+336*x^6-1200*x^5+1002*x^4+3749*x^3-9385*x^2+6250 
*x)*ln(((-16*x^4+160*x^3-600*x^2+1000*x-625)*ln(2+x)+16*x^5-160*x^4+600*x^ 
3-1001*x^2+625*x)/(16*x^4-160*x^3+600*x^2-1000*x+625))+(-32*x^7+336*x^6-12 
00*x^5+1000*x^4+3750*x^3-9375*x^2+6250*x)*ln(2+x)+32*x^8-336*x^7+1200*x^6- 
1002*x^5-3749*x^4+9385*x^3-6250*x^2),x,method=_RETURNVERBOSE)
 

Output:

ln(x-ln(((-16*x^4+160*x^3-600*x^2+1000*x-625)*ln(2+x)+16*x^5-160*x^4+600*x 
^3-1001*x^2+625*x)/(16*x^4-160*x^3+600*x^2-1000*x+625)))+x
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (29) = 58\).

Time = 0.15 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.79 \[ \int \frac {3125-9395 x+1867 x^2+8632 x^3-6351 x^4+566 x^5+832 x^6-304 x^7+32 x^8+\left (6250-3125 x-5625 x^2+4750 x^3-200 x^4-864 x^5+304 x^6-32 x^7\right ) \log (2+x)+\left (6250 x-9385 x^2+3749 x^3+1002 x^4-1200 x^5+336 x^6-32 x^7+\left (-6250+9375 x-3750 x^2-1000 x^3+1200 x^4-336 x^5+32 x^6\right ) \log (2+x)\right ) \log \left (\frac {625 x-1001 x^2+600 x^3-160 x^4+16 x^5+\left (-625+1000 x-600 x^2+160 x^3-16 x^4\right ) \log (2+x)}{625-1000 x+600 x^2-160 x^3+16 x^4}\right )}{-6250 x^2+9385 x^3-3749 x^4-1002 x^5+1200 x^6-336 x^7+32 x^8+\left (6250 x-9375 x^2+3750 x^3+1000 x^4-1200 x^5+336 x^6-32 x^7\right ) \log (2+x)+\left (6250 x-9385 x^2+3749 x^3+1002 x^4-1200 x^5+336 x^6-32 x^7+\left (-6250+9375 x-3750 x^2-1000 x^3+1200 x^4-336 x^5+32 x^6\right ) \log (2+x)\right ) \log \left (\frac {625 x-1001 x^2+600 x^3-160 x^4+16 x^5+\left (-625+1000 x-600 x^2+160 x^3-16 x^4\right ) \log (2+x)}{625-1000 x+600 x^2-160 x^3+16 x^4}\right )} \, dx=x + \log \left (-x + \log \left (\frac {16 \, x^{5} - 160 \, x^{4} + 600 \, x^{3} - 1001 \, x^{2} - {\left (16 \, x^{4} - 160 \, x^{3} + 600 \, x^{2} - 1000 \, x + 625\right )} \log \left (x + 2\right ) + 625 \, x}{16 \, x^{4} - 160 \, x^{3} + 600 \, x^{2} - 1000 \, x + 625}\right )\right ) \] Input:

integrate((((32*x^6-336*x^5+1200*x^4-1000*x^3-3750*x^2+9375*x-6250)*log(2+ 
x)-32*x^7+336*x^6-1200*x^5+1002*x^4+3749*x^3-9385*x^2+6250*x)*log(((-16*x^ 
4+160*x^3-600*x^2+1000*x-625)*log(2+x)+16*x^5-160*x^4+600*x^3-1001*x^2+625 
*x)/(16*x^4-160*x^3+600*x^2-1000*x+625))+(-32*x^7+304*x^6-864*x^5-200*x^4+ 
4750*x^3-5625*x^2-3125*x+6250)*log(2+x)+32*x^8-304*x^7+832*x^6+566*x^5-635 
1*x^4+8632*x^3+1867*x^2-9395*x+3125)/(((32*x^6-336*x^5+1200*x^4-1000*x^3-3 
750*x^2+9375*x-6250)*log(2+x)-32*x^7+336*x^6-1200*x^5+1002*x^4+3749*x^3-93 
85*x^2+6250*x)*log(((-16*x^4+160*x^3-600*x^2+1000*x-625)*log(2+x)+16*x^5-1 
60*x^4+600*x^3-1001*x^2+625*x)/(16*x^4-160*x^3+600*x^2-1000*x+625))+(-32*x 
^7+336*x^6-1200*x^5+1000*x^4+3750*x^3-9375*x^2+6250*x)*log(2+x)+32*x^8-336 
*x^7+1200*x^6-1002*x^5-3749*x^4+9385*x^3-6250*x^2),x, algorithm="fricas")
 

Output:

x + log(-x + log((16*x^5 - 160*x^4 + 600*x^3 - 1001*x^2 - (16*x^4 - 160*x^ 
3 + 600*x^2 - 1000*x + 625)*log(x + 2) + 625*x)/(16*x^4 - 160*x^3 + 600*x^ 
2 - 1000*x + 625)))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (24) = 48\).

Time = 2.18 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.59 \[ \int \frac {3125-9395 x+1867 x^2+8632 x^3-6351 x^4+566 x^5+832 x^6-304 x^7+32 x^8+\left (6250-3125 x-5625 x^2+4750 x^3-200 x^4-864 x^5+304 x^6-32 x^7\right ) \log (2+x)+\left (6250 x-9385 x^2+3749 x^3+1002 x^4-1200 x^5+336 x^6-32 x^7+\left (-6250+9375 x-3750 x^2-1000 x^3+1200 x^4-336 x^5+32 x^6\right ) \log (2+x)\right ) \log \left (\frac {625 x-1001 x^2+600 x^3-160 x^4+16 x^5+\left (-625+1000 x-600 x^2+160 x^3-16 x^4\right ) \log (2+x)}{625-1000 x+600 x^2-160 x^3+16 x^4}\right )}{-6250 x^2+9385 x^3-3749 x^4-1002 x^5+1200 x^6-336 x^7+32 x^8+\left (6250 x-9375 x^2+3750 x^3+1000 x^4-1200 x^5+336 x^6-32 x^7\right ) \log (2+x)+\left (6250 x-9385 x^2+3749 x^3+1002 x^4-1200 x^5+336 x^6-32 x^7+\left (-6250+9375 x-3750 x^2-1000 x^3+1200 x^4-336 x^5+32 x^6\right ) \log (2+x)\right ) \log \left (\frac {625 x-1001 x^2+600 x^3-160 x^4+16 x^5+\left (-625+1000 x-600 x^2+160 x^3-16 x^4\right ) \log (2+x)}{625-1000 x+600 x^2-160 x^3+16 x^4}\right )} \, dx=x + \log {\left (- x + \log {\left (\frac {16 x^{5} - 160 x^{4} + 600 x^{3} - 1001 x^{2} + 625 x + \left (- 16 x^{4} + 160 x^{3} - 600 x^{2} + 1000 x - 625\right ) \log {\left (x + 2 \right )}}{16 x^{4} - 160 x^{3} + 600 x^{2} - 1000 x + 625} \right )} \right )} \] Input:

integrate((((32*x**6-336*x**5+1200*x**4-1000*x**3-3750*x**2+9375*x-6250)*l 
n(2+x)-32*x**7+336*x**6-1200*x**5+1002*x**4+3749*x**3-9385*x**2+6250*x)*ln 
(((-16*x**4+160*x**3-600*x**2+1000*x-625)*ln(2+x)+16*x**5-160*x**4+600*x** 
3-1001*x**2+625*x)/(16*x**4-160*x**3+600*x**2-1000*x+625))+(-32*x**7+304*x 
**6-864*x**5-200*x**4+4750*x**3-5625*x**2-3125*x+6250)*ln(2+x)+32*x**8-304 
*x**7+832*x**6+566*x**5-6351*x**4+8632*x**3+1867*x**2-9395*x+3125)/(((32*x 
**6-336*x**5+1200*x**4-1000*x**3-3750*x**2+9375*x-6250)*ln(2+x)-32*x**7+33 
6*x**6-1200*x**5+1002*x**4+3749*x**3-9385*x**2+6250*x)*ln(((-16*x**4+160*x 
**3-600*x**2+1000*x-625)*ln(2+x)+16*x**5-160*x**4+600*x**3-1001*x**2+625*x 
)/(16*x**4-160*x**3+600*x**2-1000*x+625))+(-32*x**7+336*x**6-1200*x**5+100 
0*x**4+3750*x**3-9375*x**2+6250*x)*ln(2+x)+32*x**8-336*x**7+1200*x**6-1002 
*x**5-3749*x**4+9385*x**3-6250*x**2),x)
 

Output:

x + log(-x + log((16*x**5 - 160*x**4 + 600*x**3 - 1001*x**2 + 625*x + (-16 
*x**4 + 160*x**3 - 600*x**2 + 1000*x - 625)*log(x + 2))/(16*x**4 - 160*x** 
3 + 600*x**2 - 1000*x + 625)))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (29) = 58\).

Time = 0.34 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.62 \[ \int \frac {3125-9395 x+1867 x^2+8632 x^3-6351 x^4+566 x^5+832 x^6-304 x^7+32 x^8+\left (6250-3125 x-5625 x^2+4750 x^3-200 x^4-864 x^5+304 x^6-32 x^7\right ) \log (2+x)+\left (6250 x-9385 x^2+3749 x^3+1002 x^4-1200 x^5+336 x^6-32 x^7+\left (-6250+9375 x-3750 x^2-1000 x^3+1200 x^4-336 x^5+32 x^6\right ) \log (2+x)\right ) \log \left (\frac {625 x-1001 x^2+600 x^3-160 x^4+16 x^5+\left (-625+1000 x-600 x^2+160 x^3-16 x^4\right ) \log (2+x)}{625-1000 x+600 x^2-160 x^3+16 x^4}\right )}{-6250 x^2+9385 x^3-3749 x^4-1002 x^5+1200 x^6-336 x^7+32 x^8+\left (6250 x-9375 x^2+3750 x^3+1000 x^4-1200 x^5+336 x^6-32 x^7\right ) \log (2+x)+\left (6250 x-9385 x^2+3749 x^3+1002 x^4-1200 x^5+336 x^6-32 x^7+\left (-6250+9375 x-3750 x^2-1000 x^3+1200 x^4-336 x^5+32 x^6\right ) \log (2+x)\right ) \log \left (\frac {625 x-1001 x^2+600 x^3-160 x^4+16 x^5+\left (-625+1000 x-600 x^2+160 x^3-16 x^4\right ) \log (2+x)}{625-1000 x+600 x^2-160 x^3+16 x^4}\right )} \, dx=x + \log \left (-x + \log \left (16 \, x^{5} - 16 \, x^{4} {\left (\log \left (x + 2\right ) + 10\right )} + 40 \, x^{3} {\left (4 \, \log \left (x + 2\right ) + 15\right )} - x^{2} {\left (600 \, \log \left (x + 2\right ) + 1001\right )} + 125 \, x {\left (8 \, \log \left (x + 2\right ) + 5\right )} - 625 \, \log \left (x + 2\right )\right ) - 4 \, \log \left (2 \, x - 5\right )\right ) \] Input:

integrate((((32*x^6-336*x^5+1200*x^4-1000*x^3-3750*x^2+9375*x-6250)*log(2+ 
x)-32*x^7+336*x^6-1200*x^5+1002*x^4+3749*x^3-9385*x^2+6250*x)*log(((-16*x^ 
4+160*x^3-600*x^2+1000*x-625)*log(2+x)+16*x^5-160*x^4+600*x^3-1001*x^2+625 
*x)/(16*x^4-160*x^3+600*x^2-1000*x+625))+(-32*x^7+304*x^6-864*x^5-200*x^4+ 
4750*x^3-5625*x^2-3125*x+6250)*log(2+x)+32*x^8-304*x^7+832*x^6+566*x^5-635 
1*x^4+8632*x^3+1867*x^2-9395*x+3125)/(((32*x^6-336*x^5+1200*x^4-1000*x^3-3 
750*x^2+9375*x-6250)*log(2+x)-32*x^7+336*x^6-1200*x^5+1002*x^4+3749*x^3-93 
85*x^2+6250*x)*log(((-16*x^4+160*x^3-600*x^2+1000*x-625)*log(2+x)+16*x^5-1 
60*x^4+600*x^3-1001*x^2+625*x)/(16*x^4-160*x^3+600*x^2-1000*x+625))+(-32*x 
^7+336*x^6-1200*x^5+1000*x^4+3750*x^3-9375*x^2+6250*x)*log(2+x)+32*x^8-336 
*x^7+1200*x^6-1002*x^5-3749*x^4+9385*x^3-6250*x^2),x, algorithm="maxima")
 

Output:

x + log(-x + log(16*x^5 - 16*x^4*(log(x + 2) + 10) + 40*x^3*(4*log(x + 2) 
+ 15) - x^2*(600*log(x + 2) + 1001) + 125*x*(8*log(x + 2) + 5) - 625*log(x 
 + 2)) - 4*log(2*x - 5))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (29) = 58\).

Time = 1.25 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.21 \[ \int \frac {3125-9395 x+1867 x^2+8632 x^3-6351 x^4+566 x^5+832 x^6-304 x^7+32 x^8+\left (6250-3125 x-5625 x^2+4750 x^3-200 x^4-864 x^5+304 x^6-32 x^7\right ) \log (2+x)+\left (6250 x-9385 x^2+3749 x^3+1002 x^4-1200 x^5+336 x^6-32 x^7+\left (-6250+9375 x-3750 x^2-1000 x^3+1200 x^4-336 x^5+32 x^6\right ) \log (2+x)\right ) \log \left (\frac {625 x-1001 x^2+600 x^3-160 x^4+16 x^5+\left (-625+1000 x-600 x^2+160 x^3-16 x^4\right ) \log (2+x)}{625-1000 x+600 x^2-160 x^3+16 x^4}\right )}{-6250 x^2+9385 x^3-3749 x^4-1002 x^5+1200 x^6-336 x^7+32 x^8+\left (6250 x-9375 x^2+3750 x^3+1000 x^4-1200 x^5+336 x^6-32 x^7\right ) \log (2+x)+\left (6250 x-9385 x^2+3749 x^3+1002 x^4-1200 x^5+336 x^6-32 x^7+\left (-6250+9375 x-3750 x^2-1000 x^3+1200 x^4-336 x^5+32 x^6\right ) \log (2+x)\right ) \log \left (\frac {625 x-1001 x^2+600 x^3-160 x^4+16 x^5+\left (-625+1000 x-600 x^2+160 x^3-16 x^4\right ) \log (2+x)}{625-1000 x+600 x^2-160 x^3+16 x^4}\right )} \, dx=x + \log \left (x - \log \left (16 \, x^{5} - 16 \, x^{4} \log \left (x + 2\right ) - 160 \, x^{4} + 160 \, x^{3} \log \left (x + 2\right ) + 600 \, x^{3} - 600 \, x^{2} \log \left (x + 2\right ) - 1001 \, x^{2} + 1000 \, x \log \left (x + 2\right ) + 625 \, x - 625 \, \log \left (x + 2\right )\right ) + \log \left (16 \, x^{4} - 160 \, x^{3} + 600 \, x^{2} - 1000 \, x + 625\right )\right ) \] Input:

integrate((((32*x^6-336*x^5+1200*x^4-1000*x^3-3750*x^2+9375*x-6250)*log(2+ 
x)-32*x^7+336*x^6-1200*x^5+1002*x^4+3749*x^3-9385*x^2+6250*x)*log(((-16*x^ 
4+160*x^3-600*x^2+1000*x-625)*log(2+x)+16*x^5-160*x^4+600*x^3-1001*x^2+625 
*x)/(16*x^4-160*x^3+600*x^2-1000*x+625))+(-32*x^7+304*x^6-864*x^5-200*x^4+ 
4750*x^3-5625*x^2-3125*x+6250)*log(2+x)+32*x^8-304*x^7+832*x^6+566*x^5-635 
1*x^4+8632*x^3+1867*x^2-9395*x+3125)/(((32*x^6-336*x^5+1200*x^4-1000*x^3-3 
750*x^2+9375*x-6250)*log(2+x)-32*x^7+336*x^6-1200*x^5+1002*x^4+3749*x^3-93 
85*x^2+6250*x)*log(((-16*x^4+160*x^3-600*x^2+1000*x-625)*log(2+x)+16*x^5-1 
60*x^4+600*x^3-1001*x^2+625*x)/(16*x^4-160*x^3+600*x^2-1000*x+625))+(-32*x 
^7+336*x^6-1200*x^5+1000*x^4+3750*x^3-9375*x^2+6250*x)*log(2+x)+32*x^8-336 
*x^7+1200*x^6-1002*x^5-3749*x^4+9385*x^3-6250*x^2),x, algorithm="giac")
 

Output:

x + log(x - log(16*x^5 - 16*x^4*log(x + 2) - 160*x^4 + 160*x^3*log(x + 2) 
+ 600*x^3 - 600*x^2*log(x + 2) - 1001*x^2 + 1000*x*log(x + 2) + 625*x - 62 
5*log(x + 2)) + log(16*x^4 - 160*x^3 + 600*x^2 - 1000*x + 625))
 

Mupad [B] (verification not implemented)

Time = 3.17 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83 \[ \int \frac {3125-9395 x+1867 x^2+8632 x^3-6351 x^4+566 x^5+832 x^6-304 x^7+32 x^8+\left (6250-3125 x-5625 x^2+4750 x^3-200 x^4-864 x^5+304 x^6-32 x^7\right ) \log (2+x)+\left (6250 x-9385 x^2+3749 x^3+1002 x^4-1200 x^5+336 x^6-32 x^7+\left (-6250+9375 x-3750 x^2-1000 x^3+1200 x^4-336 x^5+32 x^6\right ) \log (2+x)\right ) \log \left (\frac {625 x-1001 x^2+600 x^3-160 x^4+16 x^5+\left (-625+1000 x-600 x^2+160 x^3-16 x^4\right ) \log (2+x)}{625-1000 x+600 x^2-160 x^3+16 x^4}\right )}{-6250 x^2+9385 x^3-3749 x^4-1002 x^5+1200 x^6-336 x^7+32 x^8+\left (6250 x-9375 x^2+3750 x^3+1000 x^4-1200 x^5+336 x^6-32 x^7\right ) \log (2+x)+\left (6250 x-9385 x^2+3749 x^3+1002 x^4-1200 x^5+336 x^6-32 x^7+\left (-6250+9375 x-3750 x^2-1000 x^3+1200 x^4-336 x^5+32 x^6\right ) \log (2+x)\right ) \log \left (\frac {625 x-1001 x^2+600 x^3-160 x^4+16 x^5+\left (-625+1000 x-600 x^2+160 x^3-16 x^4\right ) \log (2+x)}{625-1000 x+600 x^2-160 x^3+16 x^4}\right )} \, dx=x+\ln \left (x-\ln \left (\frac {625\,x-\ln \left (x+2\right )\,{\left (2\,x-5\right )}^4-1001\,x^2+600\,x^3-160\,x^4+16\,x^5}{{\left (2\,x-5\right )}^4}\right )\right ) \] Input:

int((log((625*x - log(x + 2)*(600*x^2 - 1000*x - 160*x^3 + 16*x^4 + 625) - 
 1001*x^2 + 600*x^3 - 160*x^4 + 16*x^5)/(600*x^2 - 1000*x - 160*x^3 + 16*x 
^4 + 625))*(6250*x - 9385*x^2 + 3749*x^3 + 1002*x^4 - 1200*x^5 + 336*x^6 - 
 32*x^7 - log(x + 2)*(3750*x^2 - 9375*x + 1000*x^3 - 1200*x^4 + 336*x^5 - 
32*x^6 + 6250)) - log(x + 2)*(3125*x + 5625*x^2 - 4750*x^3 + 200*x^4 + 864 
*x^5 - 304*x^6 + 32*x^7 - 6250) - 9395*x + 1867*x^2 + 8632*x^3 - 6351*x^4 
+ 566*x^5 + 832*x^6 - 304*x^7 + 32*x^8 + 3125)/(log(x + 2)*(6250*x - 9375* 
x^2 + 3750*x^3 + 1000*x^4 - 1200*x^5 + 336*x^6 - 32*x^7) + log((625*x - lo 
g(x + 2)*(600*x^2 - 1000*x - 160*x^3 + 16*x^4 + 625) - 1001*x^2 + 600*x^3 
- 160*x^4 + 16*x^5)/(600*x^2 - 1000*x - 160*x^3 + 16*x^4 + 625))*(6250*x - 
 9385*x^2 + 3749*x^3 + 1002*x^4 - 1200*x^5 + 336*x^6 - 32*x^7 - log(x + 2) 
*(3750*x^2 - 9375*x + 1000*x^3 - 1200*x^4 + 336*x^5 - 32*x^6 + 6250)) - 62 
50*x^2 + 9385*x^3 - 3749*x^4 - 1002*x^5 + 1200*x^6 - 336*x^7 + 32*x^8),x)
 

Output:

x + log(x - log((625*x - log(x + 2)*(2*x - 5)^4 - 1001*x^2 + 600*x^3 - 160 
*x^4 + 16*x^5)/(2*x - 5)^4))
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.28 \[ \int \frac {3125-9395 x+1867 x^2+8632 x^3-6351 x^4+566 x^5+832 x^6-304 x^7+32 x^8+\left (6250-3125 x-5625 x^2+4750 x^3-200 x^4-864 x^5+304 x^6-32 x^7\right ) \log (2+x)+\left (6250 x-9385 x^2+3749 x^3+1002 x^4-1200 x^5+336 x^6-32 x^7+\left (-6250+9375 x-3750 x^2-1000 x^3+1200 x^4-336 x^5+32 x^6\right ) \log (2+x)\right ) \log \left (\frac {625 x-1001 x^2+600 x^3-160 x^4+16 x^5+\left (-625+1000 x-600 x^2+160 x^3-16 x^4\right ) \log (2+x)}{625-1000 x+600 x^2-160 x^3+16 x^4}\right )}{-6250 x^2+9385 x^3-3749 x^4-1002 x^5+1200 x^6-336 x^7+32 x^8+\left (6250 x-9375 x^2+3750 x^3+1000 x^4-1200 x^5+336 x^6-32 x^7\right ) \log (2+x)+\left (6250 x-9385 x^2+3749 x^3+1002 x^4-1200 x^5+336 x^6-32 x^7+\left (-6250+9375 x-3750 x^2-1000 x^3+1200 x^4-336 x^5+32 x^6\right ) \log (2+x)\right ) \log \left (\frac {625 x-1001 x^2+600 x^3-160 x^4+16 x^5+\left (-625+1000 x-600 x^2+160 x^3-16 x^4\right ) \log (2+x)}{625-1000 x+600 x^2-160 x^3+16 x^4}\right )} \, dx=\mathrm {log}\left (\mathrm {log}\left (\frac {-16 \,\mathrm {log}\left (x +2\right ) x^{4}+160 \,\mathrm {log}\left (x +2\right ) x^{3}-600 \,\mathrm {log}\left (x +2\right ) x^{2}+1000 \,\mathrm {log}\left (x +2\right ) x -625 \,\mathrm {log}\left (x +2\right )+16 x^{5}-160 x^{4}+600 x^{3}-1001 x^{2}+625 x}{16 x^{4}-160 x^{3}+600 x^{2}-1000 x +625}\right )-x \right )+x \] Input:

int((((32*x^6-336*x^5+1200*x^4-1000*x^3-3750*x^2+9375*x-6250)*log(2+x)-32* 
x^7+336*x^6-1200*x^5+1002*x^4+3749*x^3-9385*x^2+6250*x)*log(((-16*x^4+160* 
x^3-600*x^2+1000*x-625)*log(2+x)+16*x^5-160*x^4+600*x^3-1001*x^2+625*x)/(1 
6*x^4-160*x^3+600*x^2-1000*x+625))+(-32*x^7+304*x^6-864*x^5-200*x^4+4750*x 
^3-5625*x^2-3125*x+6250)*log(2+x)+32*x^8-304*x^7+832*x^6+566*x^5-6351*x^4+ 
8632*x^3+1867*x^2-9395*x+3125)/(((32*x^6-336*x^5+1200*x^4-1000*x^3-3750*x^ 
2+9375*x-6250)*log(2+x)-32*x^7+336*x^6-1200*x^5+1002*x^4+3749*x^3-9385*x^2 
+6250*x)*log(((-16*x^4+160*x^3-600*x^2+1000*x-625)*log(2+x)+16*x^5-160*x^4 
+600*x^3-1001*x^2+625*x)/(16*x^4-160*x^3+600*x^2-1000*x+625))+(-32*x^7+336 
*x^6-1200*x^5+1000*x^4+3750*x^3-9375*x^2+6250*x)*log(2+x)+32*x^8-336*x^7+1 
200*x^6-1002*x^5-3749*x^4+9385*x^3-6250*x^2),x)
 

Output:

log(log(( - 16*log(x + 2)*x**4 + 160*log(x + 2)*x**3 - 600*log(x + 2)*x**2 
 + 1000*log(x + 2)*x - 625*log(x + 2) + 16*x**5 - 160*x**4 + 600*x**3 - 10 
01*x**2 + 625*x)/(16*x**4 - 160*x**3 + 600*x**2 - 1000*x + 625)) - x) + x