\(\int \frac {256 x-192 x^2+32 x^3-8 x^5+22 x^6-10 x^7+(4 x^6-2 x^7) \log (-2+x)+(-4 x^6+2 x^7) \log (x^2)+(256-192 x+32 x^2-8 x^4+22 x^5-10 x^6+(4 x^5-2 x^6) \log (-2+x)+(-4 x^5+2 x^6) \log (x^2)) \log (\frac {16-5 x^4-x^4 \log (-2+x)+x^4 \log (x^2)}{x^4})}{-32 x+16 x^2+10 x^5-5 x^6+(2 x^5-x^6) \log (-2+x)+(-2 x^5+x^6) \log (x^2)} \, dx\) [198]

Optimal result

Integrand size = 200, antiderivative size = 22 \[ \int \frac {256 x-192 x^2+32 x^3-8 x^5+22 x^6-10 x^7+\left (4 x^6-2 x^7\right ) \log (-2+x)+\left (-4 x^6+2 x^7\right ) \log \left (x^2\right )+\left (256-192 x+32 x^2-8 x^4+22 x^5-10 x^6+\left (4 x^5-2 x^6\right ) \log (-2+x)+\left (-4 x^5+2 x^6\right ) \log \left (x^2\right )\right ) \log \left (\frac {16-5 x^4-x^4 \log (-2+x)+x^4 \log \left (x^2\right )}{x^4}\right )}{-32 x+16 x^2+10 x^5-5 x^6+\left (2 x^5-x^6\right ) \log (-2+x)+\left (-2 x^5+x^6\right ) \log \left (x^2\right )} \, dx=\left (x+\log \left (-5+\frac {16}{x^4}-\log (-2+x)+\log \left (x^2\right )\right )\right )^2 \] Output:

(x+ln(16/x^4-ln(-2+x)+ln(x^2)-5))^2
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(54\) vs. \(2(22)=44\).

Time = 0.07 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.45 \[ \int \frac {256 x-192 x^2+32 x^3-8 x^5+22 x^6-10 x^7+\left (4 x^6-2 x^7\right ) \log (-2+x)+\left (-4 x^6+2 x^7\right ) \log \left (x^2\right )+\left (256-192 x+32 x^2-8 x^4+22 x^5-10 x^6+\left (4 x^5-2 x^6\right ) \log (-2+x)+\left (-4 x^5+2 x^6\right ) \log \left (x^2\right )\right ) \log \left (\frac {16-5 x^4-x^4 \log (-2+x)+x^4 \log \left (x^2\right )}{x^4}\right )}{-32 x+16 x^2+10 x^5-5 x^6+\left (2 x^5-x^6\right ) \log (-2+x)+\left (-2 x^5+x^6\right ) \log \left (x^2\right )} \, dx=2 \left (\frac {x^2}{2}+x \log \left (-5+\frac {16}{x^4}-\log (-2+x)+\log \left (x^2\right )\right )+\frac {1}{2} \log ^2\left (-5+\frac {16}{x^4}-\log (-2+x)+\log \left (x^2\right )\right )\right ) \] Input:

Integrate[(256*x - 192*x^2 + 32*x^3 - 8*x^5 + 22*x^6 - 10*x^7 + (4*x^6 - 2 
*x^7)*Log[-2 + x] + (-4*x^6 + 2*x^7)*Log[x^2] + (256 - 192*x + 32*x^2 - 8* 
x^4 + 22*x^5 - 10*x^6 + (4*x^5 - 2*x^6)*Log[-2 + x] + (-4*x^5 + 2*x^6)*Log 
[x^2])*Log[(16 - 5*x^4 - x^4*Log[-2 + x] + x^4*Log[x^2])/x^4])/(-32*x + 16 
*x^2 + 10*x^5 - 5*x^6 + (2*x^5 - x^6)*Log[-2 + x] + (-2*x^5 + x^6)*Log[x^2 
]),x]
 

Output:

2*(x^2/2 + x*Log[-5 + 16/x^4 - Log[-2 + x] + Log[x^2]] + Log[-5 + 16/x^4 - 
 Log[-2 + x] + Log[x^2]]^2/2)
 

Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {7239, 27, 25, 7237}

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.

\(\displaystyle \int \frac {-10 x^7+22 x^6-8 x^5+32 x^3-192 x^2+\left (4 x^6-2 x^7\right ) \log (x-2)+\left (2 x^7-4 x^6\right ) \log \left (x^2\right )+\left (-10 x^6+22 x^5-8 x^4+32 x^2+\left (4 x^5-2 x^6\right ) \log (x-2)+\left (2 x^6-4 x^5\right ) \log \left (x^2\right )-192 x+256\right ) \log \left (\frac {-5 x^4+x^4 (-\log (x-2))+x^4 \log \left (x^2\right )+16}{x^4}\right )+256 x}{-5 x^6+10 x^5+16 x^2+\left (2 x^5-x^6\right ) \log (x-2)+\left (x^6-2 x^5\right ) \log \left (x^2\right )-32 x} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 \left (-5 x^6+11 x^5-(x-2) x^5 \log (x-2)-4 x^4+16 x^2+(x-2) x^5 \log \left (x^2\right )-96 x+128\right ) \left (-\log \left (\frac {16}{x^4}+\log \left (x^2\right )-\log (x-2)-5\right )-x\right )}{(2-x) x \left (-5 x^4+x^4 (-\log (x-2))+x^4 \log \left (x^2\right )+16\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \int -\frac {\left (-5 x^6+(2-x) \log (x-2) x^5-(2-x) \log \left (x^2\right ) x^5+11 x^5-4 x^4+16 x^2-96 x+128\right ) \left (x+\log \left (-\log (x-2)+\log \left (x^2\right )+\frac {16}{x^4}-5\right )\right )}{(2-x) x \left (-\log (x-2) x^4+\log \left (x^2\right ) x^4-5 x^4+16\right )}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -2 \int \frac {\left (-5 x^6+(2-x) \log (x-2) x^5-(2-x) \log \left (x^2\right ) x^5+11 x^5-4 x^4+16 x^2-96 x+128\right ) \left (x+\log \left (-\log (x-2)+\log \left (x^2\right )+\frac {16}{x^4}-5\right )\right )}{(2-x) x \left (-\log (x-2) x^4+\log \left (x^2\right ) x^4-5 x^4+16\right )}dx\)

\(\Big \downarrow \) 7237

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

Input:

Int[(256*x - 192*x^2 + 32*x^3 - 8*x^5 + 22*x^6 - 10*x^7 + (4*x^6 - 2*x^7)* 
Log[-2 + x] + (-4*x^6 + 2*x^7)*Log[x^2] + (256 - 192*x + 32*x^2 - 8*x^4 + 
22*x^5 - 10*x^6 + (4*x^5 - 2*x^6)*Log[-2 + x] + (-4*x^5 + 2*x^6)*Log[x^2]) 
*Log[(16 - 5*x^4 - x^4*Log[-2 + x] + x^4*Log[x^2])/x^4])/(-32*x + 16*x^2 + 
 10*x^5 - 5*x^6 + (2*x^5 - x^6)*Log[-2 + x] + (-2*x^5 + x^6)*Log[x^2]),x]
 

Output:

(x + Log[-5 + 16/x^4 - Log[-2 + x] + Log[x^2]])^2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 7237

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si 
mp[q*(y^(m + 1)/(m + 1)), x] /;  !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

Maple [F]

\[\int \frac {\left (\left (2 x^{6}-4 x^{5}\right ) \ln \left (x^{2}\right )+\left (-2 x^{6}+4 x^{5}\right ) \ln \left (-2+x \right )-10 x^{6}+22 x^{5}-8 x^{4}+32 x^{2}-192 x +256\right ) \ln \left (\frac {x^{4} \ln \left (x^{2}\right )-x^{4} \ln \left (-2+x \right )-5 x^{4}+16}{x^{4}}\right )+\left (2 x^{7}-4 x^{6}\right ) \ln \left (x^{2}\right )+\left (-2 x^{7}+4 x^{6}\right ) \ln \left (-2+x \right )-10 x^{7}+22 x^{6}-8 x^{5}+32 x^{3}-192 x^{2}+256 x}{\left (x^{6}-2 x^{5}\right ) \ln \left (x^{2}\right )+\left (-x^{6}+2 x^{5}\right ) \ln \left (-2+x \right )-5 x^{6}+10 x^{5}+16 x^{2}-32 x}d x\]

Input:

int((((2*x^6-4*x^5)*ln(x^2)+(-2*x^6+4*x^5)*ln(-2+x)-10*x^6+22*x^5-8*x^4+32 
*x^2-192*x+256)*ln((x^4*ln(x^2)-x^4*ln(-2+x)-5*x^4+16)/x^4)+(2*x^7-4*x^6)* 
ln(x^2)+(-2*x^7+4*x^6)*ln(-2+x)-10*x^7+22*x^6-8*x^5+32*x^3-192*x^2+256*x)/ 
((x^6-2*x^5)*ln(x^2)+(-x^6+2*x^5)*ln(-2+x)-5*x^6+10*x^5+16*x^2-32*x),x)
 

Output:

int((((2*x^6-4*x^5)*ln(x^2)+(-2*x^6+4*x^5)*ln(-2+x)-10*x^6+22*x^5-8*x^4+32 
*x^2-192*x+256)*ln((x^4*ln(x^2)-x^4*ln(-2+x)-5*x^4+16)/x^4)+(2*x^7-4*x^6)* 
ln(x^2)+(-2*x^7+4*x^6)*ln(-2+x)-10*x^7+22*x^6-8*x^5+32*x^3-192*x^2+256*x)/ 
((x^6-2*x^5)*ln(x^2)+(-x^6+2*x^5)*ln(-2+x)-5*x^6+10*x^5+16*x^2-32*x),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (22) = 44\).

Time = 0.09 (sec) , antiderivative size = 67, normalized size of antiderivative = 3.05 \[ \int \frac {256 x-192 x^2+32 x^3-8 x^5+22 x^6-10 x^7+\left (4 x^6-2 x^7\right ) \log (-2+x)+\left (-4 x^6+2 x^7\right ) \log \left (x^2\right )+\left (256-192 x+32 x^2-8 x^4+22 x^5-10 x^6+\left (4 x^5-2 x^6\right ) \log (-2+x)+\left (-4 x^5+2 x^6\right ) \log \left (x^2\right )\right ) \log \left (\frac {16-5 x^4-x^4 \log (-2+x)+x^4 \log \left (x^2\right )}{x^4}\right )}{-32 x+16 x^2+10 x^5-5 x^6+\left (2 x^5-x^6\right ) \log (-2+x)+\left (-2 x^5+x^6\right ) \log \left (x^2\right )} \, dx=x^{2} + 2 \, x \log \left (\frac {x^{4} \log \left (x^{2}\right ) - x^{4} \log \left (x - 2\right ) - 5 \, x^{4} + 16}{x^{4}}\right ) + \log \left (\frac {x^{4} \log \left (x^{2}\right ) - x^{4} \log \left (x - 2\right ) - 5 \, x^{4} + 16}{x^{4}}\right )^{2} \] Input:

integrate((((2*x^6-4*x^5)*log(x^2)+(-2*x^6+4*x^5)*log(-2+x)-10*x^6+22*x^5- 
8*x^4+32*x^2-192*x+256)*log((x^4*log(x^2)-x^4*log(-2+x)-5*x^4+16)/x^4)+(2* 
x^7-4*x^6)*log(x^2)+(-2*x^7+4*x^6)*log(-2+x)-10*x^7+22*x^6-8*x^5+32*x^3-19 
2*x^2+256*x)/((x^6-2*x^5)*log(x^2)+(-x^6+2*x^5)*log(-2+x)-5*x^6+10*x^5+16* 
x^2-32*x),x, algorithm="fricas")
 

Output:

x^2 + 2*x*log((x^4*log(x^2) - x^4*log(x - 2) - 5*x^4 + 16)/x^4) + log((x^4 
*log(x^2) - x^4*log(x - 2) - 5*x^4 + 16)/x^4)^2
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (20) = 40\).

Time = 1.50 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.95 \[ \int \frac {256 x-192 x^2+32 x^3-8 x^5+22 x^6-10 x^7+\left (4 x^6-2 x^7\right ) \log (-2+x)+\left (-4 x^6+2 x^7\right ) \log \left (x^2\right )+\left (256-192 x+32 x^2-8 x^4+22 x^5-10 x^6+\left (4 x^5-2 x^6\right ) \log (-2+x)+\left (-4 x^5+2 x^6\right ) \log \left (x^2\right )\right ) \log \left (\frac {16-5 x^4-x^4 \log (-2+x)+x^4 \log \left (x^2\right )}{x^4}\right )}{-32 x+16 x^2+10 x^5-5 x^6+\left (2 x^5-x^6\right ) \log (-2+x)+\left (-2 x^5+x^6\right ) \log \left (x^2\right )} \, dx=x^{2} + 2 x \log {\left (\frac {x^{4} \log {\left (x^{2} \right )} - x^{4} \log {\left (x - 2 \right )} - 5 x^{4} + 16}{x^{4}} \right )} + \log {\left (\frac {x^{4} \log {\left (x^{2} \right )} - x^{4} \log {\left (x - 2 \right )} - 5 x^{4} + 16}{x^{4}} \right )}^{2} \] Input:

integrate((((2*x**6-4*x**5)*ln(x**2)+(-2*x**6+4*x**5)*ln(-2+x)-10*x**6+22* 
x**5-8*x**4+32*x**2-192*x+256)*ln((x**4*ln(x**2)-x**4*ln(-2+x)-5*x**4+16)/ 
x**4)+(2*x**7-4*x**6)*ln(x**2)+(-2*x**7+4*x**6)*ln(-2+x)-10*x**7+22*x**6-8 
*x**5+32*x**3-192*x**2+256*x)/((x**6-2*x**5)*ln(x**2)+(-x**6+2*x**5)*ln(-2 
+x)-5*x**6+10*x**5+16*x**2-32*x),x)
 

Output:

x**2 + 2*x*log((x**4*log(x**2) - x**4*log(x - 2) - 5*x**4 + 16)/x**4) + lo 
g((x**4*log(x**2) - x**4*log(x - 2) - 5*x**4 + 16)/x**4)**2
 

Maxima [F]

\[ \int \frac {256 x-192 x^2+32 x^3-8 x^5+22 x^6-10 x^7+\left (4 x^6-2 x^7\right ) \log (-2+x)+\left (-4 x^6+2 x^7\right ) \log \left (x^2\right )+\left (256-192 x+32 x^2-8 x^4+22 x^5-10 x^6+\left (4 x^5-2 x^6\right ) \log (-2+x)+\left (-4 x^5+2 x^6\right ) \log \left (x^2\right )\right ) \log \left (\frac {16-5 x^4-x^4 \log (-2+x)+x^4 \log \left (x^2\right )}{x^4}\right )}{-32 x+16 x^2+10 x^5-5 x^6+\left (2 x^5-x^6\right ) \log (-2+x)+\left (-2 x^5+x^6\right ) \log \left (x^2\right )} \, dx=\int { \frac {2 \, {\left (5 \, x^{7} - 11 \, x^{6} + 4 \, x^{5} - 16 \, x^{3} + 96 \, x^{2} - {\left (x^{7} - 2 \, x^{6}\right )} \log \left (x^{2}\right ) + {\left (x^{7} - 2 \, x^{6}\right )} \log \left (x - 2\right ) + {\left (5 \, x^{6} - 11 \, x^{5} + 4 \, x^{4} - 16 \, x^{2} - {\left (x^{6} - 2 \, x^{5}\right )} \log \left (x^{2}\right ) + {\left (x^{6} - 2 \, x^{5}\right )} \log \left (x - 2\right ) + 96 \, x - 128\right )} \log \left (\frac {x^{4} \log \left (x^{2}\right ) - x^{4} \log \left (x - 2\right ) - 5 \, x^{4} + 16}{x^{4}}\right ) - 128 \, x\right )}}{5 \, x^{6} - 10 \, x^{5} - 16 \, x^{2} - {\left (x^{6} - 2 \, x^{5}\right )} \log \left (x^{2}\right ) + {\left (x^{6} - 2 \, x^{5}\right )} \log \left (x - 2\right ) + 32 \, x} \,d x } \] Input:

integrate((((2*x^6-4*x^5)*log(x^2)+(-2*x^6+4*x^5)*log(-2+x)-10*x^6+22*x^5- 
8*x^4+32*x^2-192*x+256)*log((x^4*log(x^2)-x^4*log(-2+x)-5*x^4+16)/x^4)+(2* 
x^7-4*x^6)*log(x^2)+(-2*x^7+4*x^6)*log(-2+x)-10*x^7+22*x^6-8*x^5+32*x^3-19 
2*x^2+256*x)/((x^6-2*x^5)*log(x^2)+(-x^6+2*x^5)*log(-2+x)-5*x^6+10*x^5+16* 
x^2-32*x),x, algorithm="maxima")
 

Output:

2*integrate((5*x^7 - 11*x^6 + 4*x^5 - 16*x^3 + 96*x^2 - (x^7 - 2*x^6)*log( 
x^2) + (x^7 - 2*x^6)*log(x - 2) + (5*x^6 - 11*x^5 + 4*x^4 - 16*x^2 - (x^6 
- 2*x^5)*log(x^2) + (x^6 - 2*x^5)*log(x - 2) + 96*x - 128)*log((x^4*log(x^ 
2) - x^4*log(x - 2) - 5*x^4 + 16)/x^4) - 128*x)/(5*x^6 - 10*x^5 - 16*x^2 - 
 (x^6 - 2*x^5)*log(x^2) + (x^6 - 2*x^5)*log(x - 2) + 32*x), x)
 

Giac [F]

\[ \int \frac {256 x-192 x^2+32 x^3-8 x^5+22 x^6-10 x^7+\left (4 x^6-2 x^7\right ) \log (-2+x)+\left (-4 x^6+2 x^7\right ) \log \left (x^2\right )+\left (256-192 x+32 x^2-8 x^4+22 x^5-10 x^6+\left (4 x^5-2 x^6\right ) \log (-2+x)+\left (-4 x^5+2 x^6\right ) \log \left (x^2\right )\right ) \log \left (\frac {16-5 x^4-x^4 \log (-2+x)+x^4 \log \left (x^2\right )}{x^4}\right )}{-32 x+16 x^2+10 x^5-5 x^6+\left (2 x^5-x^6\right ) \log (-2+x)+\left (-2 x^5+x^6\right ) \log \left (x^2\right )} \, dx=\int { \frac {2 \, {\left (5 \, x^{7} - 11 \, x^{6} + 4 \, x^{5} - 16 \, x^{3} + 96 \, x^{2} - {\left (x^{7} - 2 \, x^{6}\right )} \log \left (x^{2}\right ) + {\left (x^{7} - 2 \, x^{6}\right )} \log \left (x - 2\right ) + {\left (5 \, x^{6} - 11 \, x^{5} + 4 \, x^{4} - 16 \, x^{2} - {\left (x^{6} - 2 \, x^{5}\right )} \log \left (x^{2}\right ) + {\left (x^{6} - 2 \, x^{5}\right )} \log \left (x - 2\right ) + 96 \, x - 128\right )} \log \left (\frac {x^{4} \log \left (x^{2}\right ) - x^{4} \log \left (x - 2\right ) - 5 \, x^{4} + 16}{x^{4}}\right ) - 128 \, x\right )}}{5 \, x^{6} - 10 \, x^{5} - 16 \, x^{2} - {\left (x^{6} - 2 \, x^{5}\right )} \log \left (x^{2}\right ) + {\left (x^{6} - 2 \, x^{5}\right )} \log \left (x - 2\right ) + 32 \, x} \,d x } \] Input:

integrate((((2*x^6-4*x^5)*log(x^2)+(-2*x^6+4*x^5)*log(-2+x)-10*x^6+22*x^5- 
8*x^4+32*x^2-192*x+256)*log((x^4*log(x^2)-x^4*log(-2+x)-5*x^4+16)/x^4)+(2* 
x^7-4*x^6)*log(x^2)+(-2*x^7+4*x^6)*log(-2+x)-10*x^7+22*x^6-8*x^5+32*x^3-19 
2*x^2+256*x)/((x^6-2*x^5)*log(x^2)+(-x^6+2*x^5)*log(-2+x)-5*x^6+10*x^5+16* 
x^2-32*x),x, algorithm="giac")
 

Output:

integrate(2*(5*x^7 - 11*x^6 + 4*x^5 - 16*x^3 + 96*x^2 - (x^7 - 2*x^6)*log( 
x^2) + (x^7 - 2*x^6)*log(x - 2) + (5*x^6 - 11*x^5 + 4*x^4 - 16*x^2 - (x^6 
- 2*x^5)*log(x^2) + (x^6 - 2*x^5)*log(x - 2) + 96*x - 128)*log((x^4*log(x^ 
2) - x^4*log(x - 2) - 5*x^4 + 16)/x^4) - 128*x)/(5*x^6 - 10*x^5 - 16*x^2 - 
 (x^6 - 2*x^5)*log(x^2) + (x^6 - 2*x^5)*log(x - 2) + 32*x), x)
 

Mupad [B] (verification not implemented)

Time = 2.87 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55 \[ \int \frac {256 x-192 x^2+32 x^3-8 x^5+22 x^6-10 x^7+\left (4 x^6-2 x^7\right ) \log (-2+x)+\left (-4 x^6+2 x^7\right ) \log \left (x^2\right )+\left (256-192 x+32 x^2-8 x^4+22 x^5-10 x^6+\left (4 x^5-2 x^6\right ) \log (-2+x)+\left (-4 x^5+2 x^6\right ) \log \left (x^2\right )\right ) \log \left (\frac {16-5 x^4-x^4 \log (-2+x)+x^4 \log \left (x^2\right )}{x^4}\right )}{-32 x+16 x^2+10 x^5-5 x^6+\left (2 x^5-x^6\right ) \log (-2+x)+\left (-2 x^5+x^6\right ) \log \left (x^2\right )} \, dx={\left (x+\ln \left (-\frac {x^4\,\ln \left (x-2\right )-x^4\,\ln \left (x^2\right )+5\,x^4-16}{x^4}\right )\right )}^2 \] Input:

int((log(x^2)*(4*x^6 - 2*x^7) - log(x - 2)*(4*x^6 - 2*x^7) - 256*x + log(- 
(x^4*log(x - 2) - x^4*log(x^2) + 5*x^4 - 16)/x^4)*(192*x - log(x - 2)*(4*x 
^5 - 2*x^6) + log(x^2)*(4*x^5 - 2*x^6) - 32*x^2 + 8*x^4 - 22*x^5 + 10*x^6 
- 256) + 192*x^2 - 32*x^3 + 8*x^5 - 22*x^6 + 10*x^7)/(32*x - log(x - 2)*(2 
*x^5 - x^6) + log(x^2)*(2*x^5 - x^6) - 16*x^2 - 10*x^5 + 5*x^6),x)
 

Output:

(x + log(-(x^4*log(x - 2) - x^4*log(x^2) + 5*x^4 - 16)/x^4))^2
 

Reduce [F]

\[ \int \frac {256 x-192 x^2+32 x^3-8 x^5+22 x^6-10 x^7+\left (4 x^6-2 x^7\right ) \log (-2+x)+\left (-4 x^6+2 x^7\right ) \log \left (x^2\right )+\left (256-192 x+32 x^2-8 x^4+22 x^5-10 x^6+\left (4 x^5-2 x^6\right ) \log (-2+x)+\left (-4 x^5+2 x^6\right ) \log \left (x^2\right )\right ) \log \left (\frac {16-5 x^4-x^4 \log (-2+x)+x^4 \log \left (x^2\right )}{x^4}\right )}{-32 x+16 x^2+10 x^5-5 x^6+\left (2 x^5-x^6\right ) \log (-2+x)+\left (-2 x^5+x^6\right ) \log \left (x^2\right )} \, dx=\text {too large to display} \] Input:

int((((2*x^6-4*x^5)*log(x^2)+(-2*x^6+4*x^5)*log(-2+x)-10*x^6+22*x^5-8*x^4+ 
32*x^2-192*x+256)*log((x^4*log(x^2)-x^4*log(-2+x)-5*x^4+16)/x^4)+(2*x^7-4* 
x^6)*log(x^2)+(-2*x^7+4*x^6)*log(-2+x)-10*x^7+22*x^6-8*x^5+32*x^3-192*x^2+ 
256*x)/((x^6-2*x^5)*log(x^2)+(-x^6+2*x^5)*log(-2+x)-5*x^6+10*x^5+16*x^2-32 
*x),x)
 

Output:

2*( - 5*int(x**6/(log(x**2)*x**5 - 2*log(x**2)*x**4 - log(x - 2)*x**5 + 2* 
log(x - 2)*x**4 - 5*x**5 + 10*x**4 + 16*x - 32),x) + 11*int(x**5/(log(x**2 
)*x**5 - 2*log(x**2)*x**4 - log(x - 2)*x**5 + 2*log(x - 2)*x**4 - 5*x**5 + 
 10*x**4 + 16*x - 32),x) - 4*int(x**4/(log(x**2)*x**5 - 2*log(x**2)*x**4 - 
 log(x - 2)*x**5 + 2*log(x - 2)*x**4 - 5*x**5 + 10*x**4 + 16*x - 32),x) + 
16*int(x**2/(log(x**2)*x**5 - 2*log(x**2)*x**4 - log(x - 2)*x**5 + 2*log(x 
 - 2)*x**4 - 5*x**5 + 10*x**4 + 16*x - 32),x) + 128*int(log((log(x**2)*x** 
4 - log(x - 2)*x**4 - 5*x**4 + 16)/x**4)/(log(x**2)*x**6 - 2*log(x**2)*x** 
5 - log(x - 2)*x**6 + 2*log(x - 2)*x**5 - 5*x**6 + 10*x**5 + 16*x**2 - 32* 
x),x) - 96*int(log((log(x**2)*x**4 - log(x - 2)*x**4 - 5*x**4 + 16)/x**4)/ 
(log(x**2)*x**5 - 2*log(x**2)*x**4 - log(x - 2)*x**5 + 2*log(x - 2)*x**4 - 
 5*x**5 + 10*x**4 + 16*x - 32),x) + int((log(x**2)*x**6)/(log(x**2)*x**5 - 
 2*log(x**2)*x**4 - log(x - 2)*x**5 + 2*log(x - 2)*x**4 - 5*x**5 + 10*x**4 
 + 16*x - 32),x) - 2*int((log(x**2)*x**5)/(log(x**2)*x**5 - 2*log(x**2)*x* 
*4 - log(x - 2)*x**5 + 2*log(x - 2)*x**4 - 5*x**5 + 10*x**4 + 16*x - 32),x 
) + int((log(x**2)*log((log(x**2)*x**4 - log(x - 2)*x**4 - 5*x**4 + 16)/x* 
*4)*x**5)/(log(x**2)*x**5 - 2*log(x**2)*x**4 - log(x - 2)*x**5 + 2*log(x - 
 2)*x**4 - 5*x**5 + 10*x**4 + 16*x - 32),x) - 2*int((log(x**2)*log((log(x* 
*2)*x**4 - log(x - 2)*x**4 - 5*x**4 + 16)/x**4)*x**4)/(log(x**2)*x**5 - 2* 
log(x**2)*x**4 - log(x - 2)*x**5 + 2*log(x - 2)*x**4 - 5*x**5 + 10*x**4...