\(\int \frac {20480+(-40960-14080 x) \log (x)}{(15129 x-5412 x^2+484 x^3) \log (x)+(-7872 x+1408 x^2) \log (x) \log (\frac {x^2}{\log (x)})+1024 x \log (x) \log ^2(\frac {x^2}{\log (x)})} \, dx\) [801]

Optimal result

Integrand size = 68, antiderivative size = 27 \[ \int \frac {20480+(-40960-14080 x) \log (x)}{\left (15129 x-5412 x^2+484 x^3\right ) \log (x)+\left (-7872 x+1408 x^2\right ) \log (x) \log \left (\frac {x^2}{\log (x)}\right )+1024 x \log (x) \log ^2\left (\frac {x^2}{\log (x)}\right )} \, dx=\frac {20}{-4+\frac {5}{16} \left (\frac {1}{2}-x\right )+x+\log \left (\frac {x^2}{\log (x)}\right )} \] Output:

4/(1/5*ln(x^2/ln(x))-123/160+11/80*x)
 

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {20480+(-40960-14080 x) \log (x)}{\left (15129 x-5412 x^2+484 x^3\right ) \log (x)+\left (-7872 x+1408 x^2\right ) \log (x) \log \left (\frac {x^2}{\log (x)}\right )+1024 x \log (x) \log ^2\left (\frac {x^2}{\log (x)}\right )} \, dx=-\frac {1280}{246-44 x-64 \log \left (\frac {x^2}{\log (x)}\right )} \] Input:

Integrate[(20480 + (-40960 - 14080*x)*Log[x])/((15129*x - 5412*x^2 + 484*x 
^3)*Log[x] + (-7872*x + 1408*x^2)*Log[x]*Log[x^2/Log[x]] + 1024*x*Log[x]*L 
og[x^2/Log[x]]^2),x]
 

Output:

-1280/(246 - 44*x - 64*Log[x^2/Log[x]])
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {7239, 27, 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.

\(\displaystyle \int \frac {(-14080 x-40960) \log (x)+20480}{1024 x \log (x) \log ^2\left (\frac {x^2}{\log (x)}\right )+\left (1408 x^2-7872 x\right ) \log (x) \log \left (\frac {x^2}{\log (x)}\right )+\left (484 x^3-5412 x^2+15129 x\right ) \log (x)} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {1280 (16-(11 x+32) \log (x))}{x \log (x) \left (-32 \log \left (\frac {x^2}{\log (x)}\right )-22 x+123\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 1280 \int \frac {16-(11 x+32) \log (x)}{x \log (x) \left (-22 x-32 \log \left (\frac {x^2}{\log (x)}\right )+123\right )^2}dx\)

\(\Big \downarrow \) 7237

\(\displaystyle -\frac {640}{-32 \log \left (\frac {x^2}{\log (x)}\right )-22 x+123}\)

Input:

Int[(20480 + (-40960 - 14080*x)*Log[x])/((15129*x - 5412*x^2 + 484*x^3)*Lo 
g[x] + (-7872*x + 1408*x^2)*Log[x]*Log[x^2/Log[x]] + 1024*x*Log[x]*Log[x^2 
/Log[x]]^2),x]
 

Output:

-640/(123 - 22*x - 32*Log[x^2/Log[x]])
 

Defintions of rubi rules used

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 [A] (verified)

Time = 1.66 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78

method	result	size
parallelrisch	\(\frac {640}{-123+32 \ln \left (\frac {x^{2}}{\ln \left (x \right )}\right )+22 x}\)	\(21\)
risch	\(-\frac {640 i}{16 \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i x^{2}}{\ln \left (x \right )}\right )^{2}-16 \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (\frac {i x^{2}}{\ln \left (x \right )}\right )-16 \pi \operatorname {csgn}\left (\frac {i x^{2}}{\ln \left (x \right )}\right )^{3}+16 \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (\frac {i x^{2}}{\ln \left (x \right )}\right )^{2}-16 \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+32 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-16 \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-22 i x -64 i \ln \left (x \right )+32 i \ln \left (\ln \left (x \right )\right )+123 i}\)	\(162\)
default	\(-\frac {10240 i}{\left (11 x \ln \left (x \right )+32 \ln \left (x \right )-16\right ) \left (16 \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (\frac {i x^{2}}{\ln \left (x \right )}\right )-16 \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i x^{2}}{\ln \left (x \right )}\right )^{2}+16 \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-32 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+16 \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-16 \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (\frac {i x^{2}}{\ln \left (x \right )}\right )^{2}+16 \pi \operatorname {csgn}\left (\frac {i x^{2}}{\ln \left (x \right )}\right )^{3}+22 i x -32 i \ln \left (\ln \left (x \right )\right )+64 i \ln \left (x \right )-123 i\right )}+\frac {640 \ln \left (x \right ) \left (11 x +32\right )}{\left (11 x \ln \left (x \right )+32 \ln \left (x \right )-16\right ) \left (-16 i \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (\frac {i x^{2}}{\ln \left (x \right )}\right )+16 i \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i x^{2}}{\ln \left (x \right )}\right )^{2}-16 i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+32 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-16 i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+16 i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (\frac {i x^{2}}{\ln \left (x \right )}\right )^{2}-16 i \pi \operatorname {csgn}\left (\frac {i x^{2}}{\ln \left (x \right )}\right )^{3}+22 x -32 \ln \left (\ln \left (x \right )\right )+64 \ln \left (x \right )-123\right )}\)	\(359\)
parts	\(-\frac {10240 i}{\left (11 x \ln \left (x \right )+32 \ln \left (x \right )-16\right ) \left (16 \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (\frac {i x^{2}}{\ln \left (x \right )}\right )-16 \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i x^{2}}{\ln \left (x \right )}\right )^{2}+16 \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-32 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+16 \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-16 \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (\frac {i x^{2}}{\ln \left (x \right )}\right )^{2}+16 \pi \operatorname {csgn}\left (\frac {i x^{2}}{\ln \left (x \right )}\right )^{3}+22 i x -32 i \ln \left (\ln \left (x \right )\right )+64 i \ln \left (x \right )-123 i\right )}+\frac {640 \ln \left (x \right ) \left (11 x +32\right )}{\left (11 x \ln \left (x \right )+32 \ln \left (x \right )-16\right ) \left (-16 i \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (\frac {i x^{2}}{\ln \left (x \right )}\right )+16 i \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i x^{2}}{\ln \left (x \right )}\right )^{2}-16 i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+32 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-16 i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+16 i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (\frac {i x^{2}}{\ln \left (x \right )}\right )^{2}-16 i \pi \operatorname {csgn}\left (\frac {i x^{2}}{\ln \left (x \right )}\right )^{3}+22 x -32 \ln \left (\ln \left (x \right )\right )+64 \ln \left (x \right )-123\right )}\)	\(359\)

Input:

int(((-14080*x-40960)*ln(x)+20480)/(1024*x*ln(x)*ln(x^2/ln(x))^2+(1408*x^2 
-7872*x)*ln(x)*ln(x^2/ln(x))+(484*x^3-5412*x^2+15129*x)*ln(x)),x,method=_R 
ETURNVERBOSE)
 

Output:

640/(-123+32*ln(x^2/ln(x))+22*x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {20480+(-40960-14080 x) \log (x)}{\left (15129 x-5412 x^2+484 x^3\right ) \log (x)+\left (-7872 x+1408 x^2\right ) \log (x) \log \left (\frac {x^2}{\log (x)}\right )+1024 x \log (x) \log ^2\left (\frac {x^2}{\log (x)}\right )} \, dx=\frac {640}{22 \, x + 32 \, \log \left (\frac {x^{2}}{\log \left (x\right )}\right ) - 123} \] Input:

integrate(((-14080*x-40960)*log(x)+20480)/(1024*x*log(x)*log(x^2/log(x))^2 
+(1408*x^2-7872*x)*log(x)*log(x^2/log(x))+(484*x^3-5412*x^2+15129*x)*log(x 
)),x, algorithm="fricas")
 

Output:

640/(22*x + 32*log(x^2/log(x)) - 123)
 

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.63 \[ \int \frac {20480+(-40960-14080 x) \log (x)}{\left (15129 x-5412 x^2+484 x^3\right ) \log (x)+\left (-7872 x+1408 x^2\right ) \log (x) \log \left (\frac {x^2}{\log (x)}\right )+1024 x \log (x) \log ^2\left (\frac {x^2}{\log (x)}\right )} \, dx=\frac {20}{\frac {11 x}{16} + \log {\left (\frac {x^{2}}{\log {\left (x \right )}} \right )} - \frac {123}{32}} \] Input:

integrate(((-14080*x-40960)*ln(x)+20480)/(1024*x*ln(x)*ln(x**2/ln(x))**2+( 
1408*x**2-7872*x)*ln(x)*ln(x**2/ln(x))+(484*x**3-5412*x**2+15129*x)*ln(x)) 
,x)
 

Output:

20/(11*x/16 + log(x**2/log(x)) - 123/32)
 

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.67 \[ \int \frac {20480+(-40960-14080 x) \log (x)}{\left (15129 x-5412 x^2+484 x^3\right ) \log (x)+\left (-7872 x+1408 x^2\right ) \log (x) \log \left (\frac {x^2}{\log (x)}\right )+1024 x \log (x) \log ^2\left (\frac {x^2}{\log (x)}\right )} \, dx=\frac {640}{22 \, x + 64 \, \log \left (x\right ) - 32 \, \log \left (\log \left (x\right )\right ) - 123} \] Input:

integrate(((-14080*x-40960)*log(x)+20480)/(1024*x*log(x)*log(x^2/log(x))^2 
+(1408*x^2-7872*x)*log(x)*log(x^2/log(x))+(484*x^3-5412*x^2+15129*x)*log(x 
)),x, algorithm="maxima")
 

Output:

640/(22*x + 64*log(x) - 32*log(log(x)) - 123)
 

Giac [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.67 \[ \int \frac {20480+(-40960-14080 x) \log (x)}{\left (15129 x-5412 x^2+484 x^3\right ) \log (x)+\left (-7872 x+1408 x^2\right ) \log (x) \log \left (\frac {x^2}{\log (x)}\right )+1024 x \log (x) \log ^2\left (\frac {x^2}{\log (x)}\right )} \, dx=\frac {640}{22 \, x + 64 \, \log \left (x\right ) - 32 \, \log \left (\log \left (x\right )\right ) - 123} \] Input:

integrate(((-14080*x-40960)*log(x)+20480)/(1024*x*log(x)*log(x^2/log(x))^2 
+(1408*x^2-7872*x)*log(x)*log(x^2/log(x))+(484*x^3-5412*x^2+15129*x)*log(x 
)),x, algorithm="giac")
 

Output:

640/(22*x + 64*log(x) - 32*log(log(x)) - 123)
 

Mupad [B] (verification not implemented)

Time = 3.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {20480+(-40960-14080 x) \log (x)}{\left (15129 x-5412 x^2+484 x^3\right ) \log (x)+\left (-7872 x+1408 x^2\right ) \log (x) \log \left (\frac {x^2}{\log (x)}\right )+1024 x \log (x) \log ^2\left (\frac {x^2}{\log (x)}\right )} \, dx=\frac {640}{22\,x+32\,\ln \left (\frac {x^2}{\ln \left (x\right )}\right )-123} \] Input:

int(-(log(x)*(14080*x + 40960) - 20480)/(log(x)*(15129*x - 5412*x^2 + 484* 
x^3) + 1024*x*log(x)*log(x^2/log(x))^2 - log(x)*log(x^2/log(x))*(7872*x - 
1408*x^2)),x)
 

Output:

640/(22*x + 32*log(x^2/log(x)) - 123)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {20480+(-40960-14080 x) \log (x)}{\left (15129 x-5412 x^2+484 x^3\right ) \log (x)+\left (-7872 x+1408 x^2\right ) \log (x) \log \left (\frac {x^2}{\log (x)}\right )+1024 x \log (x) \log ^2\left (\frac {x^2}{\log (x)}\right )} \, dx=\frac {640}{32 \,\mathrm {log}\left (\frac {x^{2}}{\mathrm {log}\left (x \right )}\right )+22 x -123} \] Input:

int(((-14080*x-40960)*log(x)+20480)/(1024*x*log(x)*log(x^2/log(x))^2+(1408 
*x^2-7872*x)*log(x)*log(x^2/log(x))+(484*x^3-5412*x^2+15129*x)*log(x)),x)
 

Output:

640/(32*log(x**2/log(x)) + 22*x - 123)