3.6.75 \(\int \frac {-2 x-x^2-2 \log (\frac {5}{4})}{4 x+4 x^2+x^3+(4+4 x+x^2) \log (\frac {5}{4})+(-4 x^2-2 x^3+(-4 x-2 x^2) \log (\frac {5}{4})) \log (x+\log (\frac {5}{4}))+(x^3+x^2 \log (\frac {5}{4})) \log ^2(x+\log (\frac {5}{4}))} \, dx\) [575]

3.6.75.1 Optimal result

Integrand size = 98, antiderivative size = 18 \begin {dmath*} \int \frac {-2 x-x^2-2 \log \left (\frac {5}{4}\right )}{4 x+4 x^2+x^3+\left (4+4 x+x^2\right ) \log \left (\frac {5}{4}\right )+\left (-4 x^2-2 x^3+\left (-4 x-2 x^2\right ) \log \left (\frac {5}{4}\right )\right ) \log \left (x+\log \left (\frac {5}{4}\right )\right )+\left (x^3+x^2 \log \left (\frac {5}{4}\right )\right ) \log ^2\left (x+\log \left (\frac {5}{4}\right )\right )} \, dx=\frac {x}{-2-x+x \log \left (x+\log \left (\frac {5}{4}\right )\right )} \end {dmath*}

output

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

3.6.75.2 Mathematica [A] (verified)

Time = 10.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \begin {dmath*} \int \frac {-2 x-x^2-2 \log \left (\frac {5}{4}\right )}{4 x+4 x^2+x^3+\left (4+4 x+x^2\right ) \log \left (\frac {5}{4}\right )+\left (-4 x^2-2 x^3+\left (-4 x-2 x^2\right ) \log \left (\frac {5}{4}\right )\right ) \log \left (x+\log \left (\frac {5}{4}\right )\right )+\left (x^3+x^2 \log \left (\frac {5}{4}\right )\right ) \log ^2\left (x+\log \left (\frac {5}{4}\right )\right )} \, dx=-\frac {x}{2+x-x \log \left (x+\log \left (\frac {5}{4}\right )\right )} \end {dmath*}

input

Integrate[(-2*x - x^2 - 2*Log[5/4])/(4*x + 4*x^2 + x^3 + (4 + 4*x + x^2)*L 
og[5/4] + (-4*x^2 - 2*x^3 + (-4*x - 2*x^2)*Log[5/4])*Log[x + Log[5/4]] + ( 
x^3 + x^2*Log[5/4])*Log[x + Log[5/4]]^2),x]
 

output

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

3.6.75.3 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.

\(\displaystyle \int \frac {-x^2-2 x-2 \log \left (\frac {5}{4}\right )}{x^3+4 x^2+\left (x^2+4 x+4\right ) \log \left (\frac {5}{4}\right )+\left (x^3+x^2 \log \left (\frac {5}{4}\right )\right ) \log ^2\left (x+\log \left (\frac {5}{4}\right )\right )+\left (-2 x^3-4 x^2+\left (-2 x^2-4 x\right ) \log \left (\frac {5}{4}\right )\right ) \log \left (x+\log \left (\frac {5}{4}\right )\right )+4 x} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-x^2-2 x-\log \left (\frac {25}{16}\right )}{\left (x+\log \left (\frac {5}{4}\right )\right ) \left (x+x \left (-\log \left (x+\log \left (\frac {5}{4}\right )\right )\right )+2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {\log ^2\left (\frac {5}{4}\right )}{\left (x+\log \left (\frac {5}{4}\right )\right ) \left (-x+x \log \left (x+\log \left (\frac {5}{4}\right )\right )-2\right )^2}-\frac {x}{\left (-x+x \log \left (x+\log \left (\frac {5}{4}\right )\right )-2\right )^2}-\frac {2 \left (1+\log \left (\frac {2}{\sqrt {5}}\right )\right )}{\left (-x+x \log \left (x+\log \left (\frac {5}{4}\right )\right )-2\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\log ^2\left (\frac {5}{4}\right ) \int \frac {1}{\left (x+\log \left (\frac {5}{4}\right )\right ) \left (\log \left (x+\log \left (\frac {5}{4}\right )\right ) x-x-2\right )^2}dx-2 \left (1+\log \left (\frac {2}{\sqrt {5}}\right )\right ) \int \frac {1}{\left (\log \left (x+\log \left (\frac {5}{4}\right )\right ) x-x-2\right )^2}dx-\int \frac {x}{\left (\log \left (x+\log \left (\frac {5}{4}\right )\right ) x-x-2\right )^2}dx\)

input

Int[(-2*x - x^2 - 2*Log[5/4])/(4*x + 4*x^2 + x^3 + (4 + 4*x + x^2)*Log[5/4 
] + (-4*x^2 - 2*x^3 + (-4*x - 2*x^2)*Log[5/4])*Log[x + Log[5/4]] + (x^3 + 
x^2*Log[5/4])*Log[x + Log[5/4]]^2),x]
 

output

$Aborted
 

3.6.75.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7239

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

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.6.75.4 Maple [A] (verified)

Time = 2.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94

method	result	size
norman	\(\frac {x}{\ln \left (\ln \left (\frac {5}{4}\right )+x \right ) x -2-x}\)	\(17\)
parallelrisch	\(\frac {x}{\ln \left (\ln \left (\frac {5}{4}\right )+x \right ) x -2-x}\)	\(17\)
risch	\(\frac {x}{x \ln \left (\ln \left (5\right )-2 \ln \left (2\right )+x \right )-x -2}\)	\(21\)
derivativedivides	\(-\frac {-\ln \left (\frac {5}{4}\right )-x +\ln \left (5\right )-2 \ln \left (2\right )}{2 \ln \left (\ln \left (\frac {5}{4}\right )+x \right ) \ln \left (2\right )-\ln \left (\ln \left (\frac {5}{4}\right )+x \right ) \ln \left (5\right )+\ln \left (\ln \left (\frac {5}{4}\right )+x \right ) \left (\ln \left (\frac {5}{4}\right )+x \right )-2 \ln \left (2\right )+\ln \left (5\right )-\ln \left (\frac {5}{4}\right )-x -2}\)	\(62\)
default	\(-\frac {-\ln \left (\frac {5}{4}\right )-x +\ln \left (5\right )-2 \ln \left (2\right )}{2 \ln \left (\ln \left (\frac {5}{4}\right )+x \right ) \ln \left (2\right )-\ln \left (\ln \left (\frac {5}{4}\right )+x \right ) \ln \left (5\right )+\ln \left (\ln \left (\frac {5}{4}\right )+x \right ) \left (\ln \left (\frac {5}{4}\right )+x \right )-2 \ln \left (2\right )+\ln \left (5\right )-\ln \left (\frac {5}{4}\right )-x -2}\)	\(62\)

input

int((-2*ln(5/4)-x^2-2*x)/((x^2*ln(5/4)+x^3)*ln(ln(5/4)+x)^2+((-2*x^2-4*x)* 
ln(5/4)-2*x^3-4*x^2)*ln(ln(5/4)+x)+(x^2+4*x+4)*ln(5/4)+x^3+4*x^2+4*x),x,me 
thod=_RETURNVERBOSE)
 

output

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

3.6.75.5 Fricas [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \begin {dmath*} \int \frac {-2 x-x^2-2 \log \left (\frac {5}{4}\right )}{4 x+4 x^2+x^3+\left (4+4 x+x^2\right ) \log \left (\frac {5}{4}\right )+\left (-4 x^2-2 x^3+\left (-4 x-2 x^2\right ) \log \left (\frac {5}{4}\right )\right ) \log \left (x+\log \left (\frac {5}{4}\right )\right )+\left (x^3+x^2 \log \left (\frac {5}{4}\right )\right ) \log ^2\left (x+\log \left (\frac {5}{4}\right )\right )} \, dx=\frac {x}{x \log \left (x + \log \left (\frac {5}{4}\right )\right ) - x - 2} \end {dmath*}

input

integrate((-2*log(5/4)-x^2-2*x)/((x^2*log(5/4)+x^3)*log(log(5/4)+x)^2+((-2 
*x^2-4*x)*log(5/4)-2*x^3-4*x^2)*log(log(5/4)+x)+(x^2+4*x+4)*log(5/4)+x^3+4 
*x^2+4*x),x, algorithm=\
 

output

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

3.6.75.6 Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \begin {dmath*} \int \frac {-2 x-x^2-2 \log \left (\frac {5}{4}\right )}{4 x+4 x^2+x^3+\left (4+4 x+x^2\right ) \log \left (\frac {5}{4}\right )+\left (-4 x^2-2 x^3+\left (-4 x-2 x^2\right ) \log \left (\frac {5}{4}\right )\right ) \log \left (x+\log \left (\frac {5}{4}\right )\right )+\left (x^3+x^2 \log \left (\frac {5}{4}\right )\right ) \log ^2\left (x+\log \left (\frac {5}{4}\right )\right )} \, dx=\frac {x}{x \log {\left (x + \log {\left (\frac {5}{4} \right )} \right )} - x - 2} \end {dmath*}

input

integrate((-2*ln(5/4)-x**2-2*x)/((x**2*ln(5/4)+x**3)*ln(ln(5/4)+x)**2+((-2 
*x**2-4*x)*ln(5/4)-2*x**3-4*x**2)*ln(ln(5/4)+x)+(x**2+4*x+4)*ln(5/4)+x**3+ 
4*x**2+4*x),x)
 

output

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

3.6.75.7 Maxima [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \begin {dmath*} \int \frac {-2 x-x^2-2 \log \left (\frac {5}{4}\right )}{4 x+4 x^2+x^3+\left (4+4 x+x^2\right ) \log \left (\frac {5}{4}\right )+\left (-4 x^2-2 x^3+\left (-4 x-2 x^2\right ) \log \left (\frac {5}{4}\right )\right ) \log \left (x+\log \left (\frac {5}{4}\right )\right )+\left (x^3+x^2 \log \left (\frac {5}{4}\right )\right ) \log ^2\left (x+\log \left (\frac {5}{4}\right )\right )} \, dx=\frac {x}{x \log \left (x + \log \left (5\right ) - 2 \, \log \left (2\right )\right ) - x - 2} \end {dmath*}

input

integrate((-2*log(5/4)-x^2-2*x)/((x^2*log(5/4)+x^3)*log(log(5/4)+x)^2+((-2 
*x^2-4*x)*log(5/4)-2*x^3-4*x^2)*log(log(5/4)+x)+(x^2+4*x+4)*log(5/4)+x^3+4 
*x^2+4*x),x, algorithm=\
 

output

x/(x*log(x + log(5) - 2*log(2)) - x - 2)
 

3.6.75.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 247 vs. \(2 (16) = 32\).

Time = 0.31 (sec) , antiderivative size = 247, normalized size of antiderivative = 13.72 \begin {dmath*} \int \frac {-2 x-x^2-2 \log \left (\frac {5}{4}\right )}{4 x+4 x^2+x^3+\left (4+4 x+x^2\right ) \log \left (\frac {5}{4}\right )+\left (-4 x^2-2 x^3+\left (-4 x-2 x^2\right ) \log \left (\frac {5}{4}\right )\right ) \log \left (x+\log \left (\frac {5}{4}\right )\right )+\left (x^3+x^2 \log \left (\frac {5}{4}\right )\right ) \log ^2\left (x+\log \left (\frac {5}{4}\right )\right )} \, dx=\frac {x^{4} + x^{3} \log \left (\frac {5}{4}\right ) + 2 \, x^{3} + 2 \, x^{2} \log \left (5\right ) - 4 \, x^{2} \log \left (2\right ) + 2 \, x^{2} \log \left (\frac {5}{4}\right ) + 2 \, x \log \left (5\right ) \log \left (\frac {5}{4}\right ) - 4 \, x \log \left (2\right ) \log \left (\frac {5}{4}\right )}{x^{4} \log \left (x + \log \left (\frac {5}{4}\right )\right ) + x^{3} \log \left (5\right ) \log \left (x + \log \left (\frac {5}{4}\right )\right ) - 2 \, x^{3} \log \left (2\right ) \log \left (x + \log \left (\frac {5}{4}\right )\right ) - x^{4} - x^{3} \log \left (5\right ) + 2 \, x^{3} \log \left (2\right ) + 2 \, x^{3} \log \left (x + \log \left (\frac {5}{4}\right )\right ) + 2 \, x^{2} \log \left (5\right ) \log \left (x + \log \left (\frac {5}{4}\right )\right ) - 4 \, x^{2} \log \left (2\right ) \log \left (x + \log \left (\frac {5}{4}\right )\right ) + 2 \, x^{2} \log \left (\frac {5}{4}\right ) \log \left (x + \log \left (\frac {5}{4}\right )\right ) + 2 \, x \log \left (5\right ) \log \left (\frac {5}{4}\right ) \log \left (x + \log \left (\frac {5}{4}\right )\right ) - 4 \, x \log \left (2\right ) \log \left (\frac {5}{4}\right ) \log \left (x + \log \left (\frac {5}{4}\right )\right ) - 4 \, x^{3} - 4 \, x^{2} \log \left (5\right ) + 8 \, x^{2} \log \left (2\right ) - 2 \, x^{2} \log \left (\frac {5}{4}\right ) - 2 \, x \log \left (5\right ) \log \left (\frac {5}{4}\right ) + 4 \, x \log \left (2\right ) \log \left (\frac {5}{4}\right ) - 4 \, x^{2} - 4 \, x \log \left (5\right ) + 8 \, x \log \left (2\right ) - 4 \, x \log \left (\frac {5}{4}\right ) - 4 \, \log \left (5\right ) \log \left (\frac {5}{4}\right ) + 8 \, \log \left (2\right ) \log \left (\frac {5}{4}\right )} \end {dmath*}

input

integrate((-2*log(5/4)-x^2-2*x)/((x^2*log(5/4)+x^3)*log(log(5/4)+x)^2+((-2 
*x^2-4*x)*log(5/4)-2*x^3-4*x^2)*log(log(5/4)+x)+(x^2+4*x+4)*log(5/4)+x^3+4 
*x^2+4*x),x, algorithm=\
 

output

(x^4 + x^3*log(5/4) + 2*x^3 + 2*x^2*log(5) - 4*x^2*log(2) + 2*x^2*log(5/4) 
 + 2*x*log(5)*log(5/4) - 4*x*log(2)*log(5/4))/(x^4*log(x + log(5/4)) + x^3 
*log(5)*log(x + log(5/4)) - 2*x^3*log(2)*log(x + log(5/4)) - x^4 - x^3*log 
(5) + 2*x^3*log(2) + 2*x^3*log(x + log(5/4)) + 2*x^2*log(5)*log(x + log(5/ 
4)) - 4*x^2*log(2)*log(x + log(5/4)) + 2*x^2*log(5/4)*log(x + log(5/4)) + 
2*x*log(5)*log(5/4)*log(x + log(5/4)) - 4*x*log(2)*log(5/4)*log(x + log(5/ 
4)) - 4*x^3 - 4*x^2*log(5) + 8*x^2*log(2) - 2*x^2*log(5/4) - 2*x*log(5)*lo 
g(5/4) + 4*x*log(2)*log(5/4) - 4*x^2 - 4*x*log(5) + 8*x*log(2) - 4*x*log(5 
/4) - 4*log(5)*log(5/4) + 8*log(2)*log(5/4))
 

3.6.75.9 Mupad [F(-1)]

Timed out. \begin {dmath*} \int \frac {-2 x-x^2-2 \log \left (\frac {5}{4}\right )}{4 x+4 x^2+x^3+\left (4+4 x+x^2\right ) \log \left (\frac {5}{4}\right )+\left (-4 x^2-2 x^3+\left (-4 x-2 x^2\right ) \log \left (\frac {5}{4}\right )\right ) \log \left (x+\log \left (\frac {5}{4}\right )\right )+\left (x^3+x^2 \log \left (\frac {5}{4}\right )\right ) \log ^2\left (x+\log \left (\frac {5}{4}\right )\right )} \, dx=\int -\frac {x^2+2\,x+2\,\ln \left (\frac {5}{4}\right )}{4\,x-\ln \left (x+\ln \left (\frac {5}{4}\right )\right )\,\left (\ln \left (\frac {5}{4}\right )\,\left (2\,x^2+4\,x\right )+4\,x^2+2\,x^3\right )+{\ln \left (x+\ln \left (\frac {5}{4}\right )\right )}^2\,\left (x^3+\ln \left (\frac {5}{4}\right )\,x^2\right )+4\,x^2+x^3+\ln \left (\frac {5}{4}\right )\,\left (x^2+4\,x+4\right )} \,d x \end {dmath*}

input

int(-(2*x + 2*log(5/4) + x^2)/(4*x - log(x + log(5/4))*(log(5/4)*(4*x + 2* 
x^2) + 4*x^2 + 2*x^3) + log(x + log(5/4))^2*(x^2*log(5/4) + x^3) + 4*x^2 + 
 x^3 + log(5/4)*(4*x + x^2 + 4)),x)
 

output

int(-(2*x + 2*log(5/4) + x^2)/(4*x - log(x + log(5/4))*(log(5/4)*(4*x + 2* 
x^2) + 4*x^2 + 2*x^3) + log(x + log(5/4))^2*(x^2*log(5/4) + x^3) + 4*x^2 + 
 x^3 + log(5/4)*(4*x + x^2 + 4)), x)