Integrand size = 157, antiderivative size = 30 \[ \int \frac {2+18 x+32 x^2+16 x^3+2 x^4+(2+2 x) \log (4 x)+\left (2-2 x-8 x^2-8 x^3-2 x^4-2 x \log (4 x)\right ) \log \left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right )}{1-x-4 x^2-4 x^3-x^4-x \log (4 x)+\left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right ) \log \left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right )} \, dx=\log \left (e^{-2 x} \left (-1+\log \left (-1+x+x^2 (2+x)^2+x \log (4 x)\right )\right )^2\right ) \]
Time = 0.06 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {2+18 x+32 x^2+16 x^3+2 x^4+(2+2 x) \log (4 x)+\left (2-2 x-8 x^2-8 x^3-2 x^4-2 x \log (4 x)\right ) \log \left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right )}{1-x-4 x^2-4 x^3-x^4-x \log (4 x)+\left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right ) \log \left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right )} \, dx=2 \left (-x+\log \left (1-\log \left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right )\right )\right ) \]
Integrate[(2 + 18*x + 32*x^2 + 16*x^3 + 2*x^4 + (2 + 2*x)*Log[4*x] + (2 - 2*x - 8*x^2 - 8*x^3 - 2*x^4 - 2*x*Log[4*x])*Log[-1 + x + 4*x^2 + 4*x^3 + x ^4 + x*Log[4*x]])/(1 - x - 4*x^2 - 4*x^3 - x^4 - x*Log[4*x] + (-1 + x + 4* x^2 + 4*x^3 + x^4 + x*Log[4*x])*Log[-1 + x + 4*x^2 + 4*x^3 + x^4 + x*Log[4 *x]]),x]
Time = 0.94 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {7292, 7293, 2009}
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.
\(\displaystyle \int \frac {2 x^4+16 x^3+32 x^2+\left (-2 x^4-8 x^3-8 x^2-2 x-2 x \log (4 x)+2\right ) \log \left (x^4+4 x^3+4 x^2+x+x \log (4 x)-1\right )+18 x+(2 x+2) \log (4 x)+2}{-x^4-4 x^3-4 x^2+\left (x^4+4 x^3+4 x^2+x+x \log (4 x)-1\right ) \log \left (x^4+4 x^3+4 x^2+x+x \log (4 x)-1\right )-x-x \log (4 x)+1} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {2 x^4+16 x^3+32 x^2+\left (-2 x^4-8 x^3-8 x^2-2 x-2 x \log (4 x)+2\right ) \log \left (x^4+4 x^3+4 x^2+x+x \log (4 x)-1\right )+18 x+(2 x+2) \log (4 x)+2}{\left (-x^4-4 x^3-4 x^2-x-x \log (4 x)+1\right ) \left (1-\log \left (x^4+4 x^3+4 x^2+x+x \log (4 x)-1\right )\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 \left (4 x^3+12 x^2+8 x+\log (4 x)+2\right )}{\left (x^4+4 x^3+4 x^2+x+x \log (4 x)-1\right ) \left (\log \left (x^4+4 x^3+4 x^2+x+x \log (4 x)-1\right )-1\right )}-2\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \log \left (1-\log \left (x^4+4 x^3+4 x^2+x+x \log (4 x)-1\right )\right )-2 x\) |
Int[(2 + 18*x + 32*x^2 + 16*x^3 + 2*x^4 + (2 + 2*x)*Log[4*x] + (2 - 2*x - 8*x^2 - 8*x^3 - 2*x^4 - 2*x*Log[4*x])*Log[-1 + x + 4*x^2 + 4*x^3 + x^4 + x *Log[4*x]])/(1 - x - 4*x^2 - 4*x^3 - x^4 - x*Log[4*x] + (-1 + x + 4*x^2 + 4*x^3 + x^4 + x*Log[4*x])*Log[-1 + x + 4*x^2 + 4*x^3 + x^4 + x*Log[4*x]]), x]
3.26.95.3.1 Defintions of rubi rules used
Time = 5.34 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.10
method | result | size |
risch | \(-2 x +2 \ln \left (\ln \left (x \ln \left (4 x \right )+x^{4}+4 x^{3}+4 x^{2}+x -1\right )-1\right )\) | \(33\) |
parallelrisch | \(-2 x +2 \ln \left (\ln \left (x \ln \left (4 x \right )+x^{4}+4 x^{3}+4 x^{2}+x -1\right )-1\right )\) | \(33\) |
int(((-2*x*ln(4*x)-2*x^4-8*x^3-8*x^2-2*x+2)*ln(x*ln(4*x)+x^4+4*x^3+4*x^2+x -1)+(2+2*x)*ln(4*x)+2*x^4+16*x^3+32*x^2+18*x+2)/((x*ln(4*x)+x^4+4*x^3+4*x^ 2+x-1)*ln(x*ln(4*x)+x^4+4*x^3+4*x^2+x-1)-x*ln(4*x)-x^4-4*x^3-4*x^2-x+1),x, method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {2+18 x+32 x^2+16 x^3+2 x^4+(2+2 x) \log (4 x)+\left (2-2 x-8 x^2-8 x^3-2 x^4-2 x \log (4 x)\right ) \log \left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right )}{1-x-4 x^2-4 x^3-x^4-x \log (4 x)+\left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right ) \log \left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right )} \, dx=-2 \, x + 2 \, \log \left (\log \left (x^{4} + 4 \, x^{3} + 4 \, x^{2} + x \log \left (4 \, x\right ) + x - 1\right ) - 1\right ) \]
integrate(((-2*x*log(4*x)-2*x^4-8*x^3-8*x^2-2*x+2)*log(x*log(4*x)+x^4+4*x^ 3+4*x^2+x-1)+(2+2*x)*log(4*x)+2*x^4+16*x^3+32*x^2+18*x+2)/((x*log(4*x)+x^4 +4*x^3+4*x^2+x-1)*log(x*log(4*x)+x^4+4*x^3+4*x^2+x-1)-x*log(4*x)-x^4-4*x^3 -4*x^2-x+1),x, algorithm=\
Time = 0.44 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {2+18 x+32 x^2+16 x^3+2 x^4+(2+2 x) \log (4 x)+\left (2-2 x-8 x^2-8 x^3-2 x^4-2 x \log (4 x)\right ) \log \left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right )}{1-x-4 x^2-4 x^3-x^4-x \log (4 x)+\left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right ) \log \left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right )} \, dx=- 2 x + 2 \log {\left (\log {\left (x^{4} + 4 x^{3} + 4 x^{2} + x \log {\left (4 x \right )} + x - 1 \right )} - 1 \right )} \]
integrate(((-2*x*ln(4*x)-2*x**4-8*x**3-8*x**2-2*x+2)*ln(x*ln(4*x)+x**4+4*x **3+4*x**2+x-1)+(2+2*x)*ln(4*x)+2*x**4+16*x**3+32*x**2+18*x+2)/((x*ln(4*x) +x**4+4*x**3+4*x**2+x-1)*ln(x*ln(4*x)+x**4+4*x**3+4*x**2+x-1)-x*ln(4*x)-x* *4-4*x**3-4*x**2-x+1),x)
Time = 0.31 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {2+18 x+32 x^2+16 x^3+2 x^4+(2+2 x) \log (4 x)+\left (2-2 x-8 x^2-8 x^3-2 x^4-2 x \log (4 x)\right ) \log \left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right )}{1-x-4 x^2-4 x^3-x^4-x \log (4 x)+\left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right ) \log \left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right )} \, dx=-2 \, x + 2 \, \log \left (\log \left (x^{4} + 4 \, x^{3} + 4 \, x^{2} + x {\left (2 \, \log \left (2\right ) + 1\right )} + x \log \left (x\right ) - 1\right ) - 1\right ) \]
integrate(((-2*x*log(4*x)-2*x^4-8*x^3-8*x^2-2*x+2)*log(x*log(4*x)+x^4+4*x^ 3+4*x^2+x-1)+(2+2*x)*log(4*x)+2*x^4+16*x^3+32*x^2+18*x+2)/((x*log(4*x)+x^4 +4*x^3+4*x^2+x-1)*log(x*log(4*x)+x^4+4*x^3+4*x^2+x-1)-x*log(4*x)-x^4-4*x^3 -4*x^2-x+1),x, algorithm=\
Time = 0.42 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {2+18 x+32 x^2+16 x^3+2 x^4+(2+2 x) \log (4 x)+\left (2-2 x-8 x^2-8 x^3-2 x^4-2 x \log (4 x)\right ) \log \left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right )}{1-x-4 x^2-4 x^3-x^4-x \log (4 x)+\left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right ) \log \left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right )} \, dx=-2 \, x + 2 \, \log \left (\log \left (x^{4} + 4 \, x^{3} + 4 \, x^{2} + x \log \left (4 \, x\right ) + x - 1\right ) - 1\right ) \]
integrate(((-2*x*log(4*x)-2*x^4-8*x^3-8*x^2-2*x+2)*log(x*log(4*x)+x^4+4*x^ 3+4*x^2+x-1)+(2+2*x)*log(4*x)+2*x^4+16*x^3+32*x^2+18*x+2)/((x*log(4*x)+x^4 +4*x^3+4*x^2+x-1)*log(x*log(4*x)+x^4+4*x^3+4*x^2+x-1)-x*log(4*x)-x^4-4*x^3 -4*x^2-x+1),x, algorithm=\
Time = 8.45 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {2+18 x+32 x^2+16 x^3+2 x^4+(2+2 x) \log (4 x)+\left (2-2 x-8 x^2-8 x^3-2 x^4-2 x \log (4 x)\right ) \log \left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right )}{1-x-4 x^2-4 x^3-x^4-x \log (4 x)+\left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right ) \log \left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right )} \, dx=2\,\ln \left (\ln \left (x+x\,\ln \left (4\,x\right )+4\,x^2+4\,x^3+x^4-1\right )-1\right )-2\,x \]
int(-(18*x + 32*x^2 + 16*x^3 + 2*x^4 + log(4*x)*(2*x + 2) - log(x + x*log( 4*x) + 4*x^2 + 4*x^3 + x^4 - 1)*(2*x + 2*x*log(4*x) + 8*x^2 + 8*x^3 + 2*x^ 4 - 2) + 2)/(x + x*log(4*x) - log(x + x*log(4*x) + 4*x^2 + 4*x^3 + x^4 - 1 )*(x + x*log(4*x) + 4*x^2 + 4*x^3 + x^4 - 1) + 4*x^2 + 4*x^3 + x^4 - 1),x)