Integrand size = 133, antiderivative size = 25 \[ \int \frac {347-273 x-59 x^2+x^3+\left (-63+64 x-x^2\right ) \log (4-4 x)+\left (80-64 x-16 x^2+(-16+16 x) \log (4-4 x)\right ) \log (-5-x+\log (4-4 x))}{320-261 x-60 x^2+x^3+\left (-64+65 x-x^2\right ) \log (4-4 x)+\left (80-64 x-16 x^2+(-16+16 x) \log (4-4 x)\right ) \log (-5-x+\log (4-4 x))} \, dx=\log \left (e^x (x-16 (4+\log (-5-x+\log (4 (1-x)))))\right ) \] Output:
ln(exp(x)*(x-16*ln(ln(4-4*x)-x-5)-64))
Time = 0.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {347-273 x-59 x^2+x^3+\left (-63+64 x-x^2\right ) \log (4-4 x)+\left (80-64 x-16 x^2+(-16+16 x) \log (4-4 x)\right ) \log (-5-x+\log (4-4 x))}{320-261 x-60 x^2+x^3+\left (-64+65 x-x^2\right ) \log (4-4 x)+\left (80-64 x-16 x^2+(-16+16 x) \log (4-4 x)\right ) \log (-5-x+\log (4-4 x))} \, dx=x+\log (64-x+16 \log (-5-x+\log (4-4 x))) \] Input:
Integrate[(347 - 273*x - 59*x^2 + x^3 + (-63 + 64*x - x^2)*Log[4 - 4*x] + (80 - 64*x - 16*x^2 + (-16 + 16*x)*Log[4 - 4*x])*Log[-5 - x + Log[4 - 4*x] ])/(320 - 261*x - 60*x^2 + x^3 + (-64 + 65*x - x^2)*Log[4 - 4*x] + (80 - 6 4*x - 16*x^2 + (-16 + 16*x)*Log[4 - 4*x])*Log[-5 - x + Log[4 - 4*x]]),x]
Output:
x + Log[64 - x + 16*Log[-5 - x + Log[4 - 4*x]]]
Time = 1.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, 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.
\(\displaystyle \int \frac {x^3-59 x^2+\left (-x^2+64 x-63\right ) \log (4-4 x)+\left (-16 x^2-64 x+(16 x-16) \log (4-4 x)+80\right ) \log (-x+\log (4-4 x)-5)-273 x+347}{x^3-60 x^2+\left (-x^2+65 x-64\right ) \log (4-4 x)+\left (-16 x^2-64 x+(16 x-16) \log (4-4 x)+80\right ) \log (-x+\log (4-4 x)-5)-261 x+320} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {x^3-59 x^2+\left (-x^2+64 x-63\right ) \log (4-4 x)+\left (-16 x^2-64 x+(16 x-16) \log (4-4 x)+80\right ) \log (-x+\log (4-4 x)-5)-273 x+347}{(1-x) (x-\log (4-4 x)+5) (-x+16 \log (-x+\log (4-4 x)-5)+64)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {x^2-12 x-x \log (4-4 x)+\log (4-4 x)+27}{(x-1) (x-\log (4-4 x)+5) (x-16 \log (-x+\log (4-4 x)-5)-64)}+1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x+\log (-x+16 \log (-x+\log (4-4 x)-5)+64)\) |
Input:
Int[(347 - 273*x - 59*x^2 + x^3 + (-63 + 64*x - x^2)*Log[4 - 4*x] + (80 - 64*x - 16*x^2 + (-16 + 16*x)*Log[4 - 4*x])*Log[-5 - x + Log[4 - 4*x]])/(32 0 - 261*x - 60*x^2 + x^3 + (-64 + 65*x - x^2)*Log[4 - 4*x] + (80 - 64*x - 16*x^2 + (-16 + 16*x)*Log[4 - 4*x])*Log[-5 - x + Log[4 - 4*x]]),x]
Output:
x + Log[64 - x + 16*Log[-5 - x + Log[4 - 4*x]]]
Time = 18.10 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84
method | result | size |
risch | \(x +\ln \left (\ln \left (\ln \left (4-4 x \right )-x -5\right )-\frac {x}{16}+4\right )\) | \(21\) |
parallelrisch | \(2+\ln \left (x -16 \ln \left (\ln \left (4-4 x \right )-x -5\right )-64\right )+x\) | \(22\) |
Input:
int((((16*x-16)*ln(4-4*x)-16*x^2-64*x+80)*ln(ln(4-4*x)-x-5)+(-x^2+64*x-63) *ln(4-4*x)+x^3-59*x^2-273*x+347)/(((16*x-16)*ln(4-4*x)-16*x^2-64*x+80)*ln( ln(4-4*x)-x-5)+(-x^2+65*x-64)*ln(4-4*x)+x^3-60*x^2-261*x+320),x,method=_RE TURNVERBOSE)
Output:
x+ln(ln(ln(4-4*x)-x-5)-1/16*x+4)
Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {347-273 x-59 x^2+x^3+\left (-63+64 x-x^2\right ) \log (4-4 x)+\left (80-64 x-16 x^2+(-16+16 x) \log (4-4 x)\right ) \log (-5-x+\log (4-4 x))}{320-261 x-60 x^2+x^3+\left (-64+65 x-x^2\right ) \log (4-4 x)+\left (80-64 x-16 x^2+(-16+16 x) \log (4-4 x)\right ) \log (-5-x+\log (4-4 x))} \, dx=x + \log \left (-x + 16 \, \log \left (-x + \log \left (-4 \, x + 4\right ) - 5\right ) + 64\right ) \] Input:
integrate((((16*x-16)*log(4-4*x)-16*x^2-64*x+80)*log(log(4-4*x)-x-5)+(-x^2 +64*x-63)*log(4-4*x)+x^3-59*x^2-273*x+347)/(((16*x-16)*log(4-4*x)-16*x^2-6 4*x+80)*log(log(4-4*x)-x-5)+(-x^2+65*x-64)*log(4-4*x)+x^3-60*x^2-261*x+320 ),x, algorithm="fricas")
Output:
x + log(-x + 16*log(-x + log(-4*x + 4) - 5) + 64)
Time = 0.40 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {347-273 x-59 x^2+x^3+\left (-63+64 x-x^2\right ) \log (4-4 x)+\left (80-64 x-16 x^2+(-16+16 x) \log (4-4 x)\right ) \log (-5-x+\log (4-4 x))}{320-261 x-60 x^2+x^3+\left (-64+65 x-x^2\right ) \log (4-4 x)+\left (80-64 x-16 x^2+(-16+16 x) \log (4-4 x)\right ) \log (-5-x+\log (4-4 x))} \, dx=x + \log {\left (- \frac {x}{16} + \log {\left (- x + \log {\left (4 - 4 x \right )} - 5 \right )} + 4 \right )} \] Input:
integrate((((16*x-16)*ln(4-4*x)-16*x**2-64*x+80)*ln(ln(4-4*x)-x-5)+(-x**2+ 64*x-63)*ln(4-4*x)+x**3-59*x**2-273*x+347)/(((16*x-16)*ln(4-4*x)-16*x**2-6 4*x+80)*ln(ln(4-4*x)-x-5)+(-x**2+65*x-64)*ln(4-4*x)+x**3-60*x**2-261*x+320 ),x)
Output:
x + log(-x/16 + log(-x + log(4 - 4*x) - 5) + 4)
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {347-273 x-59 x^2+x^3+\left (-63+64 x-x^2\right ) \log (4-4 x)+\left (80-64 x-16 x^2+(-16+16 x) \log (4-4 x)\right ) \log (-5-x+\log (4-4 x))}{320-261 x-60 x^2+x^3+\left (-64+65 x-x^2\right ) \log (4-4 x)+\left (80-64 x-16 x^2+(-16+16 x) \log (4-4 x)\right ) \log (-5-x+\log (4-4 x))} \, dx=x + \log \left (-\frac {1}{16} \, x + \log \left (i \, \pi - x + 2 \, \log \left (2\right ) + \log \left (x - 1\right ) - 5\right ) + 4\right ) \] Input:
integrate((((16*x-16)*log(4-4*x)-16*x^2-64*x+80)*log(log(4-4*x)-x-5)+(-x^2 +64*x-63)*log(4-4*x)+x^3-59*x^2-273*x+347)/(((16*x-16)*log(4-4*x)-16*x^2-6 4*x+80)*log(log(4-4*x)-x-5)+(-x^2+65*x-64)*log(4-4*x)+x^3-60*x^2-261*x+320 ),x, algorithm="maxima")
Output:
x + log(-1/16*x + log(I*pi - x + 2*log(2) + log(x - 1) - 5) + 4)
Time = 0.17 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {347-273 x-59 x^2+x^3+\left (-63+64 x-x^2\right ) \log (4-4 x)+\left (80-64 x-16 x^2+(-16+16 x) \log (4-4 x)\right ) \log (-5-x+\log (4-4 x))}{320-261 x-60 x^2+x^3+\left (-64+65 x-x^2\right ) \log (4-4 x)+\left (80-64 x-16 x^2+(-16+16 x) \log (4-4 x)\right ) \log (-5-x+\log (4-4 x))} \, dx=x + \log \left (x - 16 \, \log \left (-x + \log \left (-4 \, x + 4\right ) - 5\right ) - 64\right ) \] Input:
integrate((((16*x-16)*log(4-4*x)-16*x^2-64*x+80)*log(log(4-4*x)-x-5)+(-x^2 +64*x-63)*log(4-4*x)+x^3-59*x^2-273*x+347)/(((16*x-16)*log(4-4*x)-16*x^2-6 4*x+80)*log(log(4-4*x)-x-5)+(-x^2+65*x-64)*log(4-4*x)+x^3-60*x^2-261*x+320 ),x, algorithm="giac")
Output:
x + log(x - 16*log(-x + log(-4*x + 4) - 5) - 64)
Time = 1.54 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {347-273 x-59 x^2+x^3+\left (-63+64 x-x^2\right ) \log (4-4 x)+\left (80-64 x-16 x^2+(-16+16 x) \log (4-4 x)\right ) \log (-5-x+\log (4-4 x))}{320-261 x-60 x^2+x^3+\left (-64+65 x-x^2\right ) \log (4-4 x)+\left (80-64 x-16 x^2+(-16+16 x) \log (4-4 x)\right ) \log (-5-x+\log (4-4 x))} \, dx=x+\ln \left (\ln \left (\ln \left (4-4\,x\right )-x-5\right )-\frac {x}{16}+4\right ) \] Input:
int((273*x + log(4 - 4*x)*(x^2 - 64*x + 63) + log(log(4 - 4*x) - x - 5)*(6 4*x - log(4 - 4*x)*(16*x - 16) + 16*x^2 - 80) + 59*x^2 - x^3 - 347)/(261*x + log(4 - 4*x)*(x^2 - 65*x + 64) + log(log(4 - 4*x) - x - 5)*(64*x - log( 4 - 4*x)*(16*x - 16) + 16*x^2 - 80) + 60*x^2 - x^3 - 320),x)
Output:
x + log(log(log(4 - 4*x) - x - 5) - x/16 + 4)
Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {347-273 x-59 x^2+x^3+\left (-63+64 x-x^2\right ) \log (4-4 x)+\left (80-64 x-16 x^2+(-16+16 x) \log (4-4 x)\right ) \log (-5-x+\log (4-4 x))}{320-261 x-60 x^2+x^3+\left (-64+65 x-x^2\right ) \log (4-4 x)+\left (80-64 x-16 x^2+(-16+16 x) \log (4-4 x)\right ) \log (-5-x+\log (4-4 x))} \, dx=\mathrm {log}\left (16 \,\mathrm {log}\left (\mathrm {log}\left (-4 x +4\right )-x -5\right )-x +64\right )+x \] Input:
int((((16*x-16)*log(4-4*x)-16*x^2-64*x+80)*log(log(4-4*x)-x-5)+(-x^2+64*x- 63)*log(4-4*x)+x^3-59*x^2-273*x+347)/(((16*x-16)*log(4-4*x)-16*x^2-64*x+80 )*log(log(4-4*x)-x-5)+(-x^2+65*x-64)*log(4-4*x)+x^3-60*x^2-261*x+320),x)
Output:
log(16*log(log( - 4*x + 4) - x - 5) - x + 64) + x