Integrand size = 117, antiderivative size = 21 \[ \int \frac {-128 x+4 x^2+\left (-64-2 x^2\right ) \log (x)+\left (-64+64 x-2 x^2+2 x^3+\left (64+2 x^2\right ) \log (x)\right ) \log (-1+x+\log (x))}{-32 x+32 x^2+x^3-x^4+\left (32 x-x^3\right ) \log (x)+\left (32 x-32 x^2-x^3+x^4+\left (-32 x+x^3\right ) \log (x)\right ) \log (-1+x+\log (x))} \, dx=\log \left (\left (-\frac {32}{x}+x\right )^2 (-1+\log (-1+x+\log (x)))^2\right ) \] Output:
ln((ln(-1+ln(x)+x)-1)^2*(x-32/x)^2)
Time = 0.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int \frac {-128 x+4 x^2+\left (-64-2 x^2\right ) \log (x)+\left (-64+64 x-2 x^2+2 x^3+\left (64+2 x^2\right ) \log (x)\right ) \log (-1+x+\log (x))}{-32 x+32 x^2+x^3-x^4+\left (32 x-x^3\right ) \log (x)+\left (32 x-32 x^2-x^3+x^4+\left (-32 x+x^3\right ) \log (x)\right ) \log (-1+x+\log (x))} \, dx=2 \left (-\log (x)+\log \left (32-x^2\right )+\log (1-\log (-1+x+\log (x)))\right ) \] Input:
Integrate[(-128*x + 4*x^2 + (-64 - 2*x^2)*Log[x] + (-64 + 64*x - 2*x^2 + 2 *x^3 + (64 + 2*x^2)*Log[x])*Log[-1 + x + Log[x]])/(-32*x + 32*x^2 + x^3 - x^4 + (32*x - x^3)*Log[x] + (32*x - 32*x^2 - x^3 + x^4 + (-32*x + x^3)*Log [x])*Log[-1 + x + Log[x]]),x]
Output:
2*(-Log[x] + Log[32 - x^2] + Log[1 - Log[-1 + x + Log[x]]])
Time = 2.18 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {7292, 7276, 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.
\(\displaystyle \int \frac {4 x^2+\left (-2 x^2-64\right ) \log (x)+\left (2 x^3-2 x^2+\left (2 x^2+64\right ) \log (x)+64 x-64\right ) \log (x+\log (x)-1)-128 x}{-x^4+x^3+\left (32 x-x^3\right ) \log (x)+32 x^2+\left (x^4-x^3+\left (x^3-32 x\right ) \log (x)-32 x^2+32 x\right ) \log (x+\log (x)-1)-32 x} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-4 x^2-\left (-2 x^2-64\right ) \log (x)-\left (2 x^3-2 x^2+\left (2 x^2+64\right ) \log (x)+64 x-64\right ) \log (x+\log (x)-1)+128 x}{x \left (32-x^2\right ) (-x-\log (x)+1) (1-\log (x+\log (x)-1))}dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (\frac {2 \left (x^2+32\right )}{x \left (x^2-32\right )}+\frac {2 (x+1)}{x (x+\log (x)-1) (\log (x+\log (x)-1)-1)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \log \left (32-x^2\right )-2 \log (x)+2 \log (1-\log (x+\log (x)-1))\) |
Input:
Int[(-128*x + 4*x^2 + (-64 - 2*x^2)*Log[x] + (-64 + 64*x - 2*x^2 + 2*x^3 + (64 + 2*x^2)*Log[x])*Log[-1 + x + Log[x]])/(-32*x + 32*x^2 + x^3 - x^4 + (32*x - x^3)*Log[x] + (32*x - 32*x^2 - x^3 + x^4 + (-32*x + x^3)*Log[x])*L og[-1 + x + Log[x]]),x]
Output:
-2*Log[x] + 2*Log[32 - x^2] + 2*Log[1 - Log[-1 + x + Log[x]]]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 0.86 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19
method | result | size |
default | \(-2 \ln \left (x \right )+2 \ln \left (\ln \left (-1+\ln \left (x \right )+x \right )-1\right )+2 \ln \left (x^{2}-32\right )\) | \(25\) |
risch | \(-2 \ln \left (x \right )+2 \ln \left (\ln \left (-1+\ln \left (x \right )+x \right )-1\right )+2 \ln \left (x^{2}-32\right )\) | \(25\) |
parallelrisch | \(2 \ln \left (\ln \left (-1+\ln \left (x \right )+x \right )-1\right )+2 \ln \left (x^{2}-32\right )-2 \ln \left (x \right )-4\) | \(26\) |
Input:
int((((2*x^2+64)*ln(x)+2*x^3-2*x^2+64*x-64)*ln(-1+ln(x)+x)+(-2*x^2-64)*ln( x)+4*x^2-128*x)/(((x^3-32*x)*ln(x)+x^4-x^3-32*x^2+32*x)*ln(-1+ln(x)+x)+(-x ^3+32*x)*ln(x)-x^4+x^3+32*x^2-32*x),x,method=_RETURNVERBOSE)
Output:
-2*ln(x)+2*ln(ln(-1+ln(x)+x)-1)+2*ln(x^2-32)
Time = 0.12 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {-128 x+4 x^2+\left (-64-2 x^2\right ) \log (x)+\left (-64+64 x-2 x^2+2 x^3+\left (64+2 x^2\right ) \log (x)\right ) \log (-1+x+\log (x))}{-32 x+32 x^2+x^3-x^4+\left (32 x-x^3\right ) \log (x)+\left (32 x-32 x^2-x^3+x^4+\left (-32 x+x^3\right ) \log (x)\right ) \log (-1+x+\log (x))} \, dx=2 \, \log \left (x^{2} - 32\right ) - 2 \, \log \left (x\right ) + 2 \, \log \left (\log \left (x + \log \left (x\right ) - 1\right ) - 1\right ) \] Input:
integrate((((2*x^2+64)*log(x)+2*x^3-2*x^2+64*x-64)*log(-1+log(x)+x)+(-2*x^ 2-64)*log(x)+4*x^2-128*x)/(((x^3-32*x)*log(x)+x^4-x^3-32*x^2+32*x)*log(-1+ log(x)+x)+(-x^3+32*x)*log(x)-x^4+x^3+32*x^2-32*x),x, algorithm="fricas")
Output:
2*log(x^2 - 32) - 2*log(x) + 2*log(log(x + log(x) - 1) - 1)
Time = 0.18 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int \frac {-128 x+4 x^2+\left (-64-2 x^2\right ) \log (x)+\left (-64+64 x-2 x^2+2 x^3+\left (64+2 x^2\right ) \log (x)\right ) \log (-1+x+\log (x))}{-32 x+32 x^2+x^3-x^4+\left (32 x-x^3\right ) \log (x)+\left (32 x-32 x^2-x^3+x^4+\left (-32 x+x^3\right ) \log (x)\right ) \log (-1+x+\log (x))} \, dx=- 2 \log {\left (x \right )} + 2 \log {\left (x^{2} - 32 \right )} + 2 \log {\left (\log {\left (x + \log {\left (x \right )} - 1 \right )} - 1 \right )} \] Input:
integrate((((2*x**2+64)*ln(x)+2*x**3-2*x**2+64*x-64)*ln(-1+ln(x)+x)+(-2*x* *2-64)*ln(x)+4*x**2-128*x)/(((x**3-32*x)*ln(x)+x**4-x**3-32*x**2+32*x)*ln( -1+ln(x)+x)+(-x**3+32*x)*ln(x)-x**4+x**3+32*x**2-32*x),x)
Output:
-2*log(x) + 2*log(x**2 - 32) + 2*log(log(x + log(x) - 1) - 1)
Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {-128 x+4 x^2+\left (-64-2 x^2\right ) \log (x)+\left (-64+64 x-2 x^2+2 x^3+\left (64+2 x^2\right ) \log (x)\right ) \log (-1+x+\log (x))}{-32 x+32 x^2+x^3-x^4+\left (32 x-x^3\right ) \log (x)+\left (32 x-32 x^2-x^3+x^4+\left (-32 x+x^3\right ) \log (x)\right ) \log (-1+x+\log (x))} \, dx=2 \, \log \left (x^{2} - 32\right ) - 2 \, \log \left (x\right ) + 2 \, \log \left (\log \left (x + \log \left (x\right ) - 1\right ) - 1\right ) \] Input:
integrate((((2*x^2+64)*log(x)+2*x^3-2*x^2+64*x-64)*log(-1+log(x)+x)+(-2*x^ 2-64)*log(x)+4*x^2-128*x)/(((x^3-32*x)*log(x)+x^4-x^3-32*x^2+32*x)*log(-1+ log(x)+x)+(-x^3+32*x)*log(x)-x^4+x^3+32*x^2-32*x),x, algorithm="maxima")
Output:
2*log(x^2 - 32) - 2*log(x) + 2*log(log(x + log(x) - 1) - 1)
Time = 0.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {-128 x+4 x^2+\left (-64-2 x^2\right ) \log (x)+\left (-64+64 x-2 x^2+2 x^3+\left (64+2 x^2\right ) \log (x)\right ) \log (-1+x+\log (x))}{-32 x+32 x^2+x^3-x^4+\left (32 x-x^3\right ) \log (x)+\left (32 x-32 x^2-x^3+x^4+\left (-32 x+x^3\right ) \log (x)\right ) \log (-1+x+\log (x))} \, dx=2 \, \log \left (x^{2} - 32\right ) - 2 \, \log \left (x\right ) + 2 \, \log \left (\log \left (x + \log \left (x\right ) - 1\right ) - 1\right ) \] Input:
integrate((((2*x^2+64)*log(x)+2*x^3-2*x^2+64*x-64)*log(-1+log(x)+x)+(-2*x^ 2-64)*log(x)+4*x^2-128*x)/(((x^3-32*x)*log(x)+x^4-x^3-32*x^2+32*x)*log(-1+ log(x)+x)+(-x^3+32*x)*log(x)-x^4+x^3+32*x^2-32*x),x, algorithm="giac")
Output:
2*log(x^2 - 32) - 2*log(x) + 2*log(log(x + log(x) - 1) - 1)
Time = 4.32 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {-128 x+4 x^2+\left (-64-2 x^2\right ) \log (x)+\left (-64+64 x-2 x^2+2 x^3+\left (64+2 x^2\right ) \log (x)\right ) \log (-1+x+\log (x))}{-32 x+32 x^2+x^3-x^4+\left (32 x-x^3\right ) \log (x)+\left (32 x-32 x^2-x^3+x^4+\left (-32 x+x^3\right ) \log (x)\right ) \log (-1+x+\log (x))} \, dx=2\,\ln \left (x^2-32\right )-2\,\ln \left (x\right )+2\,\ln \left (\ln \left (x+\ln \left (x\right )-1\right )-1\right ) \] Input:
int((128*x - log(x + log(x) - 1)*(64*x - 2*x^2 + 2*x^3 + log(x)*(2*x^2 + 6 4) - 64) - 4*x^2 + log(x)*(2*x^2 + 64))/(32*x - log(x)*(32*x - x^3) + log( x + log(x) - 1)*(log(x)*(32*x - x^3) - 32*x + 32*x^2 + x^3 - x^4) - 32*x^2 - x^3 + x^4),x)
Output:
2*log(x^2 - 32) - 2*log(x) + 2*log(log(x + log(x) - 1) - 1)
Time = 0.17 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.62 \[ \int \frac {-128 x+4 x^2+\left (-64-2 x^2\right ) \log (x)+\left (-64+64 x-2 x^2+2 x^3+\left (64+2 x^2\right ) \log (x)\right ) \log (-1+x+\log (x))}{-32 x+32 x^2+x^3-x^4+\left (32 x-x^3\right ) \log (x)+\left (32 x-32 x^2-x^3+x^4+\left (-32 x+x^3\right ) \log (x)\right ) \log (-1+x+\log (x))} \, dx=2 \,\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (x \right )+x -1\right )-1\right )+2 \,\mathrm {log}\left (-4 \sqrt {2}+x \right )+2 \,\mathrm {log}\left (4 \sqrt {2}+x \right )-2 \,\mathrm {log}\left (x \right ) \] Input:
int((((2*x^2+64)*log(x)+2*x^3-2*x^2+64*x-64)*log(-1+log(x)+x)+(-2*x^2-64)* log(x)+4*x^2-128*x)/(((x^3-32*x)*log(x)+x^4-x^3-32*x^2+32*x)*log(-1+log(x) +x)+(-x^3+32*x)*log(x)-x^4+x^3+32*x^2-32*x),x)
Output:
2*(log(log(log(x) + x - 1) - 1) + log( - 4*sqrt(2) + x) + log(4*sqrt(2) + x) - log(x))