Integrand size = 167, antiderivative size = 30 \[ \int \frac {-144+24 x-72 x^2-72 x^3+\left (72-12 x+36 x^2\right ) \log (4)+\left (-144-96 x+48 x^2+(72-24 x) \log (4)\right ) \log \left (\frac {2+2 x-\log (4)}{x}\right )}{-18 x^2-6 x^3+10 x^4-2 x^5+\left (9 x^2-6 x^3+x^4\right ) \log (4)+\left (-12 x-8 x^2+4 x^3+\left (6 x-2 x^2\right ) \log (4)\right ) \log \left (\frac {2+2 x-\log (4)}{x}\right )+(-2-2 x+\log (4)) \log ^2\left (\frac {2+2 x-\log (4)}{x}\right )} \, dx=\frac {12 (-6+x)}{-3+x-\frac {\log \left (2+\frac {2}{x}-\frac {\log (4)}{x}\right )}{x}} \] Output:
4*(-6+x)/(1/3*x-1/3*ln(2-2*ln(2)/x+2/x)/x-1)
Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {-144+24 x-72 x^2-72 x^3+\left (72-12 x+36 x^2\right ) \log (4)+\left (-144-96 x+48 x^2+(72-24 x) \log (4)\right ) \log \left (\frac {2+2 x-\log (4)}{x}\right )}{-18 x^2-6 x^3+10 x^4-2 x^5+\left (9 x^2-6 x^3+x^4\right ) \log (4)+\left (-12 x-8 x^2+4 x^3+\left (6 x-2 x^2\right ) \log (4)\right ) \log \left (\frac {2+2 x-\log (4)}{x}\right )+(-2-2 x+\log (4)) \log ^2\left (\frac {2+2 x-\log (4)}{x}\right )} \, dx=-\frac {12 (-6+x) x}{3 x-x^2+\log \left (\frac {2+2 x-\log (4)}{x}\right )} \] Input:
Integrate[(-144 + 24*x - 72*x^2 - 72*x^3 + (72 - 12*x + 36*x^2)*Log[4] + ( -144 - 96*x + 48*x^2 + (72 - 24*x)*Log[4])*Log[(2 + 2*x - Log[4])/x])/(-18 *x^2 - 6*x^3 + 10*x^4 - 2*x^5 + (9*x^2 - 6*x^3 + x^4)*Log[4] + (-12*x - 8* x^2 + 4*x^3 + (6*x - 2*x^2)*Log[4])*Log[(2 + 2*x - Log[4])/x] + (-2 - 2*x + Log[4])*Log[(2 + 2*x - Log[4])/x]^2),x]
Output:
(-12*(-6 + x)*x)/(3*x - x^2 + Log[(2 + 2*x - Log[4])/x])
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.
| \(\displaystyle \int \frac {-72 x^3-72 x^2+\left (48 x^2-96 x+(72-24 x) \log (4)-144\right ) \log \left (\frac {2 x+2-\log (4)}{x}\right )+\left (36 x^2-12 x+72\right ) \log (4)+24 x-144}{-2 x^5+10 x^4-6 x^3-18 x^2+\left (4 x^3-8 x^2+\left (6 x-2 x^2\right ) \log (4)-12 x\right ) \log \left (\frac {2 x+2-\log (4)}{x}\right )+\left (x^4-6 x^3+9 x^2\right ) \log (4)+(-2 x-2+\log (4)) \log ^2\left (\frac {2 x+2-\log (4)}{x}\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {12 \left (6 x^3-3 x^2 (\log (4)-2)+x (\log (4)-2)-2 (x-3) (2 x+2-\log (4)) \log \left (\frac {2 x+2-\log (4)}{x}\right )-6 (\log (4)-2)\right )}{(2 x+2-\log (4)) \left ((x-3) x-\log \left (\frac {2 x+2-\log (4)}{x}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 12 \int \frac {6 x^3+3 (2-\log (4)) x^2-(2-\log (4)) x+2 (3-x) (2 x-\log (4)+2) \log \left (\frac {2 x-\log (4)+2}{x}\right )+6 (2-\log (4))}{(2 x-\log (4)+2) \left ((3-x) x+\log \left (\frac {2 x-\log (4)+2}{x}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 12 \int \left (\frac {2 (x-3)}{x^2-3 x-\log \left (\frac {2 x-\log (4)+2}{x}\right )}+\frac {(6-x) \left (4 x^3-2 (1+\log (4)) x^2-3 (2-\log (4)) x-\log (4)+2\right )}{(2 x-\log (4)+2) \left (-x^2+3 x+\log \left (\frac {2 x-\log (4)+2}{x}\right )\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 12 \left (-\frac {1}{2} (2-\log (4)) \int \frac {1}{\left (x^2-3 x-\log \left (\frac {2 x-\log (4)+2}{x}\right )\right )^2}dx-18 \int \frac {x}{\left (x^2-3 x-\log \left (\frac {2 x-\log (4)+2}{x}\right )\right )^2}dx+15 \int \frac {x^2}{\left (x^2-3 x-\log \left (\frac {2 x-\log (4)+2}{x}\right )\right )^2}dx-6 \int \frac {1}{x^2-3 x-\log \left (\frac {2 x-\log (4)+2}{x}\right )}dx+2 \int \frac {x}{x^2-3 x-\log \left (\frac {2 x-\log (4)+2}{x}\right )}dx+\frac {1}{2} (2-\log (4)) (14-\log (4)) \int \frac {1}{(2 x-\log (4)+2) \left (-x^2+3 x+\log \left (\frac {2 x-\log (4)+2}{x}\right )\right )^2}dx-2 \int \frac {x^3}{\left (x^2-3 x-\log \left (\frac {2 x-\log (4)+2}{x}\right )\right )^2}dx\right )\) |
Input:
Int[(-144 + 24*x - 72*x^2 - 72*x^3 + (72 - 12*x + 36*x^2)*Log[4] + (-144 - 96*x + 48*x^2 + (72 - 24*x)*Log[4])*Log[(2 + 2*x - Log[4])/x])/(-18*x^2 - 6*x^3 + 10*x^4 - 2*x^5 + (9*x^2 - 6*x^3 + x^4)*Log[4] + (-12*x - 8*x^2 + 4*x^3 + (6*x - 2*x^2)*Log[4])*Log[(2 + 2*x - Log[4])/x] + (-2 - 2*x + Log[ 4])*Log[(2 + 2*x - Log[4])/x]^2),x]
Output:
$Aborted
Time = 1.88 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07
| method | result | size |
| risch | \(\frac {12 \left (-6+x \right ) x}{x^{2}-3 x -\ln \left (\frac {-2 \ln \left (2\right )+2 x +2}{x}\right )}\) | \(32\) |
| parallelrisch | \(-\frac {-12 x^{2}+72 x}{x^{2}-3 x -\ln \left (-\frac {2 \left (\ln \left (2\right )-x -1\right )}{x}\right )}\) | \(36\) |
| norman | \(\frac {12 \ln \left (\frac {-2 \ln \left (2\right )+2 x +2}{x}\right )-36 x}{x^{2}-3 x -\ln \left (\frac {-2 \ln \left (2\right )+2 x +2}{x}\right )}\) | \(47\) |
| derivativedivides | \(-\frac {24 \left (2-2 \ln \left (2\right )\right ) \left (-1+\frac {-6 \ln \left (2\right )+6}{x}+\ln \left (2\right )\right )}{-\ln \left (2+\frac {2-2 \ln \left (2\right )}{x}\right ) \left (2+\frac {2-2 \ln \left (2\right )}{x}\right )^{2}+4 \ln \left (2\right )^{2}+6 \ln \left (2\right ) \left (2+\frac {2-2 \ln \left (2\right )}{x}\right )+4 \left (2+\frac {2-2 \ln \left (2\right )}{x}\right ) \ln \left (2+\frac {2-2 \ln \left (2\right )}{x}\right )-20 \ln \left (2\right )-4 \ln \left (2+\frac {2-2 \ln \left (2\right )}{x}\right )+4-\frac {6 \left (2-2 \ln \left (2\right )\right )}{x}}\) | \(136\) |
| default | \(-\frac {24 \left (2-2 \ln \left (2\right )\right ) \left (-1+\frac {-6 \ln \left (2\right )+6}{x}+\ln \left (2\right )\right )}{-\ln \left (2+\frac {2-2 \ln \left (2\right )}{x}\right ) \left (2+\frac {2-2 \ln \left (2\right )}{x}\right )^{2}+4 \ln \left (2\right )^{2}+6 \ln \left (2\right ) \left (2+\frac {2-2 \ln \left (2\right )}{x}\right )+4 \left (2+\frac {2-2 \ln \left (2\right )}{x}\right ) \ln \left (2+\frac {2-2 \ln \left (2\right )}{x}\right )-20 \ln \left (2\right )-4 \ln \left (2+\frac {2-2 \ln \left (2\right )}{x}\right )+4-\frac {6 \left (2-2 \ln \left (2\right )\right )}{x}}\) | \(136\) |
Input:
int(((2*(-24*x+72)*ln(2)+48*x^2-96*x-144)*ln((-2*ln(2)+2*x+2)/x)+2*(36*x^2 -12*x+72)*ln(2)-72*x^3-72*x^2+24*x-144)/((2*ln(2)-2*x-2)*ln((-2*ln(2)+2*x+ 2)/x)^2+(2*(-2*x^2+6*x)*ln(2)+4*x^3-8*x^2-12*x)*ln((-2*ln(2)+2*x+2)/x)+2*( x^4-6*x^3+9*x^2)*ln(2)-2*x^5+10*x^4-6*x^3-18*x^2),x,method=_RETURNVERBOSE)
Output:
12*(-6+x)*x/(x^2-3*x-ln((-2*ln(2)+2*x+2)/x))
Time = 0.08 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.10 \[ \int \frac {-144+24 x-72 x^2-72 x^3+\left (72-12 x+36 x^2\right ) \log (4)+\left (-144-96 x+48 x^2+(72-24 x) \log (4)\right ) \log \left (\frac {2+2 x-\log (4)}{x}\right )}{-18 x^2-6 x^3+10 x^4-2 x^5+\left (9 x^2-6 x^3+x^4\right ) \log (4)+\left (-12 x-8 x^2+4 x^3+\left (6 x-2 x^2\right ) \log (4)\right ) \log \left (\frac {2+2 x-\log (4)}{x}\right )+(-2-2 x+\log (4)) \log ^2\left (\frac {2+2 x-\log (4)}{x}\right )} \, dx=\frac {12 \, {\left (x^{2} - 6 \, x\right )}}{x^{2} - 3 \, x - \log \left (\frac {2 \, {\left (x - \log \left (2\right ) + 1\right )}}{x}\right )} \] Input:
integrate(((2*(-24*x+72)*log(2)+48*x^2-96*x-144)*log((-2*log(2)+2*x+2)/x)+ 2*(36*x^2-12*x+72)*log(2)-72*x^3-72*x^2+24*x-144)/((2*log(2)-2*x-2)*log((- 2*log(2)+2*x+2)/x)^2+(2*(-2*x^2+6*x)*log(2)+4*x^3-8*x^2-12*x)*log((-2*log( 2)+2*x+2)/x)+2*(x^4-6*x^3+9*x^2)*log(2)-2*x^5+10*x^4-6*x^3-18*x^2),x, algo rithm="fricas")
Output:
12*(x^2 - 6*x)/(x^2 - 3*x - log(2*(x - log(2) + 1)/x))
Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \[ \int \frac {-144+24 x-72 x^2-72 x^3+\left (72-12 x+36 x^2\right ) \log (4)+\left (-144-96 x+48 x^2+(72-24 x) \log (4)\right ) \log \left (\frac {2+2 x-\log (4)}{x}\right )}{-18 x^2-6 x^3+10 x^4-2 x^5+\left (9 x^2-6 x^3+x^4\right ) \log (4)+\left (-12 x-8 x^2+4 x^3+\left (6 x-2 x^2\right ) \log (4)\right ) \log \left (\frac {2+2 x-\log (4)}{x}\right )+(-2-2 x+\log (4)) \log ^2\left (\frac {2+2 x-\log (4)}{x}\right )} \, dx=\frac {- 12 x^{2} + 72 x}{- x^{2} + 3 x + \log {\left (\frac {2 x - 2 \log {\left (2 \right )} + 2}{x} \right )}} \] Input:
integrate(((2*(-24*x+72)*ln(2)+48*x**2-96*x-144)*ln((-2*ln(2)+2*x+2)/x)+2* (36*x**2-12*x+72)*ln(2)-72*x**3-72*x**2+24*x-144)/((2*ln(2)-2*x-2)*ln((-2* ln(2)+2*x+2)/x)**2+(2*(-2*x**2+6*x)*ln(2)+4*x**3-8*x**2-12*x)*ln((-2*ln(2) +2*x+2)/x)+2*(x**4-6*x**3+9*x**2)*ln(2)-2*x**5+10*x**4-6*x**3-18*x**2),x)
Output:
(-12*x**2 + 72*x)/(-x**2 + 3*x + log((2*x - 2*log(2) + 2)/x))
Time = 0.16 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {-144+24 x-72 x^2-72 x^3+\left (72-12 x+36 x^2\right ) \log (4)+\left (-144-96 x+48 x^2+(72-24 x) \log (4)\right ) \log \left (\frac {2+2 x-\log (4)}{x}\right )}{-18 x^2-6 x^3+10 x^4-2 x^5+\left (9 x^2-6 x^3+x^4\right ) \log (4)+\left (-12 x-8 x^2+4 x^3+\left (6 x-2 x^2\right ) \log (4)\right ) \log \left (\frac {2+2 x-\log (4)}{x}\right )+(-2-2 x+\log (4)) \log ^2\left (\frac {2+2 x-\log (4)}{x}\right )} \, dx=\frac {12 \, {\left (x^{2} - 6 \, x\right )}}{x^{2} - 3 \, x - \log \left (2\right ) - \log \left (x - \log \left (2\right ) + 1\right ) + \log \left (x\right )} \] Input:
integrate(((2*(-24*x+72)*log(2)+48*x^2-96*x-144)*log((-2*log(2)+2*x+2)/x)+ 2*(36*x^2-12*x+72)*log(2)-72*x^3-72*x^2+24*x-144)/((2*log(2)-2*x-2)*log((- 2*log(2)+2*x+2)/x)^2+(2*(-2*x^2+6*x)*log(2)+4*x^3-8*x^2-12*x)*log((-2*log( 2)+2*x+2)/x)+2*(x^4-6*x^3+9*x^2)*log(2)-2*x^5+10*x^4-6*x^3-18*x^2),x, algo rithm="maxima")
Output:
12*(x^2 - 6*x)/(x^2 - 3*x - log(2) - log(x - log(2) + 1) + log(x))
Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (30) = 60\).
Time = 0.38 (sec) , antiderivative size = 171, normalized size of antiderivative = 5.70 \[ \int \frac {-144+24 x-72 x^2-72 x^3+\left (72-12 x+36 x^2\right ) \log (4)+\left (-144-96 x+48 x^2+(72-24 x) \log (4)\right ) \log \left (\frac {2+2 x-\log (4)}{x}\right )}{-18 x^2-6 x^3+10 x^4-2 x^5+\left (9 x^2-6 x^3+x^4\right ) \log (4)+\left (-12 x-8 x^2+4 x^3+\left (6 x-2 x^2\right ) \log (4)\right ) \log \left (\frac {2+2 x-\log (4)}{x}\right )+(-2-2 x+\log (4)) \log ^2\left (\frac {2+2 x-\log (4)}{x}\right )} \, dx=\frac {12 \, {\left (\log \left (2\right )^{3} + \frac {6 \, {\left (x - \log \left (2\right ) + 1\right )} \log \left (2\right )^{2}}{x} - 9 \, \log \left (2\right )^{2} - \frac {12 \, {\left (x - \log \left (2\right ) + 1\right )} \log \left (2\right )}{x} + \frac {6 \, {\left (x - \log \left (2\right ) + 1\right )}}{x} + 15 \, \log \left (2\right ) - 7\right )}}{{\left (\log \left (2\right )^{2} + \frac {3 \, {\left (x - \log \left (2\right ) + 1\right )} \log \left (2\right )}{x} - \frac {{\left (x - \log \left (2\right ) + 1\right )}^{2} \log \left (\frac {2 \, {\left (x - \log \left (2\right ) + 1\right )}}{x}\right )}{x^{2}} + \frac {2 \, {\left (x - \log \left (2\right ) + 1\right )} \log \left (\frac {2 \, {\left (x - \log \left (2\right ) + 1\right )}}{x}\right )}{x} - \frac {3 \, {\left (x - \log \left (2\right ) + 1\right )}}{x} - 5 \, \log \left (2\right ) - \log \left (\frac {2 \, {\left (x - \log \left (2\right ) + 1\right )}}{x}\right ) + 4\right )} {\left (\log \left (2\right ) - 1\right )}} \] Input:
integrate(((2*(-24*x+72)*log(2)+48*x^2-96*x-144)*log((-2*log(2)+2*x+2)/x)+ 2*(36*x^2-12*x+72)*log(2)-72*x^3-72*x^2+24*x-144)/((2*log(2)-2*x-2)*log((- 2*log(2)+2*x+2)/x)^2+(2*(-2*x^2+6*x)*log(2)+4*x^3-8*x^2-12*x)*log((-2*log( 2)+2*x+2)/x)+2*(x^4-6*x^3+9*x^2)*log(2)-2*x^5+10*x^4-6*x^3-18*x^2),x, algo rithm="giac")
Output:
12*(log(2)^3 + 6*(x - log(2) + 1)*log(2)^2/x - 9*log(2)^2 - 12*(x - log(2) + 1)*log(2)/x + 6*(x - log(2) + 1)/x + 15*log(2) - 7)/((log(2)^2 + 3*(x - log(2) + 1)*log(2)/x - (x - log(2) + 1)^2*log(2*(x - log(2) + 1)/x)/x^2 + 2*(x - log(2) + 1)*log(2*(x - log(2) + 1)/x)/x - 3*(x - log(2) + 1)/x - 5 *log(2) - log(2*(x - log(2) + 1)/x) + 4)*(log(2) - 1))
Timed out. \[ \int \frac {-144+24 x-72 x^2-72 x^3+\left (72-12 x+36 x^2\right ) \log (4)+\left (-144-96 x+48 x^2+(72-24 x) \log (4)\right ) \log \left (\frac {2+2 x-\log (4)}{x}\right )}{-18 x^2-6 x^3+10 x^4-2 x^5+\left (9 x^2-6 x^3+x^4\right ) \log (4)+\left (-12 x-8 x^2+4 x^3+\left (6 x-2 x^2\right ) \log (4)\right ) \log \left (\frac {2+2 x-\log (4)}{x}\right )+(-2-2 x+\log (4)) \log ^2\left (\frac {2+2 x-\log (4)}{x}\right )} \, dx=\int \frac {\ln \left (\frac {2\,x-2\,\ln \left (2\right )+2}{x}\right )\,\left (96\,x+2\,\ln \left (2\right )\,\left (24\,x-72\right )-48\,x^2+144\right )-2\,\ln \left (2\right )\,\left (36\,x^2-12\,x+72\right )-24\,x+72\,x^2+72\,x^3+144}{{\ln \left (\frac {2\,x-2\,\ln \left (2\right )+2}{x}\right )}^2\,\left (2\,x-2\,\ln \left (2\right )+2\right )-2\,\ln \left (2\right )\,\left (x^4-6\,x^3+9\,x^2\right )+18\,x^2+6\,x^3-10\,x^4+2\,x^5+\ln \left (\frac {2\,x-2\,\ln \left (2\right )+2}{x}\right )\,\left (12\,x-2\,\ln \left (2\right )\,\left (6\,x-2\,x^2\right )+8\,x^2-4\,x^3\right )} \,d x \] Input:
int((log((2*x - 2*log(2) + 2)/x)*(96*x + 2*log(2)*(24*x - 72) - 48*x^2 + 1 44) - 2*log(2)*(36*x^2 - 12*x + 72) - 24*x + 72*x^2 + 72*x^3 + 144)/(log(( 2*x - 2*log(2) + 2)/x)^2*(2*x - 2*log(2) + 2) - 2*log(2)*(9*x^2 - 6*x^3 + x^4) + 18*x^2 + 6*x^3 - 10*x^4 + 2*x^5 + log((2*x - 2*log(2) + 2)/x)*(12*x - 2*log(2)*(6*x - 2*x^2) + 8*x^2 - 4*x^3)),x)
Output:
int((log((2*x - 2*log(2) + 2)/x)*(96*x + 2*log(2)*(24*x - 72) - 48*x^2 + 1 44) - 2*log(2)*(36*x^2 - 12*x + 72) - 24*x + 72*x^2 + 72*x^3 + 144)/(log(( 2*x - 2*log(2) + 2)/x)^2*(2*x - 2*log(2) + 2) - 2*log(2)*(9*x^2 - 6*x^3 + x^4) + 18*x^2 + 6*x^3 - 10*x^4 + 2*x^5 + log((2*x - 2*log(2) + 2)/x)*(12*x - 2*log(2)*(6*x - 2*x^2) + 8*x^2 - 4*x^3)), x)
Time = 0.16 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.57 \[ \int \frac {-144+24 x-72 x^2-72 x^3+\left (72-12 x+36 x^2\right ) \log (4)+\left (-144-96 x+48 x^2+(72-24 x) \log (4)\right ) \log \left (\frac {2+2 x-\log (4)}{x}\right )}{-18 x^2-6 x^3+10 x^4-2 x^5+\left (9 x^2-6 x^3+x^4\right ) \log (4)+\left (-12 x-8 x^2+4 x^3+\left (6 x-2 x^2\right ) \log (4)\right ) \log \left (\frac {2+2 x-\log (4)}{x}\right )+(-2-2 x+\log (4)) \log ^2\left (\frac {2+2 x-\log (4)}{x}\right )} \, dx=\frac {-12 \,\mathrm {log}\left (\frac {-2 \,\mathrm {log}\left (2\right )+2 x +2}{x}\right )+36 x}{\mathrm {log}\left (\frac {-2 \,\mathrm {log}\left (2\right )+2 x +2}{x}\right )-x^{2}+3 x} \] Input:
int(((2*(-24*x+72)*log(2)+48*x^2-96*x-144)*log((-2*log(2)+2*x+2)/x)+2*(36* x^2-12*x+72)*log(2)-72*x^3-72*x^2+24*x-144)/((2*log(2)-2*x-2)*log((-2*log( 2)+2*x+2)/x)^2+(2*(-2*x^2+6*x)*log(2)+4*x^3-8*x^2-12*x)*log((-2*log(2)+2*x +2)/x)+2*(x^4-6*x^3+9*x^2)*log(2)-2*x^5+10*x^4-6*x^3-18*x^2),x)
Output:
(12*( - log(( - 2*log(2) + 2*x + 2)/x) + 3*x))/(log(( - 2*log(2) + 2*x + 2 )/x) - x**2 + 3*x)