Integrand size = 88, antiderivative size = 22 \[ \int \frac {-4 x \log (x)+8 \log \left (\frac {4}{\log ^2(x)}\right )-2 \log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )+\left (2 x \log (x)+2 \log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )\right ) \log \left (\frac {2}{x^2+x \log ^2\left (\frac {4}{\log ^2(x)}\right )}\right )}{x \log (x)+\log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )} \, dx=2 x \log \left (\frac {2}{x \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}\right ) \] Output:
2*x*ln(2/(x+ln(4/ln(x)^2)^2)/x)
Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {-4 x \log (x)+8 \log \left (\frac {4}{\log ^2(x)}\right )-2 \log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )+\left (2 x \log (x)+2 \log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )\right ) \log \left (\frac {2}{x^2+x \log ^2\left (\frac {4}{\log ^2(x)}\right )}\right )}{x \log (x)+\log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )} \, dx=2 x \log \left (\frac {2}{x \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}\right ) \] Input:
Integrate[(-4*x*Log[x] + 8*Log[4/Log[x]^2] - 2*Log[x]*Log[4/Log[x]^2]^2 + (2*x*Log[x] + 2*Log[x]*Log[4/Log[x]^2]^2)*Log[2/(x^2 + x*Log[4/Log[x]^2]^2 )])/(x*Log[x] + Log[x]*Log[4/Log[x]^2]^2),x]
Output:
2*x*Log[2/(x*(x + Log[4/Log[x]^2]^2))]
Time = 1.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, 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.
\(\displaystyle \int \frac {\left (2 \log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )+2 x \log (x)\right ) \log \left (\frac {2}{x^2+x \log ^2\left (\frac {4}{\log ^2(x)}\right )}\right )-2 \log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )+8 \log \left (\frac {4}{\log ^2(x)}\right )-4 x \log (x)}{\log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )+x \log (x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {\left (2 \log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )+2 x \log (x)\right ) \log \left (\frac {2}{x^2+x \log ^2\left (\frac {4}{\log ^2(x)}\right )}\right )-2 \log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )+8 \log \left (\frac {4}{\log ^2(x)}\right )-4 x \log (x)}{\log (x) \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (2 \log \left (\frac {2}{x \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}\right )-\frac {2 \left (\log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )-4 \log \left (\frac {4}{\log ^2(x)}\right )+2 x \log (x)\right )}{\log (x) \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 x \log \left (\frac {2}{x \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}\right )\) |
Input:
Int[(-4*x*Log[x] + 8*Log[4/Log[x]^2] - 2*Log[x]*Log[4/Log[x]^2]^2 + (2*x*L og[x] + 2*Log[x]*Log[4/Log[x]^2]^2)*Log[2/(x^2 + x*Log[4/Log[x]^2]^2)])/(x *Log[x] + Log[x]*Log[4/Log[x]^2]^2),x]
Output:
2*x*Log[2/(x*(x + Log[4/Log[x]^2]^2))]
Time = 3.60 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05
method | result | size |
parallelrisch | \(2 x \ln \left (\frac {2}{\left (x +\ln \left (\frac {4}{\ln \left (x \right )^{2}}\right )^{2}\right ) x}\right )\) | \(23\) |
risch | \(\text {Expression too large to display}\) | \(1840\) |
Input:
int(((2*ln(x)*ln(4/ln(x)^2)^2+2*x*ln(x))*ln(2/(x*ln(4/ln(x)^2)^2+x^2))-2*l n(x)*ln(4/ln(x)^2)^2+8*ln(4/ln(x)^2)-4*x*ln(x))/(ln(x)*ln(4/ln(x)^2)^2+x*l n(x)),x,method=_RETURNVERBOSE)
Output:
2*x*ln(2/(x+ln(4/ln(x)^2)^2)/x)
Time = 0.09 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {-4 x \log (x)+8 \log \left (\frac {4}{\log ^2(x)}\right )-2 \log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )+\left (2 x \log (x)+2 \log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )\right ) \log \left (\frac {2}{x^2+x \log ^2\left (\frac {4}{\log ^2(x)}\right )}\right )}{x \log (x)+\log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )} \, dx=2 \, x \log \left (\frac {2}{x \log \left (\frac {4}{\log \left (x\right )^{2}}\right )^{2} + x^{2}}\right ) \] Input:
integrate(((2*log(x)*log(4/log(x)^2)^2+2*x*log(x))*log(2/(x*log(4/log(x)^2 )^2+x^2))-2*log(x)*log(4/log(x)^2)^2+8*log(4/log(x)^2)-4*x*log(x))/(log(x) *log(4/log(x)^2)^2+x*log(x)),x, algorithm="fricas")
Output:
2*x*log(2/(x*log(4/log(x)^2)^2 + x^2))
Time = 0.44 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {-4 x \log (x)+8 \log \left (\frac {4}{\log ^2(x)}\right )-2 \log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )+\left (2 x \log (x)+2 \log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )\right ) \log \left (\frac {2}{x^2+x \log ^2\left (\frac {4}{\log ^2(x)}\right )}\right )}{x \log (x)+\log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )} \, dx=2 x \log {\left (\frac {2}{x^{2} + x \log {\left (\frac {4}{\log {\left (x \right )}^{2}} \right )}^{2}} \right )} \] Input:
integrate(((2*ln(x)*ln(4/ln(x)**2)**2+2*x*ln(x))*ln(2/(x*ln(4/ln(x)**2)**2 +x**2))-2*ln(x)*ln(4/ln(x)**2)**2+8*ln(4/ln(x)**2)-4*x*ln(x))/(ln(x)*ln(4/ ln(x)**2)**2+x*ln(x)),x)
Output:
2*x*log(2/(x**2 + x*log(4/log(x)**2)**2))
Time = 0.16 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68 \[ \int \frac {-4 x \log (x)+8 \log \left (\frac {4}{\log ^2(x)}\right )-2 \log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )+\left (2 x \log (x)+2 \log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )\right ) \log \left (\frac {2}{x^2+x \log ^2\left (\frac {4}{\log ^2(x)}\right )}\right )}{x \log (x)+\log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )} \, dx=2 \, x \log \left (2\right ) - 2 \, x \log \left (4 \, \log \left (2\right )^{2} - 8 \, \log \left (2\right ) \log \left (\log \left (x\right )\right ) + 4 \, \log \left (\log \left (x\right )\right )^{2} + x\right ) - 2 \, x \log \left (x\right ) \] Input:
integrate(((2*log(x)*log(4/log(x)^2)^2+2*x*log(x))*log(2/(x*log(4/log(x)^2 )^2+x^2))-2*log(x)*log(4/log(x)^2)^2+8*log(4/log(x)^2)-4*x*log(x))/(log(x) *log(4/log(x)^2)^2+x*log(x)),x, algorithm="maxima")
Output:
2*x*log(2) - 2*x*log(4*log(2)^2 - 8*log(2)*log(log(x)) + 4*log(log(x))^2 + x) - 2*x*log(x)
Time = 0.79 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.77 \[ \int \frac {-4 x \log (x)+8 \log \left (\frac {4}{\log ^2(x)}\right )-2 \log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )+\left (2 x \log (x)+2 \log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )\right ) \log \left (\frac {2}{x^2+x \log ^2\left (\frac {4}{\log ^2(x)}\right )}\right )}{x \log (x)+\log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )} \, dx=2 \, x \log \left (2\right ) - 2 \, x \log \left (4 \, \log \left (2\right )^{2} - 4 \, \log \left (2\right ) \log \left (\log \left (x\right )^{2}\right ) + \log \left (\log \left (x\right )^{2}\right )^{2} + x\right ) - 2 \, x \log \left (x\right ) \] Input:
integrate(((2*log(x)*log(4/log(x)^2)^2+2*x*log(x))*log(2/(x*log(4/log(x)^2 )^2+x^2))-2*log(x)*log(4/log(x)^2)^2+8*log(4/log(x)^2)-4*x*log(x))/(log(x) *log(4/log(x)^2)^2+x*log(x)),x, algorithm="giac")
Output:
2*x*log(2) - 2*x*log(4*log(2)^2 - 4*log(2)*log(log(x)^2) + log(log(x)^2)^2 + x) - 2*x*log(x)
Time = 9.95 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {-4 x \log (x)+8 \log \left (\frac {4}{\log ^2(x)}\right )-2 \log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )+\left (2 x \log (x)+2 \log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )\right ) \log \left (\frac {2}{x^2+x \log ^2\left (\frac {4}{\log ^2(x)}\right )}\right )}{x \log (x)+\log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )} \, dx=2\,x\,\left (\ln \left (\frac {1}{x^2+x\,{\ln \left (\frac {4}{{\ln \left (x\right )}^2}\right )}^2}\right )+\ln \left (2\right )\right ) \] Input:
int((8*log(4/log(x)^2) - 2*log(4/log(x)^2)^2*log(x) - 4*x*log(x) + log(2/( x*log(4/log(x)^2)^2 + x^2))*(2*log(4/log(x)^2)^2*log(x) + 2*x*log(x)))/(lo g(4/log(x)^2)^2*log(x) + x*log(x)),x)
Output:
2*x*(log(1/(x*log(4/log(x)^2)^2 + x^2)) + log(2))
\[ \int \frac {-4 x \log (x)+8 \log \left (\frac {4}{\log ^2(x)}\right )-2 \log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )+\left (2 x \log (x)+2 \log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )\right ) \log \left (\frac {2}{x^2+x \log ^2\left (\frac {4}{\log ^2(x)}\right )}\right )}{x \log (x)+\log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )} \, dx=\int \frac {\left (2 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (\frac {4}{\mathrm {log}\left (x \right )^{2}}\right )^{2}+2 \,\mathrm {log}\left (x \right ) x \right ) \mathrm {log}\left (\frac {2}{x \mathrm {log}\left (\frac {4}{\mathrm {log}\left (x \right )^{2}}\right )^{2}+x^{2}}\right )-2 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (\frac {4}{\mathrm {log}\left (x \right )^{2}}\right )^{2}+8 \,\mathrm {log}\left (\frac {4}{\mathrm {log}\left (x \right )^{2}}\right )-4 \,\mathrm {log}\left (x \right ) x}{\mathrm {log}\left (x \right ) \mathrm {log}\left (\frac {4}{\mathrm {log}\left (x \right )^{2}}\right )^{2}+\mathrm {log}\left (x \right ) x}d x \] Input:
int(((2*log(x)*log(4/log(x)^2)^2+2*x*log(x))*log(2/(x*log(4/log(x)^2)^2+x^ 2))-2*log(x)*log(4/log(x)^2)^2+8*log(4/log(x)^2)-4*x*log(x))/(log(x)*log(4 /log(x)^2)^2+x*log(x)),x)
Output:
int(((2*log(x)*log(4/log(x)^2)^2+2*x*log(x))*log(2/(x*log(4/log(x)^2)^2+x^ 2))-2*log(x)*log(4/log(x)^2)^2+8*log(4/log(x)^2)-4*x*log(x))/(log(x)*log(4 /log(x)^2)^2+x*log(x)),x)