Integrand size = 26, antiderivative size = 24 \[ \int \frac {x^2+(4-4 x) \log (2) \log (x)-2 \log (2) \log ^2(x)}{x^2} \, dx=x+\frac {\log (2)}{2}-\frac {(-2+2 x) \log (2) \log ^2(x)}{x} \]
[Out]
Leaf count is larger than twice the leaf count of optimal. \(52\) vs. \(2(24)=48\).
Time = 0.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.17, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {14, 45, 2372, 2338, 2342, 2341} \[ \int \frac {x^2+(4-4 x) \log (2) \log (x)-2 \log (2) \log ^2(x)}{x^2} \, dx=x+\frac {\log (4) \log ^2(x)}{x}-2 \log (2) \log ^2(x)+\frac {2 \log (4) \log (x)}{x}-\frac {4 \log (2) \log (x)}{x}+\frac {2 \log (4)}{x}-\frac {4 \log (2)}{x} \]
[In]
[Out]
Rule 14
Rule 45
Rule 2338
Rule 2341
Rule 2342
Rule 2372
Rubi steps \begin{align*} \text {integral}& = \int \left (1-\frac {4 (-1+x) \log (2) \log (x)}{x^2}-\frac {\log (4) \log ^2(x)}{x^2}\right ) \, dx \\ & = x-(4 \log (2)) \int \frac {(-1+x) \log (x)}{x^2} \, dx-\log (4) \int \frac {\log ^2(x)}{x^2} \, dx \\ & = x-\frac {4 \log (2) \log (x)}{x}-4 \log (2) \log ^2(x)+\frac {\log (4) \log ^2(x)}{x}+(4 \log (2)) \int \frac {1+x \log (x)}{x^2} \, dx-(2 \log (4)) \int \frac {\log (x)}{x^2} \, dx \\ & = x+\frac {2 \log (4)}{x}-\frac {4 \log (2) \log (x)}{x}+\frac {2 \log (4) \log (x)}{x}-4 \log (2) \log ^2(x)+\frac {\log (4) \log ^2(x)}{x}+(4 \log (2)) \int \left (\frac {1}{x^2}+\frac {\log (x)}{x}\right ) \, dx \\ & = x-\frac {4 \log (2)}{x}+\frac {2 \log (4)}{x}-\frac {4 \log (2) \log (x)}{x}+\frac {2 \log (4) \log (x)}{x}-4 \log (2) \log ^2(x)+\frac {\log (4) \log ^2(x)}{x}+(4 \log (2)) \int \frac {\log (x)}{x} \, dx \\ & = x-\frac {4 \log (2)}{x}+\frac {2 \log (4)}{x}-\frac {4 \log (2) \log (x)}{x}+\frac {2 \log (4) \log (x)}{x}-2 \log (2) \log ^2(x)+\frac {\log (4) \log ^2(x)}{x} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(52\) vs. \(2(24)=48\).
Time = 0.01 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.17 \[ \int \frac {x^2+(4-4 x) \log (2) \log (x)-2 \log (2) \log ^2(x)}{x^2} \, dx=x-\frac {4 \log (2)}{x}+\frac {2 \log (4)}{x}-\frac {4 \log (2) \log (x)}{x}+\frac {2 \log (4) \log (x)}{x}-2 \log (2) \log ^2(x)+\frac {\log (4) \log ^2(x)}{x} \]
[In]
[Out]
Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71
method | result | size |
risch | \(-\frac {2 \left (-1+x \right ) \ln \left (2\right ) \ln \left (x \right )^{2}}{x}+x\) | \(17\) |
norman | \(\frac {x^{2}+2 \ln \left (2\right ) \ln \left (x \right )^{2}-2 x \ln \left (2\right ) \ln \left (x \right )^{2}}{x}\) | \(26\) |
parts | \(x -2 \ln \left (2\right ) \left (-\frac {\ln \left (x \right )^{2}}{x}-\frac {2 \ln \left (x \right )}{x}-\frac {2}{x}\right )-4 \ln \left (2\right ) \left (\frac {\ln \left (x \right )}{x}+\frac {1}{x}+\frac {\ln \left (x \right )^{2}}{2}\right )\) | \(49\) |
default | \(-2 \ln \left (2\right ) \left (-\frac {\ln \left (x \right )^{2}}{x}-\frac {2 \ln \left (x \right )}{x}-\frac {2}{x}\right )-2 \ln \left (2\right ) \ln \left (x \right )^{2}+4 \ln \left (2\right ) \left (-\frac {\ln \left (x \right )}{x}-\frac {1}{x}\right )+x\) | \(54\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {x^2+(4-4 x) \log (2) \log (x)-2 \log (2) \log ^2(x)}{x^2} \, dx=-\frac {2 \, {\left (x - 1\right )} \log \left (2\right ) \log \left (x\right )^{2} - x^{2}}{x} \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {x^2+(4-4 x) \log (2) \log (x)-2 \log (2) \log ^2(x)}{x^2} \, dx=x + \frac {\left (- 2 x \log {\left (2 \right )} + 2 \log {\left (2 \right )}\right ) \log {\left (x \right )}^{2}}{x} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (20) = 40\).
Time = 0.20 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.71 \[ \int \frac {x^2+(4-4 x) \log (2) \log (x)-2 \log (2) \log ^2(x)}{x^2} \, dx=-2 \, \log \left (2\right ) \log \left (x\right )^{2} - 4 \, {\left (\frac {\log \left (x\right )}{x} + \frac {1}{x}\right )} \log \left (2\right ) + x + \frac {2 \, {\left (\log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 2\right )} \log \left (2\right )}{x} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {x^2+(4-4 x) \log (2) \log (x)-2 \log (2) \log ^2(x)}{x^2} \, dx=2 \, {\left (\frac {\log \left (2\right )}{x} - \log \left (2\right )\right )} \log \left (x\right )^{2} + x \]
[In]
[Out]
Time = 8.43 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int \frac {x^2+(4-4 x) \log (2) \log (x)-2 \log (2) \log ^2(x)}{x^2} \, dx=x-2\,\ln \left (2\right )\,{\ln \left (x\right )}^2+\frac {2\,\ln \left (2\right )\,{\ln \left (x\right )}^2}{x} \]
[In]
[Out]