Integrand size = 33, antiderivative size = 27 \[ \int \frac {8-6 x-x \log (4)}{-4 x+6 x^2+80 x^3+x^2 \log (4)} \, dx=\log \left (8 \left (5+\frac {x+\frac {1}{4} (-4+2 x+x \log (4))}{4 x^2}\right )\right ) \]
[Out]
Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {6, 1608, 814, 642} \[ \int \frac {8-6 x-x \log (4)}{-4 x+6 x^2+80 x^3+x^2 \log (4)} \, dx=\log \left (-80 x^2-x (6+\log (4))+4\right )-2 \log (x) \]
[In]
[Out]
Rule 6
Rule 642
Rule 814
Rule 1608
Rubi steps \begin{align*} \text {integral}& = \int \frac {8+x (-6-\log (4))}{-4 x+6 x^2+80 x^3+x^2 \log (4)} \, dx \\ & = \int \frac {8+x (-6-\log (4))}{-4 x+80 x^3+x^2 (6+\log (4))} \, dx \\ & = \int \frac {8+x (-6-\log (4))}{x \left (-4+80 x^2+x (6+\log (4))\right )} \, dx \\ & = \int \left (-\frac {2}{x}+\frac {-6-160 x-\log (4)}{4-80 x^2-x (6+\log (4))}\right ) \, dx \\ & = -2 \log (x)+\int \frac {-6-160 x-\log (4)}{4-80 x^2+x (-6-\log (4))} \, dx \\ & = -2 \log (x)+\log \left (4-80 x^2-x (6+\log (4))\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {8-6 x-x \log (4)}{-4 x+6 x^2+80 x^3+x^2 \log (4)} \, dx=-2 \log (x)+\log \left (4-6 x-80 x^2-x \log (4)\right ) \]
[In]
[Out]
Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74
method | result | size |
risch | \(-2 \ln \left (x \right )+\ln \left (-2+40 x^{2}+x \left (3+\ln \left (2\right )\right )\right )\) | \(20\) |
parallelrisch | \(-2 \ln \left (x \right )+\ln \left (\frac {x \ln \left (2\right )}{40}+x^{2}+\frac {3 x}{40}-\frac {1}{20}\right )\) | \(20\) |
default | \(\ln \left (x \ln \left (2\right )+40 x^{2}+3 x -2\right )-2 \ln \left (x \right )\) | \(21\) |
norman | \(\ln \left (x \ln \left (2\right )+40 x^{2}+3 x -2\right )-2 \ln \left (x \right )\) | \(21\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {8-6 x-x \log (4)}{-4 x+6 x^2+80 x^3+x^2 \log (4)} \, dx=\log \left (40 \, x^{2} + x \log \left (2\right ) + 3 \, x - 2\right ) - 2 \, \log \left (x\right ) \]
[In]
[Out]
Time = 0.34 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {8-6 x-x \log (4)}{-4 x+6 x^2+80 x^3+x^2 \log (4)} \, dx=- 2 \log {\left (x \right )} + \log {\left (x^{2} + x \left (\frac {\log {\left (2 \right )}}{40} + \frac {3}{40}\right ) - \frac {1}{20} \right )} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {8-6 x-x \log (4)}{-4 x+6 x^2+80 x^3+x^2 \log (4)} \, dx=\log \left (40 \, x^{2} + x {\left (\log \left (2\right ) + 3\right )} - 2\right ) - 2 \, \log \left (x\right ) \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {8-6 x-x \log (4)}{-4 x+6 x^2+80 x^3+x^2 \log (4)} \, dx=\log \left ({\left | 40 \, x^{2} + x \log \left (2\right ) + 3 \, x - 2 \right |}\right ) - 2 \, \log \left ({\left | x \right |}\right ) \]
[In]
[Out]
Time = 14.88 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {8-6 x-x \log (4)}{-4 x+6 x^2+80 x^3+x^2 \log (4)} \, dx=\ln \left (6-3\,x\,\ln \left (2\right )-120\,x^2-9\,x\right )-2\,\ln \left (x\right ) \]
[In]
[Out]