Integrand size = 53, antiderivative size = 27 \[ \int \frac {-3+6 x+6 x^2+e \left (-1+2 x+2 x^2\right )+(3 x+e x) \log (2 x)}{8 x+4 x^2+2 x \log (2 x)} \, dx=\frac {1}{2} \left (6+e^5+(3+e) (x-\log (4+2 x+\log (2 x)))\right ) \]
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Time = 0.17 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {6820, 12, 6874, 6816} \[ \int \frac {-3+6 x+6 x^2+e \left (-1+2 x+2 x^2\right )+(3 x+e x) \log (2 x)}{8 x+4 x^2+2 x \log (2 x)} \, dx=\frac {1}{2} (3+e) x-\frac {1}{2} (3+e) \log (2 x+\log (2 x)+4) \]
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Rule 12
Rule 6816
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {(3+e) \left (-1+2 x+2 x^2+x \log (2 x)\right )}{2 x (4+2 x+\log (2 x))} \, dx \\ & = \frac {1}{2} (3+e) \int \frac {-1+2 x+2 x^2+x \log (2 x)}{x (4+2 x+\log (2 x))} \, dx \\ & = \frac {1}{2} (3+e) \int \left (1+\frac {-1-2 x}{x (4+2 x+\log (2 x))}\right ) \, dx \\ & = \frac {1}{2} (3+e) x+\frac {1}{2} (3+e) \int \frac {-1-2 x}{x (4+2 x+\log (2 x))} \, dx \\ & = \frac {1}{2} (3+e) x-\frac {1}{2} (3+e) \log (4+2 x+\log (2 x)) \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {-3+6 x+6 x^2+e \left (-1+2 x+2 x^2\right )+(3 x+e x) \log (2 x)}{8 x+4 x^2+2 x \log (2 x)} \, dx=\frac {1}{2} (3+e) (x-\log (4+2 x+\log (2 x))) \]
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Time = 1.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\frac {\left (3+{\mathrm e}\right ) x}{2}+\frac {\left (-2 \,{\mathrm e}-6\right ) \ln \left (\ln \left (2 x \right )+2 x +4\right )}{4}\) | \(27\) |
default | \(\frac {\left (3+{\mathrm e}\right ) x}{2}+\frac {\left (-2 \,{\mathrm e}-6\right ) \ln \left (\ln \left (2 x \right )+2 x +4\right )}{4}\) | \(27\) |
norman | \(\left (\frac {3}{2}+\frac {{\mathrm e}}{2}\right ) x +\left (-\frac {3}{2}-\frac {{\mathrm e}}{2}\right ) \ln \left (\ln \left (2 x \right )+2 x +4\right )\) | \(27\) |
risch | \(\frac {x \,{\mathrm e}}{2}+\frac {3 x}{2}-\frac {3 \ln \left (\ln \left (2 x \right )+2 x +4\right )}{2}-\frac {\ln \left (\ln \left (2 x \right )+2 x +4\right ) {\mathrm e}}{2}\) | \(36\) |
parallelrisch | \(\frac {x \,{\mathrm e}}{2}+\frac {3 x}{2}-\frac {\ln \left (\frac {\ln \left (2 x \right )}{2}+x +2\right ) {\mathrm e}}{2}-\frac {3 \ln \left (\frac {\ln \left (2 x \right )}{2}+x +2\right )}{2}\) | \(36\) |
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Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {-3+6 x+6 x^2+e \left (-1+2 x+2 x^2\right )+(3 x+e x) \log (2 x)}{8 x+4 x^2+2 x \log (2 x)} \, dx=\frac {1}{2} \, x e - \frac {1}{2} \, {\left (e + 3\right )} \log \left (2 \, x + \log \left (2 \, x\right ) + 4\right ) + \frac {3}{2} \, x \]
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Time = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {-3+6 x+6 x^2+e \left (-1+2 x+2 x^2\right )+(3 x+e x) \log (2 x)}{8 x+4 x^2+2 x \log (2 x)} \, dx=x \left (\frac {e}{2} + \frac {3}{2}\right ) - \frac {\left (e + 3\right ) \log {\left (2 x + \log {\left (2 x \right )} + 4 \right )}}{2} \]
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Time = 0.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {-3+6 x+6 x^2+e \left (-1+2 x+2 x^2\right )+(3 x+e x) \log (2 x)}{8 x+4 x^2+2 x \log (2 x)} \, dx=\frac {1}{2} \, x {\left (e + 3\right )} - \frac {1}{2} \, {\left (e + 3\right )} \log \left (2 \, x + \log \left (2\right ) + \log \left (x\right ) + 4\right ) \]
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Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {-3+6 x+6 x^2+e \left (-1+2 x+2 x^2\right )+(3 x+e x) \log (2 x)}{8 x+4 x^2+2 x \log (2 x)} \, dx=\frac {1}{2} \, x e - \frac {1}{2} \, e \log \left (2 \, x + \log \left (2 \, x\right ) + 4\right ) + \frac {3}{2} \, x - \frac {3}{2} \, \log \left (-2 \, x - \log \left (2 \, x\right ) - 4\right ) \]
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Time = 13.57 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {-3+6 x+6 x^2+e \left (-1+2 x+2 x^2\right )+(3 x+e x) \log (2 x)}{8 x+4 x^2+2 x \log (2 x)} \, dx=\left (x-\ln \left (2\,x+\ln \left (2\,x\right )+4\right )\right )\,\left (\frac {\mathrm {e}}{2}+\frac {3}{2}\right ) \]
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