Integrand size = 45, antiderivative size = 20 \[ \int \frac {-2+3 e x^2+e (-2+6 x) \log (4)+3 e \log ^2(4)}{e x^2+2 e x \log (4)+e \log ^2(4)} \, dx=x+2 \left (x+\frac {\frac {1}{e}-x}{x+\log (4)}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {27, 12, 1864} \[ \int \frac {-2+3 e x^2+e (-2+6 x) \log (4)+3 e \log ^2(4)}{e x^2+2 e x \log (4)+e \log ^2(4)} \, dx=3 x+\frac {2 (1+e \log (4))}{e (x+\log (4))} \]
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Rule 12
Rule 27
Rule 1864
Rubi steps \begin{align*} \text {integral}& = \int \frac {-2+3 e x^2+e (-2+6 x) \log (4)+3 e \log ^2(4)}{e (x+\log (4))^2} \, dx \\ & = \frac {\int \frac {-2+3 e x^2+e (-2+6 x) \log (4)+3 e \log ^2(4)}{(x+\log (4))^2} \, dx}{e} \\ & = \frac {\int \left (3 e-\frac {2 (1+e \log (4))}{(x+\log (4))^2}\right ) \, dx}{e} \\ & = 3 x+\frac {2 (1+e \log (4))}{e (x+\log (4))} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.25 \[ \int \frac {-2+3 e x^2+e (-2+6 x) \log (4)+3 e \log ^2(4)}{e x^2+2 e x \log (4)+e \log ^2(4)} \, dx=\frac {3 e (x+\log (4))+\frac {2+e \log (16)}{x+\log (4)}}{e} \]
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Time = 0.18 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.45
method | result | size |
risch | \(\frac {4 \ln \left (2\right )+6 x \ln \left (2\right )+3 x^{2}+2 \,{\mathrm e}^{-1}}{x +2 \ln \left (2\right )}\) | \(29\) |
default | \({\mathrm e}^{-1} \left (3 x \,{\mathrm e}-\frac {-4 \,{\mathrm e} \ln \left (2\right )-2}{x +2 \ln \left (2\right )}\right )\) | \(30\) |
gosper | \(-\frac {\left (12 \,{\mathrm e} \ln \left (2\right )^{2}-3 x^{2} {\mathrm e}-4 \,{\mathrm e} \ln \left (2\right )-2\right ) {\mathrm e}^{-1}}{x +2 \ln \left (2\right )}\) | \(38\) |
norman | \(\frac {3 x^{2}-2 \left (6 \,{\mathrm e} \ln \left (2\right )^{2}-2 \,{\mathrm e} \ln \left (2\right )-1\right ) {\mathrm e}^{-1}}{x +2 \ln \left (2\right )}\) | \(38\) |
parallelrisch | \(-\frac {\left (12 \,{\mathrm e} \ln \left (2\right )^{2}-3 x^{2} {\mathrm e}-4 \,{\mathrm e} \ln \left (2\right )-2\right ) {\mathrm e}^{-1}}{x +2 \ln \left (2\right )}\) | \(38\) |
meijerg | \(\frac {3 x}{1+\frac {x}{2 \ln \left (2\right )}}-\frac {{\mathrm e}^{-1} x}{2 \ln \left (2\right )^{2} \left (1+\frac {x}{2 \ln \left (2\right )}\right )}-\frac {x}{\ln \left (2\right ) \left (1+\frac {x}{2 \ln \left (2\right )}\right )}+12 \ln \left (2\right ) \left (-\frac {x}{2 \ln \left (2\right ) \left (1+\frac {x}{2 \ln \left (2\right )}\right )}+\ln \left (1+\frac {x}{2 \ln \left (2\right )}\right )\right )+6 \ln \left (2\right ) \left (\frac {x \left (\frac {3 x}{2 \ln \left (2\right )}+6\right )}{6 \ln \left (2\right ) \left (1+\frac {x}{2 \ln \left (2\right )}\right )}-2 \ln \left (1+\frac {x}{2 \ln \left (2\right )}\right )\right )\) | \(131\) |
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Time = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.70 \[ \int \frac {-2+3 e x^2+e (-2+6 x) \log (4)+3 e \log ^2(4)}{e x^2+2 e x \log (4)+e \log ^2(4)} \, dx=\frac {3 \, x^{2} e + 2 \, {\left (3 \, x + 2\right )} e \log \left (2\right ) + 2}{x e + 2 \, e \log \left (2\right )} \]
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Time = 0.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30 \[ \int \frac {-2+3 e x^2+e (-2+6 x) \log (4)+3 e \log ^2(4)}{e x^2+2 e x \log (4)+e \log ^2(4)} \, dx=3 x + \frac {2 + 4 e \log {\left (2 \right )}}{e x + 2 e \log {\left (2 \right )}} \]
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Time = 0.18 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int \frac {-2+3 e x^2+e (-2+6 x) \log (4)+3 e \log ^2(4)}{e x^2+2 e x \log (4)+e \log ^2(4)} \, dx=3 \, x + \frac {2 \, {\left (2 \, e \log \left (2\right ) + 1\right )}}{x e + 2 \, e \log \left (2\right )} \]
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Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {-2+3 e x^2+e (-2+6 x) \log (4)+3 e \log ^2(4)}{e x^2+2 e x \log (4)+e \log ^2(4)} \, dx=3 \, x + \frac {2 \, {\left (2 \, e \log \left (2\right ) + 1\right )} e^{\left (-1\right )}}{x + 2 \, \log \left (2\right )} \]
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Timed out. \[ \int \frac {-2+3 e x^2+e (-2+6 x) \log (4)+3 e \log ^2(4)}{e x^2+2 e x \log (4)+e \log ^2(4)} \, dx=\int \frac {12\,\mathrm {e}\,{\ln \left (2\right )}^2+3\,x^2\,\mathrm {e}+2\,\mathrm {e}\,\ln \left (2\right )\,\left (6\,x-2\right )-2}{\mathrm {e}\,x^2+4\,\mathrm {e}\,\ln \left (2\right )\,x+4\,\mathrm {e}\,{\ln \left (2\right )}^2} \,d x \]
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