Integrand size = 39, antiderivative size = 17 \[ \int \frac {e^x (4+x)+\left (3 e^x+e^x (12+3 x) \log (4+x)\right ) \log (2 \log (5))}{4+x} \, dx=e^x (1+3 \log (4+x) \log (2 \log (5))) \]
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Time = 0.35 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6873, 6874, 2225, 2634, 2209, 2230} \[ \int \frac {e^x (4+x)+\left (3 e^x+e^x (12+3 x) \log (4+x)\right ) \log (2 \log (5))}{4+x} \, dx=e^x+3 e^x \log (\log (25)) \log (x+4) \]
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Rule 2209
Rule 2225
Rule 2230
Rule 2634
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^x \left (x+4 \left (1+\frac {3}{4} \log (\log (25))\right )+12 \log (4+x) \log (\log (25))+3 x \log (4+x) \log (\log (25))\right )}{4+x} \, dx \\ & = \int \left (3 e^x \log (4+x) \log (\log (25))+\frac {e^x (4+x+3 \log (\log (25)))}{4+x}\right ) \, dx \\ & = (3 \log (\log (25))) \int e^x \log (4+x) \, dx+\int \frac {e^x (4+x+3 \log (\log (25)))}{4+x} \, dx \\ & = 3 e^x \log (4+x) \log (\log (25))-(3 \log (\log (25))) \int \frac {e^x}{4+x} \, dx+\int \left (e^x+\frac {3 e^x \log (\log (25))}{4+x}\right ) \, dx \\ & = -\frac {3 \operatorname {ExpIntegralEi}(4+x) \log (\log (25))}{e^4}+3 e^x \log (4+x) \log (\log (25))+(3 \log (\log (25))) \int \frac {e^x}{4+x} \, dx+\int e^x \, dx \\ & = e^x+3 e^x \log (4+x) \log (\log (25)) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {e^x (4+x)+\left (3 e^x+e^x (12+3 x) \log (4+x)\right ) \log (2 \log (5))}{4+x} \, dx=e^x (1+3 \log (4+x) \log (\log (25))) \]
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Time = 0.08 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00
method | result | size |
default | \(3 \,{\mathrm e}^{x} \ln \left (4+x \right ) \ln \left (2 \ln \left (5\right )\right )+{\mathrm e}^{x}\) | \(17\) |
parallelrisch | \(3 \,{\mathrm e}^{x} \ln \left (4+x \right ) \ln \left (2 \ln \left (5\right )\right )+{\mathrm e}^{x}\) | \(17\) |
risch | \(3 \left (\ln \left (2\right )+\ln \left (\ln \left (5\right )\right )\right ) {\mathrm e}^{x} \ln \left (4+x \right )+{\mathrm e}^{x}\) | \(18\) |
norman | \(\left (3 \ln \left (2\right )+3 \ln \left (\ln \left (5\right )\right )\right ) {\mathrm e}^{x} \ln \left (4+x \right )+{\mathrm e}^{x}\) | \(21\) |
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Time = 0.23 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {e^x (4+x)+\left (3 e^x+e^x (12+3 x) \log (4+x)\right ) \log (2 \log (5))}{4+x} \, dx=3 \, e^{x} \log \left (x + 4\right ) \log \left (2 \, \log \left (5\right )\right ) + e^{x} \]
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Time = 0.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.53 \[ \int \frac {e^x (4+x)+\left (3 e^x+e^x (12+3 x) \log (4+x)\right ) \log (2 \log (5))}{4+x} \, dx=\left (3 \log {\left (x + 4 \right )} \log {\left (\log {\left (5 \right )} \right )} + 3 \log {\left (2 \right )} \log {\left (x + 4 \right )} + 1\right ) e^{x} \]
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\[ \int \frac {e^x (4+x)+\left (3 e^x+e^x (12+3 x) \log (4+x)\right ) \log (2 \log (5))}{4+x} \, dx=\int { \frac {{\left (x + 4\right )} e^{x} + 3 \, {\left ({\left (x + 4\right )} e^{x} \log \left (x + 4\right ) + e^{x}\right )} \log \left (2 \, \log \left (5\right )\right )}{x + 4} \,d x } \]
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Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {e^x (4+x)+\left (3 e^x+e^x (12+3 x) \log (4+x)\right ) \log (2 \log (5))}{4+x} \, dx=3 \, e^{x} \log \left (x + 4\right ) \log \left (2 \, \log \left (5\right )\right ) + e^{x} \]
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Time = 8.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {e^x (4+x)+\left (3 e^x+e^x (12+3 x) \log (4+x)\right ) \log (2 \log (5))}{4+x} \, dx={\mathrm {e}}^x\,\left (3\,\ln \left (x+4\right )\,\ln \left (\ln \left (25\right )\right )+1\right ) \]
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