Integrand size = 31, antiderivative size = 28 \[ \int \left (16+3 x+6 x^2+x^3+\left (-2 x-3 x^2\right ) \log \left (e^{-x} x\right )\right ) \, dx=4 x \left (4+\left (\frac {1}{4}+\frac {x}{4}\right ) x \left (2-\log \left (e^{-x} x\right )\right )\right ) \]
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Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1607, 45, 2634, 14} \[ \int \left (16+3 x+6 x^2+x^3+\left (-2 x-3 x^2\right ) \log \left (e^{-x} x\right )\right ) \, dx=2 x^3+x^3 \left (-\log \left (e^{-x} x\right )\right )+2 x^2-x^2 \log \left (e^{-x} x\right )+16 x \]
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Rule 14
Rule 45
Rule 1607
Rule 2634
Rubi steps \begin{align*} \text {integral}& = 16 x+\frac {3 x^2}{2}+2 x^3+\frac {x^4}{4}+\int \left (-2 x-3 x^2\right ) \log \left (e^{-x} x\right ) \, dx \\ & = 16 x+\frac {3 x^2}{2}+2 x^3+\frac {x^4}{4}+\int (-2-3 x) x \log \left (e^{-x} x\right ) \, dx \\ & = 16 x+\frac {3 x^2}{2}+2 x^3+\frac {x^4}{4}-x^2 \log \left (e^{-x} x\right )-x^3 \log \left (e^{-x} x\right )-\int x \left (-1+x^2\right ) \, dx \\ & = 16 x+\frac {3 x^2}{2}+2 x^3+\frac {x^4}{4}-x^2 \log \left (e^{-x} x\right )-x^3 \log \left (e^{-x} x\right )-\int \left (-x+x^3\right ) \, dx \\ & = 16 x+2 x^2+2 x^3-x^2 \log \left (e^{-x} x\right )-x^3 \log \left (e^{-x} x\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43 \[ \int \left (16+3 x+6 x^2+x^3+\left (-2 x-3 x^2\right ) \log \left (e^{-x} x\right )\right ) \, dx=16 x+2 x^2+2 x^3-x^2 \log \left (e^{-x} x\right )-x^3 \log \left (e^{-x} x\right ) \]
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Time = 3.33 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39
method | result | size |
default | \(16 x -\ln \left (x \,{\mathrm e}^{-x}\right ) x^{3}-x^{2} \ln \left (x \,{\mathrm e}^{-x}\right )+2 x^{2}+2 x^{3}\) | \(39\) |
norman | \(16 x -\ln \left (x \,{\mathrm e}^{-x}\right ) x^{3}-x^{2} \ln \left (x \,{\mathrm e}^{-x}\right )+2 x^{2}+2 x^{3}\) | \(39\) |
parallelrisch | \(16 x -\ln \left (x \,{\mathrm e}^{-x}\right ) x^{3}-x^{2} \ln \left (x \,{\mathrm e}^{-x}\right )+2 x^{2}+2 x^{3}\) | \(39\) |
parts | \(16 x -\ln \left (x \,{\mathrm e}^{-x}\right ) x^{3}-x^{2} \ln \left (x \,{\mathrm e}^{-x}\right )+2 x^{2}+2 x^{3}\) | \(39\) |
risch | \(\left (x^{3}+x^{2}\right ) \ln \left ({\mathrm e}^{x}\right )-x^{3} \ln \left (x \right )-x^{2} \ln \left (x \right )+\frac {i \pi \,x^{3} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i x \,{\mathrm e}^{-x}\right )}{2}-\frac {i \pi \,x^{3} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{2}}{2}-\frac {i \pi \,x^{3} \operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{2}}{2}+\frac {i \pi \,x^{3} \operatorname {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{3}}{2}+\frac {i \pi \,x^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i x \,{\mathrm e}^{-x}\right )}{2}-\frac {i \pi \,x^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{2}}{2}-\frac {i \pi \,x^{2} \operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{2}}{2}+\frac {i \pi \,x^{2} \operatorname {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{3}}{2}+2 x^{2}+2 x^{3}+16 x\) | \(232\) |
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Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \left (16+3 x+6 x^2+x^3+\left (-2 x-3 x^2\right ) \log \left (e^{-x} x\right )\right ) \, dx=2 \, x^{3} + 2 \, x^{2} - {\left (x^{3} + x^{2}\right )} \log \left (x e^{\left (-x\right )}\right ) + 16 \, x \]
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Time = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \left (16+3 x+6 x^2+x^3+\left (-2 x-3 x^2\right ) \log \left (e^{-x} x\right )\right ) \, dx=2 x^{3} + 2 x^{2} + 16 x + \left (- x^{3} - x^{2}\right ) \log {\left (x e^{- x} \right )} \]
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Time = 0.19 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \left (16+3 x+6 x^2+x^3+\left (-2 x-3 x^2\right ) \log \left (e^{-x} x\right )\right ) \, dx=2 \, x^{3} + 2 \, x^{2} - {\left (x^{3} + x^{2}\right )} \log \left (x e^{\left (-x\right )}\right ) + 16 \, x \]
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Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \left (16+3 x+6 x^2+x^3+\left (-2 x-3 x^2\right ) \log \left (e^{-x} x\right )\right ) \, dx=x^{4} - x^{3} \log \left (x\right ) + 3 \, x^{3} - x^{2} \log \left (x\right ) + 2 \, x^{2} + 16 \, x \]
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Time = 13.83 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \left (16+3 x+6 x^2+x^3+\left (-2 x-3 x^2\right ) \log \left (e^{-x} x\right )\right ) \, dx=16\,x-x^2\,\ln \left (x\right )-x^3\,\ln \left (x\right )+2\,x^2+3\,x^3+x^4 \]
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