Integrand size = 38, antiderivative size = 21 \[ \int \left (8 x+6 x^2+8 x^3+e^{2 x} \left (-6 x^2-4 x^3\right )+6 x^2 \log (4)\right ) \, dx=2 x^2 \left (2+x+x \left (-e^{2 x}+x+\log (4)\right )\right ) \]
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Time = 0.06 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.43, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {6, 1607, 2227, 2207, 2225} \[ \int \left (8 x+6 x^2+8 x^3+e^{2 x} \left (-6 x^2-4 x^3\right )+6 x^2 \log (4)\right ) \, dx=2 x^4-2 e^{2 x} x^3+2 x^3 (1+\log (4))+4 x^2 \]
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Rule 6
Rule 1607
Rule 2207
Rule 2225
Rule 2227
Rubi steps \begin{align*} \text {integral}& = \int \left (8 x+8 x^3+e^{2 x} \left (-6 x^2-4 x^3\right )+x^2 (6+6 \log (4))\right ) \, dx \\ & = 4 x^2+2 x^4+2 x^3 (1+\log (4))+\int e^{2 x} \left (-6 x^2-4 x^3\right ) \, dx \\ & = 4 x^2+2 x^4+2 x^3 (1+\log (4))+\int e^{2 x} (-6-4 x) x^2 \, dx \\ & = 4 x^2+2 x^4+2 x^3 (1+\log (4))+\int \left (-6 e^{2 x} x^2-4 e^{2 x} x^3\right ) \, dx \\ & = 4 x^2+2 x^4+2 x^3 (1+\log (4))-4 \int e^{2 x} x^3 \, dx-6 \int e^{2 x} x^2 \, dx \\ & = 4 x^2-3 e^{2 x} x^2-2 e^{2 x} x^3+2 x^4+2 x^3 (1+\log (4))+6 \int e^{2 x} x \, dx+6 \int e^{2 x} x^2 \, dx \\ & = 3 e^{2 x} x+4 x^2-2 e^{2 x} x^3+2 x^4+2 x^3 (1+\log (4))-3 \int e^{2 x} \, dx-6 \int e^{2 x} x \, dx \\ & = -\frac {3 e^{2 x}}{2}+4 x^2-2 e^{2 x} x^3+2 x^4+2 x^3 (1+\log (4))+3 \int e^{2 x} \, dx \\ & = 4 x^2-2 e^{2 x} x^3+2 x^4+2 x^3 (1+\log (4)) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \left (8 x+6 x^2+8 x^3+e^{2 x} \left (-6 x^2-4 x^3\right )+6 x^2 \log (4)\right ) \, dx=2 x^2 \left (2+x^2+x \left (1-e^{2 x}+\log (4)\right )\right ) \]
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Time = 0.05 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.48
method | result | size |
norman | \(\left (4 \ln \left (2\right )+2\right ) x^{3}+4 x^{2}+2 x^{4}-2 \,{\mathrm e}^{2 x} x^{3}\) | \(31\) |
derivativedivides | \(4 x^{2}+2 x^{3}+2 x^{4}+4 x^{3} \ln \left (2\right )-2 \,{\mathrm e}^{2 x} x^{3}\) | \(33\) |
default | \(4 x^{2}+2 x^{3}+2 x^{4}+4 x^{3} \ln \left (2\right )-2 \,{\mathrm e}^{2 x} x^{3}\) | \(33\) |
risch | \(4 x^{2}+2 x^{3}+2 x^{4}+4 x^{3} \ln \left (2\right )-2 \,{\mathrm e}^{2 x} x^{3}\) | \(33\) |
parallelrisch | \(4 x^{2}+2 x^{3}+2 x^{4}+4 x^{3} \ln \left (2\right )-2 \,{\mathrm e}^{2 x} x^{3}\) | \(33\) |
parts | \(4 x^{2}+2 x^{3}+2 x^{4}+4 x^{3} \ln \left (2\right )-2 \,{\mathrm e}^{2 x} x^{3}\) | \(33\) |
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Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.52 \[ \int \left (8 x+6 x^2+8 x^3+e^{2 x} \left (-6 x^2-4 x^3\right )+6 x^2 \log (4)\right ) \, dx=2 \, x^{4} - 2 \, x^{3} e^{\left (2 \, x\right )} + 4 \, x^{3} \log \left (2\right ) + 2 \, x^{3} + 4 \, x^{2} \]
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Time = 0.05 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.38 \[ \int \left (8 x+6 x^2+8 x^3+e^{2 x} \left (-6 x^2-4 x^3\right )+6 x^2 \log (4)\right ) \, dx=2 x^{4} - 2 x^{3} e^{2 x} + x^{3} \cdot \left (2 + 4 \log {\left (2 \right )}\right ) + 4 x^{2} \]
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Time = 0.19 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.52 \[ \int \left (8 x+6 x^2+8 x^3+e^{2 x} \left (-6 x^2-4 x^3\right )+6 x^2 \log (4)\right ) \, dx=2 \, x^{4} - 2 \, x^{3} e^{\left (2 \, x\right )} + 4 \, x^{3} \log \left (2\right ) + 2 \, x^{3} + 4 \, x^{2} \]
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Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.52 \[ \int \left (8 x+6 x^2+8 x^3+e^{2 x} \left (-6 x^2-4 x^3\right )+6 x^2 \log (4)\right ) \, dx=2 \, x^{4} - 2 \, x^{3} e^{\left (2 \, x\right )} + 4 \, x^{3} \log \left (2\right ) + 2 \, x^{3} + 4 \, x^{2} \]
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Time = 7.67 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33 \[ \int \left (8 x+6 x^2+8 x^3+e^{2 x} \left (-6 x^2-4 x^3\right )+6 x^2 \log (4)\right ) \, dx=x^3\,\left (\ln \left (16\right )+2\right )-2\,x^3\,{\mathrm {e}}^{2\,x}+4\,x^2+2\,x^4 \]
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