Integrand size = 62, antiderivative size = 27 \[ \int \frac {e^x \left (16+24 x+e^4 x+12 x^2+2 x^3+\left (-8-20 x-18 x^2-7 x^3-x^4\right ) \log (7)\right )}{8+12 x+6 x^2+x^3} \, dx=-3+e^x \left (2+x \left (\frac {e^4}{x (2+x)^2}-\log (7)\right )\right ) \]
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Time = 0.11 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {6, 2230, 2208, 2209, 2207, 2225} \[ \int \frac {e^x \left (16+24 x+e^4 x+12 x^2+2 x^3+\left (-8-20 x-18 x^2-7 x^3-x^4\right ) \log (7)\right )}{8+12 x+6 x^2+x^3} \, dx=\frac {e^{x+4}}{(x+2)^2}-e^x (x+2) \log (7)+e^x (2+\log (7))+e^x \log (7) \]
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Rule 6
Rule 2207
Rule 2208
Rule 2209
Rule 2225
Rule 2230
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^x \left (16+\left (24+e^4\right ) x+12 x^2+2 x^3+\left (-8-20 x-18 x^2-7 x^3-x^4\right ) \log (7)\right )}{8+12 x+6 x^2+x^3} \, dx \\ & = \int \left (-\frac {2 e^{4+x}}{(2+x)^3}+\frac {e^{4+x}}{(2+x)^2}-e^x (2+x) \log (7)+e^x (2+\log (7))\right ) \, dx \\ & = -\left (2 \int \frac {e^{4+x}}{(2+x)^3} \, dx\right )-\log (7) \int e^x (2+x) \, dx+(2+\log (7)) \int e^x \, dx+\int \frac {e^{4+x}}{(2+x)^2} \, dx \\ & = \frac {e^{4+x}}{(2+x)^2}-\frac {e^{4+x}}{2+x}-e^x (2+x) \log (7)+e^x (2+\log (7))+\log (7) \int e^x \, dx-\int \frac {e^{4+x}}{(2+x)^2} \, dx+\int \frac {e^{4+x}}{2+x} \, dx \\ & = \frac {e^{4+x}}{(2+x)^2}+e^2 \text {Ei}(2+x)+e^x \log (7)-e^x (2+x) \log (7)+e^x (2+\log (7))-\int \frac {e^{4+x}}{2+x} \, dx \\ & = \frac {e^{4+x}}{(2+x)^2}+e^x \log (7)-e^x (2+x) \log (7)+e^x (2+\log (7)) \\ \end{align*}
Time = 5.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {e^x \left (16+24 x+e^4 x+12 x^2+2 x^3+\left (-8-20 x-18 x^2-7 x^3-x^4\right ) \log (7)\right )}{8+12 x+6 x^2+x^3} \, dx=-\frac {e^x \left (-8-e^4+x^3 \log (7)+x (-8+\log (2401))+x^2 (-2+\log (2401))\right )}{(2+x)^2} \]
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Time = 0.76 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56
method | result | size |
risch | \(-\frac {\left (\ln \left (7\right ) x^{3}+4 x^{2} \ln \left (7\right )+4 x \ln \left (7\right )-{\mathrm e}^{4}-2 x^{2}-8 x -8\right ) {\mathrm e}^{x}}{\left (2+x \right )^{2}}\) | \(42\) |
norman | \(\frac {\left ({\mathrm e}^{4}+8\right ) {\mathrm e}^{x}+\left (-4 \ln \left (7\right )+2\right ) x^{2} {\mathrm e}^{x}+\left (-4 \ln \left (7\right )+8\right ) x \,{\mathrm e}^{x}-\ln \left (7\right ) x^{3} {\mathrm e}^{x}}{\left (2+x \right )^{2}}\) | \(48\) |
gosper | \(-\frac {\left (\ln \left (7\right ) x^{3}+4 x^{2} \ln \left (7\right )+4 x \ln \left (7\right )-{\mathrm e}^{4}-2 x^{2}-8 x -8\right ) {\mathrm e}^{x}}{x^{2}+4 x +4}\) | \(49\) |
parallelrisch | \(-\frac {\ln \left (7\right ) x^{3} {\mathrm e}^{x}+4 x^{2} \ln \left (7\right ) {\mathrm e}^{x}+4 \ln \left (7\right ) x \,{\mathrm e}^{x}-{\mathrm e}^{4} {\mathrm e}^{x}-2 \,{\mathrm e}^{x} x^{2}-8 \,{\mathrm e}^{x} x -8 \,{\mathrm e}^{x}}{x^{2}+4 x +4}\) | \(62\) |
default | \(\frac {{\mathrm e}^{4} {\mathrm e}^{x}}{\left (2+x \right )^{2}}+2 \,{\mathrm e}^{x}-8 \ln \left (7\right ) \left (-\frac {{\mathrm e}^{x}}{2 \left (2+x \right )^{2}}-\frac {{\mathrm e}^{x}}{2 \left (2+x \right )}-\frac {{\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-2-x \right )}{2}\right )-\frac {20 \ln \left (7\right ) {\mathrm e}^{x}}{\left (2+x \right )^{2}}-18 \ln \left (7\right ) \left (\frac {2 \,{\mathrm e}^{x}}{2+x}+{\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-2-x \right )-\frac {2 \,{\mathrm e}^{x}}{\left (2+x \right )^{2}}\right )-7 \ln \left (7\right ) \left ({\mathrm e}^{x}-\frac {8 \,{\mathrm e}^{x}}{2+x}-2 \,{\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-2-x \right )+\frac {4 \,{\mathrm e}^{x}}{\left (2+x \right )^{2}}\right )-\ln \left (7\right ) \left ({\mathrm e}^{x} x -7 \,{\mathrm e}^{x}+\frac {24 \,{\mathrm e}^{x}}{2+x}-\frac {8 \,{\mathrm e}^{x}}{\left (2+x \right )^{2}}\right )\) | \(163\) |
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Time = 0.24 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52 \[ \int \frac {e^x \left (16+24 x+e^4 x+12 x^2+2 x^3+\left (-8-20 x-18 x^2-7 x^3-x^4\right ) \log (7)\right )}{8+12 x+6 x^2+x^3} \, dx=\frac {{\left (2 \, x^{2} - {\left (x^{3} + 4 \, x^{2} + 4 \, x\right )} \log \left (7\right ) + 8 \, x + e^{4} + 8\right )} e^{x}}{x^{2} + 4 \, x + 4} \]
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Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (20) = 40\).
Time = 0.09 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.70 \[ \int \frac {e^x \left (16+24 x+e^4 x+12 x^2+2 x^3+\left (-8-20 x-18 x^2-7 x^3-x^4\right ) \log (7)\right )}{8+12 x+6 x^2+x^3} \, dx=\frac {\left (- x^{3} \log {\left (7 \right )} - 4 x^{2} \log {\left (7 \right )} + 2 x^{2} - 4 x \log {\left (7 \right )} + 8 x + 8 + e^{4}\right ) e^{x}}{x^{2} + 4 x + 4} \]
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\[ \int \frac {e^x \left (16+24 x+e^4 x+12 x^2+2 x^3+\left (-8-20 x-18 x^2-7 x^3-x^4\right ) \log (7)\right )}{8+12 x+6 x^2+x^3} \, dx=\int { \frac {{\left (2 \, x^{3} + 12 \, x^{2} + x e^{4} - {\left (x^{4} + 7 \, x^{3} + 18 \, x^{2} + 20 \, x + 8\right )} \log \left (7\right ) + 24 \, x + 16\right )} e^{x}}{x^{3} + 6 \, x^{2} + 12 \, x + 8} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (25) = 50\).
Time = 0.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.19 \[ \int \frac {e^x \left (16+24 x+e^4 x+12 x^2+2 x^3+\left (-8-20 x-18 x^2-7 x^3-x^4\right ) \log (7)\right )}{8+12 x+6 x^2+x^3} \, dx=-\frac {x^{3} e^{x} \log \left (7\right ) + 4 \, x^{2} e^{x} \log \left (7\right ) - 2 \, x^{2} e^{x} + 4 \, x e^{x} \log \left (7\right ) - 8 \, x e^{x} - e^{\left (x + 4\right )} - 8 \, e^{x}}{x^{2} + 4 \, x + 4} \]
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Time = 8.52 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {e^x \left (16+24 x+e^4 x+12 x^2+2 x^3+\left (-8-20 x-18 x^2-7 x^3-x^4\right ) \log (7)\right )}{8+12 x+6 x^2+x^3} \, dx=\frac {{\mathrm {e}}^{x+4}}{{\left (x+2\right )}^2}-{\mathrm {e}}^x\,\left (x\,\ln \left (7\right )-2\right ) \]
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