Integrand size = 46, antiderivative size = 25 \[ \int \frac {e^2+e^{2+x}+e^{e^{\frac {2 x}{e^2}}+\frac {2 x}{e^2}} \left (6+2 e^4\right )}{3 e^2+e^6} \, dx=28+e^{e^{\frac {2 x}{e^2}}}+\frac {e^x+x}{3+e^4} \]
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Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {12, 2225, 2320} \[ \int \frac {e^2+e^{2+x}+e^{e^{\frac {2 x}{e^2}}+\frac {2 x}{e^2}} \left (6+2 e^4\right )}{3 e^2+e^6} \, dx=\frac {x}{3+e^4}+e^{e^{\frac {2 x}{e^2}}}+\frac {e^x}{3+e^4} \]
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Rule 12
Rule 2225
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (e^2+e^{2+x}+e^{e^{\frac {2 x}{e^2}}+\frac {2 x}{e^2}} \left (6+2 e^4\right )\right ) \, dx}{e^2 \left (3+e^4\right )} \\ & = \frac {x}{3+e^4}+\frac {2 \int e^{e^{\frac {2 x}{e^2}}+\frac {2 x}{e^2}} \, dx}{e^2}+\frac {\int e^{2+x} \, dx}{e^2 \left (3+e^4\right )} \\ & = \frac {e^x}{3+e^4}+\frac {x}{3+e^4}+\text {Subst}\left (\int e^x \, dx,x,e^{\frac {2 x}{e^2}}\right ) \\ & = e^{e^{\frac {2 x}{e^2}}}+\frac {e^x}{3+e^4}+\frac {x}{3+e^4} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {e^2+e^{2+x}+e^{e^{\frac {2 x}{e^2}}+\frac {2 x}{e^2}} \left (6+2 e^4\right )}{3 e^2+e^6} \, dx=\frac {e^x+e^{e^{\frac {2 x}{e^2}}} \left (3+e^4\right )+x}{3+e^4} \]
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Time = 0.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12
method | result | size |
norman | \(\frac {x}{{\mathrm e}^{4}+3}+\frac {{\mathrm e}^{x}}{{\mathrm e}^{4}+3}+{\mathrm e}^{{\mathrm e}^{2 x \,{\mathrm e}^{-2}}}\) | \(28\) |
parts | \(\frac {x}{{\mathrm e}^{4}+3}+\frac {{\mathrm e}^{x}}{{\mathrm e}^{4}+3}+{\mathrm e}^{{\mathrm e}^{2 x \,{\mathrm e}^{-2}}}\) | \(28\) |
default | \(\frac {{\mathrm e}^{2} {\mathrm e}^{x}+\frac {\left (2 \,{\mathrm e}^{4}+6\right ) {\mathrm e}^{{\mathrm e}^{2 x \,{\mathrm e}^{-2}}} {\mathrm e}^{2}}{2}+{\mathrm e}^{2} x}{{\mathrm e}^{2} {\mathrm e}^{4}+3 \,{\mathrm e}^{2}}\) | \(43\) |
parallelrisch | \(\frac {{\mathrm e}^{2} {\mathrm e}^{4} {\mathrm e}^{{\mathrm e}^{2 x \,{\mathrm e}^{-2}}}+{\mathrm e}^{2} {\mathrm e}^{x}+3 \,{\mathrm e}^{2} {\mathrm e}^{{\mathrm e}^{2 x \,{\mathrm e}^{-2}}}+{\mathrm e}^{2} x}{{\mathrm e}^{2} {\mathrm e}^{4}+3 \,{\mathrm e}^{2}}\) | \(51\) |
risch | \(\frac {{\mathrm e}^{2} x}{{\mathrm e}^{6}+3 \,{\mathrm e}^{2}}+\frac {{\mathrm e}^{6+{\mathrm e}^{2 x \,{\mathrm e}^{-2}}}}{{\mathrm e}^{6}+3 \,{\mathrm e}^{2}}+\frac {{\mathrm e}^{2+x}}{{\mathrm e}^{6}+3 \,{\mathrm e}^{2}}+\frac {3 \,{\mathrm e}^{2+{\mathrm e}^{2 x \,{\mathrm e}^{-2}}}}{{\mathrm e}^{6}+3 \,{\mathrm e}^{2}}\) | \(68\) |
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Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (20) = 40\).
Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.40 \[ \int \frac {e^2+e^{2+x}+e^{e^{\frac {2 x}{e^2}}+\frac {2 x}{e^2}} \left (6+2 e^4\right )}{3 e^2+e^6} \, dx=\frac {{\left ({\left (e^{6} + 3 \, e^{2}\right )} e^{\left ({\left (2 \, x + e^{\left (2 \, x e^{\left (-2\right )} + 2\right )}\right )} e^{\left (-2\right )}\right )} + x e^{\left (2 \, x e^{\left (-2\right )} + 2\right )} + e^{\left (2 \, x e^{\left (-2\right )} + x + 2\right )}\right )} e^{\left (-2 \, x e^{\left (-2\right )}\right )}}{e^{6} + 3 \, e^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (20) = 40\).
Time = 0.22 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.92 \[ \int \frac {e^2+e^{2+x}+e^{e^{\frac {2 x}{e^2}}+\frac {2 x}{e^2}} \left (6+2 e^4\right )}{3 e^2+e^6} \, dx=\frac {x e^{2} + e^{2} e^{x} + 3 e^{2} e^{e^{\frac {2 x}{e^{2}}}} + e^{6} e^{e^{\frac {2 x}{e^{2}}}}}{3 e^{2} + e^{6}} \]
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Time = 0.20 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \frac {e^2+e^{2+x}+e^{e^{\frac {2 x}{e^2}}+\frac {2 x}{e^2}} \left (6+2 e^4\right )}{3 e^2+e^6} \, dx=\frac {x e^{2} + {\left (e^{4} + 3\right )} e^{\left (e^{\left (2 \, x e^{\left (-2\right )}\right )} + 2\right )} + e^{\left (x + 2\right )}}{e^{6} + 3 \, e^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \frac {e^2+e^{2+x}+e^{e^{\frac {2 x}{e^2}}+\frac {2 x}{e^2}} \left (6+2 e^4\right )}{3 e^2+e^6} \, dx=\frac {x e^{2} + {\left (e^{4} + 3\right )} e^{\left (e^{\left (2 \, x e^{\left (-2\right )}\right )} + 2\right )} + e^{\left (x + 2\right )}}{e^{6} + 3 \, e^{2}} \]
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Time = 14.69 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {e^2+e^{2+x}+e^{e^{\frac {2 x}{e^2}}+\frac {2 x}{e^2}} \left (6+2 e^4\right )}{3 e^2+e^6} \, dx={\mathrm {e}}^{{\mathrm {e}}^{2\,x\,{\mathrm {e}}^{-2}}}+\frac {{\mathrm {e}}^x}{{\mathrm {e}}^4+3}+\frac {x}{{\mathrm {e}}^4+3} \]
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