Integrand size = 44, antiderivative size = 21 \[ \int e^{398+160 x+16 x^2+\frac {2 \left (e^2 x+e^4 x\right )}{e^2}} \left (18 e^4+e^2 (1458+288 x)\right ) \, dx=9 e^{2 x+2 e^2 x+16 (5+x)^2} \]
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Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {2276, 2268} \[ \int e^{398+160 x+16 x^2+\frac {2 \left (e^2 x+e^4 x\right )}{e^2}} \left (18 e^4+e^2 (1458+288 x)\right ) \, dx=9 e^{16 x^2+2 \left (81+e^2\right ) x+400} \]
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Rule 2268
Rule 2276
Rubi steps \begin{align*} \text {integral}& = \int e^{398+2 \left (81+e^2\right ) x+16 x^2} \left (18 e^2 \left (81+e^2\right )+288 e^2 x\right ) \, dx \\ & = 9 e^{400+2 \left (81+e^2\right ) x+16 x^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int e^{398+160 x+16 x^2+\frac {2 \left (e^2 x+e^4 x\right )}{e^2}} \left (18 e^4+e^2 (1458+288 x)\right ) \, dx=9 e^{400+2 \left (81+e^2\right ) x+16 x^2} \]
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Time = 0.32 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33
method | result | size |
risch | \(9 \,{\mathrm e}^{2 x \,{\mathrm e}^{-2} {\mathrm e}^{2}+2 x \,{\mathrm e}^{-2} {\mathrm e}^{4}+16 x^{2}+160 x +400}\) | \(28\) |
gosper | \(9 \,{\mathrm e}^{16 x^{2}+160 x +400} {\mathrm e}^{2 x \left ({\mathrm e}^{2}+{\mathrm e}^{4}\right ) {\mathrm e}^{-2}}\) | \(32\) |
parallelrisch | \(9 \,{\mathrm e}^{16 x^{2}+160 x +400} {\mathrm e}^{2 x \left ({\mathrm e}^{2}+{\mathrm e}^{4}\right ) {\mathrm e}^{-2}}\) | \(32\) |
norman | \(9 \,{\mathrm e}^{16 x^{2}+160 x +400} {\mathrm e}^{2 \left (x \,{\mathrm e}^{4}+{\mathrm e}^{2} x \right ) {\mathrm e}^{-2}}\) | \(35\) |
default | \({\mathrm e}^{-2} \left (-\frac {729 i {\mathrm e}^{400} {\mathrm e}^{2} \sqrt {\pi }\, {\mathrm e}^{-\frac {\left (2 \,{\mathrm e}^{2}+162\right )^{2}}{64}} \operatorname {erf}\left (4 i x +\frac {i \left (2 \,{\mathrm e}^{2}+162\right )}{8}\right )}{4}-\frac {9 i {\mathrm e}^{400} {\mathrm e}^{4} \sqrt {\pi }\, {\mathrm e}^{-\frac {\left (2 \,{\mathrm e}^{2}+162\right )^{2}}{64}} \operatorname {erf}\left (4 i x +\frac {i \left (2 \,{\mathrm e}^{2}+162\right )}{8}\right )}{4}+288 \,{\mathrm e}^{400} {\mathrm e}^{2} \left (\frac {{\mathrm e}^{16 x^{2}+\left (2 \,{\mathrm e}^{2}+162\right ) x}}{32}+\frac {i \left (2 \,{\mathrm e}^{2}+162\right ) \sqrt {\pi }\, {\mathrm e}^{-\frac {\left (2 \,{\mathrm e}^{2}+162\right )^{2}}{64}} \operatorname {erf}\left (4 i x +\frac {i \left (2 \,{\mathrm e}^{2}+162\right )}{8}\right )}{256}\right )\right )\) | \(147\) |
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Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int e^{398+160 x+16 x^2+\frac {2 \left (e^2 x+e^4 x\right )}{e^2}} \left (18 e^4+e^2 (1458+288 x)\right ) \, dx=9 \, e^{\left (16 \, x^{2} + 2 \, x e^{2} + 162 \, x + 400\right )} \]
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Time = 0.16 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.48 \[ \int e^{398+160 x+16 x^2+\frac {2 \left (e^2 x+e^4 x\right )}{e^2}} \left (18 e^4+e^2 (1458+288 x)\right ) \, dx=9 e^{\frac {2 \left (x e^{2} + x e^{4}\right )}{e^{2}}} e^{16 x^{2} + 160 x + 400} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.32 (sec) , antiderivative size = 127, normalized size of antiderivative = 6.05 \[ \int e^{398+160 x+16 x^2+\frac {2 \left (e^2 x+e^4 x\right )}{e^2}} \left (18 e^4+e^2 (1458+288 x)\right ) \, dx=-\frac {9}{4} i \, \sqrt {\pi } \operatorname {erf}\left (4 i \, x + \frac {1}{4} i \, e^{2} + \frac {81}{4} i\right ) e^{\left (-\frac {1}{16} \, {\left (e^{2} + 81\right )}^{2} + 402\right )} - \frac {729}{4} i \, \sqrt {\pi } \operatorname {erf}\left (4 i \, x + \frac {1}{4} i \, e^{2} + \frac {81}{4} i\right ) e^{\left (-\frac {1}{16} \, {\left (e^{2} + 81\right )}^{2} + 400\right )} - \frac {9}{4} \, {\left (\frac {\sqrt {\pi } {\left (16 \, x + e^{2} + 81\right )} {\left (\operatorname {erf}\left (\frac {1}{4} \, \sqrt {-{\left (16 \, x + e^{2} + 81\right )}^{2}}\right ) - 1\right )} {\left (e^{2} + 81\right )}}{\sqrt {-{\left (16 \, x + e^{2} + 81\right )}^{2}}} - 4 \, e^{\left (\frac {1}{16} \, {\left (16 \, x + e^{2} + 81\right )}^{2}\right )}\right )} e^{\left (-\frac {1}{16} \, {\left (e^{2} + 81\right )}^{2} + 400\right )} \]
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Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int e^{398+160 x+16 x^2+\frac {2 \left (e^2 x+e^4 x\right )}{e^2}} \left (18 e^4+e^2 (1458+288 x)\right ) \, dx=9 \, e^{\left (16 \, x^{2} + 2 \, x e^{2} + 162 \, x + 400\right )} \]
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Time = 15.16 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int e^{398+160 x+16 x^2+\frac {2 \left (e^2 x+e^4 x\right )}{e^2}} \left (18 e^4+e^2 (1458+288 x)\right ) \, dx=9\,{\mathrm {e}}^{162\,x}\,{\mathrm {e}}^{400}\,{\mathrm {e}}^{16\,x^2}\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^2} \]
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