Integrand size = 56, antiderivative size = 25 \[ \int \frac {e^{\frac {3+16 e^4 x-16 x^2+48 e^{3/4} x^2}{16 x}} \left (-3-16 x^2+48 e^{3/4} x^2\right )}{16 x^2} \, dx=e^{e^4-x+3 \left (e^{3/4}+\frac {1}{16 x^2}\right ) x} \]
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Time = 0.22 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {6, 12, 6838} \[ \int \frac {e^{\frac {3+16 e^4 x-16 x^2+48 e^{3/4} x^2}{16 x}} \left (-3-16 x^2+48 e^{3/4} x^2\right )}{16 x^2} \, dx=e^{\frac {48 e^{3/4} x^2-16 x^2+16 e^4 x+3}{16 x}} \]
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Rule 6
Rule 12
Rule 6838
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{\frac {3+16 e^4 x-16 x^2+48 e^{3/4} x^2}{16 x}} \left (-3+\left (-16+48 e^{3/4}\right ) x^2\right )}{16 x^2} \, dx \\ & = \frac {1}{16} \int \frac {e^{\frac {3+16 e^4 x-16 x^2+48 e^{3/4} x^2}{16 x}} \left (-3+\left (-16+48 e^{3/4}\right ) x^2\right )}{x^2} \, dx \\ & = e^{\frac {3+16 e^4 x-16 x^2+48 e^{3/4} x^2}{16 x}} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {e^{\frac {3+16 e^4 x-16 x^2+48 e^{3/4} x^2}{16 x}} \left (-3-16 x^2+48 e^{3/4} x^2\right )}{16 x^2} \, dx=e^{e^4+\frac {3}{16 x}+\left (-1+3 e^{3/4}\right ) x} \]
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Time = 0.89 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04
method | result | size |
gosper | \({\mathrm e}^{\frac {16 x \,{\mathrm e}^{4}+48 x^{2} {\mathrm e}^{\frac {3}{4}}-16 x^{2}+3}{16 x}}\) | \(26\) |
derivativedivides | \({\mathrm e}^{\frac {16 x \,{\mathrm e}^{4}+48 x^{2} {\mathrm e}^{\frac {3}{4}}-16 x^{2}+3}{16 x}}\) | \(26\) |
default | \({\mathrm e}^{\frac {16 x \,{\mathrm e}^{4}+48 x^{2} {\mathrm e}^{\frac {3}{4}}-16 x^{2}+3}{16 x}}\) | \(26\) |
norman | \({\mathrm e}^{\frac {16 x \,{\mathrm e}^{4}+48 x^{2} {\mathrm e}^{\frac {3}{4}}-16 x^{2}+3}{16 x}}\) | \(26\) |
risch | \({\mathrm e}^{\frac {16 x \,{\mathrm e}^{4}+48 x^{2} {\mathrm e}^{\frac {3}{4}}-16 x^{2}+3}{16 x}}\) | \(26\) |
parallelrisch | \({\mathrm e}^{\frac {16 x \,{\mathrm e}^{4}+48 x^{2} {\mathrm e}^{\frac {3}{4}}-16 x^{2}+3}{16 x}}\) | \(26\) |
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Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {3+16 e^4 x-16 x^2+48 e^{3/4} x^2}{16 x}} \left (-3-16 x^2+48 e^{3/4} x^2\right )}{16 x^2} \, dx=e^{\left (\frac {48 \, x^{2} e^{\frac {3}{4}} - 16 \, x^{2} + 16 \, x e^{4} + 3}{16 \, x}\right )} \]
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Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {e^{\frac {3+16 e^4 x-16 x^2+48 e^{3/4} x^2}{16 x}} \left (-3-16 x^2+48 e^{3/4} x^2\right )}{16 x^2} \, dx=e^{\frac {- x^{2} + 3 x^{2} e^{\frac {3}{4}} + x e^{4} + \frac {3}{16}}{x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (18) = 36\).
Time = 0.29 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {e^{\frac {3+16 e^4 x-16 x^2+48 e^{3/4} x^2}{16 x}} \left (-3-16 x^2+48 e^{3/4} x^2\right )}{16 x^2} \, dx=\frac {{\left (3 \, e^{\left (e^{4} + \frac {3}{4}\right )} - e^{\left (e^{4}\right )}\right )} e^{\left (3 \, x e^{\frac {3}{4}} - x + \frac {3}{16 \, x}\right )}}{3 \, e^{\frac {3}{4}} - 1} \]
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Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {3+16 e^4 x-16 x^2+48 e^{3/4} x^2}{16 x}} \left (-3-16 x^2+48 e^{3/4} x^2\right )}{16 x^2} \, dx=e^{\left (\frac {48 \, x^{2} e^{\frac {3}{4}} - 16 \, x^{2} + 16 \, x e^{4} + 3}{16 \, x}\right )} \]
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Time = 9.31 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {e^{\frac {3+16 e^4 x-16 x^2+48 e^{3/4} x^2}{16 x}} \left (-3-16 x^2+48 e^{3/4} x^2\right )}{16 x^2} \, dx={\mathrm {e}}^{-x}\,{\mathrm {e}}^{\frac {3}{16\,x}}\,{\mathrm {e}}^{3\,x\,{\mathrm {e}}^{3/4}}\,{\mathrm {e}}^{{\mathrm {e}}^4} \]
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