Integrand size = 42, antiderivative size = 28 \[ \int \frac {324 x^3+e^{4/x} (-324+162 x)+e^{2/x} \left (324 x-486 x^2\right )}{\sqrt [256]{e}} \, dx=\frac {81 x^2 \left (-e^{2/x}+x\right )^2}{\sqrt [256]{e}}+4 \log (4) \]
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Time = 0.09 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.54, number of steps used = 19, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 2258, 2237, 2241, 2245, 1607} \[ \int \frac {324 x^3+e^{4/x} (-324+162 x)+e^{2/x} \left (324 x-486 x^2\right )}{\sqrt [256]{e}} \, dx=\frac {81 x^4}{\sqrt [256]{e}}-162 e^{\frac {2}{x}-\frac {1}{256}} x^3+81 e^{\frac {4}{x}-\frac {1}{256}} x^2 \]
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Rule 12
Rule 1607
Rule 2237
Rule 2241
Rule 2245
Rule 2258
Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (324 x^3+e^{4/x} (-324+162 x)+e^{2/x} \left (324 x-486 x^2\right )\right ) \, dx}{\sqrt [256]{e}} \\ & = \frac {81 x^4}{\sqrt [256]{e}}+\frac {\int e^{4/x} (-324+162 x) \, dx}{\sqrt [256]{e}}+\frac {\int e^{2/x} \left (324 x-486 x^2\right ) \, dx}{\sqrt [256]{e}} \\ & = \frac {81 x^4}{\sqrt [256]{e}}+\frac {\int e^{2/x} (324-486 x) x \, dx}{\sqrt [256]{e}}+\frac {\int \left (-324 e^{4/x}+162 e^{4/x} x\right ) \, dx}{\sqrt [256]{e}} \\ & = \frac {81 x^4}{\sqrt [256]{e}}+\frac {\int \left (324 e^{2/x} x-486 e^{2/x} x^2\right ) \, dx}{\sqrt [256]{e}}+\frac {162 \int e^{4/x} x \, dx}{\sqrt [256]{e}}-\frac {324 \int e^{4/x} \, dx}{\sqrt [256]{e}} \\ & = -324 e^{-\frac {1}{256}+\frac {4}{x}} x+81 e^{-\frac {1}{256}+\frac {4}{x}} x^2+\frac {81 x^4}{\sqrt [256]{e}}+\frac {324 \int e^{4/x} \, dx}{\sqrt [256]{e}}+\frac {324 \int e^{2/x} x \, dx}{\sqrt [256]{e}}-\frac {486 \int e^{2/x} x^2 \, dx}{\sqrt [256]{e}}-\frac {1296 \int \frac {e^{4/x}}{x} \, dx}{\sqrt [256]{e}} \\ & = 162 e^{-\frac {1}{256}+\frac {2}{x}} x^2+81 e^{-\frac {1}{256}+\frac {4}{x}} x^2-162 e^{-\frac {1}{256}+\frac {2}{x}} x^3+\frac {81 x^4}{\sqrt [256]{e}}+\frac {1296 \operatorname {ExpIntegralEi}\left (\frac {4}{x}\right )}{\sqrt [256]{e}}+\frac {324 \int e^{2/x} \, dx}{\sqrt [256]{e}}-\frac {324 \int e^{2/x} x \, dx}{\sqrt [256]{e}}+\frac {1296 \int \frac {e^{4/x}}{x} \, dx}{\sqrt [256]{e}} \\ & = 324 e^{-\frac {1}{256}+\frac {2}{x}} x+81 e^{-\frac {1}{256}+\frac {4}{x}} x^2-162 e^{-\frac {1}{256}+\frac {2}{x}} x^3+\frac {81 x^4}{\sqrt [256]{e}}-\frac {324 \int e^{2/x} \, dx}{\sqrt [256]{e}}+\frac {648 \int \frac {e^{2/x}}{x} \, dx}{\sqrt [256]{e}} \\ & = 81 e^{-\frac {1}{256}+\frac {4}{x}} x^2-162 e^{-\frac {1}{256}+\frac {2}{x}} x^3+\frac {81 x^4}{\sqrt [256]{e}}-\frac {648 \operatorname {ExpIntegralEi}\left (\frac {2}{x}\right )}{\sqrt [256]{e}}-\frac {648 \int \frac {e^{2/x}}{x} \, dx}{\sqrt [256]{e}} \\ & = 81 e^{-\frac {1}{256}+\frac {4}{x}} x^2-162 e^{-\frac {1}{256}+\frac {2}{x}} x^3+\frac {81 x^4}{\sqrt [256]{e}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \frac {324 x^3+e^{4/x} (-324+162 x)+e^{2/x} \left (324 x-486 x^2\right )}{\sqrt [256]{e}} \, dx=\frac {81 \left (e^{2/x}-x\right )^2 x^2}{\sqrt [256]{e}} \]
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Time = 0.36 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29
method | result | size |
default | \({\mathrm e}^{-\frac {1}{256}} \left (81 x^{2} {\mathrm e}^{\frac {4}{x}}-162 x^{3} {\mathrm e}^{\frac {2}{x}}+81 x^{4}\right )\) | \(36\) |
parallelrisch | \({\mathrm e}^{-\frac {1}{256}} \left (81 x^{2} {\mathrm e}^{\frac {4}{x}}-162 x^{3} {\mathrm e}^{\frac {2}{x}}+81 x^{4}\right )\) | \(36\) |
derivativedivides | \(-{\mathrm e}^{-\frac {1}{256}} \left (-81 x^{4}+162 x^{3} {\mathrm e}^{\frac {2}{x}}-81 x^{2} {\mathrm e}^{\frac {4}{x}}\right )\) | \(37\) |
risch | \(81 \,{\mathrm e}^{-\frac {1}{256}} x^{4}+81 x^{2} {\mathrm e}^{-\frac {x -1024}{256 x}}-162 x^{3} {\mathrm e}^{-\frac {x -512}{256 x}}\) | \(37\) |
norman | \(81 \,{\mathrm e}^{-\frac {1}{256}} x^{4}+81 \,{\mathrm e}^{-\frac {1}{256}} x^{2} {\mathrm e}^{\frac {4}{x}}-162 \,{\mathrm e}^{-\frac {1}{256}} x^{3} {\mathrm e}^{\frac {2}{x}}\) | \(43\) |
parts | \(81 \,{\mathrm e}^{-\frac {1}{256}} x^{4}+81 \,{\mathrm e}^{-\frac {1}{256}} x^{2} {\mathrm e}^{\frac {4}{x}}-162 \,{\mathrm e}^{-\frac {1}{256}} x^{3} {\mathrm e}^{\frac {2}{x}}\) | \(43\) |
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Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {324 x^3+e^{4/x} (-324+162 x)+e^{2/x} \left (324 x-486 x^2\right )}{\sqrt [256]{e}} \, dx=81 \, {\left (x^{4} - 2 \, x^{3} e^{\frac {2}{x}} + x^{2} e^{\frac {4}{x}}\right )} e^{\left (-\frac {1}{256}\right )} \]
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Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57 \[ \int \frac {324 x^3+e^{4/x} (-324+162 x)+e^{2/x} \left (324 x-486 x^2\right )}{\sqrt [256]{e}} \, dx=\frac {81 x^{4}}{e^{\frac {1}{256}}} + \frac {- 162 x^{3} e^{\frac {1}{256}} e^{\frac {2}{x}} + 81 x^{2} e^{\frac {1}{256}} e^{\frac {4}{x}}}{e^{\frac {1}{128}}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.22 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {324 x^3+e^{4/x} (-324+162 x)+e^{2/x} \left (324 x-486 x^2\right )}{\sqrt [256]{e}} \, dx=81 \, {\left (x^{4} + x^{2} e^{\frac {4}{x}} + 16 \, \Gamma \left (-2, -\frac {2}{x}\right ) + 48 \, \Gamma \left (-3, -\frac {2}{x}\right )\right )} e^{\left (-\frac {1}{256}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {324 x^3+e^{4/x} (-324+162 x)+e^{2/x} \left (324 x-486 x^2\right )}{\sqrt [256]{e}} \, dx=81 \, {\left (x^{4} - 2 \, x^{3} e^{\frac {2}{x}} + x^{2} e^{\frac {4}{x}}\right )} e^{\left (-\frac {1}{256}\right )} \]
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Time = 8.95 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68 \[ \int \frac {324 x^3+e^{4/x} (-324+162 x)+e^{2/x} \left (324 x-486 x^2\right )}{\sqrt [256]{e}} \, dx=81\,x^2\,{\mathrm {e}}^{-\frac {1}{256}}\,{\left (x-{\mathrm {e}}^{2/x}\right )}^2 \]
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