Integrand size = 74, antiderivative size = 22 \[ \int \frac {e^{-6 x} \left (-6 x+e^{3 x} \left (-4-36 x-18 x^2\right )+e^{6 x} \left (-28+30 x+18 x^2\right )+\left (e^{6 x} (4-6 x)+6 e^{3 x} x\right ) \log \left (\frac {x^2}{4}\right )\right )}{x} \, dx=\left (7+e^{-3 x}+3 x-\log \left (\frac {x^2}{4}\right )\right )^2 \]
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Leaf count is larger than twice the leaf count of optimal. \(58\) vs. \(2(22)=44\).
Time = 0.75 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.64, number of steps used = 18, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {6820, 12, 6874, 2225, 2230, 2209, 2207, 2634, 6818} \[ \int \frac {e^{-6 x} \left (-6 x+e^{3 x} \left (-4-36 x-18 x^2\right )+e^{6 x} \left (-28+30 x+18 x^2\right )+\left (e^{6 x} (4-6 x)+6 e^{3 x} x\right ) \log \left (\frac {x^2}{4}\right )\right )}{x} \, dx=\left (-\log \left (x^2\right )+3 x+7+\log (4)\right )^2-2 e^{-3 x} \log \left (x^2\right )+e^{-6 x}+14 e^{-3 x}+6 e^{-3 x} x+\frac {2}{3} e^{-3 x} \log (64) \]
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Rule 12
Rule 2207
Rule 2209
Rule 2225
Rule 2230
Rule 2634
Rule 6818
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {2 e^{-6 x} \left (3 x-e^{3 x} (-2+3 x)\right ) \left (-1-e^{3 x} (7+3 x)+e^{3 x} \log \left (\frac {x^2}{4}\right )\right )}{x} \, dx \\ & = 2 \int \frac {e^{-6 x} \left (3 x-e^{3 x} (-2+3 x)\right ) \left (-1-e^{3 x} (7+3 x)+e^{3 x} \log \left (\frac {x^2}{4}\right )\right )}{x} \, dx \\ & = 2 \int \left (-3 e^{-6 x}-\frac {e^{-3 x} \left (2+18 x+9 x^2-3 x \log \left (\frac {x^2}{4}\right )\right )}{x}+\frac {(2-3 x) \left (-3 x-7 \left (1+\frac {2 \log (2)}{7}\right )+\log \left (x^2\right )\right )}{x}\right ) \, dx \\ & = -\left (2 \int \frac {e^{-3 x} \left (2+18 x+9 x^2-3 x \log \left (\frac {x^2}{4}\right )\right )}{x} \, dx\right )+2 \int \frac {(2-3 x) \left (-3 x-7 \left (1+\frac {2 \log (2)}{7}\right )+\log \left (x^2\right )\right )}{x} \, dx-6 \int e^{-6 x} \, dx \\ & = e^{-6 x}+\left (7+3 x+\log (4)-\log \left (x^2\right )\right )^2-2 \int \left (\frac {e^{-3 x} \left (2+18 x+9 x^2\right )}{x}+e^{-3 x} \log (64)-3 e^{-3 x} \log \left (x^2\right )\right ) \, dx \\ & = e^{-6 x}+\left (7+3 x+\log (4)-\log \left (x^2\right )\right )^2-2 \int \frac {e^{-3 x} \left (2+18 x+9 x^2\right )}{x} \, dx+6 \int e^{-3 x} \log \left (x^2\right ) \, dx-(2 \log (64)) \int e^{-3 x} \, dx \\ & = e^{-6 x}+\frac {2}{3} e^{-3 x} \log (64)+\left (7+3 x+\log (4)-\log \left (x^2\right )\right )^2-2 e^{-3 x} \log \left (x^2\right )-2 \int \left (18 e^{-3 x}+\frac {2 e^{-3 x}}{x}+9 e^{-3 x} x\right ) \, dx-6 \int -\frac {2 e^{-3 x}}{3 x} \, dx \\ & = e^{-6 x}+\frac {2}{3} e^{-3 x} \log (64)+\left (7+3 x+\log (4)-\log \left (x^2\right )\right )^2-2 e^{-3 x} \log \left (x^2\right )-18 \int e^{-3 x} x \, dx-36 \int e^{-3 x} \, dx \\ & = e^{-6 x}+12 e^{-3 x}+6 e^{-3 x} x+\frac {2}{3} e^{-3 x} \log (64)+\left (7+3 x+\log (4)-\log \left (x^2\right )\right )^2-2 e^{-3 x} \log \left (x^2\right )-6 \int e^{-3 x} \, dx \\ & = e^{-6 x}+14 e^{-3 x}+6 e^{-3 x} x+\frac {2}{3} e^{-3 x} \log (64)+\left (7+3 x+\log (4)-\log \left (x^2\right )\right )^2-2 e^{-3 x} \log \left (x^2\right ) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(61\) vs. \(2(22)=44\).
Time = 5.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.77 \[ \int \frac {e^{-6 x} \left (-6 x+e^{3 x} \left (-4-36 x-18 x^2\right )+e^{6 x} \left (-28+30 x+18 x^2\right )+\left (e^{6 x} (4-6 x)+6 e^{3 x} x\right ) \log \left (\frac {x^2}{4}\right )\right )}{x} \, dx=e^{-6 x}+42 x+9 x^2+2 e^{-3 x} (7+3 x)-28 \log (x)+2 \left (-e^{-3 x}-3 x\right ) \log \left (\frac {x^2}{4}\right )+\log ^2\left (\frac {x^2}{4}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(87\) vs. \(2(19)=38\).
Time = 0.32 (sec) , antiderivative size = 88, normalized size of antiderivative = 4.00
method | result | size |
default | \(2 \left (7+2 \ln \left (2\right )-\ln \left (x^{2}\right )+2 \ln \left (x \right )\right ) {\mathrm e}^{-3 x}+6 x \,{\mathrm e}^{-3 x}-4 \ln \left (x \right ) {\mathrm e}^{-3 x}+{\mathrm e}^{-6 x}+9 x^{2}+42 x -28 \ln \left (x \right )+4 \ln \left (x \right ) \ln \left (x^{2}\right )-4 \ln \left (x \right )^{2}-6 x \ln \left (x^{2}\right )+4 \ln \left (2\right ) \left (-2 \ln \left (x \right )+3 x \right )\) | \(88\) |
parts | \(2 \left (7+2 \ln \left (2\right )-\ln \left (x^{2}\right )+2 \ln \left (x \right )\right ) {\mathrm e}^{-3 x}+6 x \,{\mathrm e}^{-3 x}-4 \ln \left (x \right ) {\mathrm e}^{-3 x}+{\mathrm e}^{-6 x}+9 x^{2}+42 x -28 \ln \left (x \right )+4 \ln \left (x \right ) \ln \left (x^{2}\right )-4 \ln \left (x \right )^{2}-6 x \ln \left (x^{2}\right )+4 \ln \left (2\right ) \left (-2 \ln \left (x \right )+3 x \right )\) | \(88\) |
parallelrisch | \(\frac {\left (2+18 x^{2} {\mathrm e}^{6 x}-12 \ln \left (\frac {x^{2}}{4}\right ) {\mathrm e}^{6 x} x +2 \ln \left (\frac {x^{2}}{4}\right )^{2} {\mathrm e}^{6 x}+84 \ln \left ({\mathrm e}^{x}\right ) {\mathrm e}^{6 x}-28 \ln \left (\frac {x^{2}}{4}\right ) {\mathrm e}^{6 x}+12 x \,{\mathrm e}^{3 x}-4 \ln \left (\frac {x^{2}}{4}\right ) {\mathrm e}^{3 x}+28 \,{\mathrm e}^{3 x}\right ) {\mathrm e}^{-6 x}}{2}\) | \(91\) |
risch | \(-6 i x \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+i \pi \operatorname {csgn}\left (i x^{2}\right )^{3} {\mathrm e}^{-3 x}-2 i \pi \ln \left (x \right ) \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+3 i x \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-2 i \pi \ln \left (x \right ) \operatorname {csgn}\left (i x^{2}\right )^{3}+4 i \pi \ln \left (x \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right ) {\mathrm e}^{-3 x}-2 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2} {\mathrm e}^{-3 x}+3 i x \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-8 \ln \left (2\right ) \ln \left (x \right )+12 x \ln \left (2\right )-12 x \ln \left (x \right )+9 x^{2}+4 \ln \left (2\right ) {\mathrm e}^{-3 x}-28 \ln \left (x \right )+42 x +6 x \,{\mathrm e}^{-3 x}-4 \ln \left (x \right ) {\mathrm e}^{-3 x}+14 \,{\mathrm e}^{-3 x}+4 \ln \left (x \right )^{2}+{\mathrm e}^{-6 x}\) | \(237\) |
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Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (19) = 38\).
Time = 0.25 (sec) , antiderivative size = 69, normalized size of antiderivative = 3.14 \[ \int \frac {e^{-6 x} \left (-6 x+e^{3 x} \left (-4-36 x-18 x^2\right )+e^{6 x} \left (-28+30 x+18 x^2\right )+\left (e^{6 x} (4-6 x)+6 e^{3 x} x\right ) \log \left (\frac {x^2}{4}\right )\right )}{x} \, dx={\left (e^{\left (6 \, x\right )} \log \left (\frac {1}{4} \, x^{2}\right )^{2} + 3 \, {\left (3 \, x^{2} + 14 \, x\right )} e^{\left (6 \, x\right )} + 2 \, {\left (3 \, x + 7\right )} e^{\left (3 \, x\right )} - 2 \, {\left ({\left (3 \, x + 7\right )} e^{\left (6 \, x\right )} + e^{\left (3 \, x\right )}\right )} \log \left (\frac {1}{4} \, x^{2}\right ) + 1\right )} e^{\left (-6 \, x\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (19) = 38\).
Time = 0.20 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.55 \[ \int \frac {e^{-6 x} \left (-6 x+e^{3 x} \left (-4-36 x-18 x^2\right )+e^{6 x} \left (-28+30 x+18 x^2\right )+\left (e^{6 x} (4-6 x)+6 e^{3 x} x\right ) \log \left (\frac {x^2}{4}\right )\right )}{x} \, dx=9 x^{2} - 6 x \log {\left (\frac {x^{2}}{4} \right )} + 42 x + \left (6 x - 2 \log {\left (\frac {x^{2}}{4} \right )} + 14\right ) e^{- 3 x} - 28 \log {\left (x \right )} + \log {\left (\frac {x^{2}}{4} \right )}^{2} + e^{- 6 x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (19) = 38\).
Time = 0.20 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.86 \[ \int \frac {e^{-6 x} \left (-6 x+e^{3 x} \left (-4-36 x-18 x^2\right )+e^{6 x} \left (-28+30 x+18 x^2\right )+\left (e^{6 x} (4-6 x)+6 e^{3 x} x\right ) \log \left (\frac {x^2}{4}\right )\right )}{x} \, dx=9 \, x^{2} + 2 \, {\left (3 \, x + 1\right )} e^{\left (-3 \, x\right )} - 6 \, x \log \left (\frac {1}{4} \, x^{2}\right ) - 2 \, e^{\left (-3 \, x\right )} \log \left (\frac {1}{4} \, x^{2}\right ) + \log \left (\frac {1}{4} \, x^{2}\right )^{2} + 42 \, x + 12 \, e^{\left (-3 \, x\right )} + e^{\left (-6 \, x\right )} - 28 \, \log \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (19) = 38\).
Time = 0.29 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.68 \[ \int \frac {e^{-6 x} \left (-6 x+e^{3 x} \left (-4-36 x-18 x^2\right )+e^{6 x} \left (-28+30 x+18 x^2\right )+\left (e^{6 x} (4-6 x)+6 e^{3 x} x\right ) \log \left (\frac {x^2}{4}\right )\right )}{x} \, dx=9 \, x^{2} + 6 \, x e^{\left (-3 \, x\right )} - 6 \, x \log \left (\frac {1}{4} \, x^{2}\right ) - 2 \, e^{\left (-3 \, x\right )} \log \left (\frac {1}{4} \, x^{2}\right ) + \log \left (\frac {1}{4} \, x^{2}\right )^{2} + 42 \, x + 14 \, e^{\left (-3 \, x\right )} + e^{\left (-6 \, x\right )} - 28 \, \log \left (x\right ) \]
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Time = 13.53 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.86 \[ \int \frac {e^{-6 x} \left (-6 x+e^{3 x} \left (-4-36 x-18 x^2\right )+e^{6 x} \left (-28+30 x+18 x^2\right )+\left (e^{6 x} (4-6 x)+6 e^{3 x} x\right ) \log \left (\frac {x^2}{4}\right )\right )}{x} \, dx=30\,x+14\,{\mathrm {e}}^{-3\,x}+{\mathrm {e}}^{-6\,x}-28\,\ln \left (x\right )+{\ln \left (\frac {x^2}{4}\right )}^2+6\,x\,{\mathrm {e}}^{-3\,x}-x\,\left (6\,\ln \left (\frac {x^2}{4}\right )-12\right )-2\,{\mathrm {e}}^{-3\,x}\,\ln \left (\frac {x^2}{4}\right )+9\,x^2 \]
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