Integrand size = 35, antiderivative size = 24 \[ \int \frac {-3+e^x x^2+e^{72 \log ^2(3 x)} (-1+144 \log (3 x))}{2 x^2} \, dx=\frac {3+e^{72 \log ^2(3 x)}+e^x x}{2 x} \]
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Time = 0.04 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {12, 14, 2225, 2326} \[ \int \frac {-3+e^x x^2+e^{72 \log ^2(3 x)} (-1+144 \log (3 x))}{2 x^2} \, dx=\frac {e^x}{2}+\frac {3}{2 x}+\frac {e^{72 \log ^2(3 x)}}{2 x} \]
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Rule 12
Rule 14
Rule 2225
Rule 2326
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int \frac {-3+e^x x^2+e^{72 \log ^2(3 x)} (-1+144 \log (3 x))}{x^2} \, dx \\ & = \frac {1}{2} \int \left (\frac {-3+e^x x^2}{x^2}+\frac {e^{72 \log ^2(3 x)} (-1+144 \log (3 x))}{x^2}\right ) \, dx \\ & = \frac {1}{2} \int \frac {-3+e^x x^2}{x^2} \, dx+\frac {1}{2} \int \frac {e^{72 \log ^2(3 x)} (-1+144 \log (3 x))}{x^2} \, dx \\ & = \frac {e^{72 \log ^2(3 x)}}{2 x}+\frac {1}{2} \int \left (e^x-\frac {3}{x^2}\right ) \, dx \\ & = \frac {3}{2 x}+\frac {e^{72 \log ^2(3 x)}}{2 x}+\frac {\int e^x \, dx}{2} \\ & = \frac {e^x}{2}+\frac {3}{2 x}+\frac {e^{72 \log ^2(3 x)}}{2 x} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {-3+e^x x^2+e^{72 \log ^2(3 x)} (-1+144 \log (3 x))}{2 x^2} \, dx=\frac {1}{2} \left (e^x+\frac {3}{x}+\frac {e^{72 \log ^2(3 x)}}{x}\right ) \]
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Time = 0.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00
method | result | size |
parallelrisch | \(-\frac {-3-{\mathrm e}^{x} x -{\mathrm e}^{72 \ln \left (3 x \right )^{2}}}{2 x}\) | \(24\) |
default | \(\frac {{\mathrm e}^{72 \ln \left (3 x \right )^{2}}}{2 x}+\frac {3}{2 x}+\frac {{\mathrm e}^{x}}{2}\) | \(25\) |
parts | \(\frac {{\mathrm e}^{72 \ln \left (3 x \right )^{2}}}{2 x}+\frac {3}{2 x}+\frac {{\mathrm e}^{x}}{2}\) | \(25\) |
risch | \(\frac {{\mathrm e}^{x} x +3}{2 x}+\frac {{\mathrm e}^{72 \ln \left (3 x \right )^{2}}}{2 x}\) | \(27\) |
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Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {-3+e^x x^2+e^{72 \log ^2(3 x)} (-1+144 \log (3 x))}{2 x^2} \, dx=\frac {x e^{x} + e^{\left (72 \, \log \left (3 \, x\right )^{2}\right )} + 3}{2 \, x} \]
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Time = 0.17 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {-3+e^x x^2+e^{72 \log ^2(3 x)} (-1+144 \log (3 x))}{2 x^2} \, dx=\frac {e^{x}}{2} + \frac {e^{72 \log {\left (3 x \right )}^{2}}}{2 x} + \frac {3}{2 x} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.33 (sec) , antiderivative size = 116, normalized size of antiderivative = 4.83 \[ \int \frac {-3+e^x x^2+e^{72 \log ^2(3 x)} (-1+144 \log (3 x))}{2 x^2} \, dx=\frac {1}{16} i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (6 i \, \sqrt {2} \log \left (3 \, x\right ) - \frac {1}{24} i \, \sqrt {2}\right ) e^{\left (-\frac {1}{288}\right )} + \frac {1}{16} \, \sqrt {2} {\left (\frac {\sqrt {2} \sqrt {\frac {1}{2}} \sqrt {\pi } {\left (\operatorname {erf}\left (\frac {1}{12} \, \sqrt {\frac {1}{2}} \sqrt {-{\left (144 \, \log \left (3 \, x\right ) - 1\right )}^{2}}\right ) - 1\right )} {\left (144 \, \log \left (3 \, x\right ) - 1\right )}}{\sqrt {-{\left (144 \, \log \left (3 \, x\right ) - 1\right )}^{2}}} + 12 \, \sqrt {2} e^{\left (\frac {1}{288} \, {\left (144 \, \log \left (3 \, x\right ) - 1\right )}^{2}\right )}\right )} e^{\left (-\frac {1}{288}\right )} + \frac {3}{2 \, x} + \frac {1}{2} \, e^{x} \]
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Time = 0.30 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {-3+e^x x^2+e^{72 \log ^2(3 x)} (-1+144 \log (3 x))}{2 x^2} \, dx=\frac {x e^{x} + e^{\left (72 \, \log \left (3 \, x\right )^{2}\right )} + 3}{2 \, x} \]
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Time = 16.35 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46 \[ \int \frac {-3+e^x x^2+e^{72 \log ^2(3 x)} (-1+144 \log (3 x))}{2 x^2} \, dx=\frac {{\mathrm {e}}^x}{2}+\frac {3}{2\,x}+\frac {x^{144\,\ln \left (3\right )}\,{\mathrm {e}}^{72\,{\ln \left (x\right )}^2}\,{\mathrm {e}}^{72\,{\ln \left (3\right )}^2}}{2\,x} \]
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