Integrand size = 35, antiderivative size = 27 \[ \int \frac {2 x^3+e^{\left .\frac {3}{2}\right /x} (3+2 x) \log (2) \log ^2(\log (4))}{2 x^3} \, dx=-2+x+\frac {\left (-e^{\left .\frac {3}{2}\right /x}+x\right ) \log (2) \log ^2(\log (4))}{x} \]
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Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {12, 14, 2326} \[ \int \frac {2 x^3+e^{\left .\frac {3}{2}\right /x} (3+2 x) \log (2) \log ^2(\log (4))}{2 x^3} \, dx=x-\frac {e^{\left .\frac {3}{2}\right /x} \log (2) \log ^2(\log (4))}{x} \]
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Rule 12
Rule 14
Rule 2326
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int \frac {2 x^3+e^{\left .\frac {3}{2}\right /x} (3+2 x) \log (2) \log ^2(\log (4))}{x^3} \, dx \\ & = \frac {1}{2} \int \left (2+\frac {e^{\left .\frac {3}{2}\right /x} (3+2 x) \log (2) \log ^2(\log (4))}{x^3}\right ) \, dx \\ & = x+\frac {1}{2} \left (\log (2) \log ^2(\log (4))\right ) \int \frac {e^{\left .\frac {3}{2}\right /x} (3+2 x)}{x^3} \, dx \\ & = x-\frac {e^{\left .\frac {3}{2}\right /x} \log (2) \log ^2(\log (4))}{x} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {2 x^3+e^{\left .\frac {3}{2}\right /x} (3+2 x) \log (2) \log ^2(\log (4))}{2 x^3} \, dx=x-\frac {e^{\left .\frac {3}{2}\right /x} \log (2) \log ^2(\log (4))}{x} \]
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Time = 0.14 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85
method | result | size |
parts | \(x -\frac {\ln \left (2\right ) \ln \left (2 \ln \left (2\right )\right )^{2} {\mathrm e}^{\frac {3}{2 x}}}{x}\) | \(23\) |
parallelrisch | \(-\frac {2 \ln \left (2\right ) {\mathrm e}^{\frac {3}{2 x}} \ln \left (2 \ln \left (2\right )\right )^{2}-2 x^{2}}{2 x}\) | \(29\) |
risch | \(x -\frac {\ln \left (2\right ) \left (\ln \left (2\right )^{2}+2 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right )+\ln \left (\ln \left (2\right )\right )^{2}\right ) {\mathrm e}^{\frac {3}{2 x}}}{x}\) | \(33\) |
norman | \(\frac {x^{3}+\left (-\ln \left (2\right )^{3}-2 \ln \left (2\right )^{2} \ln \left (\ln \left (2\right )\right )-\ln \left (2\right ) \ln \left (\ln \left (2\right )\right )^{2}\right ) x \,{\mathrm e}^{\frac {3}{2 x}}}{x^{2}}\) | \(42\) |
derivativedivides | \(x -\frac {2 \ln \left (2\right )^{3} {\mathrm e}^{\frac {3}{2 x}}}{3}-\frac {2 \ln \left (2\right )^{3} \left (\frac {3 \,{\mathrm e}^{\frac {3}{2 x}}}{2 x}-{\mathrm e}^{\frac {3}{2 x}}\right )}{3}-\frac {2 \ln \left (2\right ) {\mathrm e}^{\frac {3}{2 x}} \ln \left (\ln \left (2\right )\right )^{2}}{3}-\frac {4 \ln \left (2\right )^{2} {\mathrm e}^{\frac {3}{2 x}} \ln \left (\ln \left (2\right )\right )}{3}-\frac {2 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right )^{2} \left (\frac {3 \,{\mathrm e}^{\frac {3}{2 x}}}{2 x}-{\mathrm e}^{\frac {3}{2 x}}\right )}{3}-\frac {4 \ln \left (2\right )^{2} \ln \left (\ln \left (2\right )\right ) \left (\frac {3 \,{\mathrm e}^{\frac {3}{2 x}}}{2 x}-{\mathrm e}^{\frac {3}{2 x}}\right )}{3}\) | \(129\) |
default | \(x -\frac {2 \ln \left (2\right )^{3} {\mathrm e}^{\frac {3}{2 x}}}{3}-\frac {2 \ln \left (2\right )^{3} \left (\frac {3 \,{\mathrm e}^{\frac {3}{2 x}}}{2 x}-{\mathrm e}^{\frac {3}{2 x}}\right )}{3}-\frac {2 \ln \left (2\right ) {\mathrm e}^{\frac {3}{2 x}} \ln \left (\ln \left (2\right )\right )^{2}}{3}-\frac {4 \ln \left (2\right )^{2} {\mathrm e}^{\frac {3}{2 x}} \ln \left (\ln \left (2\right )\right )}{3}-\frac {2 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right )^{2} \left (\frac {3 \,{\mathrm e}^{\frac {3}{2 x}}}{2 x}-{\mathrm e}^{\frac {3}{2 x}}\right )}{3}-\frac {4 \ln \left (2\right )^{2} \ln \left (\ln \left (2\right )\right ) \left (\frac {3 \,{\mathrm e}^{\frac {3}{2 x}}}{2 x}-{\mathrm e}^{\frac {3}{2 x}}\right )}{3}\) | \(129\) |
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Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {2 x^3+e^{\left .\frac {3}{2}\right /x} (3+2 x) \log (2) \log ^2(\log (4))}{2 x^3} \, dx=-\frac {e^{\left (\frac {3}{2 \, x}\right )} \log \left (2\right ) \log \left (2 \, \log \left (2\right )\right )^{2} - x^{2}}{x} \]
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Time = 0.08 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {2 x^3+e^{\left .\frac {3}{2}\right /x} (3+2 x) \log (2) \log ^2(\log (4))}{2 x^3} \, dx=x + \frac {\left (- \log {\left (2 \right )}^{3} - \log {\left (2 \right )} \log {\left (\log {\left (2 \right )} \right )}^{2} - 2 \log {\left (2 \right )}^{2} \log {\left (\log {\left (2 \right )} \right )}\right ) e^{\frac {3}{2 x}}}{x} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.21 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {2 x^3+e^{\left .\frac {3}{2}\right /x} (3+2 x) \log (2) \log ^2(\log (4))}{2 x^3} \, dx=-\frac {2}{3} \, e^{\left (\frac {3}{2 \, x}\right )} \log \left (2\right ) \log \left (2 \, \log \left (2\right )\right )^{2} + \frac {2}{3} \, \Gamma \left (2, -\frac {3}{2 \, x}\right ) \log \left (2\right ) \log \left (2 \, \log \left (2\right )\right )^{2} + x \]
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Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (26) = 52\).
Time = 0.25 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.00 \[ \int \frac {2 x^3+e^{\left .\frac {3}{2}\right /x} (3+2 x) \log (2) \log ^2(\log (4))}{2 x^3} \, dx=-{\left (\frac {e^{\left (\frac {3}{2 \, x}\right )} \log \left (2\right )^{3}}{x^{2}} + \frac {2 \, e^{\left (\frac {3}{2 \, x}\right )} \log \left (2\right )^{2} \log \left (\log \left (2\right )\right )}{x^{2}} + \frac {e^{\left (\frac {3}{2 \, x}\right )} \log \left (2\right ) \log \left (\log \left (2\right )\right )^{2}}{x^{2}} - 1\right )} x \]
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Time = 9.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {2 x^3+e^{\left .\frac {3}{2}\right /x} (3+2 x) \log (2) \log ^2(\log (4))}{2 x^3} \, dx=x-\frac {{\mathrm {e}}^{\frac {3}{2\,x}}\,\ln \left (2\right )\,{\ln \left (\ln \left (4\right )\right )}^2}{x} \]
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