Integrand size = 41, antiderivative size = 25 \[ \int \frac {e^{4/x} (-144-27 x)+e^{4/x} (36+9 x) \log \left (\frac {\log (4)}{x}\right )}{8 x^3} \, dx=\frac {9 e^{4/x} \left (4-\log \left (\frac {\log (4)}{x}\right )\right )}{8 x} \]
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Time = 0.17 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44, number of steps used = 15, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {12, 14, 2243, 2240, 2634, 6874, 2241} \[ \int \frac {e^{4/x} (-144-27 x)+e^{4/x} (36+9 x) \log \left (\frac {\log (4)}{x}\right )}{8 x^3} \, dx=\frac {9 e^{4/x}}{2 x}-\frac {9 e^{4/x} \log \left (\frac {\log (4)}{x}\right )}{8 x} \]
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Rule 12
Rule 14
Rule 2240
Rule 2241
Rule 2243
Rule 2634
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {1}{8} \int \frac {e^{4/x} (-144-27 x)+e^{4/x} (36+9 x) \log \left (\frac {\log (4)}{x}\right )}{x^3} \, dx \\ & = \frac {1}{8} \int \left (-\frac {144 e^{4/x}}{x^3}-\frac {27 e^{4/x}}{x^2}+\frac {36 e^{4/x} \log \left (\frac {\log (4)}{x}\right )}{x^3}+\frac {9 e^{4/x} \log \left (\frac {\log (4)}{x}\right )}{x^2}\right ) \, dx \\ & = \frac {9}{8} \int \frac {e^{4/x} \log \left (\frac {\log (4)}{x}\right )}{x^2} \, dx-\frac {27}{8} \int \frac {e^{4/x}}{x^2} \, dx+\frac {9}{2} \int \frac {e^{4/x} \log \left (\frac {\log (4)}{x}\right )}{x^3} \, dx-18 \int \frac {e^{4/x}}{x^3} \, dx \\ & = \frac {27 e^{4/x}}{32}+\frac {9 e^{4/x}}{2 x}-\frac {9 e^{4/x} \log \left (\frac {\log (4)}{x}\right )}{8 x}-\frac {9}{8} \int \frac {e^{4/x}}{4 x} \, dx+\frac {9}{2} \int \frac {e^{4/x}}{x^2} \, dx-\frac {9}{2} \int \frac {e^{4/x} (4-x)}{16 x^2} \, dx \\ & = -\frac {9 e^{4/x}}{32}+\frac {9 e^{4/x}}{2 x}-\frac {9 e^{4/x} \log \left (\frac {\log (4)}{x}\right )}{8 x}-\frac {9}{32} \int \frac {e^{4/x} (4-x)}{x^2} \, dx-\frac {9}{32} \int \frac {e^{4/x}}{x} \, dx \\ & = -\frac {9 e^{4/x}}{32}+\frac {9 e^{4/x}}{2 x}+\frac {9 \operatorname {ExpIntegralEi}\left (\frac {4}{x}\right )}{32}-\frac {9 e^{4/x} \log \left (\frac {\log (4)}{x}\right )}{8 x}-\frac {9}{32} \int \left (\frac {4 e^{4/x}}{x^2}-\frac {e^{4/x}}{x}\right ) \, dx \\ & = -\frac {9 e^{4/x}}{32}+\frac {9 e^{4/x}}{2 x}+\frac {9 \operatorname {ExpIntegralEi}\left (\frac {4}{x}\right )}{32}-\frac {9 e^{4/x} \log \left (\frac {\log (4)}{x}\right )}{8 x}+\frac {9}{32} \int \frac {e^{4/x}}{x} \, dx-\frac {9}{8} \int \frac {e^{4/x}}{x^2} \, dx \\ & = \frac {9 e^{4/x}}{2 x}-\frac {9 e^{4/x} \log \left (\frac {\log (4)}{x}\right )}{8 x} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {e^{4/x} (-144-27 x)+e^{4/x} (36+9 x) \log \left (\frac {\log (4)}{x}\right )}{8 x^3} \, dx=-\frac {9 e^{4/x} \left (-4+\log \left (\frac {\log (4)}{x}\right )\right )}{8 x} \]
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Time = 0.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24
method | result | size |
parallelrisch | \(\frac {-9 \,{\mathrm e}^{\frac {4}{x}} \ln \left (\frac {2 \ln \left (2\right )}{x}\right )+36 \,{\mathrm e}^{\frac {4}{x}}}{8 x}\) | \(31\) |
norman | \(\frac {\frac {9 x \,{\mathrm e}^{\frac {4}{x}}}{2}-\frac {9 x \,{\mathrm e}^{\frac {4}{x}} \ln \left (\frac {2 \ln \left (2\right )}{x}\right )}{8}}{x^{2}}\) | \(32\) |
risch | \(\frac {9 \,{\mathrm e}^{\frac {4}{x}} \ln \left (x \right )}{8 x}-\frac {9 \left (-8+2 \ln \left (2\right )+2 \ln \left (\ln \left (2\right )\right )\right ) {\mathrm e}^{\frac {4}{x}}}{16 x}\) | \(37\) |
default | \(\frac {\left (36-9 \ln \left (\frac {2 \ln \left (2\right )}{x}\right )-9 \ln \left (x \right )\right ) x \,{\mathrm e}^{\frac {4}{x}}+9 \,{\mathrm e}^{\frac {4}{x}} \ln \left (x \right ) x}{8 x^{2}}\) | \(42\) |
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Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {e^{4/x} (-144-27 x)+e^{4/x} (36+9 x) \log \left (\frac {\log (4)}{x}\right )}{8 x^3} \, dx=-\frac {9 \, {\left (e^{\frac {4}{x}} \log \left (\frac {2 \, \log \left (2\right )}{x}\right ) - 4 \, e^{\frac {4}{x}}\right )}}{8 \, x} \]
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Time = 0.09 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {e^{4/x} (-144-27 x)+e^{4/x} (36+9 x) \log \left (\frac {\log (4)}{x}\right )}{8 x^3} \, dx=\frac {\left (36 - 9 \log {\left (\frac {2 \log {\left (2 \right )}}{x} \right )}\right ) e^{\frac {4}{x}}}{8 x} \]
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\[ \int \frac {e^{4/x} (-144-27 x)+e^{4/x} (36+9 x) \log \left (\frac {\log (4)}{x}\right )}{8 x^3} \, dx=\int { \frac {9 \, {\left ({\left (x + 4\right )} e^{\frac {4}{x}} \log \left (\frac {2 \, \log \left (2\right )}{x}\right ) - {\left (3 \, x + 16\right )} e^{\frac {4}{x}}\right )}}{8 \, x^{3}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (21) = 42\).
Time = 0.27 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.72 \[ \int \frac {e^{4/x} (-144-27 x)+e^{4/x} (36+9 x) \log \left (\frac {\log (4)}{x}\right )}{8 x^3} \, dx=-\frac {9 \, {\left (e^{\frac {4}{x}} \log \left (2\right ) - e^{\frac {4}{x}} \log \left (x\right ) + e^{\frac {4}{x}} \log \left (\log \left (2\right )\right ) - 4 \, e^{\frac {4}{x}}\right )}}{8 \, x} \]
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Time = 9.07 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {e^{4/x} (-144-27 x)+e^{4/x} (36+9 x) \log \left (\frac {\log (4)}{x}\right )}{8 x^3} \, dx=-\frac {9\,{\mathrm {e}}^{4/x}\,\left (\ln \left (\frac {2\,\ln \left (2\right )}{x}\right )-4\right )}{8\,x} \]
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