Integrand size = 39, antiderivative size = 17 \[ \int \frac {-3-\log (x)-\log (\log (3))}{16+8 \log (x)+\log ^2(x)+(8+2 \log (x)) \log (\log (3))+\log ^2(\log (3))} \, dx=1+e^5-\frac {x}{4+\log (x)+\log (\log (3))} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.10 (sec) , antiderivative size = 86, normalized size of antiderivative = 5.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6820, 2334, 2336, 2209, 2408, 6617} \[ \int \frac {-3-\log (x)-\log (\log (3))}{16+8 \log (x)+\log ^2(x)+(8+2 \log (x)) \log (\log (3))+\log ^2(\log (3))} \, dx=-\frac {(\log (x \log (3))+3) \operatorname {ExpIntegralEi}(\log (x)+\log (\log (3))+4)}{e^4 \log (3)}+\frac {(\log (x \log (3))+4) \operatorname {ExpIntegralEi}(\log (x \log (3))+4)}{e^4 \log (3)}-\frac {\operatorname {ExpIntegralEi}(\log (x \log (3))+4)}{e^4 \log (3)}-x+\frac {x (\log (x \log (3))+3)}{\log (x)+4+\log (\log (3))} \]
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Rule 2209
Rule 2334
Rule 2336
Rule 2408
Rule 6617
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {-3-\log (x \log (3))}{\left (\log (x)+4 \left (1+\frac {1}{4} \log (\log (3))\right )\right )^2} \, dx \\ & = -\frac {\text {Ei}(4+\log (x)+\log (\log (3))) (3+\log (x \log (3)))}{e^4 \log (3)}+\frac {x (3+\log (x \log (3)))}{4+\log (x)+\log (\log (3))}+\int \left (\frac {\text {Ei}(4+\log (x \log (3)))}{e^4 x \log (3)}-\frac {1}{4+\log (x \log (3))}\right ) \, dx \\ & = -\frac {\text {Ei}(4+\log (x)+\log (\log (3))) (3+\log (x \log (3)))}{e^4 \log (3)}+\frac {x (3+\log (x \log (3)))}{4+\log (x)+\log (\log (3))}+\frac {\int \frac {\text {Ei}(4+\log (x \log (3)))}{x} \, dx}{e^4 \log (3)}-\int \frac {1}{4+\log (x \log (3))} \, dx \\ & = -\frac {\text {Ei}(4+\log (x)+\log (\log (3))) (3+\log (x \log (3)))}{e^4 \log (3)}+\frac {x (3+\log (x \log (3)))}{4+\log (x)+\log (\log (3))}-\frac {\text {Subst}\left (\int \frac {e^x}{4+x} \, dx,x,\log (x \log (3))\right )}{\log (3)}+\frac {\text {Subst}(\int \text {Ei}(4+x) \, dx,x,\log (x \log (3)))}{e^4 \log (3)} \\ & = -x-\frac {\text {Ei}(4+\log (x \log (3)))}{e^4 \log (3)}-\frac {\text {Ei}(4+\log (x)+\log (\log (3))) (3+\log (x \log (3)))}{e^4 \log (3)}+\frac {x (3+\log (x \log (3)))}{4+\log (x)+\log (\log (3))}+\frac {\text {Ei}(4+\log (x \log (3))) (4+\log (x \log (3)))}{e^4 \log (3)} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.71 \[ \int \frac {-3-\log (x)-\log (\log (3))}{16+8 \log (x)+\log ^2(x)+(8+2 \log (x)) \log (\log (3))+\log ^2(\log (3))} \, dx=-\frac {x}{4+\log (x \log (3))} \]
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Time = 0.51 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76
method | result | size |
default | \(-\frac {x}{\ln \left (x \right )+\ln \left (\ln \left (3\right )\right )+4}\) | \(13\) |
norman | \(-\frac {x}{\ln \left (x \right )+\ln \left (\ln \left (3\right )\right )+4}\) | \(13\) |
risch | \(-\frac {x}{\ln \left (x \right )+\ln \left (\ln \left (3\right )\right )+4}\) | \(13\) |
parallelrisch | \(-\frac {x}{\ln \left (x \right )+\ln \left (\ln \left (3\right )\right )+4}\) | \(13\) |
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none
Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.71 \[ \int \frac {-3-\log (x)-\log (\log (3))}{16+8 \log (x)+\log ^2(x)+(8+2 \log (x)) \log (\log (3))+\log ^2(\log (3))} \, dx=-\frac {x}{\log \left (x\right ) + \log \left (\log \left (3\right )\right ) + 4} \]
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Time = 0.08 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.71 \[ \int \frac {-3-\log (x)-\log (\log (3))}{16+8 \log (x)+\log ^2(x)+(8+2 \log (x)) \log (\log (3))+\log ^2(\log (3))} \, dx=- \frac {x}{\log {\left (x \right )} + \log {\left (\log {\left (3 \right )} \right )} + 4} \]
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none
Time = 0.32 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.71 \[ \int \frac {-3-\log (x)-\log (\log (3))}{16+8 \log (x)+\log ^2(x)+(8+2 \log (x)) \log (\log (3))+\log ^2(\log (3))} \, dx=-\frac {x}{\log \left (x\right ) + \log \left (\log \left (3\right )\right ) + 4} \]
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none
Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.71 \[ \int \frac {-3-\log (x)-\log (\log (3))}{16+8 \log (x)+\log ^2(x)+(8+2 \log (x)) \log (\log (3))+\log ^2(\log (3))} \, dx=-\frac {x}{\log \left (x\right ) + \log \left (\log \left (3\right )\right ) + 4} \]
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Time = 12.31 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.71 \[ \int \frac {-3-\log (x)-\log (\log (3))}{16+8 \log (x)+\log ^2(x)+(8+2 \log (x)) \log (\log (3))+\log ^2(\log (3))} \, dx=-\frac {x}{\ln \left (\ln \left (3\right )\right )+\ln \left (x\right )+4} \]
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