Integrand size = 55, antiderivative size = 19 \[ \int \frac {e^{-\frac {1250}{x \log (2 x)}} (-2500-2500 \log (2 x))+e^{-\frac {2500}{x \log (2 x)}} (2500+2500 \log (2 x))}{x^2 \log ^2(2 x)} \, dx=\left (1-e^{-\frac {1250}{x \log (2 x)}}\right )^2 \]
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Time = 1.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.53, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, Rules used = {6873, 12, 6874, 6838} \[ \int \frac {e^{-\frac {1250}{x \log (2 x)}} (-2500-2500 \log (2 x))+e^{-\frac {2500}{x \log (2 x)}} (2500+2500 \log (2 x))}{x^2 \log ^2(2 x)} \, dx=e^{-\frac {2500}{x \log (2 x)}}-2 e^{-\frac {1250}{x \log (2 x)}} \]
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Rule 12
Rule 6838
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {2500 e^{-\frac {2500}{x \log (2 x)}} \left (1-e^{\frac {1250}{x \log (2 x)}}\right ) (1+\log (2 x))}{x^2 \log ^2(2 x)} \, dx \\ & = 2500 \int \frac {e^{-\frac {2500}{x \log (2 x)}} \left (1-e^{\frac {1250}{x \log (2 x)}}\right ) (1+\log (2 x))}{x^2 \log ^2(2 x)} \, dx \\ & = 2500 \int \left (\frac {e^{-\frac {2500}{x \log (2 x)}} (1+\log (2 x))}{x^2 \log ^2(2 x)}-\frac {e^{-\frac {1250}{x \log (2 x)}} (1+\log (2 x))}{x^2 \log ^2(2 x)}\right ) \, dx \\ & = 2500 \int \frac {e^{-\frac {2500}{x \log (2 x)}} (1+\log (2 x))}{x^2 \log ^2(2 x)} \, dx-2500 \int \frac {e^{-\frac {1250}{x \log (2 x)}} (1+\log (2 x))}{x^2 \log ^2(2 x)} \, dx \\ & = e^{-\frac {2500}{x \log (2 x)}}-2 e^{-\frac {1250}{x \log (2 x)}} \\ \end{align*}
Time = 0.67 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.95 \[ \int \frac {e^{-\frac {1250}{x \log (2 x)}} (-2500-2500 \log (2 x))+e^{-\frac {2500}{x \log (2 x)}} (2500+2500 \log (2 x))}{x^2 \log ^2(2 x)} \, dx=-2500 \left (-\frac {e^{-\frac {2500}{x \log (2 x)}}}{2500}+\frac {e^{-\frac {1250}{x \log (2 x)}}}{1250}\right ) \]
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Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.47
method | result | size |
risch | \({\mathrm e}^{-\frac {2500}{x \ln \left (2 x \right )}}-2 \,{\mathrm e}^{-\frac {1250}{x \ln \left (2 x \right )}}\) | \(28\) |
default | \(\frac {-2 x \ln \left (2\right ) {\mathrm e}^{-\frac {1250}{\left (\ln \left (2\right )+\ln \left (x \right )\right ) x}}-2 \ln \left (x \right ) x \,{\mathrm e}^{-\frac {1250}{\left (\ln \left (2\right )+\ln \left (x \right )\right ) x}}}{x \left (\ln \left (2\right )+\ln \left (x \right )\right )}+{\mathrm e}^{-\frac {2500}{\left (\ln \left (2\right )+\ln \left (x \right )\right ) x}}\) | \(63\) |
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none
Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.42 \[ \int \frac {e^{-\frac {1250}{x \log (2 x)}} (-2500-2500 \log (2 x))+e^{-\frac {2500}{x \log (2 x)}} (2500+2500 \log (2 x))}{x^2 \log ^2(2 x)} \, dx=-2 \, e^{\left (-\frac {1250}{x \log \left (2 \, x\right )}\right )} + e^{\left (-\frac {2500}{x \log \left (2 \, x\right )}\right )} \]
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Exception generated. \[ \int \frac {e^{-\frac {1250}{x \log (2 x)}} (-2500-2500 \log (2 x))+e^{-\frac {2500}{x \log (2 x)}} (2500+2500 \log (2 x))}{x^2 \log ^2(2 x)} \, dx=\text {Exception raised: TypeError} \]
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Exception generated. \[ \int \frac {e^{-\frac {1250}{x \log (2 x)}} (-2500-2500 \log (2 x))+e^{-\frac {2500}{x \log (2 x)}} (2500+2500 \log (2 x))}{x^2 \log ^2(2 x)} \, dx=\text {Exception raised: RuntimeError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (16) = 32\).
Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.79 \[ \int \frac {e^{-\frac {1250}{x \log (2 x)}} (-2500-2500 \log (2 x))+e^{-\frac {2500}{x \log (2 x)}} (2500+2500 \log (2 x))}{x^2 \log ^2(2 x)} \, dx=-{\left (2 \, e^{\left (\frac {1250}{x \log \left (2\right ) + x \log \left (x\right )}\right )} - 1\right )} e^{\left (-\frac {2500}{x \log \left (2\right ) + x \log \left (x\right )}\right )} \]
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Time = 11.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.63 \[ \int \frac {e^{-\frac {1250}{x \log (2 x)}} (-2500-2500 \log (2 x))+e^{-\frac {2500}{x \log (2 x)}} (2500+2500 \log (2 x))}{x^2 \log ^2(2 x)} \, dx={\mathrm {e}}^{-\frac {2500}{x\,\ln \left (2\right )+x\,\ln \left (x\right )}}-2\,{\mathrm {e}}^{-\frac {1250}{x\,\ln \left (2\right )+x\,\ln \left (x\right )}} \]
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