Integrand size = 66, antiderivative size = 23 \[ \int \frac {x^4-2 x^3 \log (2)+x^2 \log ^2(2)+e^{\frac {10}{-x^2+x \log (2)}} (-20 x+10 \log (2))}{x^4-2 x^3 \log (2)+x^2 \log ^2(2)} \, dx=-e^{\frac {10}{x (-x+\log (2))}}+x-\log (5) \]
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Time = 0.58 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {1608, 27, 6820, 6838} \[ \int \frac {x^4-2 x^3 \log (2)+x^2 \log ^2(2)+e^{\frac {10}{-x^2+x \log (2)}} (-20 x+10 \log (2))}{x^4-2 x^3 \log (2)+x^2 \log ^2(2)} \, dx=x-e^{-\frac {10}{x (x-\log (2))}} \]
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Rule 27
Rule 1608
Rule 6820
Rule 6838
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^4-2 x^3 \log (2)+x^2 \log ^2(2)+e^{\frac {10}{-x^2+x \log (2)}} (-20 x+10 \log (2))}{x^2 \left (x^2-2 x \log (2)+\log ^2(2)\right )} \, dx \\ & = \int \frac {x^4-2 x^3 \log (2)+x^2 \log ^2(2)+e^{\frac {10}{-x^2+x \log (2)}} (-20 x+10 \log (2))}{x^2 (x-\log (2))^2} \, dx \\ & = \int \left (1+\frac {10 e^{-\frac {10}{x (x-\log (2))}} (-2 x+\log (2))}{x^2 (x-\log (2))^2}\right ) \, dx \\ & = x+10 \int \frac {e^{-\frac {10}{x (x-\log (2))}} (-2 x+\log (2))}{x^2 (x-\log (2))^2} \, dx \\ & = -e^{-\frac {10}{x (x-\log (2))}}+x \\ \end{align*}
Time = 0.56 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30 \[ \int \frac {x^4-2 x^3 \log (2)+x^2 \log ^2(2)+e^{\frac {10}{-x^2+x \log (2)}} (-20 x+10 \log (2))}{x^4-2 x^3 \log (2)+x^2 \log ^2(2)} \, dx=-e^{\frac {10}{x \log (2)}-\frac {10}{(x-\log (2)) \log (2)}}+x \]
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Time = 0.44 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83
method | result | size |
risch | \(-{\mathrm e}^{\frac {10}{x \left (\ln \left (2\right )-x \right )}}+x\) | \(19\) |
parts | \(x +\frac {x^{2} {\mathrm e}^{\frac {10}{x \ln \left (2\right )-x^{2}}}-x \ln \left (2\right ) {\mathrm e}^{\frac {10}{x \ln \left (2\right )-x^{2}}}}{x \left (\ln \left (2\right )-x \right )}\) | \(59\) |
norman | \(\frac {x^{2} {\mathrm e}^{\frac {10}{x \ln \left (2\right )-x^{2}}}+x \ln \left (2\right )^{2}-x^{3}-x \ln \left (2\right ) {\mathrm e}^{\frac {10}{x \ln \left (2\right )-x^{2}}}}{\left (\ln \left (2\right )-x \right ) x}\) | \(68\) |
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Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78 \[ \int \frac {x^4-2 x^3 \log (2)+x^2 \log ^2(2)+e^{\frac {10}{-x^2+x \log (2)}} (-20 x+10 \log (2))}{x^4-2 x^3 \log (2)+x^2 \log ^2(2)} \, dx=x - e^{\left (-\frac {10}{x^{2} - x \log \left (2\right )}\right )} \]
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Time = 0.12 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.52 \[ \int \frac {x^4-2 x^3 \log (2)+x^2 \log ^2(2)+e^{\frac {10}{-x^2+x \log (2)}} (-20 x+10 \log (2))}{x^4-2 x^3 \log (2)+x^2 \log ^2(2)} \, dx=x - e^{\frac {10}{- x^{2} + x \log {\left (2 \right )}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (22) = 44\).
Time = 0.34 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.48 \[ \int \frac {x^4-2 x^3 \log (2)+x^2 \log ^2(2)+e^{\frac {10}{-x^2+x \log (2)}} (-20 x+10 \log (2))}{x^4-2 x^3 \log (2)+x^2 \log ^2(2)} \, dx=2 \, {\left (\frac {\log \left (2\right )}{x - \log \left (2\right )} - \log \left (x - \log \left (2\right )\right )\right )} \log \left (2\right ) + 2 \, \log \left (2\right ) \log \left (x - \log \left (2\right )\right ) + x - \frac {2 \, \log \left (2\right )^{2}}{x - \log \left (2\right )} - e^{\left (-\frac {10}{x \log \left (2\right ) - \log \left (2\right )^{2}} + \frac {10}{x \log \left (2\right )}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78 \[ \int \frac {x^4-2 x^3 \log (2)+x^2 \log ^2(2)+e^{\frac {10}{-x^2+x \log (2)}} (-20 x+10 \log (2))}{x^4-2 x^3 \log (2)+x^2 \log ^2(2)} \, dx=x - e^{\left (-\frac {10}{x^{2} - x \log \left (2\right )}\right )} \]
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Time = 9.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {x^4-2 x^3 \log (2)+x^2 \log ^2(2)+e^{\frac {10}{-x^2+x \log (2)}} (-20 x+10 \log (2))}{x^4-2 x^3 \log (2)+x^2 \log ^2(2)} \, dx=x-{\mathrm {e}}^{\frac {10}{x\,\ln \left (2\right )-x^2}} \]
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