Integrand size = 34, antiderivative size = 18 \[ \int \frac {-1-x^2-2 x^3}{x+e^5 x^2-x^3-x^4} \, dx=-4+\log \left (-e^5-\frac {1}{x}+x+x^2\right ) \]
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Time = 0.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.28, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2099, 1601} \[ \int \frac {-1-x^2-2 x^3}{x+e^5 x^2-x^3-x^4} \, dx=\log \left (-x^3-x^2+e^5 x+1\right )-\log (x) \]
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Rule 1601
Rule 2099
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{x}+\frac {e^5-2 x-3 x^2}{1+e^5 x-x^2-x^3}\right ) \, dx \\ & = -\log (x)+\int \frac {e^5-2 x-3 x^2}{1+e^5 x-x^2-x^3} \, dx \\ & = -\log (x)+\log \left (1+e^5 x-x^2-x^3\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.28 \[ \int \frac {-1-x^2-2 x^3}{x+e^5 x^2-x^3-x^4} \, dx=-\log (x)+\log \left (1+e^5 x-x^2-x^3\right ) \]
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Time = 0.15 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11
method | result | size |
default | \(-\ln \left (x \right )+\ln \left (x^{3}-x \,{\mathrm e}^{5}+x^{2}-1\right )\) | \(20\) |
risch | \(-\ln \left (x \right )+\ln \left (x^{3}-x \,{\mathrm e}^{5}+x^{2}-1\right )\) | \(20\) |
parallelrisch | \(-\ln \left (x \right )+\ln \left (x^{3}-x \,{\mathrm e}^{5}+x^{2}-1\right )\) | \(20\) |
norman | \(-\ln \left (x \right )+\ln \left (-x^{3}+x \,{\mathrm e}^{5}-x^{2}+1\right )\) | \(23\) |
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none
Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {-1-x^2-2 x^3}{x+e^5 x^2-x^3-x^4} \, dx=\log \left (x^{3} + x^{2} - x e^{5} - 1\right ) - \log \left (x\right ) \]
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Time = 0.32 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {-1-x^2-2 x^3}{x+e^5 x^2-x^3-x^4} \, dx=- \log {\left (x \right )} + \log {\left (x^{3} + x^{2} - x e^{5} - 1 \right )} \]
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none
Time = 0.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {-1-x^2-2 x^3}{x+e^5 x^2-x^3-x^4} \, dx=\log \left (x^{3} + x^{2} - x e^{5} - 1\right ) - \log \left (x\right ) \]
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none
Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.17 \[ \int \frac {-1-x^2-2 x^3}{x+e^5 x^2-x^3-x^4} \, dx=\log \left ({\left | x^{3} + x^{2} - x e^{5} - 1 \right |}\right ) - \log \left ({\left | x \right |}\right ) \]
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Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {-1-x^2-2 x^3}{x+e^5 x^2-x^3-x^4} \, dx=\ln \left (x^3+x^2-{\mathrm {e}}^5\,x-1\right )-\ln \left (x\right ) \]
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