Integrand size = 45, antiderivative size = 24 \[ \int \frac {-12-36 x^4+e^2 \left (-18 x-6 x^2+18 x^5-18 x^6\right )}{e^2 \left (x^2+3 x^6\right )} \, dx=6 \left (\frac {2}{e^2 x}-x+\log \left (\frac {1}{3 x^3}+x\right )\right ) \]
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Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {12, 1607, 1847, 457, 78, 1600, 14} \[ \int \frac {-12-36 x^4+e^2 \left (-18 x-6 x^2+18 x^5-18 x^6\right )}{e^2 \left (x^2+3 x^6\right )} \, dx=6 \log \left (3 x^4+1\right )-6 x+\frac {12}{e^2 x}-18 \log (x) \]
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Rule 12
Rule 14
Rule 78
Rule 457
Rule 1600
Rule 1607
Rule 1847
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {-12-36 x^4+e^2 \left (-18 x-6 x^2+18 x^5-18 x^6\right )}{x^2+3 x^6} \, dx}{e^2} \\ & = \frac {\int \frac {-12-36 x^4+e^2 \left (-18 x-6 x^2+18 x^5-18 x^6\right )}{x^2 \left (1+3 x^4\right )} \, dx}{e^2} \\ & = \frac {\int \left (\frac {-18 e^2+18 e^2 x^4}{x \left (1+3 x^4\right )}+\frac {-12-6 e^2 x^2-36 x^4-18 e^2 x^6}{x^2 \left (1+3 x^4\right )}\right ) \, dx}{e^2} \\ & = \frac {\int \frac {-18 e^2+18 e^2 x^4}{x \left (1+3 x^4\right )} \, dx}{e^2}+\frac {\int \frac {-12-6 e^2 x^2-36 x^4-18 e^2 x^6}{x^2 \left (1+3 x^4\right )} \, dx}{e^2} \\ & = \frac {\text {Subst}\left (\int \frac {-18 e^2+18 e^2 x}{x (1+3 x)} \, dx,x,x^4\right )}{4 e^2}+\frac {\int \frac {-12-6 e^2 x^2}{x^2} \, dx}{e^2} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {18 e^2}{x}+\frac {72 e^2}{1+3 x}\right ) \, dx,x,x^4\right )}{4 e^2}+\frac {\int \left (-6 e^2-\frac {12}{x^2}\right ) \, dx}{e^2} \\ & = \frac {12}{e^2 x}-6 x-18 \log (x)+6 \log \left (1+3 x^4\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {-12-36 x^4+e^2 \left (-18 x-6 x^2+18 x^5-18 x^6\right )}{e^2 \left (x^2+3 x^6\right )} \, dx=-\frac {6 \left (-\frac {2}{x}+e^2 x+3 e^2 \log (x)-e^2 \log \left (1+3 x^4\right )\right )}{e^2} \]
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Time = 0.66 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08
method | result | size |
risch | \(-6 x +\frac {12 \,{\mathrm e}^{-2}}{x}-18 \ln \left (x \right )+6 \ln \left (-3 x^{4}-1\right )\) | \(26\) |
norman | \(\frac {-6 x^{2}+12 \,{\mathrm e}^{-2}}{x}-18 \ln \left (x \right )+6 \ln \left (3 x^{4}+1\right )\) | \(32\) |
default | \({\mathrm e}^{-2} \left (-6 \,{\mathrm e}^{2} x +\frac {12}{x}-18 \,{\mathrm e}^{2} \ln \left (x \right )+6 \,{\mathrm e}^{2} \ln \left (3 x^{4}+1\right )\right )\) | \(35\) |
parallelrisch | \(-\frac {{\mathrm e}^{-2} \left (18 x \,{\mathrm e}^{2} \ln \left (x \right )-6 \,{\mathrm e}^{2} \ln \left (x^{4}+\frac {1}{3}\right ) x +6 x^{2} {\mathrm e}^{2}-12\right )}{x}\) | \(37\) |
meijerg | \(\text {Expression too large to display}\) | \(718\) |
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Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46 \[ \int \frac {-12-36 x^4+e^2 \left (-18 x-6 x^2+18 x^5-18 x^6\right )}{e^2 \left (x^2+3 x^6\right )} \, dx=-\frac {6 \, {\left (x^{2} e^{2} - x e^{2} \log \left (3 \, x^{4} + 1\right ) + 3 \, x e^{2} \log \left (x\right ) - 2\right )} e^{\left (-2\right )}}{x} \]
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Time = 0.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-12-36 x^4+e^2 \left (-18 x-6 x^2+18 x^5-18 x^6\right )}{e^2 \left (x^2+3 x^6\right )} \, dx=- 6 x - 18 \log {\left (x \right )} + 6 \log {\left (3 x^{4} + 1 \right )} + \frac {12}{x e^{2}} \]
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Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \frac {-12-36 x^4+e^2 \left (-18 x-6 x^2+18 x^5-18 x^6\right )}{e^2 \left (x^2+3 x^6\right )} \, dx=-6 \, {\left (x e^{2} - e^{2} \log \left (3 \, x^{4} + 1\right ) + 3 \, e^{2} \log \left (x\right ) - \frac {2}{x}\right )} e^{\left (-2\right )} \]
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Time = 0.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38 \[ \int \frac {-12-36 x^4+e^2 \left (-18 x-6 x^2+18 x^5-18 x^6\right )}{e^2 \left (x^2+3 x^6\right )} \, dx=-6 \, {\left (x e^{2} - e^{2} \log \left (3 \, x^{4} + 1\right ) + 3 \, e^{2} \log \left ({\left | x \right |}\right ) - \frac {2}{x}\right )} e^{\left (-2\right )} \]
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Time = 0.09 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {-12-36 x^4+e^2 \left (-18 x-6 x^2+18 x^5-18 x^6\right )}{e^2 \left (x^2+3 x^6\right )} \, dx=6\,\ln \left (x^4+\frac {1}{3}\right )-6\,x-18\,\ln \left (x\right )+\frac {12\,{\mathrm {e}}^{-2}}{x} \]
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