Integrand size = 93, antiderivative size = 26 \[ \int \frac {90 x^2 \log ^2(3)+e^{x^2} \left (-300 x-300 x^3\right ) \log ^2(3)}{225+60 x^3 \log ^2(3)+100 e^{2 x^2} x^4 \log ^4(3)+4 x^6 \log ^4(3)+e^{x^2} \left (-300 x^2 \log ^2(3)-40 x^5 \log ^4(3)\right )} \, dx=\frac {3}{-3+2 \left (e^{x^2}-\frac {x}{5}\right ) x^2 \log ^2(3)} \]
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Time = 0.18 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {6820, 12, 6818} \[ \int \frac {90 x^2 \log ^2(3)+e^{x^2} \left (-300 x-300 x^3\right ) \log ^2(3)}{225+60 x^3 \log ^2(3)+100 e^{2 x^2} x^4 \log ^4(3)+4 x^6 \log ^4(3)+e^{x^2} \left (-300 x^2 \log ^2(3)-40 x^5 \log ^4(3)\right )} \, dx=-\frac {15}{2 x^3 \log ^2(3)-10 e^{x^2} x^2 \log ^2(3)+15} \]
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Rule 12
Rule 6818
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {30 x \left (3 x-10 e^{x^2} \left (1+x^2\right )\right ) \log ^2(3)}{\left (15-10 e^{x^2} x^2 \log ^2(3)+2 x^3 \log ^2(3)\right )^2} \, dx \\ & = \left (30 \log ^2(3)\right ) \int \frac {x \left (3 x-10 e^{x^2} \left (1+x^2\right )\right )}{\left (15-10 e^{x^2} x^2 \log ^2(3)+2 x^3 \log ^2(3)\right )^2} \, dx \\ & = -\frac {15}{15-10 e^{x^2} x^2 \log ^2(3)+2 x^3 \log ^2(3)} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.46 \[ \int \frac {90 x^2 \log ^2(3)+e^{x^2} \left (-300 x-300 x^3\right ) \log ^2(3)}{225+60 x^3 \log ^2(3)+100 e^{2 x^2} x^4 \log ^4(3)+4 x^6 \log ^4(3)+e^{x^2} \left (-300 x^2 \log ^2(3)-40 x^5 \log ^4(3)\right )} \, dx=-\frac {30 \log ^2(3)}{30 \log ^2(3)-20 e^{x^2} x^2 \log ^4(3)+4 x^3 \log ^4(3)} \]
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Time = 0.40 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12
method | result | size |
risch | \(-\frac {15}{2 x^{3} \ln \left (3\right )^{2}-10 \,{\mathrm e}^{x^{2}} \ln \left (3\right )^{2} x^{2}+15}\) | \(29\) |
norman | \(\frac {2 x^{3} \ln \left (3\right )^{2}-10 \,{\mathrm e}^{x^{2}} \ln \left (3\right )^{2} x^{2}}{2 x^{3} \ln \left (3\right )^{2}-10 \,{\mathrm e}^{x^{2}} \ln \left (3\right )^{2} x^{2}+15}\) | \(51\) |
parallelrisch | \(\frac {30 x^{3} \ln \left (3\right )^{2}-150 \,{\mathrm e}^{x^{2}} \ln \left (3\right )^{2} x^{2}}{30 x^{3} \ln \left (3\right )^{2}-150 \,{\mathrm e}^{x^{2}} \ln \left (3\right )^{2} x^{2}+225}\) | \(52\) |
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Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {90 x^2 \log ^2(3)+e^{x^2} \left (-300 x-300 x^3\right ) \log ^2(3)}{225+60 x^3 \log ^2(3)+100 e^{2 x^2} x^4 \log ^4(3)+4 x^6 \log ^4(3)+e^{x^2} \left (-300 x^2 \log ^2(3)-40 x^5 \log ^4(3)\right )} \, dx=-\frac {15}{2 \, x^{3} \log \left (3\right )^{2} - 10 \, x^{2} e^{\left (x^{2}\right )} \log \left (3\right )^{2} + 15} \]
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Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {90 x^2 \log ^2(3)+e^{x^2} \left (-300 x-300 x^3\right ) \log ^2(3)}{225+60 x^3 \log ^2(3)+100 e^{2 x^2} x^4 \log ^4(3)+4 x^6 \log ^4(3)+e^{x^2} \left (-300 x^2 \log ^2(3)-40 x^5 \log ^4(3)\right )} \, dx=\frac {15}{- 2 x^{3} \log {\left (3 \right )}^{2} + 10 x^{2} e^{x^{2}} \log {\left (3 \right )}^{2} - 15} \]
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Time = 0.30 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {90 x^2 \log ^2(3)+e^{x^2} \left (-300 x-300 x^3\right ) \log ^2(3)}{225+60 x^3 \log ^2(3)+100 e^{2 x^2} x^4 \log ^4(3)+4 x^6 \log ^4(3)+e^{x^2} \left (-300 x^2 \log ^2(3)-40 x^5 \log ^4(3)\right )} \, dx=-\frac {15}{2 \, x^{3} \log \left (3\right )^{2} - 10 \, x^{2} e^{\left (x^{2}\right )} \log \left (3\right )^{2} + 15} \]
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Time = 0.32 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {90 x^2 \log ^2(3)+e^{x^2} \left (-300 x-300 x^3\right ) \log ^2(3)}{225+60 x^3 \log ^2(3)+100 e^{2 x^2} x^4 \log ^4(3)+4 x^6 \log ^4(3)+e^{x^2} \left (-300 x^2 \log ^2(3)-40 x^5 \log ^4(3)\right )} \, dx=-\frac {15}{2 \, x^{3} \log \left (3\right )^{2} - 10 \, x^{2} e^{\left (x^{2}\right )} \log \left (3\right )^{2} + 15} \]
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Timed out. \[ \int \frac {90 x^2 \log ^2(3)+e^{x^2} \left (-300 x-300 x^3\right ) \log ^2(3)}{225+60 x^3 \log ^2(3)+100 e^{2 x^2} x^4 \log ^4(3)+4 x^6 \log ^4(3)+e^{x^2} \left (-300 x^2 \log ^2(3)-40 x^5 \log ^4(3)\right )} \, dx=\int \frac {90\,x^2\,{\ln \left (3\right )}^2-{\mathrm {e}}^{x^2}\,{\ln \left (3\right )}^2\,\left (300\,x^3+300\,x\right )}{60\,x^3\,{\ln \left (3\right )}^2-{\mathrm {e}}^{x^2}\,\left (40\,{\ln \left (3\right )}^4\,x^5+300\,{\ln \left (3\right )}^2\,x^2\right )+4\,x^6\,{\ln \left (3\right )}^4+100\,x^4\,{\mathrm {e}}^{2\,x^2}\,{\ln \left (3\right )}^4+225} \,d x \]
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