Integrand size = 77, antiderivative size = 26 \[ \int \frac {-256 e^{8+2 x^2}-656 x^3-1681 x^4+e^{4+x^2} \left (-256 x+1312 x^2+512 x^3\right )}{256 e^{8+2 x^2} x-1312 e^{4+x^2} x^3+1681 x^5} \, dx=\frac {x}{-e^{4+x^2}+\frac {41 x^2}{16}}-\log (x) \]
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\[ \int \frac {-256 e^{8+2 x^2}-656 x^3-1681 x^4+e^{4+x^2} \left (-256 x+1312 x^2+512 x^3\right )}{256 e^{8+2 x^2} x-1312 e^{4+x^2} x^3+1681 x^5} \, dx=\int \frac {-256 e^{8+2 x^2}-656 x^3-1681 x^4+e^{4+x^2} \left (-256 x+1312 x^2+512 x^3\right )}{256 e^{8+2 x^2} x-1312 e^{4+x^2} x^3+1681 x^5} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-256 e^{8+2 x^2}-656 x^3-1681 x^4+e^{4+x^2} \left (-256 x+1312 x^2+512 x^3\right )}{x \left (16 e^{4+x^2}-41 x^2\right )^2} \, dx \\ & = \int \left (-\frac {1}{x}+\frac {1312 x^2 \left (-1+x^2\right )}{\left (16 e^{4+x^2}-41 x^2\right )^2}+\frac {16 \left (-1+2 x^2\right )}{16 e^{4+x^2}-41 x^2}\right ) \, dx \\ & = -\log (x)+16 \int \frac {-1+2 x^2}{16 e^{4+x^2}-41 x^2} \, dx+1312 \int \frac {x^2 \left (-1+x^2\right )}{\left (16 e^{4+x^2}-41 x^2\right )^2} \, dx \\ & = -\log (x)+16 \int \left (-\frac {1}{16 e^{4+x^2}-41 x^2}+\frac {2 x^2}{16 e^{4+x^2}-41 x^2}\right ) \, dx+1312 \int \left (-\frac {x^2}{\left (16 e^{4+x^2}-41 x^2\right )^2}+\frac {x^4}{\left (16 e^{4+x^2}-41 x^2\right )^2}\right ) \, dx \\ & = -\log (x)-16 \int \frac {1}{16 e^{4+x^2}-41 x^2} \, dx+32 \int \frac {x^2}{16 e^{4+x^2}-41 x^2} \, dx-1312 \int \frac {x^2}{\left (16 e^{4+x^2}-41 x^2\right )^2} \, dx+1312 \int \frac {x^4}{\left (16 e^{4+x^2}-41 x^2\right )^2} \, dx \\ \end{align*}
Time = 1.70 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {-256 e^{8+2 x^2}-656 x^3-1681 x^4+e^{4+x^2} \left (-256 x+1312 x^2+512 x^3\right )}{256 e^{8+2 x^2} x-1312 e^{4+x^2} x^3+1681 x^5} \, dx=-\frac {16 x}{16 e^{4+x^2}-41 x^2}-\log (x) \]
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Time = 0.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96
method | result | size |
norman | \(\frac {16 x}{41 x^{2}-16 \,{\mathrm e}^{x^{2}+4}}-\ln \left (x \right )\) | \(25\) |
risch | \(\frac {16 x}{41 x^{2}-16 \,{\mathrm e}^{x^{2}+4}}-\ln \left (x \right )\) | \(25\) |
parallelrisch | \(-\frac {656 x^{2} \ln \left (x \right )-256 \ln \left (x \right ) {\mathrm e}^{x^{2}+4}-256 x}{16 \left (41 x^{2}-16 \,{\mathrm e}^{x^{2}+4}\right )}\) | \(40\) |
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Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int \frac {-256 e^{8+2 x^2}-656 x^3-1681 x^4+e^{4+x^2} \left (-256 x+1312 x^2+512 x^3\right )}{256 e^{8+2 x^2} x-1312 e^{4+x^2} x^3+1681 x^5} \, dx=-\frac {{\left (41 \, x^{2} - 16 \, e^{\left (x^{2} + 4\right )}\right )} \log \left (x\right ) - 16 \, x}{41 \, x^{2} - 16 \, e^{\left (x^{2} + 4\right )}} \]
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Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.73 \[ \int \frac {-256 e^{8+2 x^2}-656 x^3-1681 x^4+e^{4+x^2} \left (-256 x+1312 x^2+512 x^3\right )}{256 e^{8+2 x^2} x-1312 e^{4+x^2} x^3+1681 x^5} \, dx=- \frac {x}{- \frac {41 x^{2}}{16} + e^{x^{2} + 4}} - \log {\left (x \right )} \]
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Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {-256 e^{8+2 x^2}-656 x^3-1681 x^4+e^{4+x^2} \left (-256 x+1312 x^2+512 x^3\right )}{256 e^{8+2 x^2} x-1312 e^{4+x^2} x^3+1681 x^5} \, dx=\frac {16 \, x}{41 \, x^{2} - 16 \, e^{\left (x^{2} + 4\right )}} - \log \left (x\right ) \]
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Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int \frac {-256 e^{8+2 x^2}-656 x^3-1681 x^4+e^{4+x^2} \left (-256 x+1312 x^2+512 x^3\right )}{256 e^{8+2 x^2} x-1312 e^{4+x^2} x^3+1681 x^5} \, dx=-\frac {41 \, x^{2} \log \left (x\right ) - 16 \, e^{\left (x^{2} + 4\right )} \log \left (x\right ) - 16 \, x}{41 \, x^{2} - 16 \, e^{\left (x^{2} + 4\right )}} \]
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Time = 12.45 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {-256 e^{8+2 x^2}-656 x^3-1681 x^4+e^{4+x^2} \left (-256 x+1312 x^2+512 x^3\right )}{256 e^{8+2 x^2} x-1312 e^{4+x^2} x^3+1681 x^5} \, dx=-\ln \left (x\right )-\frac {16\,x}{16\,{\mathrm {e}}^{x^2+4}-41\,x^2} \]
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