Integrand size = 76, antiderivative size = 21 \[ \int \frac {96+96 e^x}{16+e^{16}+16 e^{2 x}+e^8 (8-8 x)-32 x+16 x^2+e^x \left (-32-8 e^8+32 x-32 \log (4)\right )+\left (32+8 e^8-32 x\right ) \log (4)+16 \log ^2(4)} \, dx=-\frac {6}{-1-\frac {e^8}{4}+e^x+x-\log (4)} \]
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Time = 0.13 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {6820, 12, 6818} \[ \int \frac {96+96 e^x}{16+e^{16}+16 e^{2 x}+e^8 (8-8 x)-32 x+16 x^2+e^x \left (-32-8 e^8+32 x-32 \log (4)\right )+\left (32+8 e^8-32 x\right ) \log (4)+16 \log ^2(4)} \, dx=\frac {24}{-4 x-4 e^x+e^8+4+\log (256)} \]
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Rule 12
Rule 6818
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {96 \left (1+e^x\right )}{\left (4 e^x+4 x-4 \left (1+\frac {e^8}{4}+\log (4)\right )\right )^2} \, dx \\ & = 96 \int \frac {1+e^x}{\left (4 e^x+4 x-4 \left (1+\frac {e^8}{4}+\log (4)\right )\right )^2} \, dx \\ & = \frac {24}{4+e^8-4 e^x-4 x+\log (256)} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {96+96 e^x}{16+e^{16}+16 e^{2 x}+e^8 (8-8 x)-32 x+16 x^2+e^x \left (-32-8 e^8+32 x-32 \log (4)\right )+\left (32+8 e^8-32 x\right ) \log (4)+16 \log ^2(4)} \, dx=\frac {24}{4+e^8-4 e^x-4 x+\log (256)} \]
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Time = 0.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95
method | result | size |
risch | \(\frac {24}{{\mathrm e}^{8}+8 \ln \left (2\right )-4 \,{\mathrm e}^{x}-4 x +4}\) | \(20\) |
norman | \(\frac {24}{{\mathrm e}^{8}+8 \ln \left (2\right )-4 \,{\mathrm e}^{x}-4 x +4}\) | \(22\) |
parallelrisch | \(\frac {24}{{\mathrm e}^{8}+8 \ln \left (2\right )-4 \,{\mathrm e}^{x}-4 x +4}\) | \(22\) |
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Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {96+96 e^x}{16+e^{16}+16 e^{2 x}+e^8 (8-8 x)-32 x+16 x^2+e^x \left (-32-8 e^8+32 x-32 \log (4)\right )+\left (32+8 e^8-32 x\right ) \log (4)+16 \log ^2(4)} \, dx=-\frac {24}{4 \, x - e^{8} + 4 \, e^{x} - 8 \, \log \left (2\right ) - 4} \]
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Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {96+96 e^x}{16+e^{16}+16 e^{2 x}+e^8 (8-8 x)-32 x+16 x^2+e^x \left (-32-8 e^8+32 x-32 \log (4)\right )+\left (32+8 e^8-32 x\right ) \log (4)+16 \log ^2(4)} \, dx=- \frac {6}{x + e^{x} - \frac {e^{8}}{4} - 2 \log {\left (2 \right )} - 1} \]
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Time = 0.35 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {96+96 e^x}{16+e^{16}+16 e^{2 x}+e^8 (8-8 x)-32 x+16 x^2+e^x \left (-32-8 e^8+32 x-32 \log (4)\right )+\left (32+8 e^8-32 x\right ) \log (4)+16 \log ^2(4)} \, dx=-\frac {24}{4 \, x - e^{8} + 4 \, e^{x} - 8 \, \log \left (2\right ) - 4} \]
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Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {96+96 e^x}{16+e^{16}+16 e^{2 x}+e^8 (8-8 x)-32 x+16 x^2+e^x \left (-32-8 e^8+32 x-32 \log (4)\right )+\left (32+8 e^8-32 x\right ) \log (4)+16 \log ^2(4)} \, dx=-\frac {24}{4 \, x - e^{8} + 4 \, e^{x} - 8 \, \log \left (2\right ) - 4} \]
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Time = 13.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.67 \[ \int \frac {96+96 e^x}{16+e^{16}+16 e^{2 x}+e^8 (8-8 x)-32 x+16 x^2+e^x \left (-32-8 e^8+32 x-32 \log (4)\right )+\left (32+8 e^8-32 x\right ) \log (4)+16 \log ^2(4)} \, dx=\frac {96\,\left (\frac {{\mathrm {e}}^8}{4}+\frac {\ln \left (256\right )}{4}+1\right )}{\left ({\mathrm {e}}^8+\ln \left (256\right )+4\right )\,\left ({\mathrm {e}}^8-4\,x+\ln \left (256\right )-4\,{\mathrm {e}}^x+4\right )} \]
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