Integrand size = 67, antiderivative size = 19 \[ \int \frac {-3 e^{8 x-2 x \log (2)}+e^{4 x-x \log (2)} (-64+16 \log (2))}{256-96 e^{4 x-x \log (2)} x+9 e^{8 x-2 x \log (2)} x^2} \, dx=\frac {1}{-16 e^{-x (4-\log (2))}+3 x} \]
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\[ \int \frac {-3 e^{8 x-2 x \log (2)}+e^{4 x-x \log (2)} (-64+16 \log (2))}{256-96 e^{4 x-x \log (2)} x+9 e^{8 x-2 x \log (2)} x^2} \, dx=\int \frac {-3 e^{8 x-2 x \log (2)}+e^{4 x-x \log (2)} (-64+16 \log (2))}{256-96 e^{4 x-x \log (2)} x+9 e^{8 x-2 x \log (2)} x^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-3 e^{8 x}+2^{4+x} e^{4 x} (-4+\log (2))}{\left (2^{4+x}-3 e^{4 x} x\right )^2} \, dx \\ & = \int \left (-\frac {1}{3 x^2}+\frac {2^{8+2 x} (-1-x (4-\log (2)))}{3 x^2 \left (2^{4+x}-3 e^{4 x} x\right )^2}+\frac {2^{4+x} (2+x (4-\log (2)))}{3 x^2 \left (2^{4+x}-3 e^{4 x} x\right )}\right ) \, dx \\ & = \frac {1}{3 x}+\frac {1}{3} \int \frac {2^{8+2 x} (-1-x (4-\log (2)))}{x^2 \left (2^{4+x}-3 e^{4 x} x\right )^2} \, dx+\frac {1}{3} \int \frac {2^{4+x} (2+x (4-\log (2)))}{x^2 \left (2^{4+x}-3 e^{4 x} x\right )} \, dx \\ & = \frac {1}{3 x}+\frac {1}{3} \int \frac {4^{4+x} (-1+x (-4+\log (2)))}{x^2 \left (2^{4+x}-3 e^{4 x} x\right )^2} \, dx+\frac {1}{3} \int \left (-\frac {2^{5+x}}{x^2 \left (-2^{4+x}+3 e^{4 x} x\right )}+\frac {2^{4+x} (-4+\log (2))}{x \left (-2^{4+x}+3 e^{4 x} x\right )}\right ) \, dx \\ & = \frac {1}{3 x}-\frac {1}{3} \int \frac {2^{5+x}}{x^2 \left (-2^{4+x}+3 e^{4 x} x\right )} \, dx+\frac {1}{3} \int \left (-\frac {4^{4+x}}{x^2 \left (-2^{4+x}+3 e^{4 x} x\right )^2}+\frac {4^{4+x} (-4+\log (2))}{x \left (-2^{4+x}+3 e^{4 x} x\right )^2}\right ) \, dx+\frac {1}{3} (-4+\log (2)) \int \frac {2^{4+x}}{x \left (-2^{4+x}+3 e^{4 x} x\right )} \, dx \\ & = \frac {1}{3 x}-\frac {1}{3} \int \frac {4^{4+x}}{x^2 \left (-2^{4+x}+3 e^{4 x} x\right )^2} \, dx-\frac {1}{3} \int \frac {2^{5+x}}{x^2 \left (-2^{4+x}+3 e^{4 x} x\right )} \, dx+\frac {1}{3} (-4+\log (2)) \int \frac {4^{4+x}}{x \left (-2^{4+x}+3 e^{4 x} x\right )^2} \, dx+\frac {1}{3} (-4+\log (2)) \int \frac {2^{4+x}}{x \left (-2^{4+x}+3 e^{4 x} x\right )} \, dx \\ \end{align*}
Time = 1.67 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \frac {-3 e^{8 x-2 x \log (2)}+e^{4 x-x \log (2)} (-64+16 \log (2))}{256-96 e^{4 x-x \log (2)} x+9 e^{8 x-2 x \log (2)} x^2} \, dx=-\frac {e^{4 x}}{2^{4+x}-3 e^{4 x} x} \]
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Time = 0.48 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.37
method | result | size |
risch | \(\frac {1}{3 x}+\frac {16}{3 x \left (3 \left (\frac {1}{2}\right )^{x} x \,{\mathrm e}^{4 x}-16\right )}\) | \(26\) |
norman | \(\frac {{\mathrm e}^{-x \ln \left (2\right )+3 x} {\mathrm e}^{x}}{3 x \,{\mathrm e}^{-x \ln \left (2\right )+3 x} {\mathrm e}^{x}-16}\) | \(33\) |
parallelrisch | \(\frac {{\mathrm e}^{-x \ln \left (2\right )+3 x} {\mathrm e}^{x}}{3 x \,{\mathrm e}^{-x \ln \left (2\right )+3 x} {\mathrm e}^{x}-16}\) | \(33\) |
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Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.47 \[ \int \frac {-3 e^{8 x-2 x \log (2)}+e^{4 x-x \log (2)} (-64+16 \log (2))}{256-96 e^{4 x-x \log (2)} x+9 e^{8 x-2 x \log (2)} x^2} \, dx=\frac {e^{\left (-x \log \left (2\right ) + 4 \, x\right )}}{3 \, x e^{\left (-x \log \left (2\right ) + 4 \, x\right )} - 16} \]
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Timed out. \[ \int \frac {-3 e^{8 x-2 x \log (2)}+e^{4 x-x \log (2)} (-64+16 \log (2))}{256-96 e^{4 x-x \log (2)} x+9 e^{8 x-2 x \log (2)} x^2} \, dx=\text {Timed out} \]
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Time = 0.32 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {-3 e^{8 x-2 x \log (2)}+e^{4 x-x \log (2)} (-64+16 \log (2))}{256-96 e^{4 x-x \log (2)} x+9 e^{8 x-2 x \log (2)} x^2} \, dx=\frac {e^{\left (4 \, x\right )}}{3 \, x e^{\left (4 \, x\right )} - 16 \cdot 2^{x}} \]
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Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.47 \[ \int \frac {-3 e^{8 x-2 x \log (2)}+e^{4 x-x \log (2)} (-64+16 \log (2))}{256-96 e^{4 x-x \log (2)} x+9 e^{8 x-2 x \log (2)} x^2} \, dx=\frac {e^{\left (-x \log \left (2\right ) + 4 \, x\right )}}{3 \, x e^{\left (-x \log \left (2\right ) + 4 \, x\right )} - 16} \]
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Timed out. \[ \int \frac {-3 e^{8 x-2 x \log (2)}+e^{4 x-x \log (2)} (-64+16 \log (2))}{256-96 e^{4 x-x \log (2)} x+9 e^{8 x-2 x \log (2)} x^2} \, dx=\int -\frac {3\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{6\,x-2\,x\,\ln \left (2\right )}-{\mathrm {e}}^x\,{\mathrm {e}}^{3\,x-x\,\ln \left (2\right )}\,\left (16\,\ln \left (2\right )-64\right )}{9\,x^2\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{6\,x-2\,x\,\ln \left (2\right )}-96\,x\,{\mathrm {e}}^x\,{\mathrm {e}}^{3\,x-x\,\ln \left (2\right )}+256} \,d x \]
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