Integrand size = 73, antiderivative size = 21 \[ \int \frac {e \left (192 x+96 x^2+12 x^3\right )+\left (e^5 (-96-24 x)+e \left (192 x+48 x^2\right )\right ) \log \left (e^4-2 x\right )}{\left (e^4 x^3-2 x^4\right ) \log ^3\left (e^4-2 x\right )} \, dx=\frac {3 e (4+x)^2}{x^2 \log ^2\left (e^4-2 x\right )} \]
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\[ \int \frac {e \left (192 x+96 x^2+12 x^3\right )+\left (e^5 (-96-24 x)+e \left (192 x+48 x^2\right )\right ) \log \left (e^4-2 x\right )}{\left (e^4 x^3-2 x^4\right ) \log ^3\left (e^4-2 x\right )} \, dx=\int \frac {e \left (192 x+96 x^2+12 x^3\right )+\left (e^5 (-96-24 x)+e \left (192 x+48 x^2\right )\right ) \log \left (e^4-2 x\right )}{\left (e^4 x^3-2 x^4\right ) \log ^3\left (e^4-2 x\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {e \left (192 x+96 x^2+12 x^3\right )+\left (e^5 (-96-24 x)+e \left (192 x+48 x^2\right )\right ) \log \left (e^4-2 x\right )}{\left (e^4-2 x\right ) x^3 \log ^3\left (e^4-2 x\right )} \, dx \\ & = \int \frac {12 e (4+x) \left (x (4+x)-2 \left (e^4-2 x\right ) \log \left (e^4-2 x\right )\right )}{\left (e^4-2 x\right ) x^3 \log ^3\left (e^4-2 x\right )} \, dx \\ & = (12 e) \int \frac {(4+x) \left (x (4+x)-2 \left (e^4-2 x\right ) \log \left (e^4-2 x\right )\right )}{\left (e^4-2 x\right ) x^3 \log ^3\left (e^4-2 x\right )} \, dx \\ & = (12 e) \int \left (\frac {(4+x)^2}{\left (e^4-2 x\right ) x^2 \log ^3\left (e^4-2 x\right )}-\frac {2 (4+x)}{x^3 \log ^2\left (e^4-2 x\right )}\right ) \, dx \\ & = (12 e) \int \frac {(4+x)^2}{\left (e^4-2 x\right ) x^2 \log ^3\left (e^4-2 x\right )} \, dx-(24 e) \int \frac {4+x}{x^3 \log ^2\left (e^4-2 x\right )} \, dx \\ & = (12 e) \int \left (\frac {\left (8+e^4\right )^2}{e^8 \left (e^4-2 x\right ) \log ^3\left (e^4-2 x\right )}+\frac {16}{e^4 x^2 \log ^3\left (e^4-2 x\right )}+\frac {8 \left (4+e^4\right )}{e^8 x \log ^3\left (e^4-2 x\right )}\right ) \, dx-(24 e) \int \left (\frac {4}{x^3 \log ^2\left (e^4-2 x\right )}+\frac {1}{x^2 \log ^2\left (e^4-2 x\right )}\right ) \, dx \\ & = \frac {192 \int \frac {1}{x^2 \log ^3\left (e^4-2 x\right )} \, dx}{e^3}-(24 e) \int \frac {1}{x^2 \log ^2\left (e^4-2 x\right )} \, dx-(96 e) \int \frac {1}{x^3 \log ^2\left (e^4-2 x\right )} \, dx+\frac {\left (96 \left (4+e^4\right )\right ) \int \frac {1}{x \log ^3\left (e^4-2 x\right )} \, dx}{e^7}+\frac {\left (12 \left (8+e^4\right )^2\right ) \int \frac {1}{\left (e^4-2 x\right ) \log ^3\left (e^4-2 x\right )} \, dx}{e^7} \\ & = \frac {192 \int \frac {1}{x^2 \log ^3\left (e^4-2 x\right )} \, dx}{e^3}-(24 e) \int \frac {1}{x^2 \log ^2\left (e^4-2 x\right )} \, dx-(96 e) \int \frac {1}{x^3 \log ^2\left (e^4-2 x\right )} \, dx+\frac {\left (96 \left (4+e^4\right )\right ) \int \frac {1}{x \log ^3\left (e^4-2 x\right )} \, dx}{e^7}-\frac {\left (6 \left (8+e^4\right )^2\right ) \text {Subst}\left (\int \frac {1}{x \log ^3(x)} \, dx,x,e^4-2 x\right )}{e^7} \\ & = \frac {192 \int \frac {1}{x^2 \log ^3\left (e^4-2 x\right )} \, dx}{e^3}-(24 e) \int \frac {1}{x^2 \log ^2\left (e^4-2 x\right )} \, dx-(96 e) \int \frac {1}{x^3 \log ^2\left (e^4-2 x\right )} \, dx+\frac {\left (96 \left (4+e^4\right )\right ) \int \frac {1}{x \log ^3\left (e^4-2 x\right )} \, dx}{e^7}-\frac {\left (6 \left (8+e^4\right )^2\right ) \text {Subst}\left (\int \frac {1}{x^3} \, dx,x,\log \left (e^4-2 x\right )\right )}{e^7} \\ & = \frac {3 \left (8+e^4\right )^2}{e^7 \log ^2\left (e^4-2 x\right )}+\frac {192 \int \frac {1}{x^2 \log ^3\left (e^4-2 x\right )} \, dx}{e^3}-(24 e) \int \frac {1}{x^2 \log ^2\left (e^4-2 x\right )} \, dx-(96 e) \int \frac {1}{x^3 \log ^2\left (e^4-2 x\right )} \, dx+\frac {\left (96 \left (4+e^4\right )\right ) \int \frac {1}{x \log ^3\left (e^4-2 x\right )} \, dx}{e^7} \\ \end{align*}
Time = 0.41 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {e \left (192 x+96 x^2+12 x^3\right )+\left (e^5 (-96-24 x)+e \left (192 x+48 x^2\right )\right ) \log \left (e^4-2 x\right )}{\left (e^4 x^3-2 x^4\right ) \log ^3\left (e^4-2 x\right )} \, dx=\frac {3 e (4+x)^2}{x^2 \log ^2\left (e^4-2 x\right )} \]
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Time = 1.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19
method | result | size |
risch | \(\frac {3 \,{\mathrm e} \left (x^{2}+8 x +16\right )}{x^{2} \ln \left ({\mathrm e}^{4}-2 x \right )^{2}}\) | \(25\) |
norman | \(\frac {24 x \,{\mathrm e}+3 x^{2} {\mathrm e}+48 \,{\mathrm e}}{\ln \left ({\mathrm e}^{4}-2 x \right )^{2} x^{2}}\) | \(31\) |
parallelrisch | \(\frac {12 x^{2} {\mathrm e}+96 x \,{\mathrm e}+192 \,{\mathrm e}}{4 \ln \left ({\mathrm e}^{4}-2 x \right )^{2} x^{2}}\) | \(32\) |
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Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {e \left (192 x+96 x^2+12 x^3\right )+\left (e^5 (-96-24 x)+e \left (192 x+48 x^2\right )\right ) \log \left (e^4-2 x\right )}{\left (e^4 x^3-2 x^4\right ) \log ^3\left (e^4-2 x\right )} \, dx=\frac {3 \, {\left (x^{2} + 8 \, x + 16\right )} e}{x^{2} \log \left (-2 \, x + e^{4}\right )^{2}} \]
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Time = 0.07 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.52 \[ \int \frac {e \left (192 x+96 x^2+12 x^3\right )+\left (e^5 (-96-24 x)+e \left (192 x+48 x^2\right )\right ) \log \left (e^4-2 x\right )}{\left (e^4 x^3-2 x^4\right ) \log ^3\left (e^4-2 x\right )} \, dx=\frac {3 e x^{2} + 24 e x + 48 e}{x^{2} \log {\left (- 2 x + e^{4} \right )}^{2}} \]
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Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.43 \[ \int \frac {e \left (192 x+96 x^2+12 x^3\right )+\left (e^5 (-96-24 x)+e \left (192 x+48 x^2\right )\right ) \log \left (e^4-2 x\right )}{\left (e^4 x^3-2 x^4\right ) \log ^3\left (e^4-2 x\right )} \, dx=\frac {3 \, {\left (x^{2} e + 8 \, x e + 16 \, e\right )}}{x^{2} \log \left (-2 \, x + e^{4}\right )^{2}} \]
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Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.43 \[ \int \frac {e \left (192 x+96 x^2+12 x^3\right )+\left (e^5 (-96-24 x)+e \left (192 x+48 x^2\right )\right ) \log \left (e^4-2 x\right )}{\left (e^4 x^3-2 x^4\right ) \log ^3\left (e^4-2 x\right )} \, dx=\frac {3 \, {\left (x^{2} e + 8 \, x e + 16 \, e\right )}}{x^{2} \log \left (-2 \, x + e^{4}\right )^{2}} \]
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Time = 12.20 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {e \left (192 x+96 x^2+12 x^3\right )+\left (e^5 (-96-24 x)+e \left (192 x+48 x^2\right )\right ) \log \left (e^4-2 x\right )}{\left (e^4 x^3-2 x^4\right ) \log ^3\left (e^4-2 x\right )} \, dx=\frac {3\,\mathrm {e}\,{\left (x+4\right )}^2}{x^2\,{\ln \left ({\mathrm {e}}^4-2\,x\right )}^2} \]
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