Integrand size = 148, antiderivative size = 26 \[ \int \frac {16 \log ^2\left (x^2\right )+e^{2 e^{e^{\frac {x}{\log \left (x^2\right )}}}+4 x} \left ((1+4 x) \log ^2\left (x^2\right )+e^{e^{\frac {x}{\log \left (x^2\right )}}+\frac {x}{\log \left (x^2\right )}} \left (-4 x+2 x \log \left (x^2\right )\right )\right )+e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x} \left ((8+16 x) \log ^2\left (x^2\right )+e^{e^{\frac {x}{\log \left (x^2\right )}}+\frac {x}{\log \left (x^2\right )}} \left (-16 x+8 x \log \left (x^2\right )\right )\right )}{\log ^2\left (x^2\right )} \, dx=-4+\left (4+e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x}\right )^2 x \]
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\[ \int \frac {16 \log ^2\left (x^2\right )+e^{2 e^{e^{\frac {x}{\log \left (x^2\right )}}}+4 x} \left ((1+4 x) \log ^2\left (x^2\right )+e^{e^{\frac {x}{\log \left (x^2\right )}}+\frac {x}{\log \left (x^2\right )}} \left (-4 x+2 x \log \left (x^2\right )\right )\right )+e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x} \left ((8+16 x) \log ^2\left (x^2\right )+e^{e^{\frac {x}{\log \left (x^2\right )}}+\frac {x}{\log \left (x^2\right )}} \left (-16 x+8 x \log \left (x^2\right )\right )\right )}{\log ^2\left (x^2\right )} \, dx=\int \frac {16 \log ^2\left (x^2\right )+e^{2 e^{e^{\frac {x}{\log \left (x^2\right )}}}+4 x} \left ((1+4 x) \log ^2\left (x^2\right )+e^{e^{\frac {x}{\log \left (x^2\right )}}+\frac {x}{\log \left (x^2\right )}} \left (-4 x+2 x \log \left (x^2\right )\right )\right )+e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x} \left ((8+16 x) \log ^2\left (x^2\right )+e^{e^{\frac {x}{\log \left (x^2\right )}}+\frac {x}{\log \left (x^2\right )}} \left (-16 x+8 x \log \left (x^2\right )\right )\right )}{\log ^2\left (x^2\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\left (4+e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x}\right ) \left (4+e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x}+4 e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x} x\right )+\frac {2 e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+e^{\frac {x}{\log \left (x^2\right )}}+2 x+\frac {x}{\log \left (x^2\right )}} \left (4+e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x}\right ) x \left (-2+\log \left (x^2\right )\right )}{\log ^2\left (x^2\right )}\right ) \, dx \\ & = 2 \int \frac {e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+e^{\frac {x}{\log \left (x^2\right )}}+2 x+\frac {x}{\log \left (x^2\right )}} \left (4+e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x}\right ) x \left (-2+\log \left (x^2\right )\right )}{\log ^2\left (x^2\right )} \, dx+\int \left (4+e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x}\right ) \left (4+e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x}+4 e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x} x\right ) \, dx \\ & = 2 \int \left (\frac {4 e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+e^{\frac {x}{\log \left (x^2\right )}}+2 x+\frac {x}{\log \left (x^2\right )}} x \left (-2+\log \left (x^2\right )\right )}{\log ^2\left (x^2\right )}+\frac {e^{2 e^{e^{\frac {x}{\log \left (x^2\right )}}}+e^{\frac {x}{\log \left (x^2\right )}}+4 x+\frac {x}{\log \left (x^2\right )}} x \left (-2+\log \left (x^2\right )\right )}{\log ^2\left (x^2\right )}\right ) \, dx+\int \left (4+e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x}\right ) \left (4+e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x} (1+4 x)\right ) \, dx \\ & = 2 \int \frac {e^{2 e^{e^{\frac {x}{\log \left (x^2\right )}}}+e^{\frac {x}{\log \left (x^2\right )}}+4 x+\frac {x}{\log \left (x^2\right )}} x \left (-2+\log \left (x^2\right )\right )}{\log ^2\left (x^2\right )} \, dx+8 \int \frac {e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+e^{\frac {x}{\log \left (x^2\right )}}+2 x+\frac {x}{\log \left (x^2\right )}} x \left (-2+\log \left (x^2\right )\right )}{\log ^2\left (x^2\right )} \, dx+\int \left (16+8 e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x} (1+2 x)+e^{2 \left (e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x\right )} (1+4 x)\right ) \, dx \\ & = 16 x+2 \int \left (-\frac {2 e^{2 e^{e^{\frac {x}{\log \left (x^2\right )}}}+e^{\frac {x}{\log \left (x^2\right )}}+4 x+\frac {x}{\log \left (x^2\right )}} x}{\log ^2\left (x^2\right )}+\frac {e^{2 e^{e^{\frac {x}{\log \left (x^2\right )}}}+e^{\frac {x}{\log \left (x^2\right )}}+4 x+\frac {x}{\log \left (x^2\right )}} x}{\log \left (x^2\right )}\right ) \, dx+8 \int e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x} (1+2 x) \, dx+8 \int \left (-\frac {2 e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+e^{\frac {x}{\log \left (x^2\right )}}+2 x+\frac {x}{\log \left (x^2\right )}} x}{\log ^2\left (x^2\right )}+\frac {e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+e^{\frac {x}{\log \left (x^2\right )}}+2 x+\frac {x}{\log \left (x^2\right )}} x}{\log \left (x^2\right )}\right ) \, dx+\int e^{2 \left (e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x\right )} (1+4 x) \, dx \\ & = 16 x+2 \int \frac {e^{2 e^{e^{\frac {x}{\log \left (x^2\right )}}}+e^{\frac {x}{\log \left (x^2\right )}}+4 x+\frac {x}{\log \left (x^2\right )}} x}{\log \left (x^2\right )} \, dx-4 \int \frac {e^{2 e^{e^{\frac {x}{\log \left (x^2\right )}}}+e^{\frac {x}{\log \left (x^2\right )}}+4 x+\frac {x}{\log \left (x^2\right )}} x}{\log ^2\left (x^2\right )} \, dx+8 \int \left (e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x}+2 e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x} x\right ) \, dx+8 \int \frac {e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+e^{\frac {x}{\log \left (x^2\right )}}+2 x+\frac {x}{\log \left (x^2\right )}} x}{\log \left (x^2\right )} \, dx-16 \int \frac {e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+e^{\frac {x}{\log \left (x^2\right )}}+2 x+\frac {x}{\log \left (x^2\right )}} x}{\log ^2\left (x^2\right )} \, dx+\int \left (e^{2 \left (e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x\right )}+4 e^{2 \left (e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x\right )} x\right ) \, dx \\ & = 16 x+2 \int \frac {e^{2 e^{e^{\frac {x}{\log \left (x^2\right )}}}+e^{\frac {x}{\log \left (x^2\right )}}+4 x+\frac {x}{\log \left (x^2\right )}} x}{\log \left (x^2\right )} \, dx+4 \int e^{2 \left (e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x\right )} x \, dx-4 \int \frac {e^{2 e^{e^{\frac {x}{\log \left (x^2\right )}}}+e^{\frac {x}{\log \left (x^2\right )}}+4 x+\frac {x}{\log \left (x^2\right )}} x}{\log ^2\left (x^2\right )} \, dx+8 \int e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x} \, dx+8 \int \frac {e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+e^{\frac {x}{\log \left (x^2\right )}}+2 x+\frac {x}{\log \left (x^2\right )}} x}{\log \left (x^2\right )} \, dx+16 \int e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x} x \, dx-16 \int \frac {e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+e^{\frac {x}{\log \left (x^2\right )}}+2 x+\frac {x}{\log \left (x^2\right )}} x}{\log ^2\left (x^2\right )} \, dx+\int e^{2 \left (e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x\right )} \, dx \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {16 \log ^2\left (x^2\right )+e^{2 e^{e^{\frac {x}{\log \left (x^2\right )}}}+4 x} \left ((1+4 x) \log ^2\left (x^2\right )+e^{e^{\frac {x}{\log \left (x^2\right )}}+\frac {x}{\log \left (x^2\right )}} \left (-4 x+2 x \log \left (x^2\right )\right )\right )+e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x} \left ((8+16 x) \log ^2\left (x^2\right )+e^{e^{\frac {x}{\log \left (x^2\right )}}+\frac {x}{\log \left (x^2\right )}} \left (-16 x+8 x \log \left (x^2\right )\right )\right )}{\log ^2\left (x^2\right )} \, dx=\left (4+e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x}\right )^2 x \]
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Time = 3.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62
method | result | size |
risch | \(x \,{\mathrm e}^{2 \,{\mathrm e}^{{\mathrm e}^{\frac {x}{\ln \left (x^{2}\right )}}}+4 x}+8 x \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{\frac {x}{\ln \left (x^{2}\right )}}}+2 x}+16 x\) | \(42\) |
parallelrisch | \(x \,{\mathrm e}^{2 \,{\mathrm e}^{{\mathrm e}^{\frac {x}{\ln \left (x^{2}\right )}}}+4 x}+8 x \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{\frac {x}{\ln \left (x^{2}\right )}}}+2 x}+16 x\) | \(42\) |
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Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (23) = 46\).
Time = 0.25 (sec) , antiderivative size = 108, normalized size of antiderivative = 4.15 \[ \int \frac {16 \log ^2\left (x^2\right )+e^{2 e^{e^{\frac {x}{\log \left (x^2\right )}}}+4 x} \left ((1+4 x) \log ^2\left (x^2\right )+e^{e^{\frac {x}{\log \left (x^2\right )}}+\frac {x}{\log \left (x^2\right )}} \left (-4 x+2 x \log \left (x^2\right )\right )\right )+e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x} \left ((8+16 x) \log ^2\left (x^2\right )+e^{e^{\frac {x}{\log \left (x^2\right )}}+\frac {x}{\log \left (x^2\right )}} \left (-16 x+8 x \log \left (x^2\right )\right )\right )}{\log ^2\left (x^2\right )} \, dx=x e^{\left (2 \, {\left (2 \, x e^{\left (\frac {x}{\log \left (x^{2}\right )}\right )} + e^{\left (\frac {e^{\left (\frac {x}{\log \left (x^{2}\right )}\right )} \log \left (x^{2}\right ) + x}{\log \left (x^{2}\right )}\right )}\right )} e^{\left (-\frac {x}{\log \left (x^{2}\right )}\right )}\right )} + 8 \, x e^{\left ({\left (2 \, x e^{\left (\frac {x}{\log \left (x^{2}\right )}\right )} + e^{\left (\frac {e^{\left (\frac {x}{\log \left (x^{2}\right )}\right )} \log \left (x^{2}\right ) + x}{\log \left (x^{2}\right )}\right )}\right )} e^{\left (-\frac {x}{\log \left (x^{2}\right )}\right )}\right )} + 16 \, x \]
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Exception generated. \[ \int \frac {16 \log ^2\left (x^2\right )+e^{2 e^{e^{\frac {x}{\log \left (x^2\right )}}}+4 x} \left ((1+4 x) \log ^2\left (x^2\right )+e^{e^{\frac {x}{\log \left (x^2\right )}}+\frac {x}{\log \left (x^2\right )}} \left (-4 x+2 x \log \left (x^2\right )\right )\right )+e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x} \left ((8+16 x) \log ^2\left (x^2\right )+e^{e^{\frac {x}{\log \left (x^2\right )}}+\frac {x}{\log \left (x^2\right )}} \left (-16 x+8 x \log \left (x^2\right )\right )\right )}{\log ^2\left (x^2\right )} \, dx=\text {Exception raised: TypeError} \]
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Time = 0.32 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int \frac {16 \log ^2\left (x^2\right )+e^{2 e^{e^{\frac {x}{\log \left (x^2\right )}}}+4 x} \left ((1+4 x) \log ^2\left (x^2\right )+e^{e^{\frac {x}{\log \left (x^2\right )}}+\frac {x}{\log \left (x^2\right )}} \left (-4 x+2 x \log \left (x^2\right )\right )\right )+e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x} \left ((8+16 x) \log ^2\left (x^2\right )+e^{e^{\frac {x}{\log \left (x^2\right )}}+\frac {x}{\log \left (x^2\right )}} \left (-16 x+8 x \log \left (x^2\right )\right )\right )}{\log ^2\left (x^2\right )} \, dx=x e^{\left (4 \, x + 2 \, e^{\left (e^{\left (\frac {x}{2 \, \log \left (x\right )}\right )}\right )}\right )} + 8 \, x e^{\left (2 \, x + e^{\left (e^{\left (\frac {x}{2 \, \log \left (x\right )}\right )}\right )}\right )} + 16 \, x \]
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\[ \int \frac {16 \log ^2\left (x^2\right )+e^{2 e^{e^{\frac {x}{\log \left (x^2\right )}}}+4 x} \left ((1+4 x) \log ^2\left (x^2\right )+e^{e^{\frac {x}{\log \left (x^2\right )}}+\frac {x}{\log \left (x^2\right )}} \left (-4 x+2 x \log \left (x^2\right )\right )\right )+e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x} \left ((8+16 x) \log ^2\left (x^2\right )+e^{e^{\frac {x}{\log \left (x^2\right )}}+\frac {x}{\log \left (x^2\right )}} \left (-16 x+8 x \log \left (x^2\right )\right )\right )}{\log ^2\left (x^2\right )} \, dx=\int { \frac {{\left ({\left (4 \, x + 1\right )} \log \left (x^{2}\right )^{2} + 2 \, {\left (x \log \left (x^{2}\right ) - 2 \, x\right )} e^{\left (\frac {x}{\log \left (x^{2}\right )} + e^{\left (\frac {x}{\log \left (x^{2}\right )}\right )}\right )}\right )} e^{\left (4 \, x + 2 \, e^{\left (e^{\left (\frac {x}{\log \left (x^{2}\right )}\right )}\right )}\right )} + 8 \, {\left ({\left (2 \, x + 1\right )} \log \left (x^{2}\right )^{2} + {\left (x \log \left (x^{2}\right ) - 2 \, x\right )} e^{\left (\frac {x}{\log \left (x^{2}\right )} + e^{\left (\frac {x}{\log \left (x^{2}\right )}\right )}\right )}\right )} e^{\left (2 \, x + e^{\left (e^{\left (\frac {x}{\log \left (x^{2}\right )}\right )}\right )}\right )} + 16 \, \log \left (x^{2}\right )^{2}}{\log \left (x^{2}\right )^{2}} \,d x } \]
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Time = 9.39 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58 \[ \int \frac {16 \log ^2\left (x^2\right )+e^{2 e^{e^{\frac {x}{\log \left (x^2\right )}}}+4 x} \left ((1+4 x) \log ^2\left (x^2\right )+e^{e^{\frac {x}{\log \left (x^2\right )}}+\frac {x}{\log \left (x^2\right )}} \left (-4 x+2 x \log \left (x^2\right )\right )\right )+e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x} \left ((8+16 x) \log ^2\left (x^2\right )+e^{e^{\frac {x}{\log \left (x^2\right )}}+\frac {x}{\log \left (x^2\right )}} \left (-16 x+8 x \log \left (x^2\right )\right )\right )}{\log ^2\left (x^2\right )} \, dx=16\,x+x\,{\mathrm {e}}^{2\,{\mathrm {e}}^{{\mathrm {e}}^{\frac {x}{\ln \left (x^2\right )}}}}\,{\mathrm {e}}^{4\,x}+8\,x\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^{\frac {x}{\ln \left (x^2\right )}}}} \]
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