Integrand size = 174, antiderivative size = 28 \[ \int \left (2 x+e^{4 x+4 x^2+4 e^x x^2+4 \left (x+e^x x\right ) \log ^2(x)} \left (4+8 x+e^x \left (8 x+4 x^2\right )+\left (8+8 e^x\right ) \log (x)+\left (4+e^x (4+4 x)\right ) \log ^2(x)\right )+e^{2 x+2 x^2+2 e^x x^2+2 \left (x+e^x x\right ) \log ^2(x)} \left (-2-4 x-8 x^2+e^x \left (-8 x^2-4 x^3\right )+\left (-8 x-8 e^x x\right ) \log (x)+\left (-4 x+e^x \left (-4 x-4 x^2\right )\right ) \log ^2(x)\right )\right ) \, dx=\left (e^{2 x-2 \left (-1-e^x\right ) x \left (x+\log ^2(x)\right )}-x\right )^2 \]
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\[ \int \left (2 x+e^{4 x+4 x^2+4 e^x x^2+4 \left (x+e^x x\right ) \log ^2(x)} \left (4+8 x+e^x \left (8 x+4 x^2\right )+\left (8+8 e^x\right ) \log (x)+\left (4+e^x (4+4 x)\right ) \log ^2(x)\right )+e^{2 x+2 x^2+2 e^x x^2+2 \left (x+e^x x\right ) \log ^2(x)} \left (-2-4 x-8 x^2+e^x \left (-8 x^2-4 x^3\right )+\left (-8 x-8 e^x x\right ) \log (x)+\left (-4 x+e^x \left (-4 x-4 x^2\right )\right ) \log ^2(x)\right )\right ) \, dx=\int \left (2 x+\exp \left (4 x+4 x^2+4 e^x x^2+4 \left (x+e^x x\right ) \log ^2(x)\right ) \left (4+8 x+e^x \left (8 x+4 x^2\right )+\left (8+8 e^x\right ) \log (x)+\left (4+e^x (4+4 x)\right ) \log ^2(x)\right )+\exp \left (2 x+2 x^2+2 e^x x^2+2 \left (x+e^x x\right ) \log ^2(x)\right ) \left (-2-4 x-8 x^2+e^x \left (-8 x^2-4 x^3\right )+\left (-8 x-8 e^x x\right ) \log (x)+\left (-4 x+e^x \left (-4 x-4 x^2\right )\right ) \log ^2(x)\right )\right ) \, dx \]
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Rubi steps \begin{align*} \text {integral}& = x^2+\int \exp \left (4 x+4 x^2+4 e^x x^2+4 \left (x+e^x x\right ) \log ^2(x)\right ) \left (4+8 x+e^x \left (8 x+4 x^2\right )+\left (8+8 e^x\right ) \log (x)+\left (4+e^x (4+4 x)\right ) \log ^2(x)\right ) \, dx+\int \exp \left (2 x+2 x^2+2 e^x x^2+2 \left (x+e^x x\right ) \log ^2(x)\right ) \left (-2-4 x-8 x^2+e^x \left (-8 x^2-4 x^3\right )+\left (-8 x-8 e^x x\right ) \log (x)+\left (-4 x+e^x \left (-4 x-4 x^2\right )\right ) \log ^2(x)\right ) \, dx \\ & = x^2-\frac {2 \exp \left (2 x+2 x^2+2 e^x x^2+2 \left (x+e^x x\right ) \log ^2(x)\right ) \left (x+2 x^2+e^x \left (2 x^2+x^3\right )+2 \left (x+e^x x\right ) \log (x)+\left (x+e^x \left (x+x^2\right )\right ) \log ^2(x)\right )}{1+2 x+2 e^x x+e^x x^2+\frac {2 \left (x+e^x x\right ) \log (x)}{x}+\left (1+e^x+e^x x\right ) \log ^2(x)}+\int e^{4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )} \left (4+8 x+e^x \left (8 x+4 x^2\right )+\left (8+8 e^x\right ) \log (x)+\left (4+e^x (4+4 x)\right ) \log ^2(x)\right ) \, dx \\ & = x^2-\frac {2 \exp \left (2 x+2 x^2+2 e^x x^2+2 \left (x+e^x x\right ) \log ^2(x)\right ) \left (x+2 x^2+e^x \left (2 x^2+x^3\right )+2 \left (x+e^x x\right ) \log (x)+\left (x+e^x \left (x+x^2\right )\right ) \log ^2(x)\right )}{1+2 x+2 e^x x+e^x x^2+\frac {2 \left (x+e^x x\right ) \log (x)}{x}+\left (1+e^x+e^x x\right ) \log ^2(x)}+\int \left (4 e^{4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )}+8 e^{4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )} x+4 \exp \left (x+4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )\right ) x (2+x)+8 e^{4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )} \left (1+e^x\right ) \log (x)+4 e^{4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )} \left (1+e^x+e^x x\right ) \log ^2(x)\right ) \, dx \\ & = x^2-\frac {2 \exp \left (2 x+2 x^2+2 e^x x^2+2 \left (x+e^x x\right ) \log ^2(x)\right ) \left (x+2 x^2+e^x \left (2 x^2+x^3\right )+2 \left (x+e^x x\right ) \log (x)+\left (x+e^x \left (x+x^2\right )\right ) \log ^2(x)\right )}{1+2 x+2 e^x x+e^x x^2+\frac {2 \left (x+e^x x\right ) \log (x)}{x}+\left (1+e^x+e^x x\right ) \log ^2(x)}+4 \int e^{4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )} \, dx+4 \int \exp \left (x+4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )\right ) x (2+x) \, dx+4 \int e^{4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )} \left (1+e^x+e^x x\right ) \log ^2(x) \, dx+8 \int e^{4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )} x \, dx+8 \int e^{4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )} \left (1+e^x\right ) \log (x) \, dx \\ & = x^2-\frac {2 \exp \left (2 x+2 x^2+2 e^x x^2+2 \left (x+e^x x\right ) \log ^2(x)\right ) \left (x+2 x^2+e^x \left (2 x^2+x^3\right )+2 \left (x+e^x x\right ) \log (x)+\left (x+e^x \left (x+x^2\right )\right ) \log ^2(x)\right )}{1+2 x+2 e^x x+e^x x^2+\frac {2 \left (x+e^x x\right ) \log (x)}{x}+\left (1+e^x+e^x x\right ) \log ^2(x)}+4 \int e^{4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )} \, dx+4 \int \left (2 \exp \left (x+4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )\right ) x+\exp \left (x+4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )\right ) x^2\right ) \, dx+4 \int \left (e^{4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )} \log ^2(x)+\exp \left (x+4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )\right ) (1+x) \log ^2(x)\right ) \, dx+8 \int e^{4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )} x \, dx+8 \int \left (e^{4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )} \log (x)+\exp \left (x+4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )\right ) \log (x)\right ) \, dx \\ & = x^2-\frac {2 \exp \left (2 x+2 x^2+2 e^x x^2+2 \left (x+e^x x\right ) \log ^2(x)\right ) \left (x+2 x^2+e^x \left (2 x^2+x^3\right )+2 \left (x+e^x x\right ) \log (x)+\left (x+e^x \left (x+x^2\right )\right ) \log ^2(x)\right )}{1+2 x+2 e^x x+e^x x^2+\frac {2 \left (x+e^x x\right ) \log (x)}{x}+\left (1+e^x+e^x x\right ) \log ^2(x)}+4 \int e^{4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )} \, dx+4 \int \exp \left (x+4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )\right ) x^2 \, dx+4 \int e^{4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )} \log ^2(x) \, dx+4 \int \exp \left (x+4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )\right ) (1+x) \log ^2(x) \, dx+8 \int e^{4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )} x \, dx+8 \int \exp \left (x+4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )\right ) x \, dx+8 \int e^{4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )} \log (x) \, dx+8 \int \exp \left (x+4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )\right ) \log (x) \, dx \\ & = x^2-\frac {2 \exp \left (2 x+2 x^2+2 e^x x^2+2 \left (x+e^x x\right ) \log ^2(x)\right ) \left (x+2 x^2+e^x \left (2 x^2+x^3\right )+2 \left (x+e^x x\right ) \log (x)+\left (x+e^x \left (x+x^2\right )\right ) \log ^2(x)\right )}{1+2 x+2 e^x x+e^x x^2+\frac {2 \left (x+e^x x\right ) \log (x)}{x}+\left (1+e^x+e^x x\right ) \log ^2(x)}+4 \int e^{4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )} \, dx+4 \int \exp \left (x+4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )\right ) x^2 \, dx+4 \int e^{4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )} \log ^2(x) \, dx+4 \int \left (\exp \left (x+4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )\right ) \log ^2(x)+\exp \left (x+4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )\right ) x \log ^2(x)\right ) \, dx+8 \int e^{4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )} x \, dx+8 \int \exp \left (x+4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )\right ) x \, dx+8 \int e^{4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )} \log (x) \, dx+8 \int \exp \left (x+4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )\right ) \log (x) \, dx \\ & = x^2-\frac {2 \exp \left (2 x+2 x^2+2 e^x x^2+2 \left (x+e^x x\right ) \log ^2(x)\right ) \left (x+2 x^2+e^x \left (2 x^2+x^3\right )+2 \left (x+e^x x\right ) \log (x)+\left (x+e^x \left (x+x^2\right )\right ) \log ^2(x)\right )}{1+2 x+2 e^x x+e^x x^2+\frac {2 \left (x+e^x x\right ) \log (x)}{x}+\left (1+e^x+e^x x\right ) \log ^2(x)}+4 \int e^{4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )} \, dx+4 \int \exp \left (x+4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )\right ) x^2 \, dx+4 \int e^{4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )} \log ^2(x) \, dx+4 \int \exp \left (x+4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )\right ) \log ^2(x) \, dx+4 \int \exp \left (x+4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )\right ) x \log ^2(x) \, dx+8 \int e^{4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )} x \, dx+8 \int \exp \left (x+4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )\right ) x \, dx+8 \int e^{4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )} \log (x) \, dx+8 \int \exp \left (x+4 x \left (1+x+e^x x+\log ^2(x)+e^x \log ^2(x)\right )\right ) \log (x) \, dx \\ \end{align*}
Time = 8.94 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \left (2 x+e^{4 x+4 x^2+4 e^x x^2+4 \left (x+e^x x\right ) \log ^2(x)} \left (4+8 x+e^x \left (8 x+4 x^2\right )+\left (8+8 e^x\right ) \log (x)+\left (4+e^x (4+4 x)\right ) \log ^2(x)\right )+e^{2 x+2 x^2+2 e^x x^2+2 \left (x+e^x x\right ) \log ^2(x)} \left (-2-4 x-8 x^2+e^x \left (-8 x^2-4 x^3\right )+\left (-8 x-8 e^x x\right ) \log (x)+\left (-4 x+e^x \left (-4 x-4 x^2\right )\right ) \log ^2(x)\right )\right ) \, dx=\left (e^{2 x \left (1+x+e^x x+\left (1+e^x\right ) \log ^2(x)\right )}-x\right )^2 \]
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Time = 0.65 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.86
method | result | size |
risch | \({\mathrm e}^{4 x \left ({\mathrm e}^{x} \ln \left (x \right )^{2}+\ln \left (x \right )^{2}+{\mathrm e}^{x} x +x +1\right )}-2 \,{\mathrm e}^{2 x \left ({\mathrm e}^{x} \ln \left (x \right )^{2}+\ln \left (x \right )^{2}+{\mathrm e}^{x} x +x +1\right )} x +x^{2}\) | \(52\) |
parallelrisch | \({\mathrm e}^{4 \left ({\mathrm e}^{x} x +x \right ) \ln \left (x \right )^{2}+4 \,{\mathrm e}^{x} x^{2}+4 x^{2}+4 x}-2 x \,{\mathrm e}^{2 \left ({\mathrm e}^{x} x +x \right ) \ln \left (x \right )^{2}+2 \,{\mathrm e}^{x} x^{2}+2 x^{2}+2 x}+x^{2}\) | \(58\) |
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Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (24) = 48\).
Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.32 \[ \int \left (2 x+e^{4 x+4 x^2+4 e^x x^2+4 \left (x+e^x x\right ) \log ^2(x)} \left (4+8 x+e^x \left (8 x+4 x^2\right )+\left (8+8 e^x\right ) \log (x)+\left (4+e^x (4+4 x)\right ) \log ^2(x)\right )+e^{2 x+2 x^2+2 e^x x^2+2 \left (x+e^x x\right ) \log ^2(x)} \left (-2-4 x-8 x^2+e^x \left (-8 x^2-4 x^3\right )+\left (-8 x-8 e^x x\right ) \log (x)+\left (-4 x+e^x \left (-4 x-4 x^2\right )\right ) \log ^2(x)\right )\right ) \, dx=x^{2} - 2 \, x e^{\left (2 \, x^{2} e^{x} + 2 \, {\left (x e^{x} + x\right )} \log \left (x\right )^{2} + 2 \, x^{2} + 2 \, x\right )} + e^{\left (4 \, x^{2} e^{x} + 4 \, {\left (x e^{x} + x\right )} \log \left (x\right )^{2} + 4 \, x^{2} + 4 \, x\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (24) = 48\).
Time = 1.96 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.50 \[ \int \left (2 x+e^{4 x+4 x^2+4 e^x x^2+4 \left (x+e^x x\right ) \log ^2(x)} \left (4+8 x+e^x \left (8 x+4 x^2\right )+\left (8+8 e^x\right ) \log (x)+\left (4+e^x (4+4 x)\right ) \log ^2(x)\right )+e^{2 x+2 x^2+2 e^x x^2+2 \left (x+e^x x\right ) \log ^2(x)} \left (-2-4 x-8 x^2+e^x \left (-8 x^2-4 x^3\right )+\left (-8 x-8 e^x x\right ) \log (x)+\left (-4 x+e^x \left (-4 x-4 x^2\right )\right ) \log ^2(x)\right )\right ) \, dx=x^{2} - 2 x e^{2 x^{2} e^{x} + 2 x^{2} + 2 x + 2 \left (x e^{x} + x\right ) \log {\left (x \right )}^{2}} + e^{4 x^{2} e^{x} + 4 x^{2} + 4 x + 4 \left (x e^{x} + x\right ) \log {\left (x \right )}^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (24) = 48\).
Time = 0.41 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.61 \[ \int \left (2 x+e^{4 x+4 x^2+4 e^x x^2+4 \left (x+e^x x\right ) \log ^2(x)} \left (4+8 x+e^x \left (8 x+4 x^2\right )+\left (8+8 e^x\right ) \log (x)+\left (4+e^x (4+4 x)\right ) \log ^2(x)\right )+e^{2 x+2 x^2+2 e^x x^2+2 \left (x+e^x x\right ) \log ^2(x)} \left (-2-4 x-8 x^2+e^x \left (-8 x^2-4 x^3\right )+\left (-8 x-8 e^x x\right ) \log (x)+\left (-4 x+e^x \left (-4 x-4 x^2\right )\right ) \log ^2(x)\right )\right ) \, dx=x^{2} - 2 \, x e^{\left (2 \, x e^{x} \log \left (x\right )^{2} + 2 \, x^{2} e^{x} + 2 \, x \log \left (x\right )^{2} + 2 \, x^{2} + 2 \, x\right )} + e^{\left (4 \, x e^{x} \log \left (x\right )^{2} + 4 \, x^{2} e^{x} + 4 \, x \log \left (x\right )^{2} + 4 \, x^{2} + 4 \, x\right )} \]
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\[ \int \left (2 x+e^{4 x+4 x^2+4 e^x x^2+4 \left (x+e^x x\right ) \log ^2(x)} \left (4+8 x+e^x \left (8 x+4 x^2\right )+\left (8+8 e^x\right ) \log (x)+\left (4+e^x (4+4 x)\right ) \log ^2(x)\right )+e^{2 x+2 x^2+2 e^x x^2+2 \left (x+e^x x\right ) \log ^2(x)} \left (-2-4 x-8 x^2+e^x \left (-8 x^2-4 x^3\right )+\left (-8 x-8 e^x x\right ) \log (x)+\left (-4 x+e^x \left (-4 x-4 x^2\right )\right ) \log ^2(x)\right )\right ) \, dx=\int { 4 \, {\left ({\left ({\left (x + 1\right )} e^{x} + 1\right )} \log \left (x\right )^{2} + {\left (x^{2} + 2 \, x\right )} e^{x} + 2 \, {\left (e^{x} + 1\right )} \log \left (x\right ) + 2 \, x + 1\right )} e^{\left (4 \, x^{2} e^{x} + 4 \, {\left (x e^{x} + x\right )} \log \left (x\right )^{2} + 4 \, x^{2} + 4 \, x\right )} - 2 \, {\left (2 \, {\left ({\left (x^{2} + x\right )} e^{x} + x\right )} \log \left (x\right )^{2} + 4 \, x^{2} + 2 \, {\left (x^{3} + 2 \, x^{2}\right )} e^{x} + 4 \, {\left (x e^{x} + x\right )} \log \left (x\right ) + 2 \, x + 1\right )} e^{\left (2 \, x^{2} e^{x} + 2 \, {\left (x e^{x} + x\right )} \log \left (x\right )^{2} + 2 \, x^{2} + 2 \, x\right )} + 2 \, x \,d x } \]
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Time = 10.09 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.86 \[ \int \left (2 x+e^{4 x+4 x^2+4 e^x x^2+4 \left (x+e^x x\right ) \log ^2(x)} \left (4+8 x+e^x \left (8 x+4 x^2\right )+\left (8+8 e^x\right ) \log (x)+\left (4+e^x (4+4 x)\right ) \log ^2(x)\right )+e^{2 x+2 x^2+2 e^x x^2+2 \left (x+e^x x\right ) \log ^2(x)} \left (-2-4 x-8 x^2+e^x \left (-8 x^2-4 x^3\right )+\left (-8 x-8 e^x x\right ) \log (x)+\left (-4 x+e^x \left (-4 x-4 x^2\right )\right ) \log ^2(x)\right )\right ) \, dx=x^2+{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^x\,{\ln \left (x\right )}^2}\,{\mathrm {e}}^{4\,x^2\,{\mathrm {e}}^x}\,{\mathrm {e}}^{4\,x\,{\ln \left (x\right )}^2}\,{\mathrm {e}}^{4\,x^2}-2\,x\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^x\,{\ln \left (x\right )}^2}\,{\mathrm {e}}^{2\,x^2\,{\mathrm {e}}^x}\,{\mathrm {e}}^{2\,x\,{\ln \left (x\right )}^2}\,{\mathrm {e}}^{2\,x^2} \]
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