Integrand size = 258, antiderivative size = 35 \[ \int \frac {16 e^{4+x} x+e^4 \left (8 x-56 x^2+64 x^3-8 x^5\right )+e^4 \left (32 x-32 x^2-8 x^3+8 x^4\right ) \log (4)+e^4 \left (4 x^2-2 x^3\right ) \log ^2(4)+\left (16 e^{4+x}+e^4 \left (8-56 x+64 x^2-8 x^4\right )+e^4 \left (32-32 x-8 x^2+8 x^3\right ) \log (4)+e^4 \left (4 x-2 x^2\right ) \log ^2(4)\right ) \log \left (4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)\right )}{4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)} \, dx=e^4 \left (x+\log \left (4 \left (e^x+x-(2-x)^2 \left (x-\frac {\log (4)}{2}\right )^2\right )\right )\right )^2 \]
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Time = 0.70 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.66, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {6820, 12, 6818} \[ \int \frac {16 e^{4+x} x+e^4 \left (8 x-56 x^2+64 x^3-8 x^5\right )+e^4 \left (32 x-32 x^2-8 x^3+8 x^4\right ) \log (4)+e^4 \left (4 x^2-2 x^3\right ) \log ^2(4)+\left (16 e^{4+x}+e^4 \left (8-56 x+64 x^2-8 x^4\right )+e^4 \left (32-32 x-8 x^2+8 x^3\right ) \log (4)+e^4 \left (4 x-2 x^2\right ) \log ^2(4)\right ) \log \left (4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)\right )}{4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)} \, dx=e^4 \left (\log \left (-4 x^4+16 x^3-16 x^2+4 x+4 e^x-(2-x)^2 \log ^2(4)+4 (2-x)^2 x \log (4)\right )+x\right )^2 \]
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Rule 12
Rule 6818
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {2 e^4 \left (8 e^x-4 x^4+4 x^3 \log (4)+2 x \left (-14-8 \log (4)+\log ^2(4)\right )+4 (1+\log (256))-x^2 \left (-32+\log ^2(4)+\log (256)\right )\right ) \left (x+\log \left (4 e^x+4 x-16 x^2+16 x^3-4 x^4+4 (-2+x)^2 x \log (4)-(-2+x)^2 \log ^2(4)\right )\right )}{4 e^x+4 x-16 x^2+16 x^3-4 x^4+4 (-2+x)^2 x \log (4)-(-2+x)^2 \log ^2(4)} \, dx \\ & = \left (2 e^4\right ) \int \frac {\left (8 e^x-4 x^4+4 x^3 \log (4)+2 x \left (-14-8 \log (4)+\log ^2(4)\right )+4 (1+\log (256))-x^2 \left (-32+\log ^2(4)+\log (256)\right )\right ) \left (x+\log \left (4 e^x+4 x-16 x^2+16 x^3-4 x^4+4 (-2+x)^2 x \log (4)-(-2+x)^2 \log ^2(4)\right )\right )}{4 e^x+4 x-16 x^2+16 x^3-4 x^4+4 (-2+x)^2 x \log (4)-(-2+x)^2 \log ^2(4)} \, dx \\ & = e^4 \left (x+\log \left (4 e^x+4 x-16 x^2+16 x^3-4 x^4+4 (2-x)^2 x \log (4)-(2-x)^2 \log ^2(4)\right )\right )^2 \\ \end{align*}
\[ \int \frac {16 e^{4+x} x+e^4 \left (8 x-56 x^2+64 x^3-8 x^5\right )+e^4 \left (32 x-32 x^2-8 x^3+8 x^4\right ) \log (4)+e^4 \left (4 x^2-2 x^3\right ) \log ^2(4)+\left (16 e^{4+x}+e^4 \left (8-56 x+64 x^2-8 x^4\right )+e^4 \left (32-32 x-8 x^2+8 x^3\right ) \log (4)+e^4 \left (4 x-2 x^2\right ) \log ^2(4)\right ) \log \left (4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)\right )}{4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)} \, dx=\int \frac {16 e^{4+x} x+e^4 \left (8 x-56 x^2+64 x^3-8 x^5\right )+e^4 \left (32 x-32 x^2-8 x^3+8 x^4\right ) \log (4)+e^4 \left (4 x^2-2 x^3\right ) \log ^2(4)+\left (16 e^{4+x}+e^4 \left (8-56 x+64 x^2-8 x^4\right )+e^4 \left (32-32 x-8 x^2+8 x^3\right ) \log (4)+e^4 \left (4 x-2 x^2\right ) \log ^2(4)\right ) \log \left (4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)\right )}{4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)} \, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. \(133\) vs. \(2(33)=66\).
Time = 5.23 (sec) , antiderivative size = 134, normalized size of antiderivative = 3.83
method | result | size |
risch | \(x^{2} {\mathrm e}^{4}+2 \,{\mathrm e}^{4} \ln \left (4 \,{\mathrm e}^{x}+4 \left (-x^{2}+4 x -4\right ) \ln \left (2\right )^{2}+2 \left (4 x^{3}-16 x^{2}+16 x \right ) \ln \left (2\right )-4 x^{4}+16 x^{3}-16 x^{2}+4 x \right ) x +{\mathrm e}^{4} \ln \left (4 \,{\mathrm e}^{x}+4 \left (-x^{2}+4 x -4\right ) \ln \left (2\right )^{2}+2 \left (4 x^{3}-16 x^{2}+16 x \right ) \ln \left (2\right )-4 x^{4}+16 x^{3}-16 x^{2}+4 x \right )^{2}\) | \(134\) |
parallelrisch | \(x^{2} {\mathrm e}^{4}+2 \,{\mathrm e}^{4} \ln \left (4 \,{\mathrm e}^{x}+4 \left (-x^{2}+4 x -4\right ) \ln \left (2\right )^{2}+2 \left (4 x^{3}-16 x^{2}+16 x \right ) \ln \left (2\right )-4 x^{4}+16 x^{3}-16 x^{2}+4 x \right ) x +{\mathrm e}^{4} \ln \left (4 \,{\mathrm e}^{x}+4 \left (-x^{2}+4 x -4\right ) \ln \left (2\right )^{2}+2 \left (4 x^{3}-16 x^{2}+16 x \right ) \ln \left (2\right )-4 x^{4}+16 x^{3}-16 x^{2}+4 x \right )^{2}\) | \(134\) |
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Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (31) = 62\).
Time = 0.28 (sec) , antiderivative size = 147, normalized size of antiderivative = 4.20 \[ \int \frac {16 e^{4+x} x+e^4 \left (8 x-56 x^2+64 x^3-8 x^5\right )+e^4 \left (32 x-32 x^2-8 x^3+8 x^4\right ) \log (4)+e^4 \left (4 x^2-2 x^3\right ) \log ^2(4)+\left (16 e^{4+x}+e^4 \left (8-56 x+64 x^2-8 x^4\right )+e^4 \left (32-32 x-8 x^2+8 x^3\right ) \log (4)+e^4 \left (4 x-2 x^2\right ) \log ^2(4)\right ) \log \left (4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)\right )}{4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)} \, dx=x^{2} e^{4} + 2 \, x e^{4} \log \left (-4 \, {\left ({\left (x^{2} - 4 \, x + 4\right )} e^{4} \log \left (2\right )^{2} - 2 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} e^{4} \log \left (2\right ) + {\left (x^{4} - 4 \, x^{3} + 4 \, x^{2} - x\right )} e^{4} - e^{\left (x + 4\right )}\right )} e^{\left (-4\right )}\right ) + e^{4} \log \left (-4 \, {\left ({\left (x^{2} - 4 \, x + 4\right )} e^{4} \log \left (2\right )^{2} - 2 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} e^{4} \log \left (2\right ) + {\left (x^{4} - 4 \, x^{3} + 4 \, x^{2} - x\right )} e^{4} - e^{\left (x + 4\right )}\right )} e^{\left (-4\right )}\right )^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (29) = 58\).
Time = 0.36 (sec) , antiderivative size = 133, normalized size of antiderivative = 3.80 \[ \int \frac {16 e^{4+x} x+e^4 \left (8 x-56 x^2+64 x^3-8 x^5\right )+e^4 \left (32 x-32 x^2-8 x^3+8 x^4\right ) \log (4)+e^4 \left (4 x^2-2 x^3\right ) \log ^2(4)+\left (16 e^{4+x}+e^4 \left (8-56 x+64 x^2-8 x^4\right )+e^4 \left (32-32 x-8 x^2+8 x^3\right ) \log (4)+e^4 \left (4 x-2 x^2\right ) \log ^2(4)\right ) \log \left (4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)\right )}{4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)} \, dx=x^{2} e^{4} + 2 x e^{4} \log {\left (- 4 x^{4} + 16 x^{3} - 16 x^{2} + 4 x + \left (- 4 x^{2} + 16 x - 16\right ) \log {\left (2 \right )}^{2} + \left (8 x^{3} - 32 x^{2} + 32 x\right ) \log {\left (2 \right )} + 4 e^{x} \right )} + e^{4} \log {\left (- 4 x^{4} + 16 x^{3} - 16 x^{2} + 4 x + \left (- 4 x^{2} + 16 x - 16\right ) \log {\left (2 \right )}^{2} + \left (8 x^{3} - 32 x^{2} + 32 x\right ) \log {\left (2 \right )} + 4 e^{x} \right )}^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (31) = 62\).
Time = 0.33 (sec) , antiderivative size = 138, normalized size of antiderivative = 3.94 \[ \int \frac {16 e^{4+x} x+e^4 \left (8 x-56 x^2+64 x^3-8 x^5\right )+e^4 \left (32 x-32 x^2-8 x^3+8 x^4\right ) \log (4)+e^4 \left (4 x^2-2 x^3\right ) \log ^2(4)+\left (16 e^{4+x}+e^4 \left (8-56 x+64 x^2-8 x^4\right )+e^4 \left (32-32 x-8 x^2+8 x^3\right ) \log (4)+e^4 \left (4 x-2 x^2\right ) \log ^2(4)\right ) \log \left (4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)\right )}{4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)} \, dx=x^{2} e^{4} + 4 \, x e^{4} \log \left (2\right ) + e^{4} \log \left (-x^{4} + 2 \, x^{3} {\left (\log \left (2\right ) + 2\right )} - {\left (\log \left (2\right )^{2} + 8 \, \log \left (2\right ) + 4\right )} x^{2} + {\left (4 \, \log \left (2\right )^{2} + 8 \, \log \left (2\right ) + 1\right )} x - 4 \, \log \left (2\right )^{2} + e^{x}\right )^{2} + 2 \, {\left (x e^{4} + 2 \, e^{4} \log \left (2\right )\right )} \log \left (-x^{4} + 2 \, x^{3} {\left (\log \left (2\right ) + 2\right )} - {\left (\log \left (2\right )^{2} + 8 \, \log \left (2\right ) + 4\right )} x^{2} + {\left (4 \, \log \left (2\right )^{2} + 8 \, \log \left (2\right ) + 1\right )} x - 4 \, \log \left (2\right )^{2} + e^{x}\right ) \]
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\[ \int \frac {16 e^{4+x} x+e^4 \left (8 x-56 x^2+64 x^3-8 x^5\right )+e^4 \left (32 x-32 x^2-8 x^3+8 x^4\right ) \log (4)+e^4 \left (4 x^2-2 x^3\right ) \log ^2(4)+\left (16 e^{4+x}+e^4 \left (8-56 x+64 x^2-8 x^4\right )+e^4 \left (32-32 x-8 x^2+8 x^3\right ) \log (4)+e^4 \left (4 x-2 x^2\right ) \log ^2(4)\right ) \log \left (4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)\right )}{4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)} \, dx=\int { \frac {2 \, {\left ({\left (x^{3} - 2 \, x^{2}\right )} e^{4} \log \left (2\right )^{2} - 2 \, {\left (x^{4} - x^{3} - 4 \, x^{2} + 4 \, x\right )} e^{4} \log \left (2\right ) + {\left (x^{5} - 8 \, x^{3} + 7 \, x^{2} - x\right )} e^{4} - 2 \, x e^{\left (x + 4\right )} + {\left ({\left (x^{2} - 2 \, x\right )} e^{4} \log \left (2\right )^{2} - 2 \, {\left (x^{3} - x^{2} - 4 \, x + 4\right )} e^{4} \log \left (2\right ) + {\left (x^{4} - 8 \, x^{2} + 7 \, x - 1\right )} e^{4} - 2 \, e^{\left (x + 4\right )}\right )} \log \left (-4 \, x^{4} + 16 \, x^{3} - 4 \, {\left (x^{2} - 4 \, x + 4\right )} \log \left (2\right )^{2} - 16 \, x^{2} + 8 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} \log \left (2\right ) + 4 \, x + 4 \, e^{x}\right )\right )}}{x^{4} - 4 \, x^{3} + {\left (x^{2} - 4 \, x + 4\right )} \log \left (2\right )^{2} + 4 \, x^{2} - 2 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} \log \left (2\right ) - x - e^{x}} \,d x } \]
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Time = 10.16 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.80 \[ \int \frac {16 e^{4+x} x+e^4 \left (8 x-56 x^2+64 x^3-8 x^5\right )+e^4 \left (32 x-32 x^2-8 x^3+8 x^4\right ) \log (4)+e^4 \left (4 x^2-2 x^3\right ) \log ^2(4)+\left (16 e^{4+x}+e^4 \left (8-56 x+64 x^2-8 x^4\right )+e^4 \left (32-32 x-8 x^2+8 x^3\right ) \log (4)+e^4 \left (4 x-2 x^2\right ) \log ^2(4)\right ) \log \left (4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)\right )}{4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)} \, dx={\mathrm {e}}^4\,{\left (x+\ln \left (4\,x+4\,{\mathrm {e}}^x-4\,{\ln \left (2\right )}^2\,\left (x^2-4\,x+4\right )+2\,\ln \left (2\right )\,\left (4\,x^3-16\,x^2+16\,x\right )-16\,x^2+16\,x^3-4\,x^4\right )\right )}^2 \]
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