Integrand size = 59, antiderivative size = 29 \[ \int \frac {1}{4} \left (1+32 x^3-64 e^2 x^3+32 e^4 x^3+\left (-64 e^2 x^3+64 e^4 x^3\right ) \log (3)+32 e^4 x^3 \log ^2(3)\right ) \, dx=4+\frac {x}{4}+2 \left (5+x^2 \left (x-e^2 x (1+\log (3))\right )^2\right ) \]
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Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.55, number of steps used = 8, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {6, 12, 30} \[ \int \frac {1}{4} \left (1+32 x^3-64 e^2 x^3+32 e^4 x^3+\left (-64 e^2 x^3+64 e^4 x^3\right ) \log (3)+32 e^4 x^3 \log ^2(3)\right ) \, dx=2 x^4 \left (1-2 e^2+e^4 \left (1+\log ^2(3)\right )\right )-4 e^2 \left (1-e^2\right ) x^4 \log (3)+\frac {x}{4} \]
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Rule 6
Rule 12
Rule 30
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{4} \left (1+32 e^4 x^3+\left (32-64 e^2\right ) x^3+\left (-64 e^2 x^3+64 e^4 x^3\right ) \log (3)+32 e^4 x^3 \log ^2(3)\right ) \, dx \\ & = \int \frac {1}{4} \left (1+\left (32-64 e^2+32 e^4\right ) x^3+\left (-64 e^2 x^3+64 e^4 x^3\right ) \log (3)+32 e^4 x^3 \log ^2(3)\right ) \, dx \\ & = \int \frac {1}{4} \left (1+\left (-64 e^2 x^3+64 e^4 x^3\right ) \log (3)+x^3 \left (32-64 e^2+32 e^4+32 e^4 \log ^2(3)\right )\right ) \, dx \\ & = \frac {1}{4} \int \left (1+\left (-64 e^2 x^3+64 e^4 x^3\right ) \log (3)+x^3 \left (32-64 e^2+32 e^4+32 e^4 \log ^2(3)\right )\right ) \, dx \\ & = \frac {x}{4}+2 x^4 \left (1-2 e^2+e^4 \left (1+\log ^2(3)\right )\right )+\frac {1}{4} \log (3) \int \left (-64 e^2 x^3+64 e^4 x^3\right ) \, dx \\ & = \frac {x}{4}+2 x^4 \left (1-2 e^2+e^4 \left (1+\log ^2(3)\right )\right )+\frac {1}{4} \log (3) \int \left (-64 e^2+64 e^4\right ) x^3 \, dx \\ & = \frac {x}{4}+2 x^4 \left (1-2 e^2+e^4 \left (1+\log ^2(3)\right )\right )-\left (16 e^2 \left (1-e^2\right ) \log (3)\right ) \int x^3 \, dx \\ & = \frac {x}{4}-4 e^2 \left (1-e^2\right ) x^4 \log (3)+2 x^4 \left (1-2 e^2+e^4 \left (1+\log ^2(3)\right )\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {1}{4} \left (1+32 x^3-64 e^2 x^3+32 e^4 x^3+\left (-64 e^2 x^3+64 e^4 x^3\right ) \log (3)+32 e^4 x^3 \log ^2(3)\right ) \, dx=\frac {x}{4}+2 x^4 \left (-1+e^2 (1+\log (3))\right )^2 \]
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Time = 0.04 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.55
method | result | size |
norman | \(\left (2 \,{\mathrm e}^{4} \ln \left (3\right )^{2}+4 \,{\mathrm e}^{4} \ln \left (3\right )+2 \,{\mathrm e}^{4}-4 \,{\mathrm e}^{2} \ln \left (3\right )-4 \,{\mathrm e}^{2}+2\right ) x^{4}+\frac {x}{4}\) | \(45\) |
risch | \(2 x^{4} {\mathrm e}^{4} \ln \left (3\right )^{2}+4 \,{\mathrm e}^{4} \ln \left (3\right ) x^{4}-4 \,{\mathrm e}^{2} \ln \left (3\right ) x^{4}+2 x^{4} {\mathrm e}^{4}-4 x^{4} {\mathrm e}^{2}+2 x^{4}+\frac {x}{4}\) | \(55\) |
default | \(2 x^{4} {\mathrm e}^{4} \ln \left (3\right )^{2}+4 \,{\mathrm e}^{4} \ln \left (3\right ) x^{4}-4 \,{\mathrm e}^{2} \ln \left (3\right ) x^{4}+2 x^{4} {\mathrm e}^{4}-4 x^{4} {\mathrm e}^{2}+2 x^{4}+\frac {x}{4}\) | \(59\) |
parallelrisch | \(2 x^{4} {\mathrm e}^{4} \ln \left (3\right )^{2}+4 \,{\mathrm e}^{4} \ln \left (3\right ) x^{4}-4 \,{\mathrm e}^{2} \ln \left (3\right ) x^{4}+2 x^{4} {\mathrm e}^{4}-4 x^{4} {\mathrm e}^{2}+2 x^{4}+\frac {x}{4}\) | \(59\) |
parts | \(2 x^{4} {\mathrm e}^{4} \ln \left (3\right )^{2}+4 \,{\mathrm e}^{4} \ln \left (3\right ) x^{4}-4 \,{\mathrm e}^{2} \ln \left (3\right ) x^{4}+2 x^{4} {\mathrm e}^{4}-4 x^{4} {\mathrm e}^{2}+2 x^{4}+\frac {x}{4}\) | \(59\) |
gosper | \(\frac {x \left (8 x^{3} {\mathrm e}^{4} \ln \left (3\right )^{2}+16 \,{\mathrm e}^{4} \ln \left (3\right ) x^{3}+8 x^{3} {\mathrm e}^{4}-16 \,{\mathrm e}^{2} \ln \left (3\right ) x^{3}-16 x^{3} {\mathrm e}^{2}+8 x^{3}+1\right )}{4}\) | \(60\) |
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Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (24) = 48\).
Time = 0.25 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.79 \[ \int \frac {1}{4} \left (1+32 x^3-64 e^2 x^3+32 e^4 x^3+\left (-64 e^2 x^3+64 e^4 x^3\right ) \log (3)+32 e^4 x^3 \log ^2(3)\right ) \, dx=2 \, x^{4} e^{4} \log \left (3\right )^{2} + 2 \, x^{4} e^{4} - 4 \, x^{4} e^{2} + 2 \, x^{4} + 4 \, {\left (x^{4} e^{4} - x^{4} e^{2}\right )} \log \left (3\right ) + \frac {1}{4} \, x \]
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Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.52 \[ \int \frac {1}{4} \left (1+32 x^3-64 e^2 x^3+32 e^4 x^3+\left (-64 e^2 x^3+64 e^4 x^3\right ) \log (3)+32 e^4 x^3 \log ^2(3)\right ) \, dx=x^{4} \left (- 4 e^{2} \log {\left (3 \right )} - 4 e^{2} + 2 + 2 e^{4} + 2 e^{4} \log {\left (3 \right )}^{2} + 4 e^{4} \log {\left (3 \right )}\right ) + \frac {x}{4} \]
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Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (24) = 48\).
Time = 0.23 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.79 \[ \int \frac {1}{4} \left (1+32 x^3-64 e^2 x^3+32 e^4 x^3+\left (-64 e^2 x^3+64 e^4 x^3\right ) \log (3)+32 e^4 x^3 \log ^2(3)\right ) \, dx=2 \, x^{4} e^{4} \log \left (3\right )^{2} + 2 \, x^{4} e^{4} - 4 \, x^{4} e^{2} + 2 \, x^{4} + 4 \, {\left (x^{4} e^{4} - x^{4} e^{2}\right )} \log \left (3\right ) + \frac {1}{4} \, x \]
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Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (24) = 48\).
Time = 0.29 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.79 \[ \int \frac {1}{4} \left (1+32 x^3-64 e^2 x^3+32 e^4 x^3+\left (-64 e^2 x^3+64 e^4 x^3\right ) \log (3)+32 e^4 x^3 \log ^2(3)\right ) \, dx=2 \, x^{4} e^{4} \log \left (3\right )^{2} + 2 \, x^{4} e^{4} - 4 \, x^{4} e^{2} + 2 \, x^{4} + 4 \, {\left (x^{4} e^{4} - x^{4} e^{2}\right )} \log \left (3\right ) + \frac {1}{4} \, x \]
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Time = 8.05 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int \frac {1}{4} \left (1+32 x^3-64 e^2 x^3+32 e^4 x^3+\left (-64 e^2 x^3+64 e^4 x^3\right ) \log (3)+32 e^4 x^3 \log ^2(3)\right ) \, dx=\left (2\,{\mathrm {e}}^4-4\,{\mathrm {e}}^2+2\,{\mathrm {e}}^4\,{\ln \left (3\right )}^2-\frac {\ln \left (3\right )\,\left (64\,{\mathrm {e}}^2-64\,{\mathrm {e}}^4\right )}{16}+2\right )\,x^4+\frac {x}{4} \]
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