Integrand size = 61, antiderivative size = 27 \[ \int \frac {-4 x^2+4 x^3-2 x^5+9 x^6+6 x^7+8 x^8+\left (30 x-20 x^2-5 x^3\right ) \log (4)-50 \log ^2(4)}{2 x^5} \, dx=\left (\frac {-1+x}{x}+\frac {x}{2}+x^2+\frac {5 \log (4)}{2 x^2}\right )^2 \]
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Leaf count is larger than twice the leaf count of optimal. \(58\) vs. \(2(27)=54\).
Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.15, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {12, 14} \[ \int \frac {-4 x^2+4 x^3-2 x^5+9 x^6+6 x^7+8 x^8+\left (30 x-20 x^2-5 x^3\right ) \log (4)-50 \log ^2(4)}{2 x^5} \, dx=x^4+\frac {25 \log ^2(4)}{4 x^4}+x^3-\frac {5 \log (4)}{x^3}+\frac {9 x^2}{4}+\frac {1+5 \log (4)}{x^2}-x-\frac {4-5 \log (4)}{2 x} \]
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Rule 12
Rule 14
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int \frac {-4 x^2+4 x^3-2 x^5+9 x^6+6 x^7+8 x^8+\left (30 x-20 x^2-5 x^3\right ) \log (4)-50 \log ^2(4)}{x^5} \, dx \\ & = \frac {1}{2} \int \left (-2+9 x+6 x^2+8 x^3+\frac {4-5 \log (4)}{x^2}+\frac {30 \log (4)}{x^4}-\frac {50 \log ^2(4)}{x^5}-\frac {4 (1+5 \log (4))}{x^3}\right ) \, dx \\ & = -x+\frac {9 x^2}{4}+x^3+x^4-\frac {4-5 \log (4)}{2 x}-\frac {5 \log (4)}{x^3}+\frac {25 \log ^2(4)}{4 x^4}+\frac {1+5 \log (4)}{x^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(64\) vs. \(2(27)=54\).
Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.37 \[ \int \frac {-4 x^2+4 x^3-2 x^5+9 x^6+6 x^7+8 x^8+\left (30 x-20 x^2-5 x^3\right ) \log (4)-50 \log ^2(4)}{2 x^5} \, dx=\frac {1}{2} \left (-2 x+\frac {9 x^2}{2}+2 x^3+2 x^4-\frac {10 \log (4)}{x^3}+\frac {25 \log ^2(4)}{2 x^4}+\frac {-4+5 \log (4)}{x}+\frac {2 (1+5 \log (4))}{x^2}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(52\) vs. \(2(23)=46\).
Time = 0.09 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.96
method | result | size |
norman | \(\frac {x^{7}+x^{8}+\left (-2+5 \ln \left (2\right )\right ) x^{3}+\left (10 \ln \left (2\right )+1\right ) x^{2}-x^{5}+\frac {9 x^{6}}{4}+25 \ln \left (2\right )^{2}-10 x \ln \left (2\right )}{x^{4}}\) | \(53\) |
risch | \(x^{4}+x^{3}+\frac {9 x^{2}}{4}-x +\frac {\left (10 \ln \left (2\right )-4\right ) x^{3}+\left (2+20 \ln \left (2\right )\right ) x^{2}-20 x \ln \left (2\right )+50 \ln \left (2\right )^{2}}{2 x^{4}}\) | \(53\) |
default | \(x^{4}+x^{3}+\frac {9 x^{2}}{4}-x -\frac {-10 \ln \left (2\right )+4}{2 x}-\frac {-40 \ln \left (2\right )-4}{4 x^{2}}-\frac {10 \ln \left (2\right )}{x^{3}}+\frac {25 \ln \left (2\right )^{2}}{x^{4}}\) | \(54\) |
gosper | \(\frac {4 x^{8}+4 x^{7}+9 x^{6}-4 x^{5}+20 x^{3} \ln \left (2\right )+40 x^{2} \ln \left (2\right )-8 x^{3}+100 \ln \left (2\right )^{2}-40 x \ln \left (2\right )+4 x^{2}}{4 x^{4}}\) | \(62\) |
parallelrisch | \(\frac {4 x^{8}+4 x^{7}+9 x^{6}-4 x^{5}+20 x^{3} \ln \left (2\right )+40 x^{2} \ln \left (2\right )-8 x^{3}+100 \ln \left (2\right )^{2}-40 x \ln \left (2\right )+4 x^{2}}{4 x^{4}}\) | \(62\) |
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Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (26) = 52\).
Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.15 \[ \int \frac {-4 x^2+4 x^3-2 x^5+9 x^6+6 x^7+8 x^8+\left (30 x-20 x^2-5 x^3\right ) \log (4)-50 \log ^2(4)}{2 x^5} \, dx=\frac {4 \, x^{8} + 4 \, x^{7} + 9 \, x^{6} - 4 \, x^{5} - 8 \, x^{3} + 4 \, x^{2} + 20 \, {\left (x^{3} + 2 \, x^{2} - 2 \, x\right )} \log \left (2\right ) + 100 \, \log \left (2\right )^{2}}{4 \, x^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (20) = 40\).
Time = 0.31 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.96 \[ \int \frac {-4 x^2+4 x^3-2 x^5+9 x^6+6 x^7+8 x^8+\left (30 x-20 x^2-5 x^3\right ) \log (4)-50 \log ^2(4)}{2 x^5} \, dx=x^{4} + x^{3} + \frac {9 x^{2}}{4} - x + \frac {x^{3} \left (-4 + 10 \log {\left (2 \right )}\right ) + x^{2} \cdot \left (2 + 20 \log {\left (2 \right )}\right ) - 20 x \log {\left (2 \right )} + 50 \log {\left (2 \right )}^{2}}{2 x^{4}} \]
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Time = 0.19 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.89 \[ \int \frac {-4 x^2+4 x^3-2 x^5+9 x^6+6 x^7+8 x^8+\left (30 x-20 x^2-5 x^3\right ) \log (4)-50 \log ^2(4)}{2 x^5} \, dx=x^{4} + x^{3} + \frac {9}{4} \, x^{2} - x + \frac {x^{3} {\left (5 \, \log \left (2\right ) - 2\right )} + x^{2} {\left (10 \, \log \left (2\right ) + 1\right )} - 10 \, x \log \left (2\right ) + 25 \, \log \left (2\right )^{2}}{x^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (26) = 52\).
Time = 0.26 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.96 \[ \int \frac {-4 x^2+4 x^3-2 x^5+9 x^6+6 x^7+8 x^8+\left (30 x-20 x^2-5 x^3\right ) \log (4)-50 \log ^2(4)}{2 x^5} \, dx=x^{4} + x^{3} + \frac {9}{4} \, x^{2} - x + \frac {5 \, x^{3} \log \left (2\right ) - 2 \, x^{3} + 10 \, x^{2} \log \left (2\right ) + x^{2} - 10 \, x \log \left (2\right ) + 25 \, \log \left (2\right )^{2}}{x^{4}} \]
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Time = 14.15 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.81 \[ \int \frac {-4 x^2+4 x^3-2 x^5+9 x^6+6 x^7+8 x^8+\left (30 x-20 x^2-5 x^3\right ) \log (4)-50 \log ^2(4)}{2 x^5} \, dx=\frac {\left (\ln \left (32\right )-2\right )\,x^3+\left (10\,\ln \left (2\right )+1\right )\,x^2-10\,\ln \left (2\right )\,x+25\,{\ln \left (2\right )}^2}{x^4}-x+\frac {9\,x^2}{4}+x^3+x^4 \]
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