Integrand size = 193, antiderivative size = 25 \[ \int \frac {3 x^2+8 x^3-1280 x^6-32 x^4 \log (5)+3 x^2 \log ^2(5)}{1+24 x+192 x^2+512 x^3+768 x^4+12288 x^5+49152 x^6+196608 x^8+1572864 x^9+16777216 x^{12}+\left (96 x^2+1536 x^3+6144 x^4+49152 x^6+393216 x^7+6291456 x^{10}\right ) \log (5)+\left (3+48 x+192 x^2+4608 x^4+36864 x^5+983040 x^8\right ) \log ^2(5)+\left (192 x^2+1536 x^3+81920 x^6\right ) \log ^3(5)+\left (3+24 x+3840 x^4\right ) \log ^4(5)+96 x^2 \log ^5(5)+\log ^6(5)} \, dx=\frac {x}{\left (4+\frac {1+4 x+\left (16 x^2+\log (5)\right )^2}{x}\right )^2} \]
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\[ \int \frac {3 x^2+8 x^3-1280 x^6-32 x^4 \log (5)+3 x^2 \log ^2(5)}{1+24 x+192 x^2+512 x^3+768 x^4+12288 x^5+49152 x^6+196608 x^8+1572864 x^9+16777216 x^{12}+\left (96 x^2+1536 x^3+6144 x^4+49152 x^6+393216 x^7+6291456 x^{10}\right ) \log (5)+\left (3+48 x+192 x^2+4608 x^4+36864 x^5+983040 x^8\right ) \log ^2(5)+\left (192 x^2+1536 x^3+81920 x^6\right ) \log ^3(5)+\left (3+24 x+3840 x^4\right ) \log ^4(5)+96 x^2 \log ^5(5)+\log ^6(5)} \, dx=\int \frac {3 x^2+8 x^3-1280 x^6-32 x^4 \log (5)+3 x^2 \log ^2(5)}{1+24 x+192 x^2+512 x^3+768 x^4+12288 x^5+49152 x^6+196608 x^8+1572864 x^9+16777216 x^{12}+\left (96 x^2+1536 x^3+6144 x^4+49152 x^6+393216 x^7+6291456 x^{10}\right ) \log (5)+\left (3+48 x+192 x^2+4608 x^4+36864 x^5+983040 x^8\right ) \log ^2(5)+\left (192 x^2+1536 x^3+81920 x^6\right ) \log ^3(5)+\left (3+24 x+3840 x^4\right ) \log ^4(5)+96 x^2 \log ^5(5)+\log ^6(5)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {8 x^3-1280 x^6-32 x^4 \log (5)+x^2 \left (3+3 \log ^2(5)\right )}{1+24 x+192 x^2+512 x^3+768 x^4+12288 x^5+49152 x^6+196608 x^8+1572864 x^9+16777216 x^{12}+\left (96 x^2+1536 x^3+6144 x^4+49152 x^6+393216 x^7+6291456 x^{10}\right ) \log (5)+\left (3+48 x+192 x^2+4608 x^4+36864 x^5+983040 x^8\right ) \log ^2(5)+\left (192 x^2+1536 x^3+81920 x^6\right ) \log ^3(5)+\left (3+24 x+3840 x^4\right ) \log ^4(5)+96 x^2 \log ^5(5)+\log ^6(5)} \, dx \\ & = \int \frac {8 x^3-1280 x^6-32 x^4 \log (5)+x^2 \left (3+3 \log ^2(5)\right )}{1+24 x+512 x^3+768 x^4+12288 x^5+49152 x^6+196608 x^8+1572864 x^9+16777216 x^{12}+\left (96 x^2+1536 x^3+6144 x^4+49152 x^6+393216 x^7+6291456 x^{10}\right ) \log (5)+\left (3+48 x+192 x^2+4608 x^4+36864 x^5+983040 x^8\right ) \log ^2(5)+\left (192 x^2+1536 x^3+81920 x^6\right ) \log ^3(5)+\left (3+24 x+3840 x^4\right ) \log ^4(5)+\log ^6(5)+x^2 \left (192+96 \log ^5(5)\right )} \, dx \\ & = \int \left (\frac {-10 x^2+\log (5)}{2 \left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2}+\frac {96 x^3-8 x \log (5)+16 x^2 \left (1-\log ^2(5)\right )-\log (5) \left (1+\log ^2(5)\right )}{2 \left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^3}\right ) \, dx \\ & = \frac {1}{2} \int \frac {-10 x^2+\log (5)}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2} \, dx+\frac {1}{2} \int \frac {96 x^3-8 x \log (5)+16 x^2 \left (1-\log ^2(5)\right )-\log (5) \left (1+\log ^2(5)\right )}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^3} \, dx \\ & = -\frac {3}{128 \left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2}+\frac {\int \frac {-14336 x \log (5)+16384 x^2 \left (1-\log ^2(5)\right )-256 \left (3+4 \log ^3(5)+\log (625)\right )}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^3} \, dx}{2048}+\frac {1}{2} \int \left (-\frac {10 x^2}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2}+\frac {\log (5)}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2}\right ) \, dx \\ & = -\frac {3}{128 \left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2}+\frac {\int \left (-\frac {14336 x \log (5)}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^3}-\frac {16384 x^2 \left (-1+\log ^2(5)\right )}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^3}-\frac {256 \left (3+4 \log ^3(5)+\log (625)\right )}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^3}\right ) \, dx}{2048}-5 \int \frac {x^2}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2} \, dx+\frac {1}{2} \log (5) \int \frac {1}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2} \, dx \\ & = -\frac {3}{128 \left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2}-5 \int \frac {x^2}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2} \, dx+\frac {1}{2} \log (5) \int \frac {1}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2} \, dx-(7 \log (5)) \int \frac {x}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^3} \, dx+\left (8 \left (1-\log ^2(5)\right )\right ) \int \frac {x^2}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^3} \, dx+\frac {1}{8} \left (-3-4 \log ^3(5)-\log (625)\right ) \int \frac {1}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^3} \, dx \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {3 x^2+8 x^3-1280 x^6-32 x^4 \log (5)+3 x^2 \log ^2(5)}{1+24 x+192 x^2+512 x^3+768 x^4+12288 x^5+49152 x^6+196608 x^8+1572864 x^9+16777216 x^{12}+\left (96 x^2+1536 x^3+6144 x^4+49152 x^6+393216 x^7+6291456 x^{10}\right ) \log (5)+\left (3+48 x+192 x^2+4608 x^4+36864 x^5+983040 x^8\right ) \log ^2(5)+\left (192 x^2+1536 x^3+81920 x^6\right ) \log ^3(5)+\left (3+24 x+3840 x^4\right ) \log ^4(5)+96 x^2 \log ^5(5)+\log ^6(5)} \, dx=\frac {x^3}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2} \]
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Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12
method | result | size |
default | \(\frac {x^{3}}{\left (256 x^{4}+32 x^{2} \ln \left (5\right )+\ln \left (5\right )^{2}+8 x +1\right )^{2}}\) | \(28\) |
norman | \(\frac {x^{3}}{\left (256 x^{4}+32 x^{2} \ln \left (5\right )+\ln \left (5\right )^{2}+8 x +1\right )^{2}}\) | \(28\) |
gosper | \(\frac {x^{3}}{65536 x^{8}+16384 x^{6} \ln \left (5\right )+1536 x^{4} \ln \left (5\right )^{2}+64 x^{2} \ln \left (5\right )^{3}+4096 x^{5}+\ln \left (5\right )^{4}+512 x^{3} \ln \left (5\right )+512 x^{4}+16 x \ln \left (5\right )^{2}+64 x^{2} \ln \left (5\right )+2 \ln \left (5\right )^{2}+64 x^{2}+16 x +1}\) | \(88\) |
risch | \(\frac {x^{3}}{65536 x^{8}+16384 x^{6} \ln \left (5\right )+1536 x^{4} \ln \left (5\right )^{2}+64 x^{2} \ln \left (5\right )^{3}+4096 x^{5}+\ln \left (5\right )^{4}+512 x^{3} \ln \left (5\right )+512 x^{4}+16 x \ln \left (5\right )^{2}+64 x^{2} \ln \left (5\right )+2 \ln \left (5\right )^{2}+64 x^{2}+16 x +1}\) | \(88\) |
parallelrisch | \(\frac {x^{3}}{65536 x^{8}+16384 x^{6} \ln \left (5\right )+1536 x^{4} \ln \left (5\right )^{2}+64 x^{2} \ln \left (5\right )^{3}+4096 x^{5}+\ln \left (5\right )^{4}+512 x^{3} \ln \left (5\right )+512 x^{4}+16 x \ln \left (5\right )^{2}+64 x^{2} \ln \left (5\right )+2 \ln \left (5\right )^{2}+64 x^{2}+16 x +1}\) | \(88\) |
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Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (25) = 50\).
Time = 0.25 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.12 \[ \int \frac {3 x^2+8 x^3-1280 x^6-32 x^4 \log (5)+3 x^2 \log ^2(5)}{1+24 x+192 x^2+512 x^3+768 x^4+12288 x^5+49152 x^6+196608 x^8+1572864 x^9+16777216 x^{12}+\left (96 x^2+1536 x^3+6144 x^4+49152 x^6+393216 x^7+6291456 x^{10}\right ) \log (5)+\left (3+48 x+192 x^2+4608 x^4+36864 x^5+983040 x^8\right ) \log ^2(5)+\left (192 x^2+1536 x^3+81920 x^6\right ) \log ^3(5)+\left (3+24 x+3840 x^4\right ) \log ^4(5)+96 x^2 \log ^5(5)+\log ^6(5)} \, dx=\frac {x^{3}}{65536 \, x^{8} + 4096 \, x^{5} + 64 \, x^{2} \log \left (5\right )^{3} + 512 \, x^{4} + \log \left (5\right )^{4} + 2 \, {\left (768 \, x^{4} + 8 \, x + 1\right )} \log \left (5\right )^{2} + 64 \, x^{2} + 64 \, {\left (256 \, x^{6} + 8 \, x^{3} + x^{2}\right )} \log \left (5\right ) + 16 \, x + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (20) = 40\).
Time = 4.47 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.28 \[ \int \frac {3 x^2+8 x^3-1280 x^6-32 x^4 \log (5)+3 x^2 \log ^2(5)}{1+24 x+192 x^2+512 x^3+768 x^4+12288 x^5+49152 x^6+196608 x^8+1572864 x^9+16777216 x^{12}+\left (96 x^2+1536 x^3+6144 x^4+49152 x^6+393216 x^7+6291456 x^{10}\right ) \log (5)+\left (3+48 x+192 x^2+4608 x^4+36864 x^5+983040 x^8\right ) \log ^2(5)+\left (192 x^2+1536 x^3+81920 x^6\right ) \log ^3(5)+\left (3+24 x+3840 x^4\right ) \log ^4(5)+96 x^2 \log ^5(5)+\log ^6(5)} \, dx=\frac {x^{3}}{65536 x^{8} + 16384 x^{6} \log {\left (5 \right )} + 4096 x^{5} + x^{4} \cdot \left (512 + 1536 \log {\left (5 \right )}^{2}\right ) + 512 x^{3} \log {\left (5 \right )} + x^{2} \cdot \left (64 + 64 \log {\left (5 \right )} + 64 \log {\left (5 \right )}^{3}\right ) + x \left (16 + 16 \log {\left (5 \right )}^{2}\right ) + 1 + 2 \log {\left (5 \right )}^{2} + \log {\left (5 \right )}^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (25) = 50\).
Time = 0.20 (sec) , antiderivative size = 77, normalized size of antiderivative = 3.08 \[ \int \frac {3 x^2+8 x^3-1280 x^6-32 x^4 \log (5)+3 x^2 \log ^2(5)}{1+24 x+192 x^2+512 x^3+768 x^4+12288 x^5+49152 x^6+196608 x^8+1572864 x^9+16777216 x^{12}+\left (96 x^2+1536 x^3+6144 x^4+49152 x^6+393216 x^7+6291456 x^{10}\right ) \log (5)+\left (3+48 x+192 x^2+4608 x^4+36864 x^5+983040 x^8\right ) \log ^2(5)+\left (192 x^2+1536 x^3+81920 x^6\right ) \log ^3(5)+\left (3+24 x+3840 x^4\right ) \log ^4(5)+96 x^2 \log ^5(5)+\log ^6(5)} \, dx=\frac {x^{3}}{65536 \, x^{8} + 16384 \, x^{6} \log \left (5\right ) + 512 \, {\left (3 \, \log \left (5\right )^{2} + 1\right )} x^{4} + 4096 \, x^{5} + 512 \, x^{3} \log \left (5\right ) + \log \left (5\right )^{4} + 64 \, {\left (\log \left (5\right )^{3} + \log \left (5\right ) + 1\right )} x^{2} + 16 \, {\left (\log \left (5\right )^{2} + 1\right )} x + 2 \, \log \left (5\right )^{2} + 1} \]
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Time = 0.30 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {3 x^2+8 x^3-1280 x^6-32 x^4 \log (5)+3 x^2 \log ^2(5)}{1+24 x+192 x^2+512 x^3+768 x^4+12288 x^5+49152 x^6+196608 x^8+1572864 x^9+16777216 x^{12}+\left (96 x^2+1536 x^3+6144 x^4+49152 x^6+393216 x^7+6291456 x^{10}\right ) \log (5)+\left (3+48 x+192 x^2+4608 x^4+36864 x^5+983040 x^8\right ) \log ^2(5)+\left (192 x^2+1536 x^3+81920 x^6\right ) \log ^3(5)+\left (3+24 x+3840 x^4\right ) \log ^4(5)+96 x^2 \log ^5(5)+\log ^6(5)} \, dx=\frac {x^{3}}{{\left (256 \, x^{4} + 32 \, x^{2} \log \left (5\right ) + \log \left (5\right )^{2} + 8 \, x + 1\right )}^{2}} \]
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Time = 13.95 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {3 x^2+8 x^3-1280 x^6-32 x^4 \log (5)+3 x^2 \log ^2(5)}{1+24 x+192 x^2+512 x^3+768 x^4+12288 x^5+49152 x^6+196608 x^8+1572864 x^9+16777216 x^{12}+\left (96 x^2+1536 x^3+6144 x^4+49152 x^6+393216 x^7+6291456 x^{10}\right ) \log (5)+\left (3+48 x+192 x^2+4608 x^4+36864 x^5+983040 x^8\right ) \log ^2(5)+\left (192 x^2+1536 x^3+81920 x^6\right ) \log ^3(5)+\left (3+24 x+3840 x^4\right ) \log ^4(5)+96 x^2 \log ^5(5)+\log ^6(5)} \, dx=\frac {x^3}{{\left (256\,x^4+32\,\ln \left (5\right )\,x^2+8\,x+{\ln \left (5\right )}^2+1\right )}^2} \]
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