Optimal. Leaf size=26 \[ \left (1+x-x^2-\frac {x}{3+16 x}+x^2 \log (3)\right )^2 \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(72\) vs. \(2(26)=52\).
time = 0.18, antiderivative size = 72, normalized size of antiderivative = 2.77, number of steps
used = 2, number of rules used = 1, integrand size = 106, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.009, Rules used = {2099}
\begin {gather*} x^4 (1-\log (3))^2-2 x^3 (1-\log (3))-\frac {1}{8} x^2 (7-15 \log (3))+\frac {9}{256 (16 x+3)^2}+\frac {3}{128} x (79+\log (3))+\frac {9 (61+\log (27))}{2048 (16 x+3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2099
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {9}{8 (3+16 x)^3}+6 x^2 (-1+\log (3))+4 x^3 (-1+\log (3))^2+\frac {3}{128} (79+\log (3))+\frac {1}{4} x (-7+15 \log (3))-\frac {9 (61+\log (27))}{128 (3+16 x)^2}\right ) \, dx\\ &=\frac {9}{256 (3+16 x)^2}-\frac {1}{8} x^2 (7-15 \log (3))-2 x^3 (1-\log (3))+x^4 (1-\log (3))^2+\frac {3}{128} x (79+\log (3))+\frac {9 (61+\log (27))}{2048 (3+16 x)}\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(140\) vs. \(2(26)=52\).
time = 0.08, size = 140, normalized size = 5.38 \begin {gather*} \frac {16777216 x^6 (-1+\log (3))^2-6912 x^2 \left (-2493+370 \log (3)+3 \log ^2(3)-220 \log (9)\right )+65536 x^4 \left (-407+714 \log (3)+9 \log ^2(3)-30 \log (9)\right )+98304 x^3 (248+106 \log (3)+15 \log (9))-288 x \left (-12709+1170 \log (3)+27 \log ^2(3)-900 \log (9)+512 \log (27)\right )-9 \left (-29935+270 \log (3)+81 \log ^2(3)-1080 \log (9)+1536 \log (27)\right )+2097152 x^5 \left (-13+3 \log ^2(3)+\log (59049)\right )}{65536 (3+16 x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(75\) vs.
\(2(26)=52\).
time = 0.28, size = 76, normalized size = 2.92
method | result | size |
default | \(x^{4} \ln \left (3\right )^{2}-2 x^{4} \ln \left (3\right )+x^{4}+2 x^{3} \ln \left (3\right )-2 x^{3}+\frac {15 x^{2} \ln \left (3\right )}{8}-\frac {7 x^{2}}{8}+\frac {3 x \ln \left (3\right )}{128}+\frac {237 x}{128}-\frac {4 \left (-\frac {27 \ln \left (3\right )}{8192}-\frac {549}{8192}\right )}{16 x +3}+\frac {9}{256 \left (16 x +3\right )^{2}}\) | \(76\) |
risch | \(x^{4} \ln \left (3\right )^{2}-2 x^{4} \ln \left (3\right )+2 x^{3} \ln \left (3\right )+x^{4}+\frac {15 x^{2} \ln \left (3\right )}{8}-2 x^{3}+\frac {3 x \ln \left (3\right )}{128}-\frac {7 x^{2}}{8}+\frac {237 x}{128}+\frac {\frac {\left (\frac {549}{256}+\frac {27 \ln \left (3\right )}{256}\right ) x}{128}+\frac {1719}{524288}+\frac {81 \ln \left (3\right )}{524288}}{x^{2}+\frac {3}{8} x +\frac {9}{256}}\) | \(78\) |
norman | \(\frac {\left (372+204 \ln \left (3\right )\right ) x^{3}+12 x +\left (-416+96 \ln \left (3\right )^{2}+320 \ln \left (3\right )\right ) x^{5}+\left (-407+654 \ln \left (3\right )+9 \ln \left (3\right )^{2}\right ) x^{4}+\left (146+18 \ln \left (3\right )\right ) x^{2}+\left (256 \ln \left (3\right )^{2}-512 \ln \left (3\right )+256\right ) x^{6}}{\left (16 x +3\right )^{2}}\) | \(81\) |
gosper | \(\frac {x \left (256 x^{5} \ln \left (3\right )^{2}+96 x^{4} \ln \left (3\right )^{2}-512 x^{5} \ln \left (3\right )+9 x^{3} \ln \left (3\right )^{2}+320 x^{4} \ln \left (3\right )+256 x^{5}+654 x^{3} \ln \left (3\right )-416 x^{4}+204 x^{2} \ln \left (3\right )-407 x^{3}+18 x \ln \left (3\right )+372 x^{2}+146 x +12\right )}{256 x^{2}+96 x +9}\) | \(100\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 71 vs.
\(2 (26) = 52\).
time = 0.26, size = 71, normalized size = 2.73 \begin {gather*} {\left (\log \left (3\right )^{2} - 2 \, \log \left (3\right ) + 1\right )} x^{4} + 2 \, x^{3} {\left (\log \left (3\right ) - 1\right )} + \frac {1}{8} \, x^{2} {\left (15 \, \log \left (3\right ) - 7\right )} + \frac {3}{128} \, x {\left (\log \left (3\right ) + 79\right )} + \frac {9 \, {\left (16 \, x {\left (3 \, \log \left (3\right ) + 61\right )} + 9 \, \log \left (3\right ) + 191\right )}}{2048 \, {\left (256 \, x^{2} + 96 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 100 vs.
\(2 (26) = 52\).
time = 0.35, size = 100, normalized size = 3.85 \begin {gather*} \frac {524288 \, x^{6} - 851968 \, x^{5} - 833536 \, x^{4} + 761856 \, x^{3} + 2048 \, {\left (256 \, x^{6} + 96 \, x^{5} + 9 \, x^{4}\right )} \log \left (3\right )^{2} + 347904 \, x^{2} - {\left (1048576 \, x^{6} - 655360 \, x^{5} - 1339392 \, x^{4} - 417792 \, x^{3} - 39168 \, x^{2} - 864 \, x - 81\right )} \log \left (3\right ) + 42912 \, x + 1719}{2048 \, {\left (256 \, x^{2} + 96 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 75 vs.
\(2 (20) = 40\).
time = 0.23, size = 75, normalized size = 2.88 \begin {gather*} x^{4} \left (- 2 \log {\left (3 \right )} + 1 + \log {\left (3 \right )}^{2}\right ) + x^{3} \left (-2 + 2 \log {\left (3 \right )}\right ) + x^{2} \left (- \frac {7}{8} + \frac {15 \log {\left (3 \right )}}{8}\right ) + x \left (\frac {3 \log {\left (3 \right )}}{128} + \frac {237}{128}\right ) + \frac {x \left (432 \log {\left (3 \right )} + 8784\right ) + 81 \log {\left (3 \right )} + 1719}{524288 x^{2} + 196608 x + 18432} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 74 vs.
\(2 (26) = 52\).
time = 0.41, size = 74, normalized size = 2.85 \begin {gather*} x^{4} \log \left (3\right )^{2} - 2 \, x^{4} \log \left (3\right ) + x^{4} + 2 \, x^{3} \log \left (3\right ) - 2 \, x^{3} + \frac {15}{8} \, x^{2} \log \left (3\right ) - \frac {7}{8} \, x^{2} + \frac {3}{128} \, x \log \left (3\right ) + \frac {237}{128} \, x + \frac {9 \, {\left (48 \, x \log \left (3\right ) + 976 \, x + 9 \, \log \left (3\right ) + 191\right )}}{2048 \, {\left (16 \, x + 3\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.65, size = 109, normalized size = 4.19 \begin {gather*} x^3\,\left (\frac {\ln \left (3\right )}{2}-\frac {3\,{\left (\ln \left (3\right )-1\right )}^2}{4}+\frac {3\,{\ln \left (3\right )}^2}{4}-\frac {5}{4}\right )+x^2\,\left (\frac {87\,\ln \left (3\right )}{32}+\frac {27\,{\left (\ln \left (3\right )-1\right )}^2}{64}-\frac {27\,{\ln \left (3\right )}^2}{64}-\frac {83}{64}\right )+\frac {\frac {81\,\ln \left (3\right )}{16}+x\,\left (27\,\ln \left (3\right )+549\right )+\frac {1719}{16}}{32768\,x^2+12288\,x+1152}+x^4\,{\left (\ln \left (3\right )-1\right )}^2-x\,\left (\frac {129\,\ln \left (3\right )}{256}+\frac {135\,{\left (\ln \left (3\right )-1\right )}^2}{512}-\frac {135\,{\ln \left (3\right )}^2}{512}-\frac {1083}{512}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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