Optimal. Leaf size=26 \[ \frac {1-(5-x) x^2}{-x+2 x^2+\log (6)} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(64\) vs. \(2(26)=52\).
time = 0.10, antiderivative size = 64, normalized size of antiderivative = 2.46, number of steps
used = 5, number of rules used = 5, integrand size = 65, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1694, 12, 1828,
21, 8} \begin {gather*} \frac {x}{2}-\frac {(1-8 \log (6)) (7+34 \log (6))-4 \left (x-\frac {1}{4}\right ) \left (9-16 \log ^2(6)-70 \log (6)\right )}{16 (1-8 \log (6)) \left (-2 x^2+x-\log (6)\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 21
Rule 1694
Rule 1828
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\text {Subst}\left (\int \frac {256 x^4+37 (1-8 \log (6))+32 x^2 (17+12 \log (6))-32 x (7+34 \log (6))}{2 \left (1-16 x^2-8 \log (6)\right )^2} \, dx,x,-\frac {1}{4}+x\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {256 x^4+37 (1-8 \log (6))+32 x^2 (17+12 \log (6))-32 x (7+34 \log (6))}{\left (1-16 x^2-8 \log (6)\right )^2} \, dx,x,-\frac {1}{4}+x\right )\\ &=-\frac {(1-8 \log (6)) (7+34 \log (6))+(1-4 x) \left (9-70 \log (6)-16 \log ^2(6)\right )}{16 (1-8 \log (6)) \left (x-2 x^2-\log (6)\right )}-\frac {\text {Subst}\left (\int \frac {32 x^2 (1-8 \log (6))-2 (1-8 \log (6))^2}{1-16 x^2-8 \log (6)} \, dx,x,-\frac {1}{4}+x\right )}{4 (1-8 \log (6))}\\ &=-\frac {(1-8 \log (6)) (7+34 \log (6))+(1-4 x) \left (9-70 \log (6)-16 \log ^2(6)\right )}{16 (1-8 \log (6)) \left (x-2 x^2-\log (6)\right )}+\frac {1}{2} \text {Subst}\left (\int 1 \, dx,x,-\frac {1}{4}+x\right )\\ &=\frac {x}{2}-\frac {(1-8 \log (6)) (7+34 \log (6))+(1-4 x) \left (9-70 \log (6)-16 \log ^2(6)\right )}{16 (1-8 \log (6)) \left (x-2 x^2-\log (6)\right )}\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(67\) vs. \(2(26)=52\).
time = 0.04, size = 67, normalized size = 2.58 \begin {gather*} \frac {1}{4} \left (2 x+\frac {-4+78 \log ^2(6)+\log (6) (23-2 \log (216))+x \left (9+8 \log ^2(6)+\log (46656)-4 \log (6) (19+\log (46656))\right )}{\left (-x+2 x^2+\log (6)\right ) (-1+8 \log (6))}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 32, normalized size = 1.23
method | result | size |
norman | \(\frac {x^{3}+1-\frac {5 x}{2}+\frac {5 \ln \left (6\right )}{2}}{\ln \left (6\right )+2 x^{2}-x}\) | \(27\) |
gosper | \(\frac {2 x^{3}+5 \ln \left (6\right )-5 x +2}{2 \ln \left (6\right )+4 x^{2}-2 x}\) | \(30\) |
default | \(\frac {x}{2}-\frac {\left (\ln \left (6\right )+\frac {9}{2}\right ) x -2-\frac {9 \ln \left (6\right )}{2}}{2 \left (\ln \left (6\right )+2 x^{2}-x \right )}\) | \(32\) |
risch | \(\frac {x}{2}+\frac {\frac {\left (-\frac {9}{2}-\ln \left (2\right )-\ln \left (3\right )\right ) x}{2}+\frac {9 \ln \left (3\right )}{4}+\frac {9 \ln \left (2\right )}{4}+1}{\ln \left (2\right )+\ln \left (3\right )+2 x^{2}-x}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 33, normalized size = 1.27 \begin {gather*} \frac {1}{2} \, x - \frac {x {\left (2 \, \log \left (6\right ) + 9\right )} - 9 \, \log \left (6\right ) - 4}{4 \, {\left (2 \, x^{2} - x + \log \left (6\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 34, normalized size = 1.31 \begin {gather*} \frac {4 \, x^{3} - 2 \, x^{2} - 9 \, x + 9 \, \log \left (6\right ) + 4}{4 \, {\left (2 \, x^{2} - x + \log \left (6\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.37, size = 32, normalized size = 1.23 \begin {gather*} \frac {x}{2} + \frac {x \left (-9 - 2 \log {\left (6 \right )}\right ) + 4 + 9 \log {\left (6 \right )}}{8 x^{2} - 4 x + 4 \log {\left (6 \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 33, normalized size = 1.27 \begin {gather*} \frac {1}{2} \, x - \frac {2 \, x \log \left (6\right ) + 9 \, x - 9 \, \log \left (6\right ) - 4}{4 \, {\left (2 \, x^{2} - x + \log \left (6\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 33, normalized size = 1.27 \begin {gather*} \frac {x}{2}+\frac {\frac {9\,\ln \left (6\right )}{2}-x\,\left (\ln \left (6\right )+\frac {9}{2}\right )+2}{4\,x^2-2\,x+2\,\ln \left (6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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