Optimal. Leaf size=21 \[ x-\frac {2-x}{-3+x+\frac {6 x}{\log (4)}} \]
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Rubi [A]
time = 0.07, antiderivative size = 31, normalized size of antiderivative = 1.48, number of steps
used = 5, number of rules used = 4, integrand size = 66, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {2009, 27, 6,
697} \begin {gather*} x-\frac {(12-\log (4)) \log (4)}{(6+\log (4)) (x (6+\log (4))-\log (64))} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 27
Rule 697
Rule 2009
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-6 x \log (4) (6+\log (4))+x^2 (6+\log (4))^2+4 \log (4) (3+\log (16))}{9 \log ^2(4)-6 x \log (4) (6+\log (4))+x^2 (6+\log (4))^2} \, dx\\ &=\int \frac {-6 x \log (4) (6+\log (4))+x^2 (6+\log (4))^2+4 \log (4) (3+\log (16))}{(6 x-3 \log (4)+x \log (4))^2} \, dx\\ &=\int \frac {-6 x \log (4) (6+\log (4))+x^2 (6+\log (4))^2+4 \log (4) (3+\log (16))}{(-3 \log (4)+x (6+\log (4)))^2} \, dx\\ &=\int \left (1+\frac {(12-\log (4)) \log (4)}{(3 \log (4)-x (6+\log (4)))^2}\right ) \, dx\\ &=x-\frac {(12-\log (4)) \log (4)}{(6+\log (4)) (x (6+\log (4))-\log (64))}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.04, size = 28, normalized size = 1.33 \begin {gather*} x+\frac {(-12+\log (4)) \log (4)}{(6+\log (4)) (-3 \log (4)+x (6+\log (4)))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 3.81, size = 30, normalized size = 1.43
method | result | size |
default | \(x +\frac {\ln \left (2\right ) \left (\ln \left (2\right )-6\right )}{\left (3+\ln \left (2\right )\right ) \left (x \ln \left (2\right )-3 \ln \left (2\right )+3 x \right )}\) | \(30\) |
gosper | \(\frac {x \left (3 x \ln \left (2\right )-8 \ln \left (2\right )+9 x -6\right )}{3 x \ln \left (2\right )-9 \ln \left (2\right )+9 x}\) | \(32\) |
norman | \(\frac {\left (3+\ln \left (2\right )\right ) x^{2}+\left (-\frac {8 \ln \left (2\right )}{3}-2\right ) x}{x \ln \left (2\right )-3 \ln \left (2\right )+3 x}\) | \(33\) |
risch | \(x +\frac {\ln \left (2\right )^{2}}{\left (3+\ln \left (2\right )\right ) \left (x \ln \left (2\right )-3 \ln \left (2\right )+3 x \right )}-\frac {6 \ln \left (2\right )}{\left (3+\ln \left (2\right )\right ) \left (x \ln \left (2\right )-3 \ln \left (2\right )+3 x \right )}\) | \(52\) |
meijerg | \(-\frac {3 \ln \left (2\right ) \left (-\frac {x \left (3+\ln \left (2\right )\right ) \left (-\frac {x \left (3+\ln \left (2\right )\right )}{\ln \left (2\right )}+6\right )}{9 \ln \left (2\right ) \left (1-\frac {x \left (3+\ln \left (2\right )\right )}{3 \ln \left (2\right )}\right )}-2 \ln \left (1-\frac {x \left (3+\ln \left (2\right )\right )}{3 \ln \left (2\right )}\right )\right )}{3+\ln \left (2\right )}+\frac {9 \left (-\frac {8 \ln \left (2\right )^{2}}{3}-8 \ln \left (2\right )\right ) \left (\frac {x \left (3+\ln \left (2\right )\right )}{3 \ln \left (2\right ) \left (1-\frac {x \left (3+\ln \left (2\right )\right )}{3 \ln \left (2\right )}\right )}+\ln \left (1-\frac {x \left (3+\ln \left (2\right )\right )}{3 \ln \left (2\right )}\right )\right )}{4 \ln \left (2\right )^{2}+24 \ln \left (2\right )+36}+\frac {32 \left (3+\ln \left (2\right )\right )^{2} x}{9 \left (4 \ln \left (2\right )^{2}+24 \ln \left (2\right )+36\right ) \left (1-\frac {x \left (3+\ln \left (2\right )\right )}{3 \ln \left (2\right )}\right )}+\frac {8 \left (3+\ln \left (2\right )\right )^{2} x}{3 \ln \left (2\right ) \left (4 \ln \left (2\right )^{2}+24 \ln \left (2\right )+36\right ) \left (1-\frac {x \left (3+\ln \left (2\right )\right )}{3 \ln \left (2\right )}\right )}\) | \(216\) |
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. 37 vs.
\(2 (18) = 36\).
time = 0.25, size = 37, normalized size = 1.76 \begin {gather*} x + \frac {\log \left (2\right )^{2} - 6 \, \log \left (2\right )}{{\left (\log \left (2\right )^{2} + 6 \, \log \left (2\right ) + 9\right )} x - 3 \, \log \left (2\right )^{2} - 9 \, \log \left (2\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. 57 vs.
\(2 (18) = 36\).
time = 0.39, size = 57, normalized size = 2.71 \begin {gather*} \frac {{\left (x^{2} - 3 \, x + 1\right )} \log \left (2\right )^{2} + 9 \, x^{2} + 3 \, {\left (2 \, x^{2} - 3 \, x - 2\right )} \log \left (2\right )}{{\left (x - 3\right )} \log \left (2\right )^{2} + 3 \, {\left (2 \, x - 3\right )} \log \left (2\right ) + 9 \, x} \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. 36 vs.
\(2 (14) = 28\).
time = 0.19, size = 36, normalized size = 1.71 \begin {gather*} x + \frac {- 6 \log {\left (2 \right )} + \log {\left (2 \right )}^{2}}{x \left (\log {\left (2 \right )}^{2} + 6 \log {\left (2 \right )} + 9\right ) - 9 \log {\left (2 \right )} - 3 \log {\left (2 \right )}^{2}} \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. 59 vs.
\(2 (18) = 36\).
time = 0.40, size = 59, normalized size = 2.81 \begin {gather*} \frac {x \log \left (2\right )^{2} + 6 \, x \log \left (2\right ) + 9 \, x}{\log \left (2\right )^{2} + 6 \, \log \left (2\right ) + 9} + \frac {\log \left (2\right )^{2} - 6 \, \log \left (2\right )}{{\left (x \log \left (2\right ) + 3 \, x - 3 \, \log \left (2\right )\right )} {\left (\log \left (2\right ) + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.64, size = 112, normalized size = 5.33 \begin {gather*} x-\frac {\mathrm {atan}\left (\frac {\frac {\left (18\,\ln \left (2\right )+6\,{\ln \left (2\right )}^2\right )\,\left (\ln \left (64\right )-{\ln \left (2\right )}^2\right )}{6\,\ln \left (2\right )\,\sqrt {\ln \left (64\right )-6\,\ln \left (2\right )}}-\frac {x\,\left (\ln \left (64\right )-{\ln \left (2\right )}^2\right )\,\left (2\,\ln \left (64\right )+2\,{\ln \left (2\right )}^2+18\right )}{6\,\ln \left (2\right )\,\sqrt {\ln \left (64\right )-6\,\ln \left (2\right )}}}{\ln \left (64\right )-{\ln \left (2\right )}^2}\right )\,\left (\ln \left (64\right )-{\ln \left (2\right )}^2\right )}{3\,\ln \left (2\right )\,\sqrt {\ln \left (64\right )-6\,\ln \left (2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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