Optimal. Leaf size=22 \[ \log \left (\left (x+\frac {2 x}{3 \left (3+x^2\right )}\right ) \log (22-x)\right ) \]
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Rubi [A]
time = 0.75, antiderivative size = 26, normalized size of antiderivative = 1.18, number of steps
used = 9, number of rules used = 7, integrand size = 86, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {6873, 6874,
1677, 1642, 2437, 2339, 29} \begin {gather*} -\log \left (x^2+3\right )+\log \left (3 x^2+11\right )+\log (x)+\log (\log (22-x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 1642
Rule 1677
Rule 2339
Rule 2437
Rule 6873
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-33 x-20 x^3-3 x^5-\left (-726+33 x-352 x^2+16 x^3-66 x^4+3 x^5\right ) \log (22-x)}{x \left (726-33 x+440 x^2-20 x^3+66 x^4-3 x^5\right ) \log (22-x)} \, dx\\ &=\int \left (\frac {33+16 x^2+3 x^4}{x \left (33+20 x^2+3 x^4\right )}+\frac {1}{(-22+x) \log (22-x)}\right ) \, dx\\ &=\int \frac {33+16 x^2+3 x^4}{x \left (33+20 x^2+3 x^4\right )} \, dx+\int \frac {1}{(-22+x) \log (22-x)} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {33+16 x+3 x^2}{x \left (33+20 x+3 x^2\right )} \, dx,x,x^2\right )+\text {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,22-x\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{x}-\frac {2}{3+x}+\frac {6}{11+3 x}\right ) \, dx,x,x^2\right )+\text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (22-x)\right )\\ &=\log (x)-\log \left (3+x^2\right )+\log \left (11+3 x^2\right )+\log (\log (22-x))\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.04, size = 26, normalized size = 1.18 \begin {gather*} \log (x)-\log \left (3+x^2\right )+\log \left (11+3 x^2\right )+\log (\log (22-x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.30, size = 43, normalized size = 1.95
method | result | size |
norman | \(-\ln \left (x^{2}+3\right )+\ln \left (x \right )+\ln \left (\ln \left (22-x \right )\right )+\ln \left (3 x^{2}+11\right )\) | \(27\) |
risch | \(-\ln \left (x^{2}+3\right )+\ln \left (3 x^{3}+11 x \right )+\ln \left (\ln \left (22-x \right )\right )\) | \(27\) |
derivativedivides | \(\ln \left (\ln \left (22-x \right )\right )+\ln \left (3 \left (22-x \right )^{2}-1441+132 x \right )+\ln \left (-x \right )-\ln \left (\left (22-x \right )^{2}-481+44 x \right )\) | \(43\) |
default | \(\ln \left (\ln \left (22-x \right )\right )+\ln \left (3 \left (22-x \right )^{2}-1441+132 x \right )+\ln \left (-x \right )-\ln \left (\left (22-x \right )^{2}-481+44 x \right )\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.56, size = 26, normalized size = 1.18 \begin {gather*} \log \left (3 \, x^{2} + 11\right ) - \log \left (x^{2} + 3\right ) + \log \left (x\right ) + \log \left (\log \left (-x + 22\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 26, normalized size = 1.18 \begin {gather*} \log \left (3 \, x^{3} + 11 \, x\right ) - \log \left (x^{2} + 3\right ) + \log \left (\log \left (-x + 22\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 22, normalized size = 1.00 \begin {gather*} - \log {\left (x^{2} + 3 \right )} + \log {\left (3 x^{3} + 11 x \right )} + \log {\left (\log {\left (22 - x \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 41, normalized size = 1.86 \begin {gather*} \log \left (-3 \, {\left (x - 22\right )}^{3} - 198 \, {\left (x - 22\right )}^{2} - 4367 \, x + 63888\right ) - \log \left ({\left (x - 22\right )}^{2} + 44 \, x - 481\right ) + \log \left (\log \left (-x + 22\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.47, size = 24, normalized size = 1.09 \begin {gather*} \ln \left (\ln \left (22-x\right )\right )+\ln \left (x^3+\frac {11\,x}{3}\right )-\ln \left (x^2+3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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