Optimal. Leaf size=24 \[ -x+\frac {1}{3 x^2 \left (\frac {1+x}{2}+\log (\log (5))\right )} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(59\) vs. \(2(24)=48\).
time = 0.11, antiderivative size = 59, normalized size of antiderivative = 2.46, number of steps
used = 4, number of rules used = 2, integrand size = 90, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {6, 2099}
\begin {gather*} \frac {2}{3 x^2 (1+2 \log (\log (5)))}-x+\frac {2}{3 (1+2 \log (\log (5)))^2 (x+1+2 \log (\log (5)))}-\frac {2}{3 x (1+2 \log (\log (5)))^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 2099
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4-6 x-6 x^4-3 x^5+\left (-8-12 x^3-12 x^4\right ) \log (\log (5))+x^3 \left (-3-12 \log ^2(\log (5))\right )}{3 x^3+6 x^4+3 x^5+\left (12 x^3+12 x^4\right ) \log (\log (5))+12 x^3 \log ^2(\log (5))} \, dx\\ &=\int \frac {-4-6 x-6 x^4-3 x^5+\left (-8-12 x^3-12 x^4\right ) \log (\log (5))+x^3 \left (-3-12 \log ^2(\log (5))\right )}{6 x^4+3 x^5+\left (12 x^3+12 x^4\right ) \log (\log (5))+x^3 \left (3+12 \log ^2(\log (5))\right )} \, dx\\ &=\int \left (-1+\frac {2}{3 x^2 (1+2 \log (\log (5)))^2}-\frac {4}{3 x^3 (1+2 \log (\log (5)))}-\frac {2}{3 (1+2 \log (\log (5)))^2 (1+x+2 \log (\log (5)))^2}\right ) \, dx\\ &=-x-\frac {2}{3 x (1+2 \log (\log (5)))^2}+\frac {2}{3 x^2 (1+2 \log (\log (5)))}+\frac {2}{3 (1+2 \log (\log (5)))^2 (1+x+2 \log (\log (5)))}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 35, normalized size = 1.46 \begin {gather*} -\frac {-2+3 x^4+x^3 (3+6 \log (\log (5)))}{3 x^2 (1+x+2 \log (\log (5)))} \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. \(53\) vs.
\(2(19)=38\).
time = 0.18, size = 54, normalized size = 2.25
method | result | size |
risch | \(-x +\frac {2}{3 x^{2} \left (1+2 \ln \left (\ln \left (5\right )\right )+x \right )}\) | \(20\) |
gosper | \(\frac {12 x^{2} \ln \left (\ln \left (5\right )\right )^{2}-3 x^{4}+12 \ln \left (\ln \left (5\right )\right ) x^{2}+3 x^{2}+2}{3 x^{2} \left (1+2 \ln \left (\ln \left (5\right )\right )+x \right )}\) | \(46\) |
default | \(-x +\frac {2}{3 \left (1+2 \ln \left (\ln \left (5\right )\right )\right )^{2} \left (1+2 \ln \left (\ln \left (5\right )\right )+x \right )}-\frac {2}{3 \left (1+2 \ln \left (\ln \left (5\right )\right )\right )^{2} x}+\frac {2}{3 \left (1+2 \ln \left (\ln \left (5\right )\right )\right ) x^{2}}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 23, normalized size = 0.96 \begin {gather*} -x + \frac {2}{3 \, {\left (x^{3} + x^{2} {\left (2 \, \log \left (\log \left (5\right )\right ) + 1\right )}\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. 39 vs.
\(2 (19) = 38\).
time = 0.43, size = 39, normalized size = 1.62 \begin {gather*} -\frac {3 \, x^{4} + 6 \, x^{3} \log \left (\log \left (5\right )\right ) + 3 \, x^{3} - 2}{3 \, {\left (x^{3} + 2 \, x^{2} \log \left (\log \left (5\right )\right ) + x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.24, size = 19, normalized size = 0.79 \begin {gather*} - x + \frac {2}{3 x^{3} + x^{2} \cdot \left (6 \log {\left (\log {\left (5 \right )} \right )} + 3\right )} \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. 61 vs.
\(2 (19) = 38\).
time = 0.37, size = 61, normalized size = 2.54 \begin {gather*} -x + \frac {2}{3 \, {\left (4 \, \log \left (\log \left (5\right )\right )^{2} + 4 \, \log \left (\log \left (5\right )\right ) + 1\right )} {\left (x + 2 \, \log \left (\log \left (5\right )\right ) + 1\right )}} - \frac {2 \, {\left (x - 2 \, \log \left (\log \left (5\right )\right ) - 1\right )}}{3 \, {\left (4 \, \log \left (\log \left (5\right )\right )^{2} + 4 \, \log \left (\log \left (5\right )\right ) + 1\right )} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.13, size = 25, normalized size = 1.04 \begin {gather*} \frac {2}{3\,x^3+\left (6\,\ln \left (\ln \left (5\right )\right )+3\right )\,x^2}-x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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