Optimal. Leaf size=21 \[ \frac {5}{(5+2 x+\log (5))^2-\log ^2(\log (4))} \]
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Rubi [A]
time = 0.07, antiderivative size = 28, normalized size of antiderivative = 1.33, number of steps
used = 3, number of rules used = 3, integrand size = 114, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {1694, 12, 267}
\begin {gather*} \frac {5}{4 \left (x+\frac {1}{64} (160+32 \log (5))\right )^2-\log ^2(\log (4))} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 267
Rule 1694
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\text {Subst}\left (\int -\frac {40 x}{\left (4 x^2-\log ^2(\log (4))\right )^2} \, dx,x,x+\frac {1}{64} (160+32 \log (5))\right )\\ &=-\left (40 \text {Subst}\left (\int \frac {x}{\left (4 x^2-\log ^2(\log (4))\right )^2} \, dx,x,x+\frac {1}{64} (160+32 \log (5))\right )\right )\\ &=\frac {5}{(5+2 x+\log (5))^2-\log ^2(\log (4))}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 34, normalized size = 1.62 \begin {gather*} \frac {5}{25+20 x+4 x^2+10 \log (5)+4 x \log (5)+\log ^2(5)-\log ^2(\log (4))} \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. \(49\) vs.
\(2(23)=46\).
time = 0.34, size = 50, normalized size = 2.38
method | result | size |
gosper | \(-\frac {5}{\ln \left (2 \ln \left (2\right )\right )^{2}-\ln \left (5\right )^{2}-4 x \ln \left (5\right )-4 x^{2}-10 \ln \left (5\right )-20 x -25}\) | \(37\) |
norman | \(-\frac {5}{\ln \left (2 \ln \left (2\right )\right )^{2}-\ln \left (5\right )^{2}-4 x \ln \left (5\right )-4 x^{2}-10 \ln \left (5\right )-20 x -25}\) | \(37\) |
risch | \(\frac {5}{\ln \left (5\right )^{2}+4 x \ln \left (5\right )-\ln \left (\ln \left (2\right )\right )^{2}-2 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right )-\ln \left (2\right )^{2}+4 x^{2}+10 \ln \left (5\right )+20 x +25}\) | \(48\) |
default | \(-\frac {5}{2 \ln \left (2 \ln \left (2\right )\right ) \left (2 x +5+\ln \left (2 \ln \left (2\right )\right )+\ln \left (5\right )\right )}+\frac {5}{2 \ln \left (2 \ln \left (2\right )\right ) \left (2 x +5-\ln \left (2 \ln \left (2\right )\right )+\ln \left (5\right )\right )}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 35, normalized size = 1.67 \begin {gather*} \frac {5}{4 \, x^{2} + 4 \, x {\left (\log \left (5\right ) + 5\right )} + \log \left (5\right )^{2} - \log \left (2 \, \log \left (2\right )\right )^{2} + 10 \, \log \left (5\right ) + 25} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 36, normalized size = 1.71 \begin {gather*} \frac {5}{4 \, x^{2} + 2 \, {\left (2 \, x + 5\right )} \log \left (5\right ) + \log \left (5\right )^{2} - \log \left (2 \, \log \left (2\right )\right )^{2} + 20 \, x + 25} \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. 48 vs.
\(2 (19) = 38\).
time = 0.75, size = 48, normalized size = 2.29 \begin {gather*} \frac {5}{4 x^{2} + x \left (4 \log {\left (5 \right )} + 20\right ) - \log {\left (2 \right )}^{2} - \log {\left (\log {\left (2 \right )} \right )}^{2} - 2 \log {\left (2 \right )} \log {\left (\log {\left (2 \right )} \right )} + \log {\left (5 \right )}^{2} + 10 \log {\left (5 \right )} + 25} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 36, normalized size = 1.71 \begin {gather*} \frac {5}{4 \, x^{2} + 4 \, x \log \left (5\right ) + \log \left (5\right )^{2} - \log \left (2 \, \log \left (2\right )\right )^{2} + 20 \, x + 10 \, \log \left (5\right ) + 25} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.32, size = 34, normalized size = 1.62 \begin {gather*} \frac {5}{4\,x^2+\left (4\,\ln \left (5\right )+20\right )\,x+10\,\ln \left (5\right )-{\ln \left (\ln \left (4\right )\right )}^2+{\ln \left (5\right )}^2+25} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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