Optimal. Leaf size=22 \[ \log ^2\left (\frac {x}{5-\frac {2+\log (-4+x)}{9 x}}\right ) \]
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Rubi [A]
time = 0.61, antiderivative size = 19, normalized size of antiderivative = 0.86, number of steps
used = 1, number of rules used = 4, integrand size = 66, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {6873, 6874,
6816, 6818} \begin {gather*} \log ^2\left (-\frac {9 x^2}{-45 x+\log (x-4)+2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 6816
Rule 6818
Rule 6873
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\log ^2\left (-\frac {9 x^2}{2-45 x+\log (-4+x)}\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 19, normalized size = 0.86 \begin {gather*} \log ^2\left (-\frac {9 x^2}{2-45 x+\log (-4+x)}\right ) \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. \(41\) vs.
\(2(20)=40\).
time = 2.15, size = 42, normalized size = 1.91
method | result | size |
default | \(8 \ln \left (3\right ) \ln \left (x \right )-4 \ln \left (3\right ) \ln \left (\ln \left (x -4\right )-45 x +2\right )+\ln \left (\frac {x^{2}}{-\ln \left (x -4\right )+45 x -2}\right )^{2}\) | \(42\) |
risch | \(-i \pi \ln \left (\ln \left (x -4\right )-45 x +2\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\frac {\ln \left (x -4\right )}{45}-x +\frac {2}{45}}\right )^{3}-4 \ln \left (5\right ) \ln \left (x \right )+4 \ln \left (x \right )^{2}+i \pi \ln \left (\ln \left (x -4\right )-45 x +2\right ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (\frac {i}{\frac {\ln \left (x -4\right )}{45}-x +\frac {2}{45}}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\frac {\ln \left (x -4\right )}{45}-x +\frac {2}{45}}\right )-2 i \pi \ln \left (\ln \left (x -4\right )-45 x +2\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+i \pi \ln \left (\ln \left (x -4\right )-45 x +2\right ) \mathrm {csgn}\left (i x^{2}\right )^{3}+2 i \pi \ln \left (x \right ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\frac {\ln \left (x -4\right )}{45}-x +\frac {2}{45}}\right )^{2}+4 i \pi \ln \left (x \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+i \pi \ln \left (\ln \left (x -4\right )-45 x +2\right ) \mathrm {csgn}\left (\frac {i}{\frac {\ln \left (x -4\right )}{45}-x +\frac {2}{45}}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\frac {\ln \left (x -4\right )}{45}-x +\frac {2}{45}}\right )^{2}+2 i \pi \ln \left (x \right ) \mathrm {csgn}\left (\frac {i x^{2}}{\frac {\ln \left (x -4\right )}{45}-x +\frac {2}{45}}\right )^{3}-2 i \pi \ln \left (x \right ) \mathrm {csgn}\left (i x^{2}\right )^{3}+\ln \left (-\frac {\ln \left (x -4\right )}{45}+x -\frac {2}{45}\right )^{2}+i \pi \ln \left (\ln \left (x -4\right )-45 x +2\right ) \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 i \pi \ln \left (x \right ) \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+2 \ln \left (5\right ) \ln \left (\ln \left (x -4\right )-45 x +2\right )-4 \ln \left (x \right ) \ln \left (-\frac {\ln \left (x -4\right )}{45}+x -\frac {2}{45}\right )-i \pi \ln \left (\ln \left (x -4\right )-45 x +2\right ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\frac {\ln \left (x -4\right )}{45}-x +\frac {2}{45}}\right )^{2}-2 i \pi \ln \left (x \right ) \mathrm {csgn}\left (\frac {i}{\frac {\ln \left (x -4\right )}{45}-x +\frac {2}{45}}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\frac {\ln \left (x -4\right )}{45}-x +\frac {2}{45}}\right )^{2}-2 i \pi \ln \left (x \right ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (\frac {i}{\frac {\ln \left (x -4\right )}{45}-x +\frac {2}{45}}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\frac {\ln \left (x -4\right )}{45}-x +\frac {2}{45}}\right )\) | \(536\) |
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. 73 vs.
\(2 (20) = 40\).
time = 0.32, size = 73, normalized size = 3.32 \begin {gather*} -4 \, \log \left (x\right )^{2} + 4 \, \log \left (x\right ) \log \left (-45 \, x + \log \left (x - 4\right ) + 2\right ) - \log \left (-45 \, x + \log \left (x - 4\right ) + 2\right )^{2} + 2 \, {\left (2 \, \log \left (x\right ) - \log \left (-45 \, x + \log \left (x - 4\right ) + 2\right )\right )} \log \left (\frac {9 \, x^{2}}{45 \, x - \log \left (x - 4\right ) - 2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 21, normalized size = 0.95 \begin {gather*} \log \left (\frac {9 \, x^{2}}{45 \, x - \log \left (x - 4\right ) - 2}\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.18, size = 19, normalized size = 0.86 \begin {gather*} \log {\left (- \frac {9 x^{2}}{- 45 x + \log {\left (x - 4 \right )} + 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. 72 vs.
\(2 (20) = 40\).
time = 0.43, size = 72, normalized size = 3.27 \begin {gather*} -2 \, {\left (2 \, \log \left (x\right ) - \log \left (-45 \, x + \log \left (x - 4\right ) + 2\right )\right )} \log \left (45 \, x - \log \left (x - 4\right ) - 2\right ) + 8 \, \log \left (3\right ) \log \left (x\right ) + 4 \, \log \left (x\right )^{2} - 4 \, \log \left (3\right ) \log \left (-45 \, x + \log \left (x - 4\right ) + 2\right ) - \log \left (-45 \, x + \log \left (x - 4\right ) + 2\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.21, size = 19, normalized size = 0.86 \begin {gather*} {\ln \left (-\frac {9\,x^2}{\ln \left (x-4\right )-45\,x+2}\right )}^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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