Optimal. Leaf size=26 \[ x+\log \left (4+x^2 \left (-4 \log (x)+\log \left (\frac {x}{\left (-2 x+x^2\right )^2}\right )\right )\right ) \]
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Rubi [F]
time = 0.62, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {8-14 x+7 x^2+\left (-16 x+4 x^3\right ) \log (x)+\left (4 x-x^3\right ) \log \left (\frac {1}{4 x-4 x^2+x^3}\right )}{8-4 x+\left (-8 x^2+4 x^3\right ) \log (x)+\left (2 x^2-x^3\right ) \log \left (\frac {1}{4 x-4 x^2+x^3}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8-14 x+7 x^2-x \left (-4+x^2\right ) \log \left (\frac {1}{(-2+x)^2 x}\right )+4 x \left (-4+x^2\right ) \log (x)}{(2-x) \left (4+x^2 \log \left (\frac {1}{(-2+x)^2 x}\right )-4 x^2 \log (x)\right )} \, dx\\ &=\int \left (\frac {2+x}{x}+\frac {16-8 x+10 x^2-7 x^3}{(-2+x) x \left (4+x^2 \log \left (\frac {1}{(-2+x)^2 x}\right )-4 x^2 \log (x)\right )}\right ) \, dx\\ &=\int \frac {2+x}{x} \, dx+\int \frac {16-8 x+10 x^2-7 x^3}{(-2+x) x \left (4+x^2 \log \left (\frac {1}{(-2+x)^2 x}\right )-4 x^2 \log (x)\right )} \, dx\\ &=\int \left (1+\frac {2}{x}\right ) \, dx+\int \left (-\frac {4}{4+x^2 \log \left (\frac {1}{(-2+x)^2 x}\right )-4 x^2 \log (x)}-\frac {8}{(-2+x) \left (4+x^2 \log \left (\frac {1}{(-2+x)^2 x}\right )-4 x^2 \log (x)\right )}-\frac {8}{x \left (4+x^2 \log \left (\frac {1}{(-2+x)^2 x}\right )-4 x^2 \log (x)\right )}-\frac {7 x}{4+x^2 \log \left (\frac {1}{(-2+x)^2 x}\right )-4 x^2 \log (x)}\right ) \, dx\\ &=x+2 \log (x)-4 \int \frac {1}{4+x^2 \log \left (\frac {1}{(-2+x)^2 x}\right )-4 x^2 \log (x)} \, dx-7 \int \frac {x}{4+x^2 \log \left (\frac {1}{(-2+x)^2 x}\right )-4 x^2 \log (x)} \, dx-8 \int \frac {1}{(-2+x) \left (4+x^2 \log \left (\frac {1}{(-2+x)^2 x}\right )-4 x^2 \log (x)\right )} \, dx-8 \int \frac {1}{x \left (4+x^2 \log \left (\frac {1}{(-2+x)^2 x}\right )-4 x^2 \log (x)\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.10, size = 26, normalized size = 1.00 \begin {gather*} x+\log \left (4+x^2 \log \left (\frac {1}{(-2+x)^2 x}\right )-4 x^2 \log (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 10.18, size = 211, normalized size = 8.12
method | result | size |
risch | \(2 \ln \left (x \right )+x +\ln \left (\ln \left (x -2\right )-\frac {i \left (x^{2} \pi \mathrm {csgn}\left (i \left (x -2\right )^{2}\right )^{3}-2 x^{2} \pi \,\mathrm {csgn}\left (i \left (x -2\right )\right ) \mathrm {csgn}\left (i \left (x -2\right )^{2}\right )^{2}+x^{2} \pi \mathrm {csgn}\left (i \left (x -2\right )\right )^{2} \mathrm {csgn}\left (i \left (x -2\right )^{2}\right )+x^{2} \pi \,\mathrm {csgn}\left (\frac {i}{\left (x -2\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i}{x \left (x -2\right )^{2}}\right )^{2}-x^{2} \pi \,\mathrm {csgn}\left (\frac {i}{\left (x -2\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i}{x \left (x -2\right )^{2}}\right )-x^{2} \pi \mathrm {csgn}\left (\frac {i}{x \left (x -2\right )^{2}}\right )^{3}+x^{2} \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i}{x \left (x -2\right )^{2}}\right )^{2}+10 i x^{2} \ln \left (x \right )-8 i\right )}{4 x^{2}}\right )\) | \(201\) |
default | \(2 \ln \left (x \right )+x +\ln \left (\ln \left (x \right )+\frac {i \left (x^{2} \pi \,\mathrm {csgn}\left (\frac {i}{\left (x -2\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i}{x \left (x -2\right )^{2}}\right )-x^{2} \pi \,\mathrm {csgn}\left (\frac {i}{\left (x -2\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i}{x \left (x -2\right )^{2}}\right )^{2}-x^{2} \pi \mathrm {csgn}\left (i \left (x -2\right )\right )^{2} \mathrm {csgn}\left (i \left (x -2\right )^{2}\right )+2 x^{2} \pi \,\mathrm {csgn}\left (i \left (x -2\right )\right ) \mathrm {csgn}\left (i \left (x -2\right )^{2}\right )^{2}-x^{2} \pi \mathrm {csgn}\left (i \left (x -2\right )^{2}\right )^{3}-x^{2} \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i}{x \left (x -2\right )^{2}}\right )^{2}+x^{2} \pi \mathrm {csgn}\left (\frac {i}{x \left (x -2\right )^{2}}\right )^{3}-8 i x^{2} \ln \left (x \right )-4 i x^{2} \ln \left (x -2\right )+8 i\right )}{2 x^{2}}\right )\) | \(211\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.41, size = 30, normalized size = 1.15 \begin {gather*} x + 2 \, \log \left (x\right ) + \log \left (\frac {2 \, x^{2} \log \left (x - 2\right ) + 5 \, x^{2} \log \left (x\right ) - 4}{2 \, x^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 40, normalized size = 1.54 \begin {gather*} x + 2 \, \log \left (x\right ) + \log \left (\frac {4 \, x^{2} \log \left (x\right ) - x^{2} \log \left (\frac {1}{x^{3} - 4 \, x^{2} + 4 \, x}\right ) - 4}{x^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.19, size = 36, normalized size = 1.38 \begin {gather*} x + 2 \log {\left (x \right )} + \log {\left (\log {\left (\frac {1}{x^{3} - 4 x^{2} + 4 x} \right )} + \frac {- 4 x^{2} \log {\left (x \right )} + 4}{x^{2}} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 25, normalized size = 0.96 \begin {gather*} x + \log \left (x^{2} \log \left (x^{2} - 4 \, x + 4\right ) + 5 \, x^{2} \log \left (x\right ) - 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.21, size = 34, normalized size = 1.31 \begin {gather*} x+\ln \left (\frac {1}{x^2}\right )+2\,\ln \left (x\right )+\ln \left (x^2\,\ln \left (\frac {1}{x\,{\left (x-2\right )}^2}\right )-4\,x^2\,\ln \left (x\right )+4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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