Optimal. Leaf size=30 \[ \frac {1}{\log \left (\frac {1-4 x}{5 \left (1-x+\frac {5 (-x+\log (x))}{x}\right )}\right )} \]
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Rubi [F]
time = 3.57, 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 {-5+20 x+17 x^2+(5-40 x) \log (x)}{\left (4 x^2-15 x^3-4 x^4+\left (-5 x+20 x^2\right ) \log (x)\right ) \log ^2\left (\frac {x-4 x^2}{-20 x-5 x^2+25 \log (x)}\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 {-5+20 x+17 x^2+(5-40 x) \log (x)}{(1-4 x) x \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (-\frac {(1-4 x) x}{5 \left (4 x+x^2-5 \log (x)\right )}\right )} \, dx\\ &=\int \left (\frac {-5+20 x+17 x^2+5 \log (x)-40 x \log (x)}{x \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )}-\frac {4 \left (-5+20 x+17 x^2+5 \log (x)-40 x \log (x)\right )}{(-1+4 x) \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )}\right ) \, dx\\ &=-\left (4 \int \frac {-5+20 x+17 x^2+5 \log (x)-40 x \log (x)}{(-1+4 x) \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )} \, dx\right )+\int \frac {-5+20 x+17 x^2+5 \log (x)-40 x \log (x)}{x \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )} \, dx\\ &=-\left (4 \int \left (-\frac {5}{(-1+4 x) \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )}+\frac {20 x}{(-1+4 x) \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )}+\frac {17 x^2}{(-1+4 x) \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )}+\frac {5 \log (x)}{(-1+4 x) \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )}-\frac {40 x \log (x)}{(-1+4 x) \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )}\right ) \, dx\right )+\int \left (\frac {20}{\left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )}-\frac {5}{x \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )}+\frac {17 x}{\left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )}-\frac {40 \log (x)}{\left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )}+\frac {5 \log (x)}{x \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )}\right ) \, dx\\ &=-\left (5 \int \frac {1}{x \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )} \, dx\right )+5 \int \frac {\log (x)}{x \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )} \, dx+17 \int \frac {x}{\left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )} \, dx+20 \int \frac {1}{\left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )} \, dx+20 \int \frac {1}{(-1+4 x) \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )} \, dx-20 \int \frac {\log (x)}{(-1+4 x) \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )} \, dx-40 \int \frac {\log (x)}{\left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )} \, dx-68 \int \frac {x^2}{(-1+4 x) \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )} \, dx-80 \int \frac {x}{(-1+4 x) \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )} \, dx+160 \int \frac {x \log (x)}{(-1+4 x) \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )} \, dx\\ &=-\left (5 \int \frac {1}{x \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )} \, dx\right )+5 \int \frac {\log (x)}{x \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )} \, dx+17 \int \frac {x}{\left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )} \, dx+20 \int \frac {1}{\left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )} \, dx+20 \int \frac {1}{(-1+4 x) \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )} \, dx-20 \int \frac {\log (x)}{(-1+4 x) \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )} \, dx-40 \int \frac {\log (x)}{\left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )} \, dx-68 \int \left (\frac {1}{16 \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )}+\frac {x}{4 \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )}+\frac {1}{16 (-1+4 x) \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )}\right ) \, dx-80 \int \left (\frac {1}{4 \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )}+\frac {1}{4 (-1+4 x) \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )}\right ) \, dx+160 \int \left (\frac {\log (x)}{4 \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )}+\frac {\log (x)}{4 (-1+4 x) \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )}\right ) \, dx\\ &=-\left (\frac {17}{4} \int \frac {1}{\left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )} \, dx\right )-\frac {17}{4} \int \frac {1}{(-1+4 x) \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )} \, dx-5 \int \frac {1}{x \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )} \, dx+5 \int \frac {\log (x)}{x \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )} \, dx-20 \int \frac {\log (x)}{(-1+4 x) \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )} \, dx+40 \int \frac {\log (x)}{(-1+4 x) \left (4 x+x^2-5 \log (x)\right ) \log ^2\left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \log (x))}\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.09, size = 25, normalized size = 0.83 \begin {gather*} \frac {1}{\log \left (\frac {x (-1+4 x)}{5 (x (4+x)-5 \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 = 8.66, size = 368, normalized size = 12.27
method | result | size |
risch | \(\frac {2 i}{\pi \,\mathrm {csgn}\left (i \left (x -\frac {1}{4}\right )\right ) \mathrm {csgn}\left (\frac {i}{-x^{2}+5 \ln \left (x \right )-4 x}\right ) \mathrm {csgn}\left (\frac {i \left (x -\frac {1}{4}\right )}{-x^{2}+5 \ln \left (x \right )-4 x}\right )-\pi \,\mathrm {csgn}\left (i \left (x -\frac {1}{4}\right )\right ) \mathrm {csgn}\left (\frac {i \left (x -\frac {1}{4}\right )}{-x^{2}+5 \ln \left (x \right )-4 x}\right )^{2}+\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i \left (x -\frac {1}{4}\right )}{-x^{2}+5 \ln \left (x \right )-4 x}\right ) \mathrm {csgn}\left (\frac {i x \left (x -\frac {1}{4}\right )}{-x^{2}+5 \ln \left (x \right )-4 x}\right )-\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x \left (x -\frac {1}{4}\right )}{-x^{2}+5 \ln \left (x \right )-4 x}\right )^{2}+\pi \,\mathrm {csgn}\left (\frac {i}{-x^{2}+5 \ln \left (x \right )-4 x}\right ) \mathrm {csgn}\left (\frac {i \left (x -\frac {1}{4}\right )}{-x^{2}+5 \ln \left (x \right )-4 x}\right )^{2}-\pi \mathrm {csgn}\left (\frac {i \left (x -\frac {1}{4}\right )}{-x^{2}+5 \ln \left (x \right )-4 x}\right )^{3}+\pi \,\mathrm {csgn}\left (\frac {i \left (x -\frac {1}{4}\right )}{-x^{2}+5 \ln \left (x \right )-4 x}\right ) \mathrm {csgn}\left (\frac {i x \left (x -\frac {1}{4}\right )}{-x^{2}+5 \ln \left (x \right )-4 x}\right )^{2}-\pi \mathrm {csgn}\left (\frac {i x \left (x -\frac {1}{4}\right )}{-x^{2}+5 \ln \left (x \right )-4 x}\right )^{3}-2 i \ln \left (x^{2}-5 \ln \left (x \right )+4 x \right )-2 i \ln \left (5\right )+2 i \ln \left (x \right )+4 i \ln \left (2\right )+2 i \ln \left (x -\frac {1}{4}\right )}\) | \(362\) |
default | \(\frac {2 i}{2 \pi \mathrm {csgn}\left (\frac {i x \left (x -\frac {1}{4}\right )}{-\frac {x^{2}}{5}+\ln \left (x \right )-\frac {4 x}{5}}\right )^{2}+\pi \,\mathrm {csgn}\left (i \left (x -\frac {1}{4}\right )\right ) \mathrm {csgn}\left (\frac {i}{-\frac {x^{2}}{5}+\ln \left (x \right )-\frac {4 x}{5}}\right ) \mathrm {csgn}\left (\frac {i \left (x -\frac {1}{4}\right )}{-\frac {x^{2}}{5}+\ln \left (x \right )-\frac {4 x}{5}}\right )-\pi \,\mathrm {csgn}\left (i \left (x -\frac {1}{4}\right )\right ) \mathrm {csgn}\left (\frac {i \left (x -\frac {1}{4}\right )}{-\frac {x^{2}}{5}+\ln \left (x \right )-\frac {4 x}{5}}\right )^{2}-\pi \,\mathrm {csgn}\left (\frac {i}{-\frac {x^{2}}{5}+\ln \left (x \right )-\frac {4 x}{5}}\right ) \mathrm {csgn}\left (\frac {i \left (x -\frac {1}{4}\right )}{-\frac {x^{2}}{5}+\ln \left (x \right )-\frac {4 x}{5}}\right )^{2}+\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i \left (x -\frac {1}{4}\right )}{-\frac {x^{2}}{5}+\ln \left (x \right )-\frac {4 x}{5}}\right ) \mathrm {csgn}\left (\frac {i x \left (x -\frac {1}{4}\right )}{-\frac {x^{2}}{5}+\ln \left (x \right )-\frac {4 x}{5}}\right )-\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x \left (x -\frac {1}{4}\right )}{-\frac {x^{2}}{5}+\ln \left (x \right )-\frac {4 x}{5}}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i \left (x -\frac {1}{4}\right )}{-\frac {x^{2}}{5}+\ln \left (x \right )-\frac {4 x}{5}}\right )^{3}-\pi \,\mathrm {csgn}\left (\frac {i \left (x -\frac {1}{4}\right )}{-\frac {x^{2}}{5}+\ln \left (x \right )-\frac {4 x}{5}}\right ) \mathrm {csgn}\left (\frac {i x \left (x -\frac {1}{4}\right )}{-\frac {x^{2}}{5}+\ln \left (x \right )-\frac {4 x}{5}}\right )^{2}-\pi \mathrm {csgn}\left (\frac {i x \left (x -\frac {1}{4}\right )}{-\frac {x^{2}}{5}+\ln \left (x \right )-\frac {4 x}{5}}\right )^{3}-2 \pi +2 i \ln \left (x \right )-4 i \ln \left (5\right )-2 i \ln \left (-\frac {x^{2}}{5}+\ln \left (x \right )-\frac {4 x}{5}\right )+4 i \ln \left (2\right )+2 i \ln \left (x -\frac {1}{4}\right )}\) | \(368\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.65, size = 31, normalized size = 1.03 \begin {gather*} -\frac {1}{\log \left (5\right ) + \log \left (x^{2} + 4 \, x - 5 \, \log \left (x\right )\right ) - \log \left (4 \, x - 1\right ) - \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 27, normalized size = 0.90 \begin {gather*} \frac {1}{\log \left (\frac {4 \, x^{2} - x}{5 \, {\left (x^{2} + 4 \, x - 5 \, \log \left (x\right )\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.20, size = 22, normalized size = 0.73 \begin {gather*} \frac {1}{\log {\left (\frac {- 4 x^{2} + x}{- 5 x^{2} - 20 x + 25 \log {\left (x \right )}} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 31, normalized size = 1.03 \begin {gather*} -\frac {1}{\log \left (5 \, x^{2} + 20 \, x - 25 \, \log \left (x\right )\right ) - \log \left (4 \, x - 1\right ) - \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.51, size = 27, normalized size = 0.90 \begin {gather*} \frac {1}{\ln \left (-\frac {x-4\,x^2}{20\,x-25\,\ln \left (x\right )+5\,x^2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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