Optimal. Leaf size=33 \[ \frac {\left (x-x^2+\log \left (\frac {15}{x}\right )\right ) \log \left (\frac {1+x}{4}\right )}{16 x (5+x)} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(132\) vs. \(2(33)=66\).
time = 2.18, antiderivative size = 132, normalized size of antiderivative = 4.00, number of steps
used = 53, number of rules used = 24, integrand size = 95, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.253, Rules used = {6873, 12,
6874, 46, 78, 90, 2404, 2338, 2354, 2438, 907, 2341, 2351, 31, 2465, 2442, 36, 29, 2439, 2441, 2440,
75, 2637, 2463} \begin {gather*} \frac {\log \left (\frac {15}{x}\right ) \log \left (\frac {x}{4}+\frac {1}{4}\right )}{16 x (x+5)}-\frac {1}{400} \log \left (\frac {x+5}{4}\right ) \log \left (\frac {x}{4}+\frac {1}{4}\right )+\frac {\log \left (\frac {x}{4}+\frac {1}{4}\right )}{80 x}+\frac {3 \log \left (\frac {x}{4}+\frac {1}{4}\right )}{8 (x+5)}+\frac {\log (4)-\log (x+1)}{80 x}-\frac {1}{16} \log (x+1)-\frac {1}{400} (\log (4)-\log (x+1)) \log \left (\frac {x+5}{4}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 29
Rule 31
Rule 36
Rule 46
Rule 75
Rule 78
Rule 90
Rule 907
Rule 2338
Rule 2341
Rule 2351
Rule 2354
Rule 2404
Rule 2438
Rule 2439
Rule 2440
Rule 2441
Rule 2442
Rule 2463
Rule 2465
Rule 2637
Rule 6873
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 x^2-4 x^3-x^4+\left (5 x+x^2\right ) \log \left (\frac {15}{x}\right )+\left (-5-6 x-7 x^2-6 x^3+\left (-5-7 x-2 x^2\right ) \log \left (\frac {15}{x}\right )\right ) \log \left (\frac {1+x}{4}\right )}{16 x^2 (1+x) (5+x)^2} \, dx\\ &=\frac {1}{16} \int \frac {5 x^2-4 x^3-x^4+\left (5 x+x^2\right ) \log \left (\frac {15}{x}\right )+\left (-5-6 x-7 x^2-6 x^3+\left (-5-7 x-2 x^2\right ) \log \left (\frac {15}{x}\right )\right ) \log \left (\frac {1+x}{4}\right )}{x^2 (1+x) (5+x)^2} \, dx\\ &=\frac {1}{16} \int \left (\frac {5}{(1+x) (5+x)^2}-\frac {4 x}{(1+x) (5+x)^2}-\frac {x^2}{(1+x) (5+x)^2}+\frac {\log \left (\frac {15}{x}\right )}{x \left (5+6 x+x^2\right )}+\frac {\log \left (\frac {1}{4}+\frac {x}{4}\right ) \left (-5-x-6 x^2-5 \log \left (\frac {15}{x}\right )-2 x \log \left (\frac {15}{x}\right )\right )}{x^2 (5+x)^2}\right ) \, dx\\ &=-\left (\frac {1}{16} \int \frac {x^2}{(1+x) (5+x)^2} \, dx\right )+\frac {1}{16} \int \frac {\log \left (\frac {15}{x}\right )}{x \left (5+6 x+x^2\right )} \, dx+\frac {1}{16} \int \frac {\log \left (\frac {1}{4}+\frac {x}{4}\right ) \left (-5-x-6 x^2-5 \log \left (\frac {15}{x}\right )-2 x \log \left (\frac {15}{x}\right )\right )}{x^2 (5+x)^2} \, dx-\frac {1}{4} \int \frac {x}{(1+x) (5+x)^2} \, dx+\frac {5}{16} \int \frac {1}{(1+x) (5+x)^2} \, dx\\ &=-\left (\frac {1}{16} \int \left (\frac {1}{16 (1+x)}-\frac {25}{4 (5+x)^2}+\frac {15}{16 (5+x)}\right ) \, dx\right )+\frac {1}{16} \int \left (\frac {\log \left (\frac {15}{x}\right )}{5 x}-\frac {\log \left (\frac {15}{x}\right )}{4 (1+x)}+\frac {\log \left (\frac {15}{x}\right )}{20 (5+x)}\right ) \, dx+\frac {1}{16} \int \left (\frac {\left (-5-x-6 x^2\right ) \log \left (\frac {1}{4}+\frac {x}{4}\right )}{x^2 (5+x)^2}+\frac {(-5-2 x) \log \left (\frac {1}{4}+\frac {x}{4}\right ) \log \left (\frac {15}{x}\right )}{x^2 (5+x)^2}\right ) \, dx-\frac {1}{4} \int \left (-\frac {1}{16 (1+x)}+\frac {5}{4 (5+x)^2}+\frac {1}{16 (5+x)}\right ) \, dx+\frac {5}{16} \int \left (\frac {1}{16 (1+x)}-\frac {1}{4 (5+x)^2}-\frac {1}{16 (5+x)}\right ) \, dx\\ &=\frac {1}{32} \log (1+x)-\frac {3}{32} \log (5+x)+\frac {1}{320} \int \frac {\log \left (\frac {15}{x}\right )}{5+x} \, dx+\frac {1}{80} \int \frac {\log \left (\frac {15}{x}\right )}{x} \, dx-\frac {1}{64} \int \frac {\log \left (\frac {15}{x}\right )}{1+x} \, dx+\frac {1}{16} \int \frac {\left (-5-x-6 x^2\right ) \log \left (\frac {1}{4}+\frac {x}{4}\right )}{x^2 (5+x)^2} \, dx+\frac {1}{16} \int \frac {(-5-2 x) \log \left (\frac {1}{4}+\frac {x}{4}\right ) \log \left (\frac {15}{x}\right )}{x^2 (5+x)^2} \, dx\\ &=\frac {1}{320} \log \left (1+\frac {x}{5}\right ) \log \left (\frac {15}{x}\right )+\frac {\log \left (\frac {1}{4}+\frac {x}{4}\right ) \log \left (\frac {15}{x}\right )}{16 x (5+x)}-\frac {1}{160} \log ^2\left (\frac {15}{x}\right )+\frac {1}{32} \log (1+x)-\frac {1}{64} \log \left (\frac {15}{x}\right ) \log (1+x)-\frac {3}{32} \log (5+x)+\frac {1}{320} \int \frac {\log \left (1+\frac {x}{5}\right )}{x} \, dx-\frac {1}{64} \int \frac {\log (1+x)}{x} \, dx+\frac {1}{16} \int \left (-\frac {\log \left (\frac {1}{4}+\frac {x}{4}\right )}{5 x^2}+\frac {\log \left (\frac {1}{4}+\frac {x}{4}\right )}{25 x}-\frac {6 \log \left (\frac {1}{4}+\frac {x}{4}\right )}{(5+x)^2}-\frac {\log \left (\frac {1}{4}+\frac {x}{4}\right )}{25 (5+x)}\right ) \, dx-\frac {1}{16} \int \frac {\log \left (\frac {15}{x}\right )}{x \left (5+6 x+x^2\right )} \, dx-\frac {1}{16} \int \frac {\log (4)-\log (1+x)}{x^2 (5+x)} \, dx\\ &=\frac {1}{320} \log \left (1+\frac {x}{5}\right ) \log \left (\frac {15}{x}\right )+\frac {\log \left (\frac {1}{4}+\frac {x}{4}\right ) \log \left (\frac {15}{x}\right )}{16 x (5+x)}-\frac {1}{160} \log ^2\left (\frac {15}{x}\right )+\frac {1}{32} \log (1+x)-\frac {1}{64} \log \left (\frac {15}{x}\right ) \log (1+x)-\frac {3}{32} \log (5+x)+\frac {\text {Li}_2(-x)}{64}-\frac {\text {Li}_2\left (-\frac {x}{5}\right )}{320}+\frac {1}{400} \int \frac {\log \left (\frac {1}{4}+\frac {x}{4}\right )}{x} \, dx-\frac {1}{400} \int \frac {\log \left (\frac {1}{4}+\frac {x}{4}\right )}{5+x} \, dx-\frac {1}{80} \int \frac {\log \left (\frac {1}{4}+\frac {x}{4}\right )}{x^2} \, dx-\frac {1}{16} \int \left (\frac {\log \left (\frac {15}{x}\right )}{5 x}-\frac {\log \left (\frac {15}{x}\right )}{4 (1+x)}+\frac {\log \left (\frac {15}{x}\right )}{20 (5+x)}\right ) \, dx-\frac {1}{16} \int \left (\frac {\log (4)-\log (1+x)}{5 x^2}-\frac {\log (4)-\log (1+x)}{25 x}+\frac {\log (4)-\log (1+x)}{25 (5+x)}\right ) \, dx-\frac {3}{8} \int \frac {\log \left (\frac {1}{4}+\frac {x}{4}\right )}{(5+x)^2} \, dx\\ &=\frac {\log \left (\frac {1}{4}+\frac {x}{4}\right )}{80 x}+\frac {3 \log \left (\frac {1}{4}+\frac {x}{4}\right )}{8 (5+x)}+\frac {1}{320} \log \left (1+\frac {x}{5}\right ) \log \left (\frac {15}{x}\right )+\frac {\log \left (\frac {1}{4}+\frac {x}{4}\right ) \log \left (\frac {15}{x}\right )}{16 x (5+x)}-\frac {1}{160} \log ^2\left (\frac {15}{x}\right )-\frac {1}{400} \log (4) \log (x)+\frac {1}{32} \log (1+x)-\frac {1}{64} \log \left (\frac {15}{x}\right ) \log (1+x)-\frac {1}{400} \log \left (\frac {1}{4}+\frac {x}{4}\right ) \log \left (\frac {5+x}{4}\right )-\frac {3}{32} \log (5+x)+\frac {\text {Li}_2(-x)}{64}-\frac {\text {Li}_2\left (-\frac {x}{5}\right )}{320}+\frac {\int \frac {\log \left (\frac {5+x}{4}\right )}{\frac {1}{4}+\frac {x}{4}} \, dx}{1600}+\frac {1}{400} \int \frac {\log (4)-\log (1+x)}{x} \, dx-\frac {1}{400} \int \frac {\log (4)-\log (1+x)}{5+x} \, dx+\frac {1}{400} \int \frac {\log (1+x)}{x} \, dx-\frac {1}{320} \int \frac {1}{\left (\frac {1}{4}+\frac {x}{4}\right ) x} \, dx-\frac {1}{320} \int \frac {\log \left (\frac {15}{x}\right )}{5+x} \, dx-\frac {1}{80} \int \frac {\log \left (\frac {15}{x}\right )}{x} \, dx-\frac {1}{80} \int \frac {\log (4)-\log (1+x)}{x^2} \, dx+\frac {1}{64} \int \frac {\log \left (\frac {15}{x}\right )}{1+x} \, dx-\frac {3}{32} \int \frac {1}{\left (\frac {1}{4}+\frac {x}{4}\right ) (5+x)} \, dx\\ &=\frac {\log \left (\frac {1}{4}+\frac {x}{4}\right )}{80 x}+\frac {3 \log \left (\frac {1}{4}+\frac {x}{4}\right )}{8 (5+x)}+\frac {\log \left (\frac {1}{4}+\frac {x}{4}\right ) \log \left (\frac {15}{x}\right )}{16 x (5+x)}+\frac {\log (4)-\log (1+x)}{80 x}+\frac {1}{32} \log (1+x)-\frac {1}{400} \log \left (\frac {1}{4}+\frac {x}{4}\right ) \log \left (\frac {5+x}{4}\right )-\frac {1}{400} (\log (4)-\log (1+x)) \log \left (\frac {5+x}{4}\right )-\frac {3}{32} \log (5+x)+\frac {21 \text {Li}_2(-x)}{1600}-\frac {\text {Li}_2\left (-\frac {x}{5}\right )}{320}-\frac {1}{400} \int \frac {\log (1+x)}{x} \, dx-\frac {1}{400} \int \frac {\log \left (\frac {5+x}{4}\right )}{1+x} \, dx+\frac {1}{400} \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,\frac {1}{4}+\frac {x}{4}\right )+\frac {1}{320} \int \frac {1}{\frac {1}{4}+\frac {x}{4}} \, dx-\frac {1}{320} \int \frac {\log \left (1+\frac {x}{5}\right )}{x} \, dx-\frac {1}{80} \int \frac {1}{x} \, dx+\frac {1}{80} \int \frac {1}{x (1+x)} \, dx+\frac {1}{64} \int \frac {\log (1+x)}{x} \, dx-\frac {3}{128} \int \frac {1}{\frac {1}{4}+\frac {x}{4}} \, dx+\frac {3}{32} \int \frac {1}{5+x} \, dx\\ &=\frac {\log \left (\frac {1}{4}+\frac {x}{4}\right )}{80 x}+\frac {3 \log \left (\frac {1}{4}+\frac {x}{4}\right )}{8 (5+x)}+\frac {\log \left (\frac {1}{4}+\frac {x}{4}\right ) \log \left (\frac {15}{x}\right )}{16 x (5+x)}-\frac {\log (x)}{80}+\frac {\log (4)-\log (1+x)}{80 x}-\frac {1}{20} \log (1+x)-\frac {1}{400} \log \left (\frac {1}{4}+\frac {x}{4}\right ) \log \left (\frac {5+x}{4}\right )-\frac {1}{400} (\log (4)-\log (1+x)) \log \left (\frac {5+x}{4}\right )-\frac {1}{400} \text {Li}_2\left (\frac {1}{4} (-1-x)\right )-\frac {1}{400} \text {Subst}\left (\int \frac {\log \left (1+\frac {x}{4}\right )}{x} \, dx,x,1+x\right )+\frac {1}{80} \int \frac {1}{x} \, dx-\frac {1}{80} \int \frac {1}{1+x} \, dx\\ &=\frac {\log \left (\frac {1}{4}+\frac {x}{4}\right )}{80 x}+\frac {3 \log \left (\frac {1}{4}+\frac {x}{4}\right )}{8 (5+x)}+\frac {\log \left (\frac {1}{4}+\frac {x}{4}\right ) \log \left (\frac {15}{x}\right )}{16 x (5+x)}+\frac {\log (4)-\log (1+x)}{80 x}-\frac {1}{16} \log (1+x)-\frac {1}{400} \log \left (\frac {1}{4}+\frac {x}{4}\right ) \log \left (\frac {5+x}{4}\right )-\frac {1}{400} (\log (4)-\log (1+x)) \log \left (\frac {5+x}{4}\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.05, size = 38, normalized size = 1.15 \begin {gather*} \frac {1}{16} \left (\frac {\left (6 x+\log \left (\frac {15}{x}\right )\right ) \log \left (\frac {1+x}{4}\right )}{x (5+x)}-\log (1+x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 39.98, size = 40, normalized size = 1.21
method | result | size |
risch | \(\frac {\left (2 \ln \left (3\right )+2 \ln \left (5\right )+12 x -2 \ln \left (x \right )\right ) \ln \left (\frac {x}{4}+\frac {1}{4}\right )}{32 \left (5+x \right ) x}-\frac {\ln \left (x +1\right )}{16}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.84, size = 54, normalized size = 1.64 \begin {gather*} -\frac {12 \, x \log \left (2\right ) + 2 \, {\left (\log \left (5\right ) + \log \left (3\right )\right )} \log \left (2\right ) + {\left (x^{2} - x - \log \left (5\right ) - \log \left (3\right ) + \log \left (x\right )\right )} \log \left (x + 1\right ) - 2 \, \log \left (2\right ) \log \left (x\right )}{16 \, {\left (x^{2} + 5 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 32, normalized size = 0.97 \begin {gather*} -\frac {{\left (x^{2} - x - \log \left (\frac {15}{x}\right )\right )} \log \left (\frac {1}{4} \, x + \frac {1}{4}\right )}{16 \, {\left (x^{2} + 5 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.34, size = 32, normalized size = 0.97 \begin {gather*} \frac {\left (6 x + \log {\left (\frac {15}{x} \right )}\right ) \log {\left (\frac {x}{4} + \frac {1}{4} \right )}}{16 x^{2} + 80 x} - \frac {\log {\left (16 x + 16 \right )}}{16} \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. 91 vs.
\(2 (31) = 62\).
time = 0.44, size = 91, normalized size = 2.76 \begin {gather*} \frac {1}{80} \, {\left ({\left (\frac {1}{x + 5} - \frac {1}{x}\right )} \log \left (x\right ) - \frac {\log \left (15\right ) - 30}{x + 5} + \frac {\log \left (15\right )}{x}\right )} \log \left (x + 1\right ) - \frac {1}{40} \, {\left (\frac {\log \left (2\right )}{x + 5} - \frac {\log \left (2\right )}{x}\right )} \log \left (x\right ) - \frac {\log \left (15\right ) \log \left (2\right )}{40 \, x} + \frac {\log \left (15\right ) \log \left (2\right ) - 30 \, \log \left (2\right )}{40 \, {\left (x + 5\right )}} - \frac {1}{16} \, \log \left (x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.66, size = 46, normalized size = 1.39 \begin {gather*} \frac {3\,\ln \left (\frac {x}{4}+\frac {1}{4}\right )}{8\,\left (x+5\right )}-\frac {\ln \left (x+1\right )}{16}+\frac {\ln \left (\frac {x}{4}+\frac {1}{4}\right )\,\ln \left (\frac {15}{x}\right )}{16\,\left (x^2+5\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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