Optimal. Leaf size=18 \[ 4+\log ^2\left (\frac {x}{\left (-9+x^4-\log (3)\right )^2}\right ) \]
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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 2.22, antiderivative size = 1299, normalized size of antiderivative = 72.17, number of
steps used = 104, number of rules used = 16, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6,
1607, 2608, 2604, 2404, 2338, 2375, 2438, 2465, 2439, 266, 2463, 2437, 2441, 2440, 2352}
\begin {gather*} -\log ^2(x)+2 \log \left (\frac {x}{x^8-18 x^4+\log ^2(3)+2 \left (9-x^4\right ) \log (3)+81}\right ) \log (x)+4 \log \left (1-\frac {x^4}{9+\log (3)}\right ) \log (x)+2 \log (9+\log (3)) \log (x)-4 \log ^2\left (i \sqrt [4]{9+\log (3)}-x\right )-4 \log ^2\left (x+i \sqrt [4]{9+\log (3)}\right )-4 \log ^2\left (\sqrt [4]{9+\log (3)}-x\right )-4 \log ^2\left (x+\sqrt [4]{9+\log (3)}\right )+4 \log \left (-\frac {i x}{\sqrt [4]{9+\log (3)}}\right ) \log \left (i \sqrt [4]{9+\log (3)}-x\right )-4 \log \left (\frac {x}{x^8-18 x^4+\log ^2(3)+2 \left (9-x^4\right ) \log (3)+81}\right ) \log \left (i \sqrt [4]{9+\log (3)}-x\right )-8 \log \left (-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (x-\sqrt [4]{9+\log (3)}\right )}{\sqrt [4]{9+\log (3)}}\right ) \log \left (i \sqrt [4]{9+\log (3)}-x\right )+4 \log \left (\frac {i x}{\sqrt [4]{9+\log (3)}}\right ) \log \left (x+i \sqrt [4]{9+\log (3)}\right )-4 \log \left (\frac {x}{x^8-18 x^4+\log ^2(3)+2 \left (9-x^4\right ) \log (3)+81}\right ) \log \left (x+i \sqrt [4]{9+\log (3)}\right )-8 \log \left (-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \left (x-\sqrt [4]{9+\log (3)}\right )}{\sqrt [4]{9+\log (3)}}\right ) \log \left (x+i \sqrt [4]{9+\log (3)}\right )-8 \log \left (\frac {i \left (x-i \sqrt [4]{9+\log (3)}\right )}{2 \sqrt [4]{9+\log (3)}}\right ) \log \left (x+i \sqrt [4]{9+\log (3)}\right )-8 \log \left (i \sqrt [4]{9+\log (3)}-x\right ) \log \left (-\frac {i \left (x+i \sqrt [4]{9+\log (3)}\right )}{2 \sqrt [4]{9+\log (3)}}\right )-4 \log \left (\frac {x}{x^8-18 x^4+\log ^2(3)+2 \left (9-x^4\right ) \log (3)+81}\right ) \log \left (\sqrt [4]{9+\log (3)}-x\right )-8 \log \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (x-i \sqrt [4]{9+\log (3)}\right )}{\sqrt [4]{9+\log (3)}}\right ) \log \left (\sqrt [4]{9+\log (3)}-x\right )-8 \log \left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \left (x+i \sqrt [4]{9+\log (3)}\right )}{\sqrt [4]{9+\log (3)}}\right ) \log \left (\sqrt [4]{9+\log (3)}-x\right )-4 \log \left (\frac {x}{x^8-18 x^4+\log ^2(3)+2 \left (9-x^4\right ) \log (3)+81}\right ) \log \left (x+\sqrt [4]{9+\log (3)}\right )-8 \log \left (-\frac {x-\sqrt [4]{9+\log (3)}}{2 \sqrt [4]{9+\log (3)}}\right ) \log \left (x+\sqrt [4]{9+\log (3)}\right )-8 \log \left (-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \left (x-i \sqrt [4]{9+\log (3)}\right )}{\sqrt [4]{9+\log (3)}}\right ) \log \left (x+\sqrt [4]{9+\log (3)}\right )-8 \log \left (-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (x+i \sqrt [4]{9+\log (3)}\right )}{\sqrt [4]{9+\log (3)}}\right ) \log \left (x+\sqrt [4]{9+\log (3)}\right )-8 \log \left (\sqrt [4]{9+\log (3)}-x\right ) \log \left (\frac {x+\sqrt [4]{9+\log (3)}}{2 \sqrt [4]{9+\log (3)}}\right )-8 \log \left (i \sqrt [4]{9+\log (3)}-x\right ) \log \left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \left (x+\sqrt [4]{9+\log (3)}\right )}{\sqrt [4]{9+\log (3)}}\right )-8 \log \left (x+i \sqrt [4]{9+\log (3)}\right ) \log \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (x+\sqrt [4]{9+\log (3)}\right )}{\sqrt [4]{9+\log (3)}}\right )+\text {Li}_2\left (\frac {x^4}{9+\log (3)}\right )-4 \text {Li}_2\left (-\frac {x}{\sqrt [4]{9+\log (3)}}\right )-4 \text {Li}_2\left (\frac {x}{\sqrt [4]{9+\log (3)}}\right )-8 \text {Li}_2\left (\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (i-\frac {x}{\sqrt [4]{9+\log (3)}}\right )\right )-8 \text {Li}_2\left (\frac {1}{2} \left (1-\frac {x}{\sqrt [4]{9+\log (3)}}\right )\right )-8 \text {Li}_2\left (\left (\frac {1}{2}-\frac {i}{2}\right ) \left (1-\frac {x}{\sqrt [4]{9+\log (3)}}\right )\right )-8 \text {Li}_2\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1-\frac {x}{\sqrt [4]{9+\log (3)}}\right )\right )-8 \text {Li}_2\left (\frac {1}{2} \left (1-\frac {i x}{\sqrt [4]{9+\log (3)}}\right )\right )-8 \text {Li}_2\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1-\frac {i x}{\sqrt [4]{9+\log (3)}}\right )\right )+4 \text {Li}_2\left (1-\frac {i x}{\sqrt [4]{9+\log (3)}}\right )-8 \text {Li}_2\left (\frac {1}{2} \left (\frac {i x}{\sqrt [4]{9+\log (3)}}+1\right )\right )-8 \text {Li}_2\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {i x}{\sqrt [4]{9+\log (3)}}+1\right )\right )+4 \text {Li}_2\left (\frac {i x}{\sqrt [4]{9+\log (3)}}+1\right )-8 \text {Li}_2\left (\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (\frac {x}{\sqrt [4]{9+\log (3)}}+i\right )\right )-8 \text {Li}_2\left (\frac {1}{2} \left (\frac {x}{\sqrt [4]{9+\log (3)}}+1\right )\right )-8 \text {Li}_2\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {x}{\sqrt [4]{9+\log (3)}}+1\right )\right )-8 \text {Li}_2\left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \left (x+\sqrt [4]{9+\log (3)}\right )}{\sqrt [4]{9+\log (3)}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 266
Rule 1607
Rule 2338
Rule 2352
Rule 2375
Rule 2404
Rule 2437
Rule 2438
Rule 2439
Rule 2440
Rule 2441
Rule 2463
Rule 2465
Rule 2604
Rule 2608
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (18+14 x^4+2 \log (3)\right ) \log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{-x^5+x (9+\log (3))} \, dx\\ &=\int \frac {\left (18+14 x^4+2 \log (3)\right ) \log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{x \left (9-x^4+\log (3)\right )} \, dx\\ &=\int \left (\frac {2 \log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{x}-\frac {16 x^3 \log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{-9+x^4-\log (3)}\right ) \, dx\\ &=2 \int \frac {\log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{x} \, dx-16 \int \frac {x^3 \log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{-9+x^4-\log (3)} \, dx\\ &=2 \log (x) \log \left (\frac {x}{81-18 x^4+x^8+2 \left (9-x^4\right ) \log (3)+\log ^2(3)}\right )-2 \int \frac {\left (81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)\right ) \left (-\frac {x \left (-72 x^3+8 x^7-8 x^3 \log (3)\right )}{\left (81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)\right )^2}+\frac {1}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right ) \log (x)}{x} \, dx-16 \int \left (\frac {x \log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{2 \left (x^2-\sqrt {9+\log (3)}\right )}+\frac {x \log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{2 \left (x^2+\sqrt {9+\log (3)}\right )}\right ) \, dx\\ &=2 \log (x) \log \left (\frac {x}{81-18 x^4+x^8+2 \left (9-x^4\right ) \log (3)+\log ^2(3)}\right )-2 \int \left (\frac {\log (x)}{x}-\frac {8 x^3 \log (x)}{-9+x^4-\log (3)}\right ) \, dx-8 \int \frac {x \log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{x^2-\sqrt {9+\log (3)}} \, dx-8 \int \frac {x \log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{x^2+\sqrt {9+\log (3)}} \, dx\\ &=2 \log (x) \log \left (\frac {x}{81-18 x^4+x^8+2 \left (9-x^4\right ) \log (3)+\log ^2(3)}\right )-2 \int \frac {\log (x)}{x} \, dx-8 \int \left (-\frac {\log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{2 \left (-x+i \sqrt [4]{9+\log (3)}\right )}+\frac {\log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{2 \left (x+i \sqrt [4]{9+\log (3)}\right )}\right ) \, dx-8 \int \left (-\frac {\log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{2 \left (-x+\sqrt [4]{9+\log (3)}\right )}+\frac {\log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{2 \left (x+\sqrt [4]{9+\log (3)}\right )}\right ) \, dx+16 \int \frac {x^3 \log (x)}{-9+x^4-\log (3)} \, dx\\ &=-\log ^2(x)+2 \log (x) \log \left (\frac {x}{81-18 x^4+x^8+2 \left (9-x^4\right ) \log (3)+\log ^2(3)}\right )+4 \log (x) \log \left (1-\frac {x^4}{9+\log (3)}\right )-4 \int \frac {\log \left (1+\frac {x^4}{-9-\log (3)}\right )}{x} \, dx+4 \int \frac {\log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{-x+i \sqrt [4]{9+\log (3)}} \, dx-4 \int \frac {\log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{x+i \sqrt [4]{9+\log (3)}} \, dx+4 \int \frac {\log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{-x+\sqrt [4]{9+\log (3)}} \, dx-4 \int \frac {\log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{x+\sqrt [4]{9+\log (3)}} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.38, size = 1485, normalized size = 82.50 \begin {gather*} \text {Too large to display} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.52, size = 32, normalized size = 1.78
method | result | size |
norman | \(\ln \left (\frac {x}{\ln \left (3\right )^{2}+\left (-2 x^{4}+18\right ) \ln \left (3\right )+x^{8}-18 x^{4}+81}\right )^{2}\) | \(32\) |
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. 82 vs.
\(2 (18) = 36\).
time = 0.48, size = 82, normalized size = 4.56 \begin {gather*} -4 \, \log \left (x^{4} - \log \left (3\right ) - 9\right )^{2} + 4 \, \log \left (x^{4} - \log \left (3\right ) - 9\right ) \log \left (x\right ) - \log \left (x\right )^{2} - 2 \, {\left (2 \, \log \left (x^{4} - \log \left (3\right ) - 9\right ) - \log \left (x\right )\right )} \log \left (\frac {x}{x^{8} - 18 \, x^{4} - 2 \, {\left (x^{4} - 9\right )} \log \left (3\right ) + \log \left (3\right )^{2} + 81}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 30, normalized size = 1.67 \begin {gather*} \log \left (\frac {x}{x^{8} - 18 \, x^{4} - 2 \, {\left (x^{4} - 9\right )} \log \left (3\right ) + \log \left (3\right )^{2} + 81}\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 29, normalized size = 1.61 \begin {gather*} \log {\left (\frac {x}{x^{8} - 18 x^{4} + \left (18 - 2 x^{4}\right ) \log {\left (3 \right )} + \log {\left (3 \right )}^{2} + 81} \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. 64 vs.
\(2 (18) = 36\).
time = 0.40, size = 64, normalized size = 3.56 \begin {gather*} 2 \, {\left (2 \, \log \left (x^{4} - \log \left (3\right ) - 9\right ) - \log \left (x\right )\right )} \log \left (x^{8} - 2 \, x^{4} \log \left (3\right ) - 18 \, x^{4} + \log \left (3\right )^{2} + 18 \, \log \left (3\right ) + 81\right ) - 4 \, \log \left (x^{4} - \log \left (3\right ) - 9\right )^{2} + \log \left (x\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.18, size = 32, normalized size = 1.78 \begin {gather*} {\ln \left (\frac {x}{{\ln \left (3\right )}^2-\ln \left (3\right )\,\left (2\,x^4-18\right )-18\,x^4+x^8+81}\right )}^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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