Optimal. Leaf size=34 \[ 2+\frac {2}{4+x^2-\frac {\left (-x+\log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )^2}{x^2}} \]
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Rubi [F]
time = 2.15, 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 x^2+\left (4 x^2-4 x^5\right ) \log (x)+\left (8 x+\left (-4 x+4 x^2\right ) \log (x)\right ) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )}{\left (9 x^4+6 x^6+x^8\right ) \log (x)+\left (12 x^3+4 x^5\right ) \log (x) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\left (-2 x^2-2 x^4\right ) \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^3\left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\log (x) \log ^4\left (\frac {3 \log (5) \log ^2(x)}{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 {4 x \left (-2 \left (x-\log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )-\log (x) \left (x \left (-1+x^3\right )-(-1+x) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )\right )}{\log (x) \left (3 x^2+x^4+2 x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )^2} \, dx\\ &=4 \int \frac {x \left (-2 \left (x-\log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )-\log (x) \left (x \left (-1+x^3\right )-(-1+x) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )\right )}{\log (x) \left (3 x^2+x^4+2 x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )^2} \, dx\\ &=4 \int \left (-\frac {x \left (2 x-x \log (x)+3 x^2 \log (x)+2 x^4 \log (x)-2 \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\log (x) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )+x \log (x) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )}{\log (x) \left (3 x^2+x^4+2 x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )^2}+\frac {x}{3 x^2+x^4+2 x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )}\right ) \, dx\\ &=-\left (4 \int \frac {x \left (2 x-x \log (x)+3 x^2 \log (x)+2 x^4 \log (x)-2 \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\log (x) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )+x \log (x) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )}{\log (x) \left (3 x^2+x^4+2 x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )^2} \, dx\right )+4 \int \frac {x}{3 x^2+x^4+2 x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )} \, dx\\ &=4 \int \frac {x}{3 x^2+x^4+2 x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )} \, dx-4 \int \left (-\frac {x^2}{\left (3 x^2+x^4+2 x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )^2}+\frac {3 x^3}{\left (3 x^2+x^4+2 x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )^2}+\frac {2 x^5}{\left (3 x^2+x^4+2 x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )^2}+\frac {2 x^2}{\log (x) \left (3 x^2+x^4+2 x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )^2}+\frac {x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )}{\left (3 x^2+x^4+2 x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )^2}+\frac {x^2 \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )}{\left (3 x^2+x^4+2 x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )^2}-\frac {2 x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )}{\log (x) \left (3 x^2+x^4+2 x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )^2}\right ) \, dx\\ &=4 \int \frac {x^2}{\left (3 x^2+x^4+2 x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )^2} \, dx-4 \int \frac {x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )}{\left (3 x^2+x^4+2 x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )^2} \, dx-4 \int \frac {x^2 \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )}{\left (3 x^2+x^4+2 x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )^2} \, dx+4 \int \frac {x}{3 x^2+x^4+2 x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )} \, dx-8 \int \frac {x^5}{\left (3 x^2+x^4+2 x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )^2} \, dx-8 \int \frac {x^2}{\log (x) \left (3 x^2+x^4+2 x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )^2} \, dx+8 \int \frac {x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )}{\log (x) \left (3 x^2+x^4+2 x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )^2} \, dx-12 \int \frac {x^3}{\left (3 x^2+x^4+2 x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.05, size = 47, normalized size = 1.38 \begin {gather*} -\frac {2 x^2}{-3 x^2-x^4-2 x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\log ^2\left (\frac {3 \log (5) \log ^2(x)}{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 = 132.57, size = 1723, normalized size = 50.68
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1723\) |
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. 91 vs.
\(2 (34) = 68\).
time = 0.57, size = 91, normalized size = 2.68 \begin {gather*} \frac {2 \, x^{2}}{x^{4} + 3 \, x^{2} + 2 \, x {\left (\log \left (3\right ) + \log \left (\log \left (5\right )\right )\right )} - \log \left (3\right )^{2} - 2 \, {\left (x - \log \left (3\right ) - \log \left (\log \left (5\right )\right )\right )} \log \left (x\right ) - \log \left (x\right )^{2} - 2 \, \log \left (3\right ) \log \left (\log \left (5\right )\right ) - \log \left (\log \left (5\right )\right )^{2} + 4 \, {\left (x - \log \left (3\right ) + \log \left (x\right ) - \log \left (\log \left (5\right )\right )\right )} \log \left (\log \left (x\right )\right ) - 4 \, \log \left (\log \left (x\right )\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 47, normalized size = 1.38 \begin {gather*} \frac {2 \, x^{2}}{x^{4} + 3 \, x^{2} + 2 \, x \log \left (\frac {3 \, \log \left (5\right ) \log \left (x\right )^{2}}{x}\right ) - \log \left (\frac {3 \, \log \left (5\right ) \log \left (x\right )^{2}}{x}\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.23, size = 46, normalized size = 1.35 \begin {gather*} - \frac {2 x^{2}}{- x^{4} - 3 x^{2} - 2 x \log {\left (\frac {3 \log {\left (5 \right )} \log {\left (x \right )}^{2}}{x} \right )} + \log {\left (\frac {3 \log {\left (5 \right )} \log {\left (x \right )}^{2}}{x} \right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.77, size = 65, normalized size = 1.91 \begin {gather*} \frac {2 \, x^{2}}{x^{4} + 3 \, x^{2} + 2 \, x \log \left (3 \, \log \left (5\right ) \log \left (x\right )^{2}\right ) - \log \left (3 \, \log \left (5\right ) \log \left (x\right )^{2}\right )^{2} - 2 \, x \log \left (x\right ) + 2 \, \log \left (3 \, \log \left (5\right ) \log \left (x\right )^{2}\right ) \log \left (x\right ) - \log \left (x\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\ln \left (x\right )\,\left (4\,x^2-4\,x^5\right )+\ln \left (\frac {3\,\ln \left (5\right )\,{\ln \left (x\right )}^2}{x}\right )\,\left (8\,x-\ln \left (x\right )\,\left (4\,x-4\,x^2\right )\right )-8\,x^2-4\,x\,{\ln \left (\frac {3\,\ln \left (5\right )\,{\ln \left (x\right )}^2}{x}\right )}^2\,\ln \left (x\right )}{\ln \left (x\right )\,{\ln \left (\frac {3\,\ln \left (5\right )\,{\ln \left (x\right )}^2}{x}\right )}^4-4\,x\,\ln \left (x\right )\,{\ln \left (\frac {3\,\ln \left (5\right )\,{\ln \left (x\right )}^2}{x}\right )}^3-\ln \left (x\right )\,\left (2\,x^4+2\,x^2\right )\,{\ln \left (\frac {3\,\ln \left (5\right )\,{\ln \left (x\right )}^2}{x}\right )}^2+\ln \left (x\right )\,\left (4\,x^5+12\,x^3\right )\,\ln \left (\frac {3\,\ln \left (5\right )\,{\ln \left (x\right )}^2}{x}\right )+\ln \left (x\right )\,\left (x^8+6\,x^6+9\,x^4\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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