Optimal. Leaf size=29 \[ \frac {x}{\log \left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \]
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Rubi [F]
time = 4.35, 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 {-2 x \log (x)+\left (3 x+x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (6+2 \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )+\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log \left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )}{\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log ^2\left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\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 {2 x \log (x)-x \left (3+\log ^2(x)\right ) \log \left (3+\log ^2(x)\right )-2 \left (3+\log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )+\left (3+\log ^2(x)\right ) \log \left (3+\log ^2(x)\right ) \left (x-\left (-5+\log \left (\frac {15}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )\right ) \log \left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )}{\left (3+\log ^2(x)\right ) \log \left (3+\log ^2(x)\right ) \left (x-\left (-5+\log \left (\frac {15}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \, dx\\ &=\int \left (\frac {-2 x \log (x)+3 x \log \left (3+\log ^2(x)\right )+x \log ^2(x) \log \left (3+\log ^2(x)\right )+6 \log ^2\left (3+\log ^2(x)\right )+2 \log ^2(x) \log ^2\left (3+\log ^2(x)\right )}{\left (3+\log ^2(x)\right ) \log \left (3+\log ^2(x)\right ) \left (-x-5 \log \left (3+\log ^2(x)\right )+\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )}+\frac {1}{\log \left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )}\right ) \, dx\\ &=\int \frac {-2 x \log (x)+3 x \log \left (3+\log ^2(x)\right )+x \log ^2(x) \log \left (3+\log ^2(x)\right )+6 \log ^2\left (3+\log ^2(x)\right )+2 \log ^2(x) \log ^2\left (3+\log ^2(x)\right )}{\left (3+\log ^2(x)\right ) \log \left (3+\log ^2(x)\right ) \left (-x-5 \log \left (3+\log ^2(x)\right )+\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \, dx+\int \frac {1}{\log \left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \, dx\\ &=\int \left (-\frac {3 x}{\left (3+\log ^2(x)\right ) \left (x+5 \log \left (3+\log ^2(x)\right )-\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )}-\frac {x \log ^2(x)}{\left (3+\log ^2(x)\right ) \left (x+5 \log \left (3+\log ^2(x)\right )-\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )}-\frac {2 x \log (x)}{\left (3+\log ^2(x)\right ) \log \left (3+\log ^2(x)\right ) \left (-x-5 \log \left (3+\log ^2(x)\right )+\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )}+\frac {6 \log \left (3+\log ^2(x)\right )}{\left (3+\log ^2(x)\right ) \left (-x-5 \log \left (3+\log ^2(x)\right )+\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )}+\frac {2 \log ^2(x) \log \left (3+\log ^2(x)\right )}{\left (3+\log ^2(x)\right ) \left (-x-5 \log \left (3+\log ^2(x)\right )+\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )}\right ) \, dx+\int \frac {1}{\log \left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \, dx\\ &=-\left (2 \int \frac {x \log (x)}{\left (3+\log ^2(x)\right ) \log \left (3+\log ^2(x)\right ) \left (-x-5 \log \left (3+\log ^2(x)\right )+\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \, dx\right )+2 \int \frac {\log ^2(x) \log \left (3+\log ^2(x)\right )}{\left (3+\log ^2(x)\right ) \left (-x-5 \log \left (3+\log ^2(x)\right )+\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \, dx-3 \int \frac {x}{\left (3+\log ^2(x)\right ) \left (x+5 \log \left (3+\log ^2(x)\right )-\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \, dx+6 \int \frac {\log \left (3+\log ^2(x)\right )}{\left (3+\log ^2(x)\right ) \left (-x-5 \log \left (3+\log ^2(x)\right )+\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \, dx-\int \frac {x \log ^2(x)}{\left (3+\log ^2(x)\right ) \left (x+5 \log \left (3+\log ^2(x)\right )-\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \, dx+\int \frac {1}{\log \left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.16, size = 27, normalized size = 0.93 \begin {gather*} \frac {x}{\log \left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\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 = 2.83, size = 928, normalized size = 32.00
method | result | size |
risch | \(\text {Expression too large to display}\) | \(928\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.59, size = 49, normalized size = 1.69 \begin {gather*} \frac {x}{\log \left ({\left (\log \left (5\right ) + \log \left (3\right ) - 2 \, \log \left (2\right ) - 5\right )} \log \left (\log \left (x\right )^{2} + 3\right ) - 2 \, \log \left (\log \left (x\right )^{2} + 3\right ) \log \left (x\right ) - x\right ) - \log \left (\log \left (\log \left (x\right )^{2} + 3\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 81 vs.
\(2 (27) = 54\).
time = 0.35, size = 81, normalized size = 2.79 \begin {gather*} \frac {x}{\log \left (\frac {{\left (\log \left (5\right ) + \log \left (\frac {3}{4 \, x^{2}}\right ) - 5\right )} \log \left (\frac {1}{4} \, \log \left (\frac {3}{4}\right )^{2} - \frac {1}{2} \, \log \left (\frac {3}{4}\right ) \log \left (\frac {3}{4 \, x^{2}}\right ) + \frac {1}{4} \, \log \left (\frac {3}{4 \, x^{2}}\right )^{2} + 3\right ) - x}{\log \left (\frac {1}{4} \, \log \left (\frac {3}{4}\right )^{2} - \frac {1}{2} \, \log \left (\frac {3}{4}\right ) \log \left (\frac {3}{4 \, x^{2}}\right ) + \frac {1}{4} \, \log \left (\frac {3}{4 \, x^{2}}\right )^{2} + 3\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 81.91, size = 36, normalized size = 1.24 \begin {gather*} \frac {x}{\log {\left (\frac {- x + \left (- 2 \log {\left (x \right )} - 5 + \log {\left (\frac {3}{4} \right )} + \log {\left (5 \right )}\right ) \log {\left (\log {\left (x \right )}^{2} + 3 \right )}}{\log {\left (\log {\left (x \right )}^{2} + 3 \right )}} \right )}} \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. 71 vs.
\(2 (27) = 54\).
time = 2.52, size = 71, normalized size = 2.45 \begin {gather*} \frac {x}{\log \left (\log \left (5\right ) \log \left (\log \left (x\right )^{2} + 3\right ) + \log \left (3\right ) \log \left (\log \left (x\right )^{2} + 3\right ) - 2 \, \log \left (2\right ) \log \left (\log \left (x\right )^{2} + 3\right ) - 2 \, \log \left (\log \left (x\right )^{2} + 3\right ) \log \left (x\right ) - x - 5 \, \log \left (\log \left (x\right )^{2} + 3\right )\right ) - \log \left (\log \left (\log \left (x\right )^{2} + 3\right )\right )} \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 (-\frac {x-\ln \left ({\ln \left (x\right )}^2+3\right )\,\left (\ln \left (5\right )+\ln \left (\frac {3}{4\,x^2}\right )-5\right )}{\ln \left ({\ln \left (x\right )}^2+3\right )}\right )\,\left ({\ln \left ({\ln \left (x\right )}^2+3\right )}^2\,\left (\left (\ln \left (5\right )+\ln \left (\frac {3}{4\,x^2}\right )-5\right )\,{\ln \left (x\right )}^2+3\,\ln \left (5\right )+3\,\ln \left (\frac {3}{4\,x^2}\right )-15\right )-\ln \left ({\ln \left (x\right )}^2+3\right )\,\left (x\,{\ln \left (x\right )}^2+3\,x\right )\right )-2\,x\,\ln \left (x\right )+\ln \left ({\ln \left (x\right )}^2+3\right )\,\left (x\,{\ln \left (x\right )}^2+3\,x\right )+{\ln \left ({\ln \left (x\right )}^2+3\right )}^2\,\left (2\,{\ln \left (x\right )}^2+6\right )}{{\ln \left (-\frac {x-\ln \left ({\ln \left (x\right )}^2+3\right )\,\left (\ln \left (5\right )+\ln \left (\frac {3}{4\,x^2}\right )-5\right )}{\ln \left ({\ln \left (x\right )}^2+3\right )}\right )}^2\,\left ({\ln \left ({\ln \left (x\right )}^2+3\right )}^2\,\left (\left (\ln \left (5\right )+\ln \left (\frac {3}{4\,x^2}\right )-5\right )\,{\ln \left (x\right )}^2+3\,\ln \left (5\right )+3\,\ln \left (\frac {3}{4\,x^2}\right )-15\right )-\ln \left ({\ln \left (x\right )}^2+3\right )\,\left (x\,{\ln \left (x\right )}^2+3\,x\right )\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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