Optimal. Leaf size=22 \[ \frac {2+\left (25+x+\frac {x}{\log ^2\left (\frac {4}{x^2}\right )}\right ) \log (x)}{x} \]
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Rubi [A] time = 0.73, antiderivative size = 26, normalized size of antiderivative = 1.18, number of steps used = 18, number of rules used = 9, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.180, Rules used = {6688, 14, 43, 2302, 30, 6742, 2304, 2366, 12} \begin {gather*} \frac {\log (x)}{\log ^2\left (\frac {4}{x^2}\right )}+\frac {2}{x}+\frac {25 \log (x)}{x}+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 30
Rule 43
Rule 2302
Rule 2304
Rule 2366
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {23+x+\frac {x}{\log ^2\left (\frac {4}{x^2}\right )}-25 \log (x)+\frac {4 x \log (x)}{\log ^3\left (\frac {4}{x^2}\right )}}{x^2} \, dx\\ &=\int \left (\frac {x+23 \log ^2\left (\frac {4}{x^2}\right )+x \log ^2\left (\frac {4}{x^2}\right )}{x^2 \log ^2\left (\frac {4}{x^2}\right )}+\frac {\left (4 x-25 \log ^3\left (\frac {4}{x^2}\right )\right ) \log (x)}{x^2 \log ^3\left (\frac {4}{x^2}\right )}\right ) \, dx\\ &=\int \frac {x+23 \log ^2\left (\frac {4}{x^2}\right )+x \log ^2\left (\frac {4}{x^2}\right )}{x^2 \log ^2\left (\frac {4}{x^2}\right )} \, dx+\int \frac {\left (4 x-25 \log ^3\left (\frac {4}{x^2}\right )\right ) \log (x)}{x^2 \log ^3\left (\frac {4}{x^2}\right )} \, dx\\ &=\int \frac {23+x+\frac {x}{\log ^2\left (\frac {4}{x^2}\right )}}{x^2} \, dx+\int \frac {\left (-25+\frac {4 x}{\log ^3\left (\frac {4}{x^2}\right )}\right ) \log (x)}{x^2} \, dx\\ &=\int \left (\frac {23+x}{x^2}+\frac {1}{x \log ^2\left (\frac {4}{x^2}\right )}\right ) \, dx+\int \left (-\frac {25 \log (x)}{x^2}+\frac {4 \log (x)}{x \log ^3\left (\frac {4}{x^2}\right )}\right ) \, dx\\ &=4 \int \frac {\log (x)}{x \log ^3\left (\frac {4}{x^2}\right )} \, dx-25 \int \frac {\log (x)}{x^2} \, dx+\int \frac {23+x}{x^2} \, dx+\int \frac {1}{x \log ^2\left (\frac {4}{x^2}\right )} \, dx\\ &=\frac {25}{x}+\frac {25 \log (x)}{x}+\frac {\log (x)}{\log ^2\left (\frac {4}{x^2}\right )}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log \left (\frac {4}{x^2}\right )\right )-4 \int \frac {1}{4 x \log ^2\left (\frac {4}{x^2}\right )} \, dx+\int \left (\frac {23}{x^2}+\frac {1}{x}\right ) \, dx\\ &=\frac {2}{x}+\frac {1}{2 \log \left (\frac {4}{x^2}\right )}+\log (x)+\frac {25 \log (x)}{x}+\frac {\log (x)}{\log ^2\left (\frac {4}{x^2}\right )}-\int \frac {1}{x \log ^2\left (\frac {4}{x^2}\right )} \, dx\\ &=\frac {2}{x}+\frac {1}{2 \log \left (\frac {4}{x^2}\right )}+\log (x)+\frac {25 \log (x)}{x}+\frac {\log (x)}{\log ^2\left (\frac {4}{x^2}\right )}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log \left (\frac {4}{x^2}\right )\right )\\ &=\frac {2}{x}+\log (x)+\frac {25 \log (x)}{x}+\frac {\log (x)}{\log ^2\left (\frac {4}{x^2}\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.04, size = 24, normalized size = 1.09 \begin {gather*} \frac {2}{x}+\left (1+\frac {25}{x}+\frac {1}{\log ^2\left (\frac {4}{x^2}\right )}\right ) \log (x) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.85, size = 55, normalized size = 2.50 \begin {gather*} -\frac {{\left (x + 25\right )} \log \left (\frac {4}{x^{2}}\right )^{3} - 2 \, {\left (25 \, \log \relax (2) + 2\right )} \log \left (\frac {4}{x^{2}}\right )^{2} - 2 \, x \log \relax (2) + x \log \left (\frac {4}{x^{2}}\right )}{2 \, x \log \left (\frac {4}{x^{2}}\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 36, normalized size = 1.64 \begin {gather*} \frac {\log \relax (x)}{4 \, {\left (\log \relax (2)^{2} - 2 \, \log \relax (2) \log \relax (x) + \log \relax (x)^{2}\right )}} + \frac {25 \, \log \relax (x)}{x} + \frac {2}{x} + \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.27, size = 83, normalized size = 3.77
method | result | size |
risch | \(\frac {25 \ln \relax (x )}{x}+\frac {x \ln \relax (x )+2}{x}+\frac {4 \ln \relax (x )}{\left (-i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2}+2 i \pi \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right )-i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}-4 \ln \relax (2)+4 \ln \relax (x )\right )^{2}}\) | \(83\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 26, normalized size = 1.18 \begin {gather*} \frac {25 \, \log \relax (x)}{x} + \frac {2}{x} + \frac {\log \relax (x)}{\log \left (\frac {4}{x^{2}}\right )^{2}} + \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.26, size = 74, normalized size = 3.36 \begin {gather*} \ln \relax (x)-\frac {1}{4\,\ln \left (\frac {4}{x^2}\right )}+\frac {25\,\ln \relax (x)}{x}+\frac {\frac {\ln \left (\frac {4}{x^2}\right )}{4}+\ln \relax (x)}{4\,{\ln \relax (x)}^2-4\,\ln \relax (x)\,\left (\ln \left (\frac {4}{x^2}\right )+2\,\ln \relax (x)\right )+{\left (\ln \left (\frac {4}{x^2}\right )+2\,\ln \relax (x)\right )}^2}+\frac {2}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.33, size = 37, normalized size = 1.68 \begin {gather*} \log {\relax (x )} + \frac {\log {\relax (x )}}{4 \log {\relax (x )}^{2} - 8 \log {\relax (2 )} \log {\relax (x )} + 4 \log {\relax (2 )}^{2}} + \frac {25 \log {\relax (x )}}{x} + \frac {2}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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