Integrand size = 17, antiderivative size = 11 \[ \int \frac {6 \log (x)+6 \log \left (-x^2\right )}{x} \, dx=\left (\log (x)+\log \left (-x^2\right )\right )^2 \]
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Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.73, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {14, 2338} \[ \int \frac {6 \log (x)+6 \log \left (-x^2\right )}{x} \, dx=\frac {3}{2} \log ^2\left (-x^2\right )+3 \log ^2(x) \]
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Rule 14
Rule 2338
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {6 \log (x)}{x}+\frac {6 \log \left (-x^2\right )}{x}\right ) \, dx \\ & = 6 \int \frac {\log (x)}{x} \, dx+6 \int \frac {\log \left (-x^2\right )}{x} \, dx \\ & = 3 \log ^2(x)+\frac {3}{2} \log ^2\left (-x^2\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.73 \[ \int \frac {6 \log (x)+6 \log \left (-x^2\right )}{x} \, dx=3 \log ^2(x)+\frac {3}{2} \log ^2\left (-x^2\right ) \]
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Time = 0.06 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.64
method | result | size |
default | \(\frac {3 \ln \left (-x^{2}\right )^{2}}{2}+3 \ln \left (x \right )^{2}\) | \(18\) |
norman | \(-3 \ln \left (x \right )^{2}+6 \ln \left (x \right ) \ln \left (-x^{2}\right )\) | \(18\) |
parts | \(\frac {3 \ln \left (-x^{2}\right )^{2}}{2}+3 \ln \left (x \right )^{2}\) | \(18\) |
risch | \(9 \ln \left (x \right )^{2}-3 i \pi \left (\operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-\operatorname {csgn}\left (i x^{2}\right )^{3}+2 \operatorname {csgn}\left (i x^{2}\right )^{2}-2\right ) \ln \left (x \right )\) | \(69\) |
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Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.82 \[ \int \frac {6 \log (x)+6 \log \left (-x^2\right )}{x} \, dx=\frac {3}{2} i \, \pi \log \left (-x^{2}\right ) + \frac {9}{4} \, \log \left (-x^{2}\right )^{2} \]
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Result contains complex when optimal does not.
Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.27 \[ \int \frac {6 \log (x)+6 \log \left (-x^2\right )}{x} \, dx=9 \log {\left (x \right )}^{2} + 6 i \pi \log {\left (x \right )} \]
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none
Time = 0.19 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {6 \log (x)+6 \log \left (-x^2\right )}{x} \, dx={\left (\log \left (-x^{2}\right ) + \log \left (x\right )\right )}^{2} \]
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09 \[ \int \frac {6 \log (x)+6 \log \left (-x^2\right )}{x} \, dx=6 i \, \pi \log \left (x\right ) + 9 \, \log \left (x\right )^{2} \]
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Time = 8.15 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.45 \[ \int \frac {6 \log (x)+6 \log \left (-x^2\right )}{x} \, dx=\ln \left (x\right )\,\left (\ln \left (x^{12}\right )-3\,\ln \left (x\right )+\pi \,6{}\mathrm {i}\right ) \]
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