Integrand size = 10, antiderivative size = 37 \[ \int \frac {\cot ^{-1}\left (a x^2\right )}{x} \, dx=-\frac {1}{4} i \operatorname {PolyLog}\left (2,-\frac {i}{a x^2}\right )+\frac {1}{4} i \operatorname {PolyLog}\left (2,\frac {i}{a x^2}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4945, 4941, 2438} \[ \int \frac {\cot ^{-1}\left (a x^2\right )}{x} \, dx=\frac {1}{4} i \operatorname {PolyLog}\left (2,\frac {i}{a x^2}\right )-\frac {1}{4} i \operatorname {PolyLog}\left (2,-\frac {i}{a x^2}\right ) \]
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Rule 2438
Rule 4941
Rule 4945
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\cot ^{-1}(a x)}{x} \, dx,x,x^2\right ) \\ & = \frac {1}{4} i \text {Subst}\left (\int \frac {\log \left (1-\frac {i}{a x}\right )}{x} \, dx,x,x^2\right )-\frac {1}{4} i \text {Subst}\left (\int \frac {\log \left (1+\frac {i}{a x}\right )}{x} \, dx,x,x^2\right ) \\ & = -\frac {1}{4} i \operatorname {PolyLog}\left (2,-\frac {i}{a x^2}\right )+\frac {1}{4} i \operatorname {PolyLog}\left (2,\frac {i}{a x^2}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^{-1}\left (a x^2\right )}{x} \, dx=-\frac {1}{4} i \operatorname {PolyLog}\left (2,-\frac {i}{a x^2}\right )+\frac {1}{4} i \operatorname {PolyLog}\left (2,\frac {i}{a x^2}\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.28 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.54
method | result | size |
default | \(\ln \left (x \right ) \operatorname {arccot}\left (a \,x^{2}\right )+\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{4}+1\right )}{\sum }\frac {\ln \left (x \right ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}}}{2 a}\) | \(57\) |
parts | \(\ln \left (x \right ) \operatorname {arccot}\left (a \,x^{2}\right )+\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{4}+1\right )}{\sum }\frac {\ln \left (x \right ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}}}{2 a}\) | \(57\) |
risch | \(\frac {\pi \ln \left (x \right )}{2}+\frac {i \ln \left (x \right ) \ln \left (1-i x \sqrt {-i a}\right )}{2}+\frac {i \ln \left (x \right ) \ln \left (1+i x \sqrt {-i a}\right )}{2}-\frac {i \ln \left (x \right ) \ln \left (-i a \,x^{2}+1\right )}{2}+\frac {i \operatorname {dilog}\left (1-i x \sqrt {-i a}\right )}{2}+\frac {i \operatorname {dilog}\left (1+i x \sqrt {-i a}\right )}{2}-\frac {i \ln \left (x \right ) \ln \left (1+i x \sqrt {i a}\right )}{2}-\frac {i \ln \left (x \right ) \ln \left (1-i x \sqrt {i a}\right )}{2}+\frac {i \ln \left (x \right ) \ln \left (i a \,x^{2}+1\right )}{2}-\frac {i \operatorname {dilog}\left (1+i x \sqrt {i a}\right )}{2}-\frac {i \operatorname {dilog}\left (1-i x \sqrt {i a}\right )}{2}\) | \(173\) |
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\[ \int \frac {\cot ^{-1}\left (a x^2\right )}{x} \, dx=\int { \frac {\operatorname {arccot}\left (a x^{2}\right )}{x} \,d x } \]
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\[ \int \frac {\cot ^{-1}\left (a x^2\right )}{x} \, dx=\int \frac {\operatorname {acot}{\left (a x^{2} \right )}}{x}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (23) = 46\).
Time = 0.33 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.84 \[ \int \frac {\cot ^{-1}\left (a x^2\right )}{x} \, dx=\frac {1}{8} \, \pi \log \left (a^{2} x^{4} + 1\right ) - \frac {1}{2} \, \arctan \left (a x^{2}\right ) \log \left (a x^{2}\right ) + \operatorname {arccot}\left (a x^{2}\right ) \log \left (x\right ) + \arctan \left (a x^{2}\right ) \log \left (x\right ) + \frac {1}{4} i \, {\rm Li}_2\left (i \, a x^{2} + 1\right ) - \frac {1}{4} i \, {\rm Li}_2\left (-i \, a x^{2} + 1\right ) \]
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\[ \int \frac {\cot ^{-1}\left (a x^2\right )}{x} \, dx=\int { \frac {\operatorname {arccot}\left (a x^{2}\right )}{x} \,d x } \]
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Timed out. \[ \int \frac {\cot ^{-1}\left (a x^2\right )}{x} \, dx=\int \frac {\mathrm {acot}\left (a\,x^2\right )}{x} \,d x \]
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