Optimal. Leaf size=37 \[ \frac{1}{4} i \text{PolyLog}\left (2,\frac{i}{a x^2}\right )-\frac{1}{4} i \text{PolyLog}\left (2,-\frac{i}{a x^2}\right ) \]
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Rubi [A] time = 0.0338291, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5032, 4849, 2391} \[ \frac{1}{4} i \text{PolyLog}\left (2,\frac{i}{a x^2}\right )-\frac{1}{4} i \text{PolyLog}\left (2,-\frac{i}{a x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 5032
Rule 4849
Rule 2391
Rubi steps
\begin{align*} \int \frac{\cot ^{-1}\left (a x^2\right )}{x} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\cot ^{-1}(a x)}{x} \, dx,x,x^2\right )\\ &=\frac{1}{4} i \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i}{a x}\right )}{x} \, dx,x,x^2\right )-\frac{1}{4} i \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{i}{a x}\right )}{x} \, dx,x,x^2\right )\\ &=-\frac{1}{4} i \text{Li}_2\left (-\frac{i}{a x^2}\right )+\frac{1}{4} i \text{Li}_2\left (\frac{i}{a x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0056303, size = 37, normalized size = 1. \[ \frac{1}{4} i \text{PolyLog}\left (2,\frac{i}{a x^2}\right )-\frac{1}{4} i \text{PolyLog}\left (2,-\frac{i}{a x^2}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.117, size = 57, normalized size = 1.5 \begin{align*} \ln \left ( x \right ){\rm arccot} \left (a{x}^{2}\right )+{\frac{1}{2\,a}\sum _{{\it \_R1}={\it RootOf} \left ({a}^{2}{{\it \_Z}}^{4}+1 \right ) }{\frac{1}{{{\it \_R1}}^{2}} \left ( \ln \left ( x \right ) \ln \left ({\frac{{\it \_R1}-x}{{\it \_R1}}} \right ) +{\it dilog} \left ({\frac{{\it \_R1}-x}{{\it \_R1}}} \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.61403, size = 108, normalized size = 2.92 \begin{align*} -\frac{1}{2} i \, \arctan \left (a x^{2}\right ) \arctan \left (0, a\right ) + \frac{1}{8} \, \pi \log \left (a^{2} x^{4} + 1\right ) - \frac{1}{2} \, \arctan \left (a x^{2}\right ) \log \left (x^{2}{\left | a \right |}\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{arccot}\left (a x^{2}\right )}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acot}{\left (a x^{2} \right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arccot}\left (a x^{2}\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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