Optimal. Leaf size=59 \[ -\frac{1}{2} a^2 \log \left (a^2 x^2+1\right )+a^2 \log (x)-\frac{1}{2} a^2 \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{2 x^2}+\frac{a \cot ^{-1}(a x)}{x} \]
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Rubi [A] time = 0.087978, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {4853, 4919, 266, 36, 29, 31, 4885} \[ -\frac{1}{2} a^2 \log \left (a^2 x^2+1\right )+a^2 \log (x)-\frac{1}{2} a^2 \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{2 x^2}+\frac{a \cot ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
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Rule 4853
Rule 4919
Rule 266
Rule 36
Rule 29
Rule 31
Rule 4885
Rubi steps
\begin{align*} \int \frac{\cot ^{-1}(a x)^2}{x^3} \, dx &=-\frac{\cot ^{-1}(a x)^2}{2 x^2}-a \int \frac{\cot ^{-1}(a x)}{x^2 \left (1+a^2 x^2\right )} \, dx\\ &=-\frac{\cot ^{-1}(a x)^2}{2 x^2}-a \int \frac{\cot ^{-1}(a x)}{x^2} \, dx+a^3 \int \frac{\cot ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac{a \cot ^{-1}(a x)}{x}-\frac{1}{2} a^2 \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{2 x^2}+a^2 \int \frac{1}{x \left (1+a^2 x^2\right )} \, dx\\ &=\frac{a \cot ^{-1}(a x)}{x}-\frac{1}{2} a^2 \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{2 x^2}+\frac{1}{2} a^2 \operatorname{Subst}\left (\int \frac{1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )\\ &=\frac{a \cot ^{-1}(a x)}{x}-\frac{1}{2} a^2 \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{2 x^2}+\frac{1}{2} a^2 \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )-\frac{1}{2} a^4 \operatorname{Subst}\left (\int \frac{1}{1+a^2 x} \, dx,x,x^2\right )\\ &=\frac{a \cot ^{-1}(a x)}{x}-\frac{1}{2} a^2 \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{2 x^2}+a^2 \log (x)-\frac{1}{2} a^2 \log \left (1+a^2 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0167775, size = 56, normalized size = 0.95 \[ -\frac{1}{2} a^2 \log \left (a^2 x^2+1\right )+\frac{\left (-a^2 x^2-1\right ) \cot ^{-1}(a x)^2}{2 x^2}+a^2 \log (x)+\frac{a \cot ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 68, normalized size = 1.2 \begin{align*} -{\frac{ \left ({\rm arccot} \left (ax\right ) \right ) ^{2}}{2\,{x}^{2}}}+{a}^{2}{\rm arccot} \left (ax\right )\arctan \left ( ax \right ) +{\frac{a{\rm arccot} \left (ax\right )}{x}}-{\frac{{a}^{2}\ln \left ({a}^{2}{x}^{2}+1 \right ) }{2}}+{a}^{2}\ln \left ( ax \right ) +{\frac{{a}^{2} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47291, size = 76, normalized size = 1.29 \begin{align*} \frac{1}{2} \,{\left (\arctan \left (a x\right )^{2} - \log \left (a^{2} x^{2} + 1\right ) + 2 \, \log \left (x\right )\right )} a^{2} +{\left (a \arctan \left (a x\right ) + \frac{1}{x}\right )} a \operatorname{arccot}\left (a x\right ) - \frac{\operatorname{arccot}\left (a x\right )^{2}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02672, size = 143, normalized size = 2.42 \begin{align*} -\frac{a^{2} x^{2} \log \left (a^{2} x^{2} + 1\right ) - 2 \, a^{2} x^{2} \log \left (x\right ) - 2 \, a x \operatorname{arccot}\left (a x\right ) +{\left (a^{2} x^{2} + 1\right )} \operatorname{arccot}\left (a x\right )^{2}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.695179, size = 53, normalized size = 0.9 \begin{align*} a^{2} \log{\left (x \right )} - \frac{a^{2} \log{\left (a^{2} x^{2} + 1 \right )}}{2} - \frac{a^{2} \operatorname{acot}^{2}{\left (a x \right )}}{2} + \frac{a \operatorname{acot}{\left (a x \right )}}{x} - \frac{\operatorname{acot}^{2}{\left (a x \right )}}{2 x^{2}} \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\right )^{2}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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