Optimal. Leaf size=111 \[ -\frac{i \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{3 a^3}+\frac{x}{3 a^2}-\frac{\tan ^{-1}(a x)}{3 a^3}-\frac{i \cot ^{-1}(a x)^2}{3 a^3}+\frac{2 \log \left (\frac{2}{1+i a x}\right ) \cot ^{-1}(a x)}{3 a^3}+\frac{1}{3} x^3 \cot ^{-1}(a x)^2+\frac{x^2 \cot ^{-1}(a x)}{3 a} \]
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Rubi [A] time = 0.140106, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8, Rules used = {4853, 4917, 321, 203, 4921, 4855, 2402, 2315} \[ -\frac{i \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{3 a^3}+\frac{x}{3 a^2}-\frac{\tan ^{-1}(a x)}{3 a^3}-\frac{i \cot ^{-1}(a x)^2}{3 a^3}+\frac{2 \log \left (\frac{2}{1+i a x}\right ) \cot ^{-1}(a x)}{3 a^3}+\frac{1}{3} x^3 \cot ^{-1}(a x)^2+\frac{x^2 \cot ^{-1}(a x)}{3 a} \]
Antiderivative was successfully verified.
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Rule 4853
Rule 4917
Rule 321
Rule 203
Rule 4921
Rule 4855
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int x^2 \cot ^{-1}(a x)^2 \, dx &=\frac{1}{3} x^3 \cot ^{-1}(a x)^2+\frac{1}{3} (2 a) \int \frac{x^3 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac{1}{3} x^3 \cot ^{-1}(a x)^2+\frac{2 \int x \cot ^{-1}(a x) \, dx}{3 a}-\frac{2 \int \frac{x \cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{3 a}\\ &=\frac{x^2 \cot ^{-1}(a x)}{3 a}-\frac{i \cot ^{-1}(a x)^2}{3 a^3}+\frac{1}{3} x^3 \cot ^{-1}(a x)^2+\frac{1}{3} \int \frac{x^2}{1+a^2 x^2} \, dx+\frac{2 \int \frac{\cot ^{-1}(a x)}{i-a x} \, dx}{3 a^2}\\ &=\frac{x}{3 a^2}+\frac{x^2 \cot ^{-1}(a x)}{3 a}-\frac{i \cot ^{-1}(a x)^2}{3 a^3}+\frac{1}{3} x^3 \cot ^{-1}(a x)^2+\frac{2 \cot ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{3 a^3}-\frac{\int \frac{1}{1+a^2 x^2} \, dx}{3 a^2}+\frac{2 \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{3 a^2}\\ &=\frac{x}{3 a^2}+\frac{x^2 \cot ^{-1}(a x)}{3 a}-\frac{i \cot ^{-1}(a x)^2}{3 a^3}+\frac{1}{3} x^3 \cot ^{-1}(a x)^2-\frac{\tan ^{-1}(a x)}{3 a^3}+\frac{2 \cot ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{3 a^3}-\frac{(2 i) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{3 a^3}\\ &=\frac{x}{3 a^2}+\frac{x^2 \cot ^{-1}(a x)}{3 a}-\frac{i \cot ^{-1}(a x)^2}{3 a^3}+\frac{1}{3} x^3 \cot ^{-1}(a x)^2-\frac{\tan ^{-1}(a x)}{3 a^3}+\frac{2 \cot ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{3 a^3}-\frac{i \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{3 a^3}\\ \end{align*}
Mathematica [A] time = 0.262658, size = 76, normalized size = 0.68 \[ \frac{-i \text{PolyLog}\left (2,e^{2 i \cot ^{-1}(a x)}\right )+\left (a^3 x^3-i\right ) \cot ^{-1}(a x)^2+\cot ^{-1}(a x) \left (a^2 x^2+2 \log \left (1-e^{2 i \cot ^{-1}(a x)}\right )+1\right )+a x}{3 a^3} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.112, size = 213, normalized size = 1.9 \begin{align*}{\frac{{x}^{3} \left ({\rm arccot} \left (ax\right ) \right ) ^{2}}{3}}+{\frac{{x}^{2}{\rm arccot} \left (ax\right )}{3\,a}}-{\frac{{\rm arccot} \left (ax\right )\ln \left ({a}^{2}{x}^{2}+1 \right ) }{3\,{a}^{3}}}+{\frac{x}{3\,{a}^{2}}}-{\frac{\arctan \left ( ax \right ) }{3\,{a}^{3}}}-{\frac{{\frac{i}{12}} \left ( \ln \left ( ax-i \right ) \right ) ^{2}}{{a}^{3}}}-{\frac{{\frac{i}{6}}\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{a}^{3}}}+{\frac{{\frac{i}{6}}\ln \left ( ax-i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{{a}^{3}}}-{\frac{{\frac{i}{6}}{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{a}^{3}}}+{\frac{{\frac{i}{12}} \left ( \ln \left ( ax+i \right ) \right ) ^{2}}{{a}^{3}}}+{\frac{{\frac{i}{6}}\ln \left ( ax+i \right ) \ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{a}^{3}}}-{\frac{{\frac{i}{6}}\ln \left ( ax+i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{{a}^{3}}}+{\frac{{\frac{i}{6}}{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{12} \, x^{3} \arctan \left (1, a x\right )^{2} - \frac{1}{48} \, x^{3} \log \left (a^{2} x^{2} + 1\right )^{2} + \int \frac{36 \, a^{2} x^{4} \arctan \left (1, a x\right )^{2} + 4 \, a^{2} x^{4} \log \left (a^{2} x^{2} + 1\right ) + 8 \, a x^{3} \arctan \left (1, a x\right ) + 36 \, x^{2} \arctan \left (1, a x\right )^{2} + 3 \,{\left (a^{2} x^{4} + x^{2}\right )} \log \left (a^{2} x^{2} + 1\right )^{2}}{48 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} \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 (x^{2} \operatorname{arccot}\left (a x\right )^{2}, 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 x^{2} \operatorname{acot}^{2}{\left (a x \right )}\, 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 x^{2} \operatorname{arccot}\left (a x\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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