Optimal. Leaf size=67 \[ \frac{i \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{a}+x \cot ^{-1}(a x)^2+\frac{i \cot ^{-1}(a x)^2}{a}-\frac{2 \log \left (\frac{2}{1+i a x}\right ) \cot ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.074329, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {4847, 4921, 4855, 2402, 2315} \[ \frac{i \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{a}+x \cot ^{-1}(a x)^2+\frac{i \cot ^{-1}(a x)^2}{a}-\frac{2 \log \left (\frac{2}{1+i a x}\right ) \cot ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Rule 4847
Rule 4921
Rule 4855
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \cot ^{-1}(a x)^2 \, dx &=x \cot ^{-1}(a x)^2+(2 a) \int \frac{x \cot ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac{i \cot ^{-1}(a x)^2}{a}+x \cot ^{-1}(a x)^2-2 \int \frac{\cot ^{-1}(a x)}{i-a x} \, dx\\ &=\frac{i \cot ^{-1}(a x)^2}{a}+x \cot ^{-1}(a x)^2-\frac{2 \cot ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{a}-2 \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=\frac{i \cot ^{-1}(a x)^2}{a}+x \cot ^{-1}(a x)^2-\frac{2 \cot ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{a}+\frac{(2 i) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{a}\\ &=\frac{i \cot ^{-1}(a x)^2}{a}+x \cot ^{-1}(a x)^2-\frac{2 \cot ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{a}+\frac{i \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0758582, size = 56, normalized size = 0.84 \[ \frac{i \text{PolyLog}\left (2,e^{2 i \cot ^{-1}(a x)}\right )+\cot ^{-1}(a x) \left ((a x+i) \cot ^{-1}(a x)-2 \log \left (1-e^{2 i \cot ^{-1}(a x)}\right )\right )}{a} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.164, size = 136, normalized size = 2. \begin{align*} x \left ({\rm arccot} \left (ax\right ) \right ) ^{2}+{\frac{i \left ({\rm arccot} \left (ax\right ) \right ) ^{2}}{a}}+{\frac{2\,i}{a}{\it polylog} \left ( 2,-{(ax+i){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) }+{\frac{2\,i}{a}{\it polylog} \left ( 2,{(ax+i){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) }-2\,{\frac{{\rm arccot} \left (ax\right )}{a}\ln \left ( 1+{\frac{ax+i}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) }-2\,{\frac{{\rm arccot} \left (ax\right )}{a}\ln \left ( 1-{\frac{ax+i}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) } \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}{4} \, x \arctan \left (1, a x\right )^{2} + 12 \, a^{2} \int \frac{x^{2} \arctan \left (\frac{1}{a x}\right )^{2}}{16 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + a^{2} \int \frac{x^{2} \log \left (a^{2} x^{2} + 1\right )^{2}}{16 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + 4 \, a^{2} \int \frac{x^{2} \log \left (a^{2} x^{2} + 1\right )}{16 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} - \frac{1}{16} \, x \log \left (a^{2} x^{2} + 1\right )^{2} + \frac{\arctan \left (a x\right )^{3}}{4 \, a} + \frac{3 \, \arctan \left (a x\right )^{2} \arctan \left (\frac{1}{a x}\right )}{4 \, a} + \frac{3 \, \arctan \left (a x\right ) \arctan \left (\frac{1}{a x}\right )^{2}}{4 \, a} + 8 \, a \int \frac{x \arctan \left (\frac{1}{a x}\right )}{16 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + \int \frac{\log \left (a^{2} x^{2} + 1\right )^{2}}{16 \,{\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 (\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 \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 \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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