Optimal. Leaf size=103 \[ \frac{3 i \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{2 a^2}+\frac{\cot ^{-1}(a x)^3}{2 a^2}+\frac{3 i \cot ^{-1}(a x)^2}{2 a^2}-\frac{3 \log \left (\frac{2}{1+i a x}\right ) \cot ^{-1}(a x)}{a^2}+\frac{1}{2} x^2 \cot ^{-1}(a x)^3+\frac{3 x \cot ^{-1}(a x)^2}{2 a} \]
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Rubi [A] time = 0.170855, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {4853, 4917, 4847, 4921, 4855, 2402, 2315, 4885} \[ \frac{3 i \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{2 a^2}+\frac{\cot ^{-1}(a x)^3}{2 a^2}+\frac{3 i \cot ^{-1}(a x)^2}{2 a^2}-\frac{3 \log \left (\frac{2}{1+i a x}\right ) \cot ^{-1}(a x)}{a^2}+\frac{1}{2} x^2 \cot ^{-1}(a x)^3+\frac{3 x \cot ^{-1}(a x)^2}{2 a} \]
Antiderivative was successfully verified.
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Rule 4853
Rule 4917
Rule 4847
Rule 4921
Rule 4855
Rule 2402
Rule 2315
Rule 4885
Rubi steps
\begin{align*} \int x \cot ^{-1}(a x)^3 \, dx &=\frac{1}{2} x^2 \cot ^{-1}(a x)^3+\frac{1}{2} (3 a) \int \frac{x^2 \cot ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=\frac{1}{2} x^2 \cot ^{-1}(a x)^3+\frac{3 \int \cot ^{-1}(a x)^2 \, dx}{2 a}-\frac{3 \int \frac{\cot ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{2 a}\\ &=\frac{3 x \cot ^{-1}(a x)^2}{2 a}+\frac{\cot ^{-1}(a x)^3}{2 a^2}+\frac{1}{2} x^2 \cot ^{-1}(a x)^3+3 \int \frac{x \cot ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac{3 i \cot ^{-1}(a x)^2}{2 a^2}+\frac{3 x \cot ^{-1}(a x)^2}{2 a}+\frac{\cot ^{-1}(a x)^3}{2 a^2}+\frac{1}{2} x^2 \cot ^{-1}(a x)^3-\frac{3 \int \frac{\cot ^{-1}(a x)}{i-a x} \, dx}{a}\\ &=\frac{3 i \cot ^{-1}(a x)^2}{2 a^2}+\frac{3 x \cot ^{-1}(a x)^2}{2 a}+\frac{\cot ^{-1}(a x)^3}{2 a^2}+\frac{1}{2} x^2 \cot ^{-1}(a x)^3-\frac{3 \cot ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{a^2}-\frac{3 \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a}\\ &=\frac{3 i \cot ^{-1}(a x)^2}{2 a^2}+\frac{3 x \cot ^{-1}(a x)^2}{2 a}+\frac{\cot ^{-1}(a x)^3}{2 a^2}+\frac{1}{2} x^2 \cot ^{-1}(a x)^3-\frac{3 \cot ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{a^2}+\frac{(3 i) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{a^2}\\ &=\frac{3 i \cot ^{-1}(a x)^2}{2 a^2}+\frac{3 x \cot ^{-1}(a x)^2}{2 a}+\frac{\cot ^{-1}(a x)^3}{2 a^2}+\frac{1}{2} x^2 \cot ^{-1}(a x)^3-\frac{3 \cot ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{a^2}+\frac{3 i \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{2 a^2}\\ \end{align*}
Mathematica [A] time = 0.0857338, size = 76, normalized size = 0.74 \[ \frac{3 i \text{PolyLog}\left (2,e^{2 i \cot ^{-1}(a x)}\right )+\cot ^{-1}(a x) \left (\left (a^2 x^2+1\right ) \cot ^{-1}(a x)^2+3 (a x+i) \cot ^{-1}(a x)-6 \log \left (1-e^{2 i \cot ^{-1}(a x)}\right )\right )}{2 a^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.316, size = 162, normalized size = 1.6 \begin{align*}{\frac{{x}^{2} \left ({\rm arccot} \left (ax\right ) \right ) ^{3}}{2}}+{\frac{ \left ({\rm arccot} \left (ax\right ) \right ) ^{3}}{2\,{a}^{2}}}+{\frac{3\,x \left ({\rm arccot} \left (ax\right ) \right ) ^{2}}{2\,a}}+{\frac{{\frac{3\,i}{2}} \left ({\rm arccot} \left (ax\right ) \right ) ^{2}}{{a}^{2}}}-3\,{\frac{{\rm arccot} \left (ax\right )}{{a}^{2}}\ln \left ( 1-{\frac{ax+i}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) }-3\,{\frac{{\rm arccot} \left (ax\right )}{{a}^{2}}\ln \left ( 1+{\frac{ax+i}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) }+{\frac{3\,i}{{a}^{2}}{\it polylog} \left ( 2,-{(ax+i){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) }+{\frac{3\,i}{{a}^{2}}{\it polylog} \left ( 2,{(ax+i){\frac{1}{\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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \operatorname{arccot}\left (a x\right )^{3}, 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 \operatorname{acot}^{3}{\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 \operatorname{arccot}\left (a x\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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