Optimal. Leaf size=96 \[ -\frac{3 \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )}{2 a}+\frac{3 i \cot ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{a}+x \cot ^{-1}(a x)^3+\frac{i \cot ^{-1}(a x)^3}{a}-\frac{3 \log \left (\frac{2}{1+i a x}\right ) \cot ^{-1}(a x)^2}{a} \]
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Rubi [A] time = 0.149912, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 6, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {4847, 4921, 4855, 4885, 4995, 6610} \[ -\frac{3 \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )}{2 a}+\frac{3 i \cot ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{a}+x \cot ^{-1}(a x)^3+\frac{i \cot ^{-1}(a x)^3}{a}-\frac{3 \log \left (\frac{2}{1+i a x}\right ) \cot ^{-1}(a x)^2}{a} \]
Antiderivative was successfully verified.
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Rule 4847
Rule 4921
Rule 4855
Rule 4885
Rule 4995
Rule 6610
Rubi steps
\begin{align*} \int \cot ^{-1}(a x)^3 \, dx &=x \cot ^{-1}(a x)^3+(3 a) \int \frac{x \cot ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=\frac{i \cot ^{-1}(a x)^3}{a}+x \cot ^{-1}(a x)^3-3 \int \frac{\cot ^{-1}(a x)^2}{i-a x} \, dx\\ &=\frac{i \cot ^{-1}(a x)^3}{a}+x \cot ^{-1}(a x)^3-\frac{3 \cot ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{a}-6 \int \frac{\cot ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=\frac{i \cot ^{-1}(a x)^3}{a}+x \cot ^{-1}(a x)^3-\frac{3 \cot ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{a}+\frac{3 i \cot ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{a}+3 i \int \frac{\text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=\frac{i \cot ^{-1}(a x)^3}{a}+x \cot ^{-1}(a x)^3-\frac{3 \cot ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{a}+\frac{3 i \cot ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{a}-\frac{3 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )}{2 a}\\ \end{align*}
Mathematica [A] time = 0.104136, size = 90, normalized size = 0.94 \[ -\frac{3 i \cot ^{-1}(a x) \text{PolyLog}\left (2,e^{-2 i \cot ^{-1}(a x)}\right )}{a}-\frac{3 \text{PolyLog}\left (3,e^{-2 i \cot ^{-1}(a x)}\right )}{2 a}+x \cot ^{-1}(a x)^3-\frac{i \cot ^{-1}(a x)^3}{a}-\frac{3 \cot ^{-1}(a x)^2 \log \left (1-e^{-2 i \cot ^{-1}(a x)}\right )}{a} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.163, size = 199, normalized size = 2.1 \begin{align*} x \left ({\rm arccot} \left (ax\right ) \right ) ^{3}+{\frac{i \left ({\rm arccot} \left (ax\right ) \right ) ^{3}}{a}}-3\,{\frac{ \left ({\rm arccot} \left (ax\right ) \right ) ^{2}}{a}\ln \left ( 1-{\frac{ax+i}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) }-3\,{\frac{ \left ({\rm arccot} \left (ax\right ) \right ) ^{2}}{a}\ln \left ( 1+{\frac{ax+i}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) }+{\frac{6\,i{\rm arccot} \left (ax\right )}{a}{\it polylog} \left ( 2,{(ax+i){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) }+{\frac{6\,i{\rm arccot} \left (ax\right )}{a}{\it polylog} \left ( 2,-{(ax+i){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) }-6\,{\frac{1}{a}{\it polylog} \left ( 3,-{\frac{ax+i}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) }-6\,{\frac{1}{a}{\it polylog} \left ( 3,{\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}{8} \, x \arctan \left (1, a x\right )^{3} - \frac{3}{32} \, x \arctan \left (1, a x\right ) \log \left (a^{2} x^{2} + 1\right )^{2} + \frac{21 \, \arctan \left (a x\right )^{2} \arctan \left (\frac{1}{a x}\right )^{2}}{16 \, a} + \frac{7 \, \arctan \left (a x\right ) \arctan \left (\frac{1}{a x}\right )^{3}}{8 \, a} + 28 \, a^{2} \int \frac{x^{2} \arctan \left (\frac{1}{a x}\right )^{3}}{32 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + 3 \, a^{2} \int \frac{x^{2} \arctan \left (\frac{1}{a x}\right ) \log \left (a^{2} x^{2} + 1\right )^{2}}{32 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + 12 \, a^{2} \int \frac{x^{2} \arctan \left (\frac{1}{a x}\right ) \log \left (a^{2} x^{2} + 1\right )}{32 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + 12 \, a \int \frac{x \arctan \left (\frac{1}{a x}\right )^{2}}{32 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} - 3 \, a \int \frac{x \log \left (a^{2} x^{2} + 1\right )^{2}}{32 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + \frac{7 \,{\left (a \arctan \left (a x\right )^{4} + 4 \, a \arctan \left (a x\right )^{3} \arctan \left (\frac{1}{a x}\right )\right )}}{32 \, a^{2}} + 3 \, \int \frac{\arctan \left (\frac{1}{a x}\right ) \log \left (a^{2} x^{2} + 1\right )^{2}}{32 \,{\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 )^{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 \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 \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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