Optimal. Leaf size=96 \[ -\frac {3 \text {Li}_3\left (1-\frac {2}{i a x+1}\right )}{2 a}+\frac {3 i \text {Li}_2\left (1-\frac {2}{i a x+1}\right ) \cot ^{-1}(a x)}{a}+x \cot ^{-1}(a x)^3+\frac {i \cot ^{-1}(a x)^3}{a}-\frac {3 \log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)^2}{a} \]
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Rubi [A] time = 0.15, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, 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 4855
Rule 4885
Rule 4921
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*}
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Mathematica [A] time = 0.12, size = 90, normalized size = 0.94 \[ -\frac {3 i \cot ^{-1}(a x) \text {Li}_2\left (e^{-2 i \cot ^{-1}(a x)}\right )}{a}-\frac {3 \text {Li}_3\left (e^{-2 i \cot ^{-1}(a x)}\right )}{2 a}+x \cot ^{-1}(a x)^3-\frac {i \cot ^{-1}(a x)^3}{a}-\frac {3 \cot ^{-1}(a x)^2 \log \left (1-e^{-2 i \cot ^{-1}(a x)}\right )}{a} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {arccot}\left (a x\right )^{3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {arccot}\left (a x\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.45, size = 199, normalized size = 2.07 \[ x \mathrm {arccot}\left (a x \right )^{3}+\frac {i \mathrm {arccot}\left (a x \right )^{3}}{a}-\frac {3 \mathrm {arccot}\left (a x \right )^{2} \ln \left (1+\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )}{a}-\frac {3 \mathrm {arccot}\left (a x \right )^{2} \ln \left (1-\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )}{a}+\frac {6 i \mathrm {arccot}\left (a x \right ) \polylog \left (2, -\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )}{a}+\frac {6 i \mathrm {arccot}\left (a x \right ) \polylog \left (2, \frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )}{a}-\frac {6 \polylog \left (3, \frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )}{a}-\frac {6 \polylog \left (3, -\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{8} \, x \arctan \left (1, a x\right )^{3} - \frac {3}{32} \, x \arctan \left (1, a x\right ) \log \left (a^{2} x^{2} + 1\right )^{2} + \frac {21 \, \arctan \left (a x\right )^{2} \arctan \left (\frac {1}{a x}\right )^{2}}{16 \, a} + \frac {7 \, \arctan \left (a x\right ) \arctan \left (\frac {1}{a x}\right )^{3}}{8 \, a} + 28 \, a^{2} \int \frac {x^{2} \arctan \left (\frac {1}{a x}\right )^{3}}{32 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} + 3 \, a^{2} \int \frac {x^{2} \arctan \left (\frac {1}{a x}\right ) \log \left (a^{2} x^{2} + 1\right )^{2}}{32 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} + 12 \, a^{2} \int \frac {x^{2} \arctan \left (\frac {1}{a x}\right ) \log \left (a^{2} x^{2} + 1\right )}{32 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} + 12 \, a \int \frac {x \arctan \left (\frac {1}{a x}\right )^{2}}{32 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} - 3 \, a \int \frac {x \log \left (a^{2} x^{2} + 1\right )^{2}}{32 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} + \frac {7 \, {\left (a \arctan \left (a x\right )^{4} + 4 \, a \arctan \left (a x\right )^{3} \arctan \left (\frac {1}{a x}\right )\right )}}{32 \, a^{2}} + 3 \, \int \frac {\arctan \left (\frac {1}{a x}\right ) \log \left (a^{2} x^{2} + 1\right )^{2}}{32 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {acot}\left (a\,x\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {acot}^{3}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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