Optimal. Leaf size=103 \[ \frac {3 i \text {Li}_2\left (1-\frac {2}{i a x+1}\right )}{2 a^2}+\frac {\cot ^{-1}(a x)^3}{2 a^2}+\frac {3 i \cot ^{-1}(a x)^2}{2 a^2}-\frac {3 \log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{a^2}+\frac {1}{2} x^2 \cot ^{-1}(a x)^3+\frac {3 x \cot ^{-1}(a x)^2}{2 a} \]
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Rubi [A] time = 0.17, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, 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 2315
Rule 2402
Rule 4847
Rule 4853
Rule 4855
Rule 4885
Rule 4917
Rule 4921
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*}
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Mathematica [A] time = 0.10, size = 76, normalized size = 0.74 \[ \frac {\cot ^{-1}(a x) \left (\left (a^2 x^2+1\right ) \cot ^{-1}(a x)^2+3 (a x+i) \cot ^{-1}(a x)-6 \log \left (1-e^{2 i \cot ^{-1}(a x)}\right )\right )+3 i \text {Li}_2\left (e^{2 i \cot ^{-1}(a x)}\right )}{2 a^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \operatorname {arccot}\left (a x\right )^{3}, x\right ) \]
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 \[ \int x \operatorname {arccot}\left (a x\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.73, size = 162, normalized size = 1.57 \[ \frac {x^{2} \mathrm {arccot}\left (a x \right )^{3}}{2}+\frac {\mathrm {arccot}\left (a x \right )^{3}}{2 a^{2}}+\frac {3 x \mathrm {arccot}\left (a x \right )^{2}}{2 a}+\frac {3 i \mathrm {arccot}\left (a x \right )^{2}}{2 a^{2}}-\frac {3 \,\mathrm {arccot}\left (a x \right ) \ln \left (1-\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )}{a^{2}}-\frac {3 \,\mathrm {arccot}\left (a x \right ) \ln \left (1+\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )}{a^{2}}+\frac {3 i \polylog \left (2, -\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )}{a^{2}}+\frac {3 i \polylog \left (2, \frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int x\,{\mathrm {acot}\left (a\,x\right )}^3 \,d x \]
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 \[ \int x \operatorname {acot}^{3}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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