Optimal. Leaf size=178 \[ \frac {3}{4} i \text {Li}_4\left (1-\frac {2 i}{a x+i}\right )-\frac {3}{4} i \text {Li}_4\left (1-\frac {2 a x}{a x+i}\right )-\frac {3}{2} i \text {Li}_2\left (1-\frac {2 i}{a x+i}\right ) \cot ^{-1}(a x)^2+\frac {3}{2} i \text {Li}_2\left (1-\frac {2 a x}{a x+i}\right ) \cot ^{-1}(a x)^2-\frac {3}{2} \text {Li}_3\left (1-\frac {2 i}{a x+i}\right ) \cot ^{-1}(a x)+\frac {3}{2} \text {Li}_3\left (1-\frac {2 a x}{a x+i}\right ) \cot ^{-1}(a x)+2 \cot ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac {2}{1+i a x}\right ) \]
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Rubi [A] time = 0.33, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4851, 4989, 4885, 4993, 4997, 6610} \[ \frac {3}{4} i \text {PolyLog}\left (4,1-\frac {2 i}{a x+i}\right )-\frac {3}{4} i \text {PolyLog}\left (4,1-\frac {2 a x}{a x+i}\right )-\frac {3}{2} i \cot ^{-1}(a x)^2 \text {PolyLog}\left (2,1-\frac {2 i}{a x+i}\right )+\frac {3}{2} i \cot ^{-1}(a x)^2 \text {PolyLog}\left (2,1-\frac {2 a x}{a x+i}\right )-\frac {3}{2} \cot ^{-1}(a x) \text {PolyLog}\left (3,1-\frac {2 i}{a x+i}\right )+\frac {3}{2} \cot ^{-1}(a x) \text {PolyLog}\left (3,1-\frac {2 a x}{a x+i}\right )+2 \cot ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac {2}{1+i a x}\right ) \]
Antiderivative was successfully verified.
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Rule 4851
Rule 4885
Rule 4989
Rule 4993
Rule 4997
Rule 6610
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}(a x)^3}{x} \, dx &=2 \cot ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac {2}{1+i a x}\right )+(6 a) \int \frac {\cot ^{-1}(a x)^2 \coth ^{-1}\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=2 \cot ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac {2}{1+i a x}\right )-(3 a) \int \frac {\cot ^{-1}(a x)^2 \log \left (\frac {2 i}{i+a x}\right )}{1+a^2 x^2} \, dx+(3 a) \int \frac {\cot ^{-1}(a x)^2 \log \left (\frac {2 a x}{i+a x}\right )}{1+a^2 x^2} \, dx\\ &=2 \cot ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac {2}{1+i a x}\right )-\frac {3}{2} i \cot ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2 i}{i+a x}\right )+\frac {3}{2} i \cot ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2 a x}{i+a x}\right )-(3 i a) \int \frac {\cot ^{-1}(a x) \text {Li}_2\left (1-\frac {2 i}{i+a x}\right )}{1+a^2 x^2} \, dx+(3 i a) \int \frac {\cot ^{-1}(a x) \text {Li}_2\left (1-\frac {2 a x}{i+a x}\right )}{1+a^2 x^2} \, dx\\ &=2 \cot ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac {2}{1+i a x}\right )-\frac {3}{2} i \cot ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2 i}{i+a x}\right )+\frac {3}{2} i \cot ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2 a x}{i+a x}\right )-\frac {3}{2} \cot ^{-1}(a x) \text {Li}_3\left (1-\frac {2 i}{i+a x}\right )+\frac {3}{2} \cot ^{-1}(a x) \text {Li}_3\left (1-\frac {2 a x}{i+a x}\right )-\frac {1}{2} (3 a) \int \frac {\text {Li}_3\left (1-\frac {2 i}{i+a x}\right )}{1+a^2 x^2} \, dx+\frac {1}{2} (3 a) \int \frac {\text {Li}_3\left (1-\frac {2 a x}{i+a x}\right )}{1+a^2 x^2} \, dx\\ &=2 \cot ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac {2}{1+i a x}\right )-\frac {3}{2} i \cot ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2 i}{i+a x}\right )+\frac {3}{2} i \cot ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2 a x}{i+a x}\right )-\frac {3}{2} \cot ^{-1}(a x) \text {Li}_3\left (1-\frac {2 i}{i+a x}\right )+\frac {3}{2} \cot ^{-1}(a x) \text {Li}_3\left (1-\frac {2 a x}{i+a x}\right )+\frac {3}{4} i \text {Li}_4\left (1-\frac {2 i}{i+a x}\right )-\frac {3}{4} i \text {Li}_4\left (1-\frac {2 a x}{i+a x}\right )\\ \end {align*}
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Mathematica [A] time = 0.08, size = 180, normalized size = 1.01 \[ \frac {1}{64} i \left (-96 \cot ^{-1}(a x)^2 \text {Li}_2\left (e^{-2 i \cot ^{-1}(a x)}\right )-96 \cot ^{-1}(a x)^2 \text {Li}_2\left (-e^{2 i \cot ^{-1}(a x)}\right )+96 i \cot ^{-1}(a x) \text {Li}_3\left (e^{-2 i \cot ^{-1}(a x)}\right )-96 i \cot ^{-1}(a x) \text {Li}_3\left (-e^{2 i \cot ^{-1}(a x)}\right )+48 \text {Li}_4\left (e^{-2 i \cot ^{-1}(a x)}\right )+48 \text {Li}_4\left (-e^{2 i \cot ^{-1}(a x)}\right )-32 \cot ^{-1}(a x)^4+64 i \cot ^{-1}(a x)^3 \log \left (1-e^{-2 i \cot ^{-1}(a x)}\right )-64 i \cot ^{-1}(a x)^3 \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )+\pi ^4\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arccot}\left (a x\right )^{3}}{x}, x\right ) \]
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 \[ \int \frac {\operatorname {arccot}\left (a x\right )^{3}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.57, size = 1050, normalized size = 5.90 \[ \text {result too large to display} \]
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 \[ \int \frac {\operatorname {arccot}\left (a x\right )^{3}}{x}\,{d x} \]
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 \[ \int \frac {{\mathrm {acot}\left (a\,x\right )}^3}{x} \,d x \]
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 \[ \int \frac {\operatorname {acot}^{3}{\left (a x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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