Optimal. Leaf size=178 \[ 2 \cot ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac {2}{1+i a x}\right )-\frac {3}{2} i \cot ^{-1}(a x)^2 \text {PolyLog}\left (2,1-\frac {2 i}{i+a x}\right )+\frac {3}{2} i \cot ^{-1}(a x)^2 \text {PolyLog}\left (2,1-\frac {2 a x}{i+a x}\right )-\frac {3}{2} \cot ^{-1}(a x) \text {PolyLog}\left (3,1-\frac {2 i}{i+a x}\right )+\frac {3}{2} \cot ^{-1}(a x) \text {PolyLog}\left (3,1-\frac {2 a x}{i+a x}\right )+\frac {3}{4} i \text {PolyLog}\left (4,1-\frac {2 i}{i+a x}\right )-\frac {3}{4} i \text {PolyLog}\left (4,1-\frac {2 a x}{i+a x}\right ) \]
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Rubi [A]
time = 0.23, 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 = {4943, 5109,
5005, 5113, 5117, 6745} \begin {gather*} \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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 4943
Rule 5005
Rule 5109
Rule 5113
Rule 5117
Rule 6745
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 = 0.06, size = 180, normalized size = 1.01 \begin {gather*} \frac {1}{64} i \left (\pi ^4-32 \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 )-96 \cot ^{-1}(a x)^2 \text {PolyLog}\left (2,e^{-2 i \cot ^{-1}(a x)}\right )-96 \cot ^{-1}(a x)^2 \text {PolyLog}\left (2,-e^{2 i \cot ^{-1}(a x)}\right )+96 i \cot ^{-1}(a x) \text {PolyLog}\left (3,e^{-2 i \cot ^{-1}(a x)}\right )-96 i \cot ^{-1}(a x) \text {PolyLog}\left (3,-e^{2 i \cot ^{-1}(a x)}\right )+48 \text {PolyLog}\left (4,e^{-2 i \cot ^{-1}(a x)}\right )+48 \text {PolyLog}\left (4,-e^{2 i \cot ^{-1}(a x)}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.91, size = 1050, normalized size = 5.90
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1050\) |
default | \(\text {Expression too large to display}\) | \(1050\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {acot}^{3}{\left (a x \right )}}{x}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\mathrm {acot}\left (a\,x\right )}^3}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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