Optimal. Leaf size=167 \[ \frac {1}{2} a^3 \text {Li}_3\left (\frac {2}{1-i a x}-1\right )+i a^3 \text {Li}_2\left (\frac {2}{1-i a x}-1\right ) \cot ^{-1}(a x)-a^3 \log (x)+\frac {1}{3} i a^3 \cot ^{-1}(a x)^3+\frac {1}{2} a^3 \cot ^{-1}(a x)^2+a^3 \log \left (2-\frac {2}{1-i a x}\right ) \cot ^{-1}(a x)^2-\frac {a^2 \cot ^{-1}(a x)}{x}+\frac {1}{2} a^3 \log \left (a^2 x^2+1\right )-\frac {\cot ^{-1}(a x)^3}{3 x^3}+\frac {a \cot ^{-1}(a x)^2}{2 x^2} \]
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Rubi [A] time = 0.34, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {4853, 4919, 266, 36, 29, 31, 4885, 4925, 4869, 4993, 6610} \[ \frac {1}{2} a^3 \text {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )+i a^3 \cot ^{-1}(a x) \text {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+\frac {1}{2} a^3 \log \left (a^2 x^2+1\right )-a^3 \log (x)+\frac {1}{3} i a^3 \cot ^{-1}(a x)^3+\frac {1}{2} a^3 \cot ^{-1}(a x)^2-\frac {a^2 \cot ^{-1}(a x)}{x}+a^3 \log \left (2-\frac {2}{1-i a x}\right ) \cot ^{-1}(a x)^2+\frac {a \cot ^{-1}(a x)^2}{2 x^2}-\frac {\cot ^{-1}(a x)^3}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 4853
Rule 4869
Rule 4885
Rule 4919
Rule 4925
Rule 4993
Rule 6610
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}(a x)^3}{x^4} \, dx &=-\frac {\cot ^{-1}(a x)^3}{3 x^3}-a \int \frac {\cot ^{-1}(a x)^2}{x^3 \left (1+a^2 x^2\right )} \, dx\\ &=-\frac {\cot ^{-1}(a x)^3}{3 x^3}-a \int \frac {\cot ^{-1}(a x)^2}{x^3} \, dx+a^3 \int \frac {\cot ^{-1}(a x)^2}{x \left (1+a^2 x^2\right )} \, dx\\ &=\frac {a \cot ^{-1}(a x)^2}{2 x^2}+\frac {1}{3} i a^3 \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^3}{3 x^3}+a^2 \int \frac {\cot ^{-1}(a x)}{x^2 \left (1+a^2 x^2\right )} \, dx+\left (i a^3\right ) \int \frac {\cot ^{-1}(a x)^2}{x (i+a x)} \, dx\\ &=\frac {a \cot ^{-1}(a x)^2}{2 x^2}+\frac {1}{3} i a^3 \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^3}{3 x^3}+a^3 \cot ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )+a^2 \int \frac {\cot ^{-1}(a x)}{x^2} \, dx-a^4 \int \frac {\cot ^{-1}(a x)}{1+a^2 x^2} \, dx+\left (2 a^4\right ) \int \frac {\cot ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac {a^2 \cot ^{-1}(a x)}{x}+\frac {1}{2} a^3 \cot ^{-1}(a x)^2+\frac {a \cot ^{-1}(a x)^2}{2 x^2}+\frac {1}{3} i a^3 \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^3}{3 x^3}+a^3 \cot ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )+i a^3 \cot ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )-a^3 \int \frac {1}{x \left (1+a^2 x^2\right )} \, dx+\left (i a^4\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac {a^2 \cot ^{-1}(a x)}{x}+\frac {1}{2} a^3 \cot ^{-1}(a x)^2+\frac {a \cot ^{-1}(a x)^2}{2 x^2}+\frac {1}{3} i a^3 \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^3}{3 x^3}+a^3 \cot ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )+i a^3 \cot ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )+\frac {1}{2} a^3 \text {Li}_3\left (-1+\frac {2}{1-i a x}\right )-\frac {1}{2} a^3 \operatorname {Subst}\left (\int \frac {1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {a^2 \cot ^{-1}(a x)}{x}+\frac {1}{2} a^3 \cot ^{-1}(a x)^2+\frac {a \cot ^{-1}(a x)^2}{2 x^2}+\frac {1}{3} i a^3 \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^3}{3 x^3}+a^3 \cot ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )+i a^3 \cot ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )+\frac {1}{2} a^3 \text {Li}_3\left (-1+\frac {2}{1-i a x}\right )-\frac {1}{2} a^3 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{2} a^5 \operatorname {Subst}\left (\int \frac {1}{1+a^2 x} \, dx,x,x^2\right )\\ &=-\frac {a^2 \cot ^{-1}(a x)}{x}+\frac {1}{2} a^3 \cot ^{-1}(a x)^2+\frac {a \cot ^{-1}(a x)^2}{2 x^2}+\frac {1}{3} i a^3 \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^3}{3 x^3}-a^3 \log (x)+\frac {1}{2} a^3 \log \left (1+a^2 x^2\right )+a^3 \cot ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )+i a^3 \cot ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )+\frac {1}{2} a^3 \text {Li}_3\left (-1+\frac {2}{1-i a x}\right )\\ \end {align*}
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Mathematica [A] time = 0.26, size = 151, normalized size = 0.90 \[ \frac {1}{6} \left (-6 i a^3 \cot ^{-1}(a x) \text {Li}_2\left (-e^{2 i \cot ^{-1}(a x)}\right )+3 a^3 \text {Li}_3\left (-e^{2 i \cot ^{-1}(a x)}\right )-2 i a^3 \cot ^{-1}(a x)^3+3 a^3 \cot ^{-1}(a x)^2+6 a^3 \cot ^{-1}(a x)^2 \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )-\frac {6 a^2 \cot ^{-1}(a x)}{x}-6 a^3 \log \left (\frac {1}{\sqrt {\frac {1}{a^2 x^2}+1}}\right )-\frac {2 \cot ^{-1}(a x)^3}{x^3}+\frac {3 a \cot ^{-1}(a x)^2}{x^2}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arccot}\left (a x\right )^{3}}{x^{4}}, x\right ) \]
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 \[ \int \frac {\operatorname {arccot}\left (a x\right )^{3}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 5.04, size = 5029, normalized size = 30.11 \[ \text {output too large to display} \]
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 \[ \int \frac {{\mathrm {acot}\left (a\,x\right )}^3}{x^4} \,d x \]
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 \[ \int \frac {\operatorname {acot}^{3}{\left (a x \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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