Optimal. Leaf size=152 \[ -i a^4 \text {Li}_2\left (\frac {2}{1-i a x}-1\right )+\frac {1}{4} a^4 \tan ^{-1}(a x)+\frac {1}{4} a^4 \cot ^{-1}(a x)^3-i a^4 \cot ^{-1}(a x)^2-2 a^4 \log \left (2-\frac {2}{1-i a x}\right ) \cot ^{-1}(a x)+\frac {a^3}{4 x}-\frac {3 a^3 \cot ^{-1}(a x)^2}{4 x}-\frac {a^2 \cot ^{-1}(a x)}{4 x^2}-\frac {\cot ^{-1}(a x)^3}{4 x^4}+\frac {a \cot ^{-1}(a x)^2}{4 x^3} \]
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Rubi [A] time = 0.42, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {4853, 4919, 325, 203, 4925, 4869, 2447, 4885} \[ -i a^4 \text {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )-\frac {a^2 \cot ^{-1}(a x)}{4 x^2}+\frac {a^3}{4 x}+\frac {1}{4} a^4 \tan ^{-1}(a x)+\frac {1}{4} a^4 \cot ^{-1}(a x)^3-i a^4 \cot ^{-1}(a x)^2-\frac {3 a^3 \cot ^{-1}(a x)^2}{4 x}-2 a^4 \log \left (2-\frac {2}{1-i a x}\right ) \cot ^{-1}(a x)+\frac {a \cot ^{-1}(a x)^2}{4 x^3}-\frac {\cot ^{-1}(a x)^3}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 203
Rule 325
Rule 2447
Rule 4853
Rule 4869
Rule 4885
Rule 4919
Rule 4925
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}(a x)^3}{x^5} \, dx &=-\frac {\cot ^{-1}(a x)^3}{4 x^4}-\frac {1}{4} (3 a) \int \frac {\cot ^{-1}(a x)^2}{x^4 \left (1+a^2 x^2\right )} \, dx\\ &=-\frac {\cot ^{-1}(a x)^3}{4 x^4}-\frac {1}{4} (3 a) \int \frac {\cot ^{-1}(a x)^2}{x^4} \, dx+\frac {1}{4} \left (3 a^3\right ) \int \frac {\cot ^{-1}(a x)^2}{x^2 \left (1+a^2 x^2\right )} \, dx\\ &=\frac {a \cot ^{-1}(a x)^2}{4 x^3}-\frac {\cot ^{-1}(a x)^3}{4 x^4}+\frac {1}{2} a^2 \int \frac {\cot ^{-1}(a x)}{x^3 \left (1+a^2 x^2\right )} \, dx+\frac {1}{4} \left (3 a^3\right ) \int \frac {\cot ^{-1}(a x)^2}{x^2} \, dx-\frac {1}{4} \left (3 a^5\right ) \int \frac {\cot ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=\frac {a \cot ^{-1}(a x)^2}{4 x^3}-\frac {3 a^3 \cot ^{-1}(a x)^2}{4 x}+\frac {1}{4} a^4 \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^3}{4 x^4}+\frac {1}{2} a^2 \int \frac {\cot ^{-1}(a x)}{x^3} \, dx-\frac {1}{2} a^4 \int \frac {\cot ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx-\frac {1}{2} \left (3 a^4\right ) \int \frac {\cot ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx\\ &=-\frac {a^2 \cot ^{-1}(a x)}{4 x^2}-i a^4 \cot ^{-1}(a x)^2+\frac {a \cot ^{-1}(a x)^2}{4 x^3}-\frac {3 a^3 \cot ^{-1}(a x)^2}{4 x}+\frac {1}{4} a^4 \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^3}{4 x^4}-\frac {1}{4} a^3 \int \frac {1}{x^2 \left (1+a^2 x^2\right )} \, dx-\frac {1}{2} \left (i a^4\right ) \int \frac {\cot ^{-1}(a x)}{x (i+a x)} \, dx-\frac {1}{2} \left (3 i a^4\right ) \int \frac {\cot ^{-1}(a x)}{x (i+a x)} \, dx\\ &=\frac {a^3}{4 x}-\frac {a^2 \cot ^{-1}(a x)}{4 x^2}-i a^4 \cot ^{-1}(a x)^2+\frac {a \cot ^{-1}(a x)^2}{4 x^3}-\frac {3 a^3 \cot ^{-1}(a x)^2}{4 x}+\frac {1}{4} a^4 \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^3}{4 x^4}-2 a^4 \cot ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )+\frac {1}{4} a^5 \int \frac {1}{1+a^2 x^2} \, dx-\frac {1}{2} a^5 \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx-\frac {1}{2} \left (3 a^5\right ) \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx\\ &=\frac {a^3}{4 x}-\frac {a^2 \cot ^{-1}(a x)}{4 x^2}-i a^4 \cot ^{-1}(a x)^2+\frac {a \cot ^{-1}(a x)^2}{4 x^3}-\frac {3 a^3 \cot ^{-1}(a x)^2}{4 x}+\frac {1}{4} a^4 \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^3}{4 x^4}+\frac {1}{4} a^4 \tan ^{-1}(a x)-2 a^4 \cot ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )-i a^4 \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )\\ \end {align*}
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Mathematica [A] time = 0.28, size = 126, normalized size = 0.83 \[ \frac {4 i a^4 x^4 \text {Li}_2\left (-e^{2 i \cot ^{-1}(a x)}\right )+\left (a^4 x^4-1\right ) \cot ^{-1}(a x)^3+a^3 x^3-a^2 x^2 \cot ^{-1}(a x) \left (a^2 x^2+8 a^2 x^2 \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )+1\right )+\left (4 i a^4 x^4-3 a^3 x^3+a x\right ) \cot ^{-1}(a x)^2}{4 x^4} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arccot}\left (a x\right )^{3}}{x^{5}}, x\right ) \]
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 \[ \int \frac {\operatorname {arccot}\left (a x\right )^{3}}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.59, size = 158, normalized size = 1.04 \[ -\frac {\mathrm {arccot}\left (a x \right )^{3}}{4 x^{4}}+\frac {a^{4} \mathrm {arccot}\left (a x \right )^{3}}{4}+i a^{4} \mathrm {arccot}\left (a x \right )^{2}-\frac {a^{4} \mathrm {arccot}\left (a x \right )}{4}-\frac {3 a^{3} \mathrm {arccot}\left (a x \right )^{2}}{4 x}+\frac {i a^{4}}{4}+\frac {a^{3}}{4 x}-\frac {a^{2} \mathrm {arccot}\left (a x \right )}{4 x^{2}}+\frac {a \mathrm {arccot}\left (a x \right )^{2}}{4 x^{3}}-2 a^{4} \mathrm {arccot}\left (a x \right ) \ln \left (\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}+1\right )+i a^{4} \polylog \left (2, -\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}\right ) \]
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 \[ \int \frac {{\mathrm {acot}\left (a\,x\right )}^3}{x^5} \,d x \]
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 \[ \int \frac {\operatorname {acot}^{3}{\left (a x \right )}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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