Integrand size = 10, antiderivative size = 152 \[ \int \frac {\cot ^{-1}(a x)^3}{x^5} \, dx=\frac {a^3}{4 x}-\frac {a^2 \cot ^{-1}(a x)}{4 x^2}-i a^4 \cot ^{-1}(a x)^2+\frac {a \cot ^{-1}(a x)^2}{4 x^3}-\frac {3 a^3 \cot ^{-1}(a x)^2}{4 x}+\frac {1}{4} a^4 \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^3}{4 x^4}+\frac {1}{4} a^4 \arctan (a x)-2 a^4 \cot ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )-i a^4 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right ) \]
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Time = 0.30 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {4947, 5039, 331, 209, 5045, 4989, 2497, 5005} \[ \int \frac {\cot ^{-1}(a x)^3}{x^5} \, dx=\frac {1}{4} a^4 \arctan (a x)-i a^4 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )+\frac {1}{4} a^4 \cot ^{-1}(a x)^3-i a^4 \cot ^{-1}(a x)^2-2 a^4 \log \left (2-\frac {2}{1-i a x}\right ) \cot ^{-1}(a x)+\frac {a^3}{4 x}-\frac {3 a^3 \cot ^{-1}(a x)^2}{4 x}-\frac {a^2 \cot ^{-1}(a x)}{4 x^2}-\frac {\cot ^{-1}(a x)^3}{4 x^4}+\frac {a \cot ^{-1}(a x)^2}{4 x^3} \]
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Rule 209
Rule 331
Rule 2497
Rule 4947
Rule 4989
Rule 5005
Rule 5039
Rule 5045
Rubi steps \begin{align*} \text {integral}& = -\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 \arctan (a x)-2 a^4 \cot ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )-i a^4 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right ) \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.83 \[ \int \frac {\cot ^{-1}(a x)^3}{x^5} \, dx=\frac {a^3 x^3+\left (a x-3 a^3 x^3+4 i a^4 x^4\right ) \cot ^{-1}(a x)^2+\left (-1+a^4 x^4\right ) \cot ^{-1}(a x)^3-a^2 x^2 \cot ^{-1}(a x) \left (1+a^2 x^2+8 a^2 x^2 \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )\right )+4 i a^4 x^4 \operatorname {PolyLog}\left (2,-e^{2 i \cot ^{-1}(a x)}\right )}{4 x^4} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 10.74 (sec) , antiderivative size = 852, normalized size of antiderivative = 5.61
method | result | size |
parts | \(\text {Expression too large to display}\) | \(852\) |
derivativedivides | \(\text {Expression too large to display}\) | \(855\) |
default | \(\text {Expression too large to display}\) | \(855\) |
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\[ \int \frac {\cot ^{-1}(a x)^3}{x^5} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )^{3}}{x^{5}} \,d x } \]
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\[ \int \frac {\cot ^{-1}(a x)^3}{x^5} \, dx=\int \frac {\operatorname {acot}^{3}{\left (a x \right )}}{x^{5}}\, dx \]
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Timed out. \[ \int \frac {\cot ^{-1}(a x)^3}{x^5} \, dx=\text {Timed out} \]
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\[ \int \frac {\cot ^{-1}(a x)^3}{x^5} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )^{3}}{x^{5}} \,d x } \]
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Timed out. \[ \int \frac {\cot ^{-1}(a x)^3}{x^5} \, dx=\int \frac {{\mathrm {acot}\left (a\,x\right )}^3}{x^5} \,d x \]
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