Optimal. Leaf size=105 \[ \frac {3}{2} i a^2 \text {Li}_2\left (\frac {2}{1-i a x}-1\right )-\frac {1}{2} a^2 \cot ^{-1}(a x)^3+\frac {3}{2} i a^2 \cot ^{-1}(a x)^2+3 a^2 \log \left (2-\frac {2}{1-i a x}\right ) \cot ^{-1}(a x)-\frac {\cot ^{-1}(a x)^3}{2 x^2}+\frac {3 a \cot ^{-1}(a x)^2}{2 x} \]
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Rubi [A] time = 0.20, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4853, 4919, 4925, 4869, 2447, 4885} \[ \frac {3}{2} i a^2 \text {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )-\frac {1}{2} a^2 \cot ^{-1}(a x)^3+\frac {3}{2} i a^2 \cot ^{-1}(a x)^2+3 a^2 \log \left (2-\frac {2}{1-i a x}\right ) \cot ^{-1}(a x)-\frac {\cot ^{-1}(a x)^3}{2 x^2}+\frac {3 a \cot ^{-1}(a x)^2}{2 x} \]
Antiderivative was successfully verified.
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Rule 2447
Rule 4853
Rule 4869
Rule 4885
Rule 4919
Rule 4925
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}(a x)^3}{x^3} \, dx &=-\frac {\cot ^{-1}(a x)^3}{2 x^2}-\frac {1}{2} (3 a) \int \frac {\cot ^{-1}(a x)^2}{x^2 \left (1+a^2 x^2\right )} \, dx\\ &=-\frac {\cot ^{-1}(a x)^3}{2 x^2}-\frac {1}{2} (3 a) \int \frac {\cot ^{-1}(a x)^2}{x^2} \, dx+\frac {1}{2} \left (3 a^3\right ) \int \frac {\cot ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=\frac {3 a \cot ^{-1}(a x)^2}{2 x}-\frac {1}{2} a^2 \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^3}{2 x^2}+\left (3 a^2\right ) \int \frac {\cot ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx\\ &=\frac {3}{2} i a^2 \cot ^{-1}(a x)^2+\frac {3 a \cot ^{-1}(a x)^2}{2 x}-\frac {1}{2} a^2 \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^3}{2 x^2}+\left (3 i a^2\right ) \int \frac {\cot ^{-1}(a x)}{x (i+a x)} \, dx\\ &=\frac {3}{2} i a^2 \cot ^{-1}(a x)^2+\frac {3 a \cot ^{-1}(a x)^2}{2 x}-\frac {1}{2} a^2 \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^3}{2 x^2}+3 a^2 \cot ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )+\left (3 a^3\right ) \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx\\ &=\frac {3}{2} i a^2 \cot ^{-1}(a x)^2+\frac {3 a \cot ^{-1}(a x)^2}{2 x}-\frac {1}{2} a^2 \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^3}{2 x^2}+3 a^2 \cot ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )+\frac {3}{2} i a^2 \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )\\ \end {align*}
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Mathematica [A] time = 0.18, size = 90, normalized size = 0.86 \[ -\frac {3}{2} i a^2 \text {Li}_2\left (-e^{2 i \cot ^{-1}(a x)}\right )-\frac {\cot ^{-1}(a x) \left (\left (a^2 x^2+1\right ) \cot ^{-1}(a x)^2-6 a^2 x^2 \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )+3 i a x (a x+i) \cot ^{-1}(a x)\right )}{2 x^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arccot}\left (a x\right )^{3}}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 29, normalized size = 0.28 \[ -\frac {1}{2} \, a \arctan \left (\frac {1}{a x}\right )^{3} - \frac {\arctan \left (\frac {1}{a x}\right )^{3}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.76, size = 109, normalized size = 1.04 \[ -\frac {\mathrm {arccot}\left (a x \right )^{3}}{2 x^{2}}-\frac {a^{2} \mathrm {arccot}\left (a x \right )^{3}}{2}-\frac {3 i a^{2} \mathrm {arccot}\left (a x \right )^{2}}{2}+\frac {3 a \mathrm {arccot}\left (a x \right )^{2}}{2 x}+3 a^{2} \mathrm {arccot}\left (a x \right ) \ln \left (\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}+1\right )-\frac {3 i a^{2} \polylog \left (2, -\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}\right )}{2} \]
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 \[ \int \frac {{\mathrm {acot}\left (a\,x\right )}^3}{x^3} \,d x \]
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 \[ \int \frac {\operatorname {acot}^{3}{\left (a x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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