Optimal. Leaf size=66 \[ -i a \text {Li}_2\left (\frac {2}{1-i a x}-1\right )-i a \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{x}-2 a \log \left (2-\frac {2}{1-i a x}\right ) \cot ^{-1}(a x) \]
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Rubi [A] time = 0.11, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4853, 4925, 4869, 2447} \[ -i a \text {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )-i a \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{x}-2 a \log \left (2-\frac {2}{1-i a x}\right ) \cot ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 2447
Rule 4853
Rule 4869
Rule 4925
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}(a x)^2}{x^2} \, dx &=-\frac {\cot ^{-1}(a x)^2}{x}-(2 a) \int \frac {\cot ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx\\ &=-i a \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{x}-(2 i a) \int \frac {\cot ^{-1}(a x)}{x (i+a x)} \, dx\\ &=-i a \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{x}-2 a \cot ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )-\left (2 a^2\right ) \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx\\ &=-i a \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{x}-2 a \cot ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )-i a \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 64, normalized size = 0.97 \[ a \left (i \text {Li}_2\left (-e^{2 i \cot ^{-1}(a x)}\right )-\frac {\cot ^{-1}(a x)^2}{a x}+i \cot ^{-1}(a x)^2-2 \cot ^{-1}(a x) \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arccot}\left (a x\right )^{2}}{x^{2}}, x\right ) \]
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 \[ \int \frac {\operatorname {arccot}\left (a x\right )^{2}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.14, size = 234, normalized size = 3.55 \[ -\frac {\mathrm {arccot}\left (a x \right )^{2}}{x}-2 a \,\mathrm {arccot}\left (a x \right ) \ln \left (a x \right )+a \,\mathrm {arccot}\left (a x \right ) \ln \left (a^{2} x^{2}+1\right )+i a \ln \left (a x \right ) \ln \left (i a x +1\right )-i a \ln \left (a x \right ) \ln \left (-i a x +1\right )+i a \dilog \left (i a x +1\right )-i a \dilog \left (-i a x +1\right )-\frac {i a \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )}{2}+\frac {i a \ln \left (a x -i\right )^{2}}{4}+\frac {i a \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )}{2}+\frac {i a \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )}{2}+\frac {i a \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )}{2}-\frac {i a \ln \left (a x +i\right )^{2}}{4}-\frac {i a \dilog \left (\frac {i \left (a x -i\right )}{2}\right )}{2}-\frac {i a \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )}{2} \]
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 = -0.02 \[ \int \frac {{\mathrm {acot}\left (a\,x\right )}^2}{x^2} \,d x \]
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 \[ \int \frac {\operatorname {acot}^{2}{\left (a x \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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