Optimal. Leaf size=72 \[ -\frac {1}{2} i \text {Li}_2\left (\frac {2}{1-i x}-1\right )-\frac {\cot ^{-1}(x)}{2 x^2}+\frac {1}{2 x}+\frac {1}{2} \tan ^{-1}(x)-\frac {1}{2} i \cot ^{-1}(x)^2-\log \left (2-\frac {2}{1-i x}\right ) \cot ^{-1}(x) \]
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Rubi [A] time = 0.11, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {4919, 4853, 325, 203, 4925, 4869, 2447} \[ -\frac {1}{2} i \text {PolyLog}\left (2,-1+\frac {2}{1-i x}\right )-\frac {\cot ^{-1}(x)}{2 x^2}+\frac {1}{2 x}+\frac {1}{2} \tan ^{-1}(x)-\frac {1}{2} i \cot ^{-1}(x)^2-\log \left (2-\frac {2}{1-i x}\right ) \cot ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 203
Rule 325
Rule 2447
Rule 4853
Rule 4869
Rule 4919
Rule 4925
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}(x)}{x^3 \left (1+x^2\right )} \, dx &=\int \frac {\cot ^{-1}(x)}{x^3} \, dx-\int \frac {\cot ^{-1}(x)}{x \left (1+x^2\right )} \, dx\\ &=-\frac {\cot ^{-1}(x)}{2 x^2}-\frac {1}{2} i \cot ^{-1}(x)^2-i \int \frac {\cot ^{-1}(x)}{x (i+x)} \, dx-\frac {1}{2} \int \frac {1}{x^2 \left (1+x^2\right )} \, dx\\ &=\frac {1}{2 x}-\frac {\cot ^{-1}(x)}{2 x^2}-\frac {1}{2} i \cot ^{-1}(x)^2-\cot ^{-1}(x) \log \left (2-\frac {2}{1-i x}\right )+\frac {1}{2} \int \frac {1}{1+x^2} \, dx-\int \frac {\log \left (2-\frac {2}{1-i x}\right )}{1+x^2} \, dx\\ &=\frac {1}{2 x}-\frac {\cot ^{-1}(x)}{2 x^2}-\frac {1}{2} i \cot ^{-1}(x)^2+\frac {1}{2} \tan ^{-1}(x)-\cot ^{-1}(x) \log \left (2-\frac {2}{1-i x}\right )-\frac {1}{2} i \text {Li}_2\left (-1+\frac {2}{1-i x}\right )\\ \end {align*}
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Mathematica [C] time = 0.07, size = 280, normalized size = 3.89 \[ \frac {\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-x^2\right )}{2 x}+\frac {1}{4} i \text {Li}_2\left (-\frac {1}{2} i (i-x)\right )+\frac {1}{2} i \text {Li}_2\left (-\frac {i}{x}\right )-\frac {1}{2} i \text {Li}_2\left (\frac {i}{x}\right )-\frac {1}{4} i \text {Li}_2\left (-\frac {1}{2} i (x+i)\right )-\frac {\cot ^{-1}(x)}{2 x^2}-\frac {1}{8} i \log ^2(-x+i)+\frac {1}{8} i \log ^2(x+i)+\frac {1}{4} i \log \left (-\frac {-x+i}{x}\right ) \log (-x+i)+\frac {1}{4} i \log \left (-\frac {1}{2} i (x+i)\right ) \log (-x+i)-\frac {1}{4} i \log \left (\frac {x+i}{x}\right ) \log (-x+i)-\frac {1}{4} i \log \left (-\frac {1}{2} i (-x+i)\right ) \log (x+i)+\frac {1}{4} i \log \left (-\frac {-x+i}{x}\right ) \log (x+i)-\frac {1}{4} i \log (x+i) \log \left (\frac {x+i}{x}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arccot}\relax (x)}{x^{5} + x^{3}}, x\right ) \]
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 \[ \int \frac {\operatorname {arccot}\relax (x)}{{\left (x^{2} + 1\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 180, normalized size = 2.50 \[ -\frac {\mathrm {arccot}\relax (x )}{2 x^{2}}-\mathrm {arccot}\relax (x ) \ln \relax (x )+\frac {\mathrm {arccot}\relax (x ) \ln \left (x^{2}+1\right )}{2}+\frac {i \dilog \left (-\frac {i \left (x +i\right )}{2}\right )}{4}-\frac {i \ln \left (x +i\right ) \ln \left (\frac {i \left (x -i\right )}{2}\right )}{4}-\frac {i \ln \relax (x ) \ln \left (-i x +1\right )}{2}+\frac {i \ln \relax (x ) \ln \left (i x +1\right )}{2}-\frac {i \ln \left (x -i\right ) \ln \left (x^{2}+1\right )}{4}-\frac {i \dilog \left (\frac {i \left (x -i\right )}{2}\right )}{4}-\frac {i \ln \left (x +i\right )^{2}}{8}+\frac {i \dilog \left (i x +1\right )}{2}+\frac {1}{2 x}+\frac {\arctan \relax (x )}{2}-\frac {i \dilog \left (-i x +1\right )}{2}+\frac {i \ln \left (x -i\right ) \ln \left (-\frac {i \left (x +i\right )}{2}\right )}{4}+\frac {i \ln \left (x -i\right )^{2}}{8}+\frac {i \ln \left (x +i\right ) \ln \left (x^{2}+1\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arccot}\relax (x)}{{\left (x^{2} + 1\right )} x^{3}}\,{d x} \]
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 \[ \int \frac {\mathrm {acot}\relax (x)}{x^3\,\left (x^2+1\right )} \,d x \]
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 \[ \int \frac {\operatorname {acot}{\relax (x )}}{x^{3} \left (x^{2} + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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