Optimal. Leaf size=72 \[ \frac {1}{2 x}-\frac {\cot ^{-1}(x)}{2 x^2}-\frac {1}{2} i \cot ^{-1}(x)^2+\frac {\text {ArcTan}(x)}{2}-\cot ^{-1}(x) \log \left (2-\frac {2}{1-i x}\right )-\frac {1}{2} i \text {PolyLog}\left (2,-1+\frac {2}{1-i x}\right ) \]
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Rubi [A]
time = 0.09, 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 = {5039, 4947,
331, 209, 5045, 4989, 2497} \begin {gather*} \frac {\text {ArcTan}(x)}{2}-\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} i \cot ^{-1}(x)^2-\log \left (2-\frac {2}{1-i x}\right ) \cot ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 331
Rule 2497
Rule 4947
Rule 4989
Rule 5039
Rule 5045
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] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.04, size = 280, normalized size = 3.89 \begin {gather*} -\frac {\cot ^{-1}(x)}{2 x^2}+\frac {\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-x^2\right )}{2 x}-\frac {1}{8} i \log ^2(i-x)+\frac {1}{4} i \log (i-x) \log \left (-\frac {i-x}{x}\right )+\frac {1}{4} i \log (i-x) \log \left (-\frac {1}{2} i (i+x)\right )-\frac {1}{4} i \log \left (-\frac {1}{2} i (i-x)\right ) \log (i+x)+\frac {1}{4} i \log \left (-\frac {i-x}{x}\right ) \log (i+x)+\frac {1}{8} i \log ^2(i+x)-\frac {1}{4} i \log (i-x) \log \left (\frac {i+x}{x}\right )-\frac {1}{4} i \log (i+x) \log \left (\frac {i+x}{x}\right )+\frac {1}{4} i \text {PolyLog}\left (2,-\frac {1}{2} i (i-x)\right )+\frac {1}{2} i \text {PolyLog}\left (2,-\frac {i}{x}\right )-\frac {1}{2} i \text {PolyLog}\left (2,\frac {i}{x}\right )-\frac {1}{4} i \text {PolyLog}\left (2,-\frac {1}{2} i (i+x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 179 vs. \(2 (58 ) = 116\).
time = 0.14, size = 180, normalized size = 2.50
method | result | size |
risch | \(\frac {\pi \ln \left (x^{2}+1\right )}{4}-\frac {i \ln \left (-i x \right )}{4}-\frac {\pi }{4 x^{2}}-\frac {\pi \ln \left (-i x \right )}{2}+\frac {i \dilog \left (\frac {1}{2}-\frac {i x}{2}\right )}{4}+\frac {i \ln \left (i x +1\right )^{2}}{8}-\frac {i \ln \left (-i x +1\right )^{2}}{8}-\frac {i \dilog \left (-i x +1\right )}{2}-\frac {i \ln \left (\frac {1}{2}+\frac {i x}{2}\right ) \ln \left (-i x +1\right )}{4}-\frac {i \ln \left (i x +1\right )}{4 x^{2}}+\frac {i \dilog \left (i x +1\right )}{2}+\frac {1}{2 x}+\frac {i \ln \left (\frac {1}{2}-\frac {i x}{2}\right ) \ln \left (i x +1\right )}{4}+\frac {i \ln \left (i x \right )}{4}+\frac {i \ln \left (-i x +1\right )}{4 x^{2}}+\frac {\arctan \left (x \right )}{2}-\frac {i \dilog \left (\frac {1}{2}+\frac {i x}{2}\right )}{4}\) | \(174\) |
default | \(-\frac {\mathrm {arccot}\left (x \right )}{2 x^{2}}-\mathrm {arccot}\left (x \right ) \ln \left (x \right )+\frac {\mathrm {arccot}\left (x \right ) \ln \left (x^{2}+1\right )}{2}-\frac {i \ln \left (x -i\right ) \ln \left (x^{2}+1\right )}{4}+\frac {i \ln \left (i+x \right ) \ln \left (x^{2}+1\right )}{4}-\frac {i \ln \left (i+x \right )^{2}}{8}+\frac {i \ln \left (x -i\right ) \ln \left (-\frac {i \left (i+x \right )}{2}\right )}{4}-\frac {i \dilog \left (-i x +1\right )}{2}+\frac {i \ln \left (x \right ) \ln \left (i x +1\right )}{2}+\frac {i \dilog \left (i x +1\right )}{2}-\frac {i \dilog \left (\frac {i \left (x -i\right )}{2}\right )}{4}+\frac {1}{2 x}+\frac {\arctan \left (x \right )}{2}-\frac {i \ln \left (i+x \right ) \ln \left (\frac {i \left (x -i\right )}{2}\right )}{4}-\frac {i \ln \left (x \right ) \ln \left (-i x +1\right )}{2}+\frac {i \dilog \left (-\frac {i \left (i+x \right )}{2}\right )}{4}+\frac {i \ln \left (x -i\right )^{2}}{8}\) | \(180\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {acot}{\left (x \right )}}{x^{3} \left (x^{2} + 1\right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\mathrm {acot}\left (x\right )}{x^3\,\left (x^2+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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