Integrand size = 15, antiderivative size = 212 \[ \int \frac {\cot ^{-1}(c x)}{x^2 \left (1+x^2\right )} \, dx=-\frac {\cot ^{-1}(c x)}{x}-\frac {1}{2} i \arctan (x) \log \left (1-\frac {i}{c x}\right )+\frac {1}{2} i \arctan (x) \log \left (1+\frac {i}{c x}\right )-c \log (x)+\frac {1}{2} i \arctan (x) \log \left (-\frac {2 i (i-c x)}{(1-c) (1-i x)}\right )-\frac {1}{2} i \arctan (x) \log \left (-\frac {2 i (i+c x)}{(1+c) (1-i x)}\right )+\frac {1}{2} c \log \left (1+c^2 x^2\right )+\frac {1}{4} \operatorname {PolyLog}\left (2,1+\frac {2 i (i-c x)}{(1-c) (1-i x)}\right )-\frac {1}{4} \operatorname {PolyLog}\left (2,1+\frac {2 i (i+c x)}{(1+c) (1-i x)}\right ) \]
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Time = 0.38 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.267, Rules used = {5039, 4947, 272, 36, 29, 31, 5029, 209, 2520, 266, 6820, 12, 4996, 4940, 2438, 4966, 2449, 2352, 2497} \[ \int \frac {\cot ^{-1}(c x)}{x^2 \left (1+x^2\right )} \, dx=-\frac {1}{2} i \arctan (x) \log \left (1-\frac {i}{c x}\right )+\frac {1}{2} i \arctan (x) \log \left (1+\frac {i}{c x}\right )+\frac {1}{2} i \arctan (x) \log \left (-\frac {2 i (-c x+i)}{(1-c) (1-i x)}\right )-\frac {1}{2} i \arctan (x) \log \left (-\frac {2 i (c x+i)}{(c+1) (1-i x)}\right )+\frac {1}{2} c \log \left (c^2 x^2+1\right )+\frac {1}{4} \operatorname {PolyLog}\left (2,\frac {2 i (i-c x)}{(1-c) (1-i x)}+1\right )-\frac {1}{4} \operatorname {PolyLog}\left (2,\frac {2 i (c x+i)}{(c+1) (1-i x)}+1\right )-c \log (x)-\frac {\cot ^{-1}(c x)}{x} \]
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Rule 12
Rule 29
Rule 31
Rule 36
Rule 209
Rule 266
Rule 272
Rule 2352
Rule 2438
Rule 2449
Rule 2497
Rule 2520
Rule 4940
Rule 4947
Rule 4966
Rule 4996
Rule 5029
Rule 5039
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {\cot ^{-1}(c x)}{x^2} \, dx-\int \frac {\cot ^{-1}(c x)}{1+x^2} \, dx \\ & = -\frac {\cot ^{-1}(c x)}{x}-\frac {1}{2} i \int \frac {\log \left (1-\frac {i}{c x}\right )}{1+x^2} \, dx+\frac {1}{2} i \int \frac {\log \left (1+\frac {i}{c x}\right )}{1+x^2} \, dx-c \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx \\ & = -\frac {\cot ^{-1}(c x)}{x}-\frac {1}{2} i \arctan (x) \log \left (1-\frac {i}{c x}\right )+\frac {1}{2} i \arctan (x) \log \left (1+\frac {i}{c x}\right )-\frac {\int \frac {\arctan (x)}{\left (1-\frac {i}{c x}\right ) x^2} \, dx}{2 c}-\frac {\int \frac {\arctan (x)}{\left (1+\frac {i}{c x}\right ) x^2} \, dx}{2 c}-\frac {1}{2} c \text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right ) \\ & = -\frac {\cot ^{-1}(c x)}{x}-\frac {1}{2} i \arctan (x) \log \left (1-\frac {i}{c x}\right )+\frac {1}{2} i \arctan (x) \log \left (1+\frac {i}{c x}\right )-\frac {\int \frac {c \arctan (x)}{x (-i+c x)} \, dx}{2 c}-\frac {\int \frac {c \arctan (x)}{x (i+c x)} \, dx}{2 c}-\frac {1}{2} c \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{2} c^3 \text {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right ) \\ & = -\frac {\cot ^{-1}(c x)}{x}-\frac {1}{2} i \arctan (x) \log \left (1-\frac {i}{c x}\right )+\frac {1}{2} i \arctan (x) \log \left (1+\frac {i}{c x}\right )-c \log (x)+\frac {1}{2} c \log \left (1+c^2 x^2\right )-\frac {1}{2} \int \frac {\arctan (x)}{x (-i+c x)} \, dx-\frac {1}{2} \int \frac {\arctan (x)}{x (i+c x)} \, dx \\ & = -\frac {\cot ^{-1}(c x)}{x}-\frac {1}{2} i \arctan (x) \log \left (1-\frac {i}{c x}\right )+\frac {1}{2} i \arctan (x) \log \left (1+\frac {i}{c x}\right )-c \log (x)+\frac {1}{2} c \log \left (1+c^2 x^2\right )-\frac {1}{2} \int \left (\frac {i \arctan (x)}{x}-\frac {i c \arctan (x)}{-i+c x}\right ) \, dx-\frac {1}{2} \int \left (-\frac {i \arctan (x)}{x}+\frac {i c \arctan (x)}{i+c x}\right ) \, dx \\ & = -\frac {\cot ^{-1}(c x)}{x}-\frac {1}{2} i \arctan (x) \log \left (1-\frac {i}{c x}\right )+\frac {1}{2} i \arctan (x) \log \left (1+\frac {i}{c x}\right )-c \log (x)+\frac {1}{2} c \log \left (1+c^2 x^2\right )+\frac {1}{2} (i c) \int \frac {\arctan (x)}{-i+c x} \, dx-\frac {1}{2} (i c) \int \frac {\arctan (x)}{i+c x} \, dx \\ & = -\frac {\cot ^{-1}(c x)}{x}-\frac {1}{2} i \arctan (x) \log \left (1-\frac {i}{c x}\right )+\frac {1}{2} i \arctan (x) \log \left (1+\frac {i}{c x}\right )-c \log (x)+\frac {1}{2} i \arctan (x) \log \left (-\frac {2 i (i-c x)}{(1-c) (1-i x)}\right )-\frac {1}{2} i \arctan (x) \log \left (-\frac {2 i (i+c x)}{(1+c) (1-i x)}\right )+\frac {1}{2} c \log \left (1+c^2 x^2\right )-\frac {1}{2} i \int \frac {\log \left (\frac {2 (-i+c x)}{(-i+i c) (1-i x)}\right )}{1+x^2} \, dx+\frac {1}{2} i \int \frac {\log \left (\frac {2 (i+c x)}{(i+i c) (1-i x)}\right )}{1+x^2} \, dx \\ & = -\frac {\cot ^{-1}(c x)}{x}-\frac {1}{2} i \arctan (x) \log \left (1-\frac {i}{c x}\right )+\frac {1}{2} i \arctan (x) \log \left (1+\frac {i}{c x}\right )-c \log (x)+\frac {1}{2} i \arctan (x) \log \left (-\frac {2 i (i-c x)}{(1-c) (1-i x)}\right )-\frac {1}{2} i \arctan (x) \log \left (-\frac {2 i (i+c x)}{(1+c) (1-i x)}\right )+\frac {1}{2} c \log \left (1+c^2 x^2\right )+\frac {1}{4} \operatorname {PolyLog}\left (2,1+\frac {2 i (i-c x)}{(1-c) (1-i x)}\right )-\frac {1}{4} \operatorname {PolyLog}\left (2,1+\frac {2 i (i+c x)}{(1+c) (1-i x)}\right ) \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(619\) vs. \(2(212)=424\).
Time = 1.42 (sec) , antiderivative size = 619, normalized size of antiderivative = 2.92 \[ \int \frac {\cot ^{-1}(c x)}{x^2 \left (1+x^2\right )} \, dx=-\frac {\cot ^{-1}(c x)}{x}-c \log \left (\frac {1}{\sqrt {1+\frac {1}{c^2 x^2}}}\right )-\frac {c \left (2 \arccos \left (\frac {1+c^2}{-1+c^2}\right ) \text {arctanh}\left (\frac {\sqrt {-c^2}}{c x}\right )-4 \cot ^{-1}(c x) \text {arctanh}\left (\frac {c x}{\sqrt {-c^2}}\right )-\left (\arccos \left (\frac {1+c^2}{-1+c^2}\right )-2 i \text {arctanh}\left (\frac {\sqrt {-c^2}}{c x}\right )\right ) \log \left (-\frac {2 \left (c^2+i \sqrt {-c^2}\right ) (-i+c x)}{\left (-1+c^2\right ) \left (\sqrt {-c^2}-c x\right )}\right )-\left (\arccos \left (\frac {1+c^2}{-1+c^2}\right )+2 i \text {arctanh}\left (\frac {\sqrt {-c^2}}{c x}\right )\right ) \log \left (\frac {2 i \left (i c^2+\sqrt {-c^2}\right ) (i+c x)}{\left (-1+c^2\right ) \left (\sqrt {-c^2}-c x\right )}\right )+\left (\arccos \left (\frac {1+c^2}{-1+c^2}\right )-2 i \text {arctanh}\left (\frac {\sqrt {-c^2}}{c x}\right )+2 i \text {arctanh}\left (\frac {c x}{\sqrt {-c^2}}\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {-c^2} e^{-i \cot ^{-1}(c x)}}{\sqrt {-1+c^2} \sqrt {-1-c^2+\left (-1+c^2\right ) \cos \left (2 \cot ^{-1}(c x)\right )}}\right )+\left (\arccos \left (\frac {1+c^2}{-1+c^2}\right )+2 i \text {arctanh}\left (\frac {\sqrt {-c^2}}{c x}\right )-2 i \text {arctanh}\left (\frac {c x}{\sqrt {-c^2}}\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {-c^2} e^{i \cot ^{-1}(c x)}}{\sqrt {-1+c^2} \sqrt {-1-c^2+\left (-1+c^2\right ) \cos \left (2 \cot ^{-1}(c x)\right )}}\right )+i \left (-\operatorname {PolyLog}\left (2,\frac {\left (1+c^2-2 i \sqrt {-c^2}\right ) \left (\sqrt {-c^2}+c x\right )}{\left (-1+c^2\right ) \left (\sqrt {-c^2}-c x\right )}\right )+\operatorname {PolyLog}\left (2,\frac {\left (1+c^2+2 i \sqrt {-c^2}\right ) \left (\sqrt {-c^2}+c x\right )}{\left (-1+c^2\right ) \left (\sqrt {-c^2}-c x\right )}\right )\right )\right )}{4 \sqrt {-c^2}} \]
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Time = 0.84 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.21
method | result | size |
risch | \(-\frac {\pi \arctan \left (x \right )}{2}-\frac {\pi }{2 x}+\frac {\ln \left (-i c x +1\right ) \ln \left (\frac {-i c x -c}{-c -1}\right )}{4}+\frac {\operatorname {dilog}\left (\frac {-i c x -c}{-c -1}\right )}{4}-\frac {\ln \left (-i c x +1\right ) \ln \left (\frac {-i c x +c}{c -1}\right )}{4}-\frac {\operatorname {dilog}\left (\frac {-i c x +c}{c -1}\right )}{4}-\frac {c \ln \left (-i c x \right )}{2}+\frac {c \ln \left (-i c x +1\right )}{2}+\frac {i \ln \left (-i c x +1\right )}{2 x}+\frac {\ln \left (i c x +1\right ) \ln \left (\frac {i c x -c}{-c -1}\right )}{4}+\frac {\operatorname {dilog}\left (\frac {i c x -c}{-c -1}\right )}{4}-\frac {\ln \left (i c x +1\right ) \ln \left (\frac {i c x +c}{c -1}\right )}{4}-\frac {\operatorname {dilog}\left (\frac {i c x +c}{c -1}\right )}{4}-\frac {c \ln \left (i c x \right )}{2}+\frac {c \ln \left (i c x +1\right )}{2}-\frac {i \ln \left (i c x +1\right )}{2 x}\) | \(257\) |
parts | \(-\frac {\operatorname {arccot}\left (c x \right )}{x}-\operatorname {arccot}\left (c x \right ) \arctan \left (x \right )+c \left (-\ln \left (x \right )+\frac {\ln \left (c^{2} x^{2}+1\right )}{2}-\frac {i \arctan \left (x \right ) \ln \left (1-\frac {\left (c -1\right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (1+c \right )}\right )}{2 c}-\frac {\arctan \left (x \right )^{2}}{2 c}-\frac {\operatorname {polylog}\left (2, \frac {\left (c -1\right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (1+c \right )}\right )}{4 c}+\frac {i \ln \left (1-\frac {\left (1+c \right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (c -1\right )}\right ) \arctan \left (x \right )}{2 c -2}-\frac {i \ln \left (1-\frac {\left (1+c \right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (c -1\right )}\right ) \arctan \left (x \right )}{2 c \left (c -1\right )}+\frac {\arctan \left (x \right )^{2}}{2 c -2}+\frac {\operatorname {polylog}\left (2, \frac {\left (1+c \right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (c -1\right )}\right )}{4 c -4}-\frac {\arctan \left (x \right )^{2}}{2 c \left (c -1\right )}-\frac {\operatorname {polylog}\left (2, \frac {\left (1+c \right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (c -1\right )}\right )}{4 c \left (c -1\right )}\right )\) | \(287\) |
derivativedivides | \(c \left (-\frac {\operatorname {arccot}\left (c x \right ) \arctan \left (x \right )}{c}-\frac {\operatorname {arccot}\left (c x \right )}{c x}+c^{3} \left (-\frac {\ln \left (x \right )-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}}{c^{3}}-\frac {i \arctan \left (x \right ) \ln \left (1-\frac {\left (c -1\right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (1+c \right )}\right )}{2 c^{4}}-\frac {\arctan \left (x \right )^{2}}{2 c^{4}}-\frac {\operatorname {polylog}\left (2, \frac {\left (c -1\right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (1+c \right )}\right )}{4 c^{4}}+\frac {i \ln \left (1-\frac {\left (1+c \right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (c -1\right )}\right ) \arctan \left (x \right )}{2 c^{3} \left (c -1\right )}-\frac {i \ln \left (1-\frac {\left (1+c \right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (c -1\right )}\right ) \arctan \left (x \right )}{2 c^{4} \left (c -1\right )}+\frac {\arctan \left (x \right )^{2}}{2 c^{3} \left (c -1\right )}+\frac {\operatorname {polylog}\left (2, \frac {\left (1+c \right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (c -1\right )}\right )}{4 c^{3} \left (c -1\right )}-\frac {\arctan \left (x \right )^{2}}{2 c^{4} \left (c -1\right )}-\frac {\operatorname {polylog}\left (2, \frac {\left (1+c \right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (c -1\right )}\right )}{4 c^{4} \left (c -1\right )}\right )\right )\) | \(310\) |
default | \(c \left (-\frac {\operatorname {arccot}\left (c x \right ) \arctan \left (x \right )}{c}-\frac {\operatorname {arccot}\left (c x \right )}{c x}+c^{3} \left (-\frac {\ln \left (x \right )-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}}{c^{3}}-\frac {i \arctan \left (x \right ) \ln \left (1-\frac {\left (c -1\right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (1+c \right )}\right )}{2 c^{4}}-\frac {\arctan \left (x \right )^{2}}{2 c^{4}}-\frac {\operatorname {polylog}\left (2, \frac {\left (c -1\right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (1+c \right )}\right )}{4 c^{4}}+\frac {i \ln \left (1-\frac {\left (1+c \right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (c -1\right )}\right ) \arctan \left (x \right )}{2 c^{3} \left (c -1\right )}-\frac {i \ln \left (1-\frac {\left (1+c \right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (c -1\right )}\right ) \arctan \left (x \right )}{2 c^{4} \left (c -1\right )}+\frac {\arctan \left (x \right )^{2}}{2 c^{3} \left (c -1\right )}+\frac {\operatorname {polylog}\left (2, \frac {\left (1+c \right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (c -1\right )}\right )}{4 c^{3} \left (c -1\right )}-\frac {\arctan \left (x \right )^{2}}{2 c^{4} \left (c -1\right )}-\frac {\operatorname {polylog}\left (2, \frac {\left (1+c \right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (c -1\right )}\right )}{4 c^{4} \left (c -1\right )}\right )\right )\) | \(310\) |
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\[ \int \frac {\cot ^{-1}(c x)}{x^2 \left (1+x^2\right )} \, dx=\int { \frac {\operatorname {arccot}\left (c x\right )}{{\left (x^{2} + 1\right )} x^{2}} \,d x } \]
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\[ \int \frac {\cot ^{-1}(c x)}{x^2 \left (1+x^2\right )} \, dx=\int \frac {\operatorname {acot}{\left (c x \right )}}{x^{2} \left (x^{2} + 1\right )}\, dx \]
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Time = 0.32 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.86 \[ \int \frac {\cot ^{-1}(c x)}{x^2 \left (1+x^2\right )} \, dx=-{\left (\frac {1}{x} + \arctan \left (x\right )\right )} \operatorname {arccot}\left (c x\right ) - \frac {1}{2} \, \arctan \left (c x\right ) \arctan \left (x\right ) + \frac {1}{2} \, \arctan \left (x\right ) \arctan \left (\frac {c x}{c - 1}, -\frac {1}{c - 1}\right ) + \frac {1}{2} \, c \log \left (c^{2} x^{2} + 1\right ) - c \log \left (x\right ) - \frac {1}{8} \, \log \left (x^{2} + 1\right ) \log \left (\frac {c^{2} x^{2} + 1}{c^{2} + 2 \, c + 1}\right ) + \frac {1}{8} \, \log \left (x^{2} + 1\right ) \log \left (\frac {c^{2} x^{2} + 1}{c^{2} - 2 \, c + 1}\right ) - \frac {1}{4} \, {\rm Li}_2\left (\frac {i \, c x + c}{c + 1}\right ) - \frac {1}{4} \, {\rm Li}_2\left (-\frac {i \, c x - c}{c + 1}\right ) + \frac {1}{4} \, {\rm Li}_2\left (\frac {i \, c x + c}{c - 1}\right ) + \frac {1}{4} \, {\rm Li}_2\left (-\frac {i \, c x - c}{c - 1}\right ) \]
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\[ \int \frac {\cot ^{-1}(c x)}{x^2 \left (1+x^2\right )} \, dx=\int { \frac {\operatorname {arccot}\left (c x\right )}{{\left (x^{2} + 1\right )} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\cot ^{-1}(c x)}{x^2 \left (1+x^2\right )} \, dx=\int \frac {\mathrm {acot}\left (c\,x\right )}{x^2\,\left (x^2+1\right )} \,d x \]
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