Optimal. Leaf size=188 \[ -\frac {1}{2} i \text {Li}_2\left (1-\frac {2}{1-i c x}\right )+\frac {1}{4} i \text {Li}_2\left (1-\frac {2 i c (i-x)}{(1-c) (1-i c x)}\right )+\frac {1}{4} i \text {Li}_2\left (\frac {2 i c (x+i)}{(c+1) (1-i c x)}+1\right )+\log \left (\frac {2}{1-i c x}\right ) \left (-\cot ^{-1}(c x)\right )+\frac {1}{2} \log \left (\frac {2 i c (-x+i)}{(1-c) (1-i c x)}\right ) \cot ^{-1}(c x)+\frac {1}{2} \log \left (-\frac {2 i c (x+i)}{(c+1) (1-i c x)}\right ) \cot ^{-1}(c x) \]
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Rubi [A] time = 0.18, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {4929, 4857, 2402, 2315, 2447} \[ -\frac {1}{2} i \text {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )+\frac {1}{4} i \text {PolyLog}\left (2,1-\frac {2 i c (-x+i)}{(1-c) (1-i c x)}\right )+\frac {1}{4} i \text {PolyLog}\left (2,1+\frac {2 i c (x+i)}{(c+1) (1-i c x)}\right )+\log \left (\frac {2}{1-i c x}\right ) \left (-\cot ^{-1}(c x)\right )+\frac {1}{2} \log \left (\frac {2 i c (-x+i)}{(1-c) (1-i c x)}\right ) \cot ^{-1}(c x)+\frac {1}{2} \log \left (-\frac {2 i c (x+i)}{(c+1) (1-i c x)}\right ) \cot ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2402
Rule 2447
Rule 4857
Rule 4929
Rubi steps
\begin {align*} \int \frac {x \cot ^{-1}(c x)}{1+x^2} \, dx &=\int \left (-\frac {\cot ^{-1}(c x)}{2 (i-x)}+\frac {\cot ^{-1}(c x)}{2 (i+x)}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {\cot ^{-1}(c x)}{i-x} \, dx\right )+\frac {1}{2} \int \frac {\cot ^{-1}(c x)}{i+x} \, dx\\ &=-\cot ^{-1}(c x) \log \left (\frac {2}{1-i c x}\right )+\frac {1}{2} \cot ^{-1}(c x) \log \left (\frac {2 i c (i-x)}{(1-c) (1-i c x)}\right )+\frac {1}{2} \cot ^{-1}(c x) \log \left (-\frac {2 i c (i+x)}{(1+c) (1-i c x)}\right )-2 \left (\frac {1}{2} c \int \frac {\log \left (\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx\right )+\frac {1}{2} c \int \frac {\log \left (\frac {2 c (i-x)}{(-i+i c) (1-i c x)}\right )}{1+c^2 x^2} \, dx+\frac {1}{2} c \int \frac {\log \left (\frac {2 c (i+x)}{(i+i c) (1-i c x)}\right )}{1+c^2 x^2} \, dx\\ &=-\cot ^{-1}(c x) \log \left (\frac {2}{1-i c x}\right )+\frac {1}{2} \cot ^{-1}(c x) \log \left (\frac {2 i c (i-x)}{(1-c) (1-i c x)}\right )+\frac {1}{2} \cot ^{-1}(c x) \log \left (-\frac {2 i c (i+x)}{(1+c) (1-i c x)}\right )+\frac {1}{4} i \text {Li}_2\left (1-\frac {2 i c (i-x)}{(1-c) (1-i c x)}\right )+\frac {1}{4} i \text {Li}_2\left (1+\frac {2 i c (i+x)}{(1+c) (1-i c x)}\right )-2 \left (\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i c x}\right )\right )\\ &=-\cot ^{-1}(c x) \log \left (\frac {2}{1-i c x}\right )+\frac {1}{2} \cot ^{-1}(c x) \log \left (\frac {2 i c (i-x)}{(1-c) (1-i c x)}\right )+\frac {1}{2} \cot ^{-1}(c x) \log \left (-\frac {2 i c (i+x)}{(1+c) (1-i c x)}\right )-\frac {1}{2} i \text {Li}_2\left (1-\frac {2}{1-i c x}\right )+\frac {1}{4} i \text {Li}_2\left (1-\frac {2 i c (i-x)}{(1-c) (1-i c x)}\right )+\frac {1}{4} i \text {Li}_2\left (1+\frac {2 i c (i+x)}{(1+c) (1-i c x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.09, size = 343, normalized size = 1.82 \[ -\frac {1}{4} i \text {Li}_2\left (\frac {i c (i-x)}{1-c}\right )+\frac {1}{4} i \text {Li}_2\left (-\frac {i c (i-x)}{c+1}\right )+\frac {1}{4} i \text {Li}_2\left (\frac {i c (x+i)}{1-c}\right )-\frac {1}{4} i \text {Li}_2\left (-\frac {i c (x+i)}{c+1}\right )-\frac {1}{4} i \log (-x+i) \log \left (-\frac {i (-c x+i)}{1-c}\right )-\frac {1}{4} i \log (x+i) \log \left (-\frac {i (-c x+i)}{c+1}\right )+\frac {1}{4} i \log (-x+i) \log \left (-\frac {-c x+i}{c x}\right )+\frac {1}{4} i \log (x+i) \log \left (-\frac {-c x+i}{c x}\right )+\frac {1}{4} i \log (x+i) \log \left (-\frac {i (c x+i)}{1-c}\right )+\frac {1}{4} i \log (-x+i) \log \left (-\frac {i (c x+i)}{c+1}\right )-\frac {1}{4} i \log (-x+i) \log \left (\frac {c x+i}{c x}\right )-\frac {1}{4} i \log (x+i) \log \left (\frac {c x+i}{c x}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 2.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x \operatorname {arccot}\left (c x\right )}{x^{2} + 1}, x\right ) \]
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 \[ \int \frac {x \operatorname {arccot}\left (c x\right )}{x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 284, normalized size = 1.51 \[ \frac {\ln \left (c^{2} x^{2}+c^{2}\right ) \mathrm {arccot}\left (c x \right )}{2}+\frac {i \ln \left (c x -i\right ) \ln \left (\frac {i \left (c x -i\right )-c -1}{-c -1}\right )}{4}+\frac {i \ln \left (c x -i\right ) \ln \left (\frac {i \left (c x -i\right )+c -1}{-1+c}\right )}{4}-\frac {i \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+c^{2}\right )}{4}+\frac {i \dilog \left (\frac {i \left (c x -i\right )-c -1}{-c -1}\right )}{4}+\frac {i \dilog \left (\frac {i \left (c x -i\right )+c -1}{-1+c}\right )}{4}-\frac {i \ln \left (c x +i\right ) \ln \left (\frac {-i \left (c x +i\right )-c -1}{-c -1}\right )}{4}-\frac {i \ln \left (c x +i\right ) \ln \left (\frac {-i \left (c x +i\right )+c -1}{-1+c}\right )}{4}+\frac {i \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+c^{2}\right )}{4}-\frac {i \dilog \left (\frac {-i \left (c x +i\right )+c -1}{-1+c}\right )}{4}-\frac {i \dilog \left (\frac {-i \left (c x +i\right )-c -1}{-c -1}\right )}{4} \]
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 \[ \int \frac {x \operatorname {arccot}\left (c x\right )}{x^{2} + 1}\,{d x} \]
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 \[ \int \frac {x\,\mathrm {acot}\left (c\,x\right )}{x^2+1} \,d x \]
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 \[ \int \frac {x \operatorname {acot}{\left (c x \right )}}{x^{2} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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