Integrand size = 21, antiderivative size = 21 \[ \int \frac {\cot ^{-1}(c+(1-i c) \cot (a+b x))}{x} \, dx=\text {Int}\left (\frac {\cot ^{-1}(c+(1-i c) \cot (a+b x))}{x},x\right ) \]
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Not integrable
Time = 0.09 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\cot ^{-1}(c+(1-i c) \cot (a+b x))}{x} \, dx=\int \frac {\cot ^{-1}(c+(1-i c) \cot (a+b x))}{x} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\cot ^{-1}(c+(1-i c) \cot (a+b x))}{x} \, dx \\ \end{align*}
Not integrable
Time = 0.33 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {\cot ^{-1}(c+(1-i c) \cot (a+b x))}{x} \, dx=\int \frac {\cot ^{-1}(c+(1-i c) \cot (a+b x))}{x} \, dx \]
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Not integrable
Time = 0.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24
\[\int \frac {\pi -\operatorname {arccot}\left (-c -\left (-i c +1\right ) \cot \left (b x +a \right )\right )}{x}d x\]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.00 \[ \int \frac {\cot ^{-1}(c+(1-i c) \cot (a+b x))}{x} \, dx=\int { \frac {\pi - \operatorname {arccot}\left (-{\left (-i \, c + 1\right )} \cot \left (b x + a\right ) - c\right )}{x} \,d x } \]
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Timed out. \[ \int \frac {\cot ^{-1}(c+(1-i c) \cot (a+b x))}{x} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\cot ^{-1}(c+(1-i c) \cot (a+b x))}{x} \, dx=\text {Exception raised: ValueError} \]
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Not integrable
Time = 2.38 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33 \[ \int \frac {\cot ^{-1}(c+(1-i c) \cot (a+b x))}{x} \, dx=\int { \frac {\pi - \operatorname {arccot}\left (-{\left (-i \, c + 1\right )} \cot \left (b x + a\right ) - c\right )}{x} \,d x } \]
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Not integrable
Time = 1.43 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {\cot ^{-1}(c+(1-i c) \cot (a+b x))}{x} \, dx=\int \frac {\Pi +\mathrm {acot}\left (c-\mathrm {cot}\left (a+b\,x\right )\,\left (-1+c\,1{}\mathrm {i}\right )\right )}{x} \,d x \]
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