Integrand size = 15, antiderivative size = 15 \[ \int \frac {\coth ^{-1}(c+d \cot (a+b x))}{x} \, dx=\text {Int}\left (\frac {\coth ^{-1}(c+d \cot (a+b x))}{x},x\right ) \]
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Not integrable
Time = 0.11 (sec) , antiderivative size = 15, 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 = {} \[ \int \frac {\coth ^{-1}(c+d \cot (a+b x))}{x} \, dx=\int \frac {\coth ^{-1}(c+d \cot (a+b x))}{x} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\coth ^{-1}(c+d \cot (a+b x))}{x} \, dx \\ \end{align*}
Not integrable
Time = 0.34 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {\coth ^{-1}(c+d \cot (a+b x))}{x} \, dx=\int \frac {\coth ^{-1}(c+d \cot (a+b x))}{x} \, dx \]
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Not integrable
Time = 0.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00
\[\int \frac {\operatorname {arccoth}\left (c +d \cot \left (b x +a \right )\right )}{x}d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {\coth ^{-1}(c+d \cot (a+b x))}{x} \, dx=\int { \frac {\operatorname {arcoth}\left (d \cot \left (b x + a\right ) + c\right )}{x} \,d x } \]
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Not integrable
Time = 0.98 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {\coth ^{-1}(c+d \cot (a+b x))}{x} \, dx=\int \frac {\operatorname {acoth}{\left (c + d \cot {\left (a + b x \right )} \right )}}{x}\, dx \]
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Not integrable
Time = 3.57 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {\coth ^{-1}(c+d \cot (a+b x))}{x} \, dx=\int { \frac {\operatorname {arcoth}\left (d \cot \left (b x + a\right ) + c\right )}{x} \,d x } \]
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Not integrable
Time = 0.71 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {\coth ^{-1}(c+d \cot (a+b x))}{x} \, dx=\int { \frac {\operatorname {arcoth}\left (d \cot \left (b x + a\right ) + c\right )}{x} \,d x } \]
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Not integrable
Time = 6.15 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {\coth ^{-1}(c+d \cot (a+b x))}{x} \, dx=\int \frac {\mathrm {acoth}\left (c+d\,\mathrm {cot}\left (a+b\,x\right )\right )}{x} \,d x \]
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