Integrand size = 21, antiderivative size = 21 \[ \int \frac {\coth ^{-1}(1+i d-d \tan (a+b x))}{x} \, dx=\text {Int}\left (\frac {\coth ^{-1}(1+i d-d \tan (a+b x))}{x},x\right ) \]
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Not integrable
Time = 0.07 (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 = {} \[ \int \frac {\coth ^{-1}(1+i d-d \tan (a+b x))}{x} \, dx=\int \frac {\coth ^{-1}(1+i d-d \tan (a+b x))}{x} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\coth ^{-1}(1+i d-d \tan (a+b x))}{x} \, dx \\ \end{align*}
Not integrable
Time = 0.62 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {\coth ^{-1}(1+i d-d \tan (a+b x))}{x} \, dx=\int \frac {\coth ^{-1}(1+i d-d \tan (a+b x))}{x} \, dx \]
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Not integrable
Time = 0.13 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90
\[\int \frac {\operatorname {arccoth}\left (1+i d -d \tan \left (b x +a \right )\right )}{x}d x\]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.71 \[ \int \frac {\coth ^{-1}(1+i d-d \tan (a+b x))}{x} \, dx=\int { \frac {\operatorname {arcoth}\left (-d \tan \left (b x + a\right ) + i \, d + 1\right )}{x} \,d x } \]
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Not integrable
Time = 0.76 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {\coth ^{-1}(1+i d-d \tan (a+b x))}{x} \, dx=\int \frac {\operatorname {acoth}{\left (- d \tan {\left (a + b x \right )} + i d + 1 \right )}}{x}\, dx \]
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Not integrable
Time = 4.39 (sec) , antiderivative size = 141, normalized size of antiderivative = 6.71 \[ \int \frac {\coth ^{-1}(1+i d-d \tan (a+b x))}{x} \, dx=\int { \frac {\operatorname {arcoth}\left (-d \tan \left (b x + a\right ) + i \, d + 1\right )}{x} \,d x } \]
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Not integrable
Time = 0.85 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {\coth ^{-1}(1+i d-d \tan (a+b x))}{x} \, dx=\int { \frac {\operatorname {arcoth}\left (-d \tan \left (b x + a\right ) + i \, d + 1\right )}{x} \,d x } \]
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Not integrable
Time = 4.82 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int \frac {\coth ^{-1}(1+i d-d \tan (a+b x))}{x} \, dx=\int \frac {\mathrm {acoth}\left (1-d\,\mathrm {tan}\left (a+b\,x\right )+d\,1{}\mathrm {i}\right )}{x} \,d x \]
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