Integrand size = 15, antiderivative size = 15 \[ \int \frac {\coth ^{-1}(\tan (a+b x))}{e+f x} \, dx=\text {Int}\left (\frac {\coth ^{-1}(\tan (a+b x))}{e+f x},x\right ) \]
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Not integrable
Time = 0.03 (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 = {} \[ \int \frac {\coth ^{-1}(\tan (a+b x))}{e+f x} \, dx=\int \frac {\coth ^{-1}(\tan (a+b x))}{e+f x} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\coth ^{-1}(\tan (a+b x))}{e+f x} \, dx \\ \end{align*}
Not integrable
Time = 0.49 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {\coth ^{-1}(\tan (a+b x))}{e+f x} \, dx=\int \frac {\coth ^{-1}(\tan (a+b x))}{e+f x} \, dx \]
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Not integrable
Time = 0.16 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00
\[\int \frac {\operatorname {arccoth}\left (\tan \left (b x +a \right )\right )}{f x +e}d x\]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {\coth ^{-1}(\tan (a+b x))}{e+f x} \, dx=\int { \frac {\operatorname {arcoth}\left (\tan \left (b x + a\right )\right )}{f x + e} \,d x } \]
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Not integrable
Time = 0.53 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {\coth ^{-1}(\tan (a+b x))}{e+f x} \, dx=\int \frac {\operatorname {acoth}{\left (\tan {\left (a + b x \right )} \right )}}{e + f x}\, dx \]
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Not integrable
Time = 1.54 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {\coth ^{-1}(\tan (a+b x))}{e+f x} \, dx=\int { \frac {\operatorname {arcoth}\left (\tan \left (b x + a\right )\right )}{f x + e} \,d x } \]
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Not integrable
Time = 0.99 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {\coth ^{-1}(\tan (a+b x))}{e+f x} \, dx=\int { \frac {\operatorname {arcoth}\left (\tan \left (b x + a\right )\right )}{f x + e} \,d x } \]
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Not integrable
Time = 4.42 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {\coth ^{-1}(\tan (a+b x))}{e+f x} \, dx=\int \frac {\mathrm {acoth}\left (\mathrm {tan}\left (a+b\,x\right )\right )}{e+f\,x} \,d x \]
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