Integrand size = 13, antiderivative size = 64 \[ \int \frac {\coth ^{-1}(\tanh (a+b x))^n}{x} \, dx=\frac {\coth ^{-1}(\tanh (a+b x))^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,-\frac {\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{(1+n) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )} \]
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Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2195} \[ \int \frac {\coth ^{-1}(\tanh (a+b x))^n}{x} \, dx=\frac {\coth ^{-1}(\tanh (a+b x))^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,-\frac {\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{(n+1) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )} \]
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Rule 2195
Rubi steps \begin{align*} \text {integral}& = \frac {\coth ^{-1}(\tanh (a+b x))^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,-\frac {\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{(1+n) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.94 \[ \int \frac {\coth ^{-1}(\tanh (a+b x))^n}{x} \, dx=\frac {\coth ^{-1}(\tanh (a+b x))^n \left (\frac {\coth ^{-1}(\tanh (a+b x))}{b x}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,-n,1-n,1-\frac {\coth ^{-1}(\tanh (a+b x))}{b x}\right )}{n} \]
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\[\int \frac {\operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n}}{x}d x\]
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\[ \int \frac {\coth ^{-1}(\tanh (a+b x))^n}{x} \, dx=\int { \frac {\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{n}}{x} \,d x } \]
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\[ \int \frac {\coth ^{-1}(\tanh (a+b x))^n}{x} \, dx=\int \frac {\operatorname {acoth}^{n}{\left (\tanh {\left (a + b x \right )} \right )}}{x}\, dx \]
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\[ \int \frac {\coth ^{-1}(\tanh (a+b x))^n}{x} \, dx=\int { \frac {\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{n}}{x} \,d x } \]
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\[ \int \frac {\coth ^{-1}(\tanh (a+b x))^n}{x} \, dx=\int { \frac {\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{n}}{x} \,d x } \]
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Timed out. \[ \int \frac {\coth ^{-1}(\tanh (a+b x))^n}{x} \, dx=\int \frac {{\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^n}{x} \,d x \]
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