Optimal. Leaf size=64 \[ \frac{\coth ^{-1}(\tanh (a+b x))^{n+1} \text{Hypergeometric2F1}\left (1,n+1,n+2,-\frac{\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{(n+1) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )} \]
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Rubi [A] time = 0.0208313, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2164} \[ \frac{\coth ^{-1}(\tanh (a+b x))^{n+1} \, _2F_1\left (1,n+1;n+2;-\frac{\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{(n+1) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )} \]
Antiderivative was successfully verified.
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Rule 2164
Rubi steps
\begin{align*} \int \frac{\coth ^{-1}(\tanh (a+b x))^n}{x} \, dx &=\frac{\coth ^{-1}(\tanh (a+b x))^{1+n} \, _2F_1\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*}
Mathematica [A] time = 0.0811981, size = 60, normalized size = 0.94 \[ \frac{\coth ^{-1}(\tanh (a+b x))^n \left (\frac{\coth ^{-1}(\tanh (a+b x))}{b x}\right )^{-n} \text{Hypergeometric2F1}\left (-n,-n,1-n,1-\frac{\coth ^{-1}(\tanh (a+b x))}{b x}\right )}{n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.855, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\rm arccoth} \left (\tanh \left ( bx+a \right ) \right ) \right ) ^{n}}{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{n}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{n}}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acoth}^{n}{\left (\tanh{\left (a + b x \right )} \right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{n}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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