Optimal. Leaf size=71 \[ \frac{b \coth ^{-1}(\tanh (a+b x))^n \text{Hypergeometric2F1}\left (1,n,n+1,-\frac{\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac{\coth ^{-1}(\tanh (a+b x))^n}{x} \]
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Rubi [A] time = 0.0412581, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2168, 2164} \[ \frac{b \coth ^{-1}(\tanh (a+b x))^n \, _2F_1\left (1,n;n+1;-\frac{\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac{\coth ^{-1}(\tanh (a+b x))^n}{x} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 2164
Rubi steps
\begin{align*} \int \frac{\coth ^{-1}(\tanh (a+b x))^n}{x^2} \, dx &=-\frac{\coth ^{-1}(\tanh (a+b x))^n}{x}+(b n) \int \frac{\coth ^{-1}(\tanh (a+b x))^{-1+n}}{x} \, dx\\ &=-\frac{\coth ^{-1}(\tanh (a+b x))^n}{x}+\frac{b \coth ^{-1}(\tanh (a+b x))^n \, _2F_1\left (1,n;1+n;-\frac{\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{b x-\coth ^{-1}(\tanh (a+b x))}\\ \end{align*}
Mathematica [A] time = 0.0404399, size = 67, normalized size = 0.94 \[ \frac{\coth ^{-1}(\tanh (a+b x))^n \left (\frac{\coth ^{-1}(\tanh (a+b x))}{b x}\right )^{-n} \text{Hypergeometric2F1}\left (1-n,-n,2-n,1-\frac{\coth ^{-1}(\tanh (a+b x))}{b x}\right )}{(n-1) x} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.964, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\rm arccoth} \left (\tanh \left ( bx+a \right ) \right ) \right ) ^{n}}{{x}^{2}}}\, 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^{2}}\,{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^{2}}, 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^{2}}\, 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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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