Optimal. Leaf size=79 \[ \frac{x^m \left (\frac{b x}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )^{-m} \tanh ^{-1}(\tanh (a+b x))^{n+1} \text{Hypergeometric2F1}\left (-m,n+1,n+2,-\frac{\tanh ^{-1}(\tanh (a+b x))}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{b (n+1)} \]
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Rubi [A] time = 0.0432839, antiderivative size = 79, 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 = {2173} \[ \frac{x^m \left (\frac{b x}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )^{-m} \tanh ^{-1}(\tanh (a+b x))^{n+1} \, _2F_1\left (-m,n+1;n+2;-\frac{\tanh ^{-1}(\tanh (a+b x))}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{b (n+1)} \]
Antiderivative was successfully verified.
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Rule 2173
Rubi steps
\begin{align*} \int x^m \tanh ^{-1}(\tanh (a+b x))^n \, dx &=\frac{x^m \left (\frac{b x}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )^{-m} \tanh ^{-1}(\tanh (a+b x))^{1+n} \, _2F_1\left (-m,1+n;2+n;-\frac{\tanh ^{-1}(\tanh (a+b x))}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{b (1+n)}\\ \end{align*}
Mathematica [A] time = 0.128212, size = 71, normalized size = 0.9 \[ \frac{x^{m+1} \tanh ^{-1}(\tanh (a+b x))^n \left (\frac{b x}{\tanh ^{-1}(\tanh (a+b x))-b x}+1\right )^{-n} \text{Hypergeometric2F1}\left (m+1,-n,m+2,-\frac{b x}{\tanh ^{-1}(\tanh (a+b x))-b x}\right )}{m+1} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.392, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{n}\, 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 x^{m} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{n}\,{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 (x^{m} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{n}, 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 x^{m} \operatorname{atanh}^{n}{\left (\tanh{\left (a + b x \right )} \right )}\, 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 x^{m} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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