Optimal. Leaf size=53 \[ -\frac{x^{m+1} \text{Hypergeometric2F1}\left (1,m+1,m+2,\frac{b x}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{(m+1) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )} \]
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Rubi [A] time = 0.0284971, antiderivative size = 53, 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} \[ -\frac{x^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{b x}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{(m+1) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )} \]
Antiderivative was successfully verified.
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Rule 2164
Rubi steps
\begin{align*} \int \frac{x^m}{\coth ^{-1}(\tanh (a+b x))} \, dx &=-\frac{x^{1+m} \, _2F_1\left (1,1+m;2+m;\frac{b x}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{(1+m) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}\\ \end{align*}
Mathematica [A] time = 0.0842978, size = 51, normalized size = 0.96 \[ \frac{x^{m+1} \text{Hypergeometric2F1}\left (1,m+1,m+2,-\frac{b x}{\coth ^{-1}(\tanh (a+b x))-b x}\right )}{(m+1) \left (\coth ^{-1}(\tanh (a+b x))-b x\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.346, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{{\rm arccoth} \left (\tanh \left ( bx+a \right ) \right )}}\, 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{x^{m}}{\operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )}\,{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{x^{m}}{\operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )}, 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{x^{m}}{\operatorname{acoth}{\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 \frac{x^{m}}{\operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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