Optimal. Leaf size=65 \[ -\frac{x^m \text{Hypergeometric2F1}\left (1,m,m+1,\frac{b x}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}-\frac{x^m}{b \tanh ^{-1}(\tanh (a+b x))} \]
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Rubi [A] time = 0.0360839, antiderivative size = 65, 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} \[ -\frac{x^m \, _2F_1\left (1,m;m+1;\frac{b x}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}-\frac{x^m}{b \tanh ^{-1}(\tanh (a+b x))} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 2164
Rubi steps
\begin{align*} \int \frac{x^m}{\tanh ^{-1}(\tanh (a+b x))^2} \, dx &=-\frac{x^m}{b \tanh ^{-1}(\tanh (a+b x))}+\frac{m \int \frac{x^{-1+m}}{\tanh ^{-1}(\tanh (a+b x))} \, dx}{b}\\ &=-\frac{x^m}{b \tanh ^{-1}(\tanh (a+b x))}-\frac{x^m \, _2F_1\left (1,m;1+m;\frac{b x}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ \end{align*}
Mathematica [A] time = 0.48889, size = 51, normalized size = 0.78 \[ \frac{x^{m+1} \text{Hypergeometric2F1}\left (2,m+1,m+2,-\frac{b x}{\tanh ^{-1}(\tanh (a+b x))-b x}\right )}{(m+1) \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.622, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{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{x^{m}}{\operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{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{x^{m}}{\operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{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{x^{m}}{\operatorname{atanh}^{2}{\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{artanh}\left (\tanh \left (b x + a\right )\right )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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