Optimal. Leaf size=101 \[ \frac{b^2 n \tanh ^{-1}(\tanh (a+b x))^{n-1} \text{Hypergeometric2F1}\left (1,n-1,n,-\frac{\tanh ^{-1}(\tanh (a+b x))}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}-\frac{\tanh ^{-1}(\tanh (a+b x))^n}{2 x^2}-\frac{b n \tanh ^{-1}(\tanh (a+b x))^{n-1}}{2 x} \]
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Rubi [A] time = 0.0668799, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2168, 2164} \[ \frac{b^2 n \tanh ^{-1}(\tanh (a+b x))^{n-1} \, _2F_1\left (1,n-1;n;-\frac{\tanh ^{-1}(\tanh (a+b x))}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}-\frac{\tanh ^{-1}(\tanh (a+b x))^n}{2 x^2}-\frac{b n \tanh ^{-1}(\tanh (a+b x))^{n-1}}{2 x} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 2164
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(\tanh (a+b x))^n}{x^3} \, dx &=-\frac{\tanh ^{-1}(\tanh (a+b x))^n}{2 x^2}+\frac{1}{2} (b n) \int \frac{\tanh ^{-1}(\tanh (a+b x))^{-1+n}}{x^2} \, dx\\ &=-\frac{b n \tanh ^{-1}(\tanh (a+b x))^{-1+n}}{2 x}-\frac{\tanh ^{-1}(\tanh (a+b x))^n}{2 x^2}-\frac{1}{2} \left (b^2 (1-n) n\right ) \int \frac{\tanh ^{-1}(\tanh (a+b x))^{-2+n}}{x} \, dx\\ &=-\frac{b n \tanh ^{-1}(\tanh (a+b x))^{-1+n}}{2 x}-\frac{\tanh ^{-1}(\tanh (a+b x))^n}{2 x^2}+\frac{b^2 n \tanh ^{-1}(\tanh (a+b x))^{-1+n} \, _2F_1\left (1,-1+n;n;-\frac{\tanh ^{-1}(\tanh (a+b x))}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ \end{align*}
Mathematica [A] time = 0.0520349, size = 67, normalized size = 0.66 \[ \frac{\tanh ^{-1}(\tanh (a+b x))^n \left (\frac{\tanh ^{-1}(\tanh (a+b x))}{b x}\right )^{-n} \text{Hypergeometric2F1}\left (2-n,-n,3-n,1-\frac{\tanh ^{-1}(\tanh (a+b x))}{b x}\right )}{(n-2) x^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.913, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{n}}{{x}^{3}}}\, 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{artanh}\left (\tanh \left (b x + a\right )\right )^{n}}{x^{3}}\,{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{artanh}\left (\tanh \left (b x + a\right )\right )^{n}}{x^{3}}, 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{atanh}^{n}{\left (\tanh{\left (a + b x \right )} \right )}}{x^{3}}\, 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{artanh}\left (\tanh \left (b x + a\right )\right )^{n}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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