Optimal. Leaf size=48 \[ \frac{x \coth ^{-1}(\tanh (a+b x))^{n+1}}{b (n+1)}-\frac{\coth ^{-1}(\tanh (a+b x))^{n+2}}{b^2 (n+1) (n+2)} \]
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Rubi [A] time = 0.0200901, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {2168, 2157, 30} \[ \frac{x \coth ^{-1}(\tanh (a+b x))^{n+1}}{b (n+1)}-\frac{\coth ^{-1}(\tanh (a+b x))^{n+2}}{b^2 (n+1) (n+2)} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 2157
Rule 30
Rubi steps
\begin{align*} \int x \coth ^{-1}(\tanh (a+b x))^n \, dx &=\frac{x \coth ^{-1}(\tanh (a+b x))^{1+n}}{b (1+n)}-\frac{\int \coth ^{-1}(\tanh (a+b x))^{1+n} \, dx}{b (1+n)}\\ &=\frac{x \coth ^{-1}(\tanh (a+b x))^{1+n}}{b (1+n)}-\frac{\operatorname{Subst}\left (\int x^{1+n} \, dx,x,\coth ^{-1}(\tanh (a+b x))\right )}{b^2 (1+n)}\\ &=\frac{x \coth ^{-1}(\tanh (a+b x))^{1+n}}{b (1+n)}-\frac{\coth ^{-1}(\tanh (a+b x))^{2+n}}{b^2 (1+n) (2+n)}\\ \end{align*}
Mathematica [A] time = 0.0428894, size = 41, normalized size = 0.85 \[ \frac{\left (b (n+2) x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^{n+1}}{b^2 (n+1) (n+2)} \]
Antiderivative was successfully verified.
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Maple [C] time = 15.611, size = 71611, normalized size = 1491.9 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.79195, size = 138, normalized size = 2.88 \begin{align*} \frac{{\left (4 \, b^{2}{\left (n + 1\right )} x^{2} + \pi ^{2} + 4 i \, \pi a - 4 \, a^{2} - 2 \,{\left (i \, \pi b n - 2 \, a b n\right )} x\right )}{\left (\cosh \left (-n \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )\right ) - \sinh \left (-n \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )\right )\right )}}{{\left (2^{n + 2} n^{2} + 3 \cdot 2^{n + 2} n + 2^{n + 3}\right )} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.60396, size = 489, normalized size = 10.19 \begin{align*} \frac{{\left (4 \, a b n x + 4 \,{\left (b^{2} n + b^{2}\right )} x^{2} + \pi ^{2} - 4 \, a^{2}\right )}{\left (b^{2} x^{2} + 2 \, a b x + \frac{1}{4} \, \pi ^{2} + a^{2}\right )}^{\frac{1}{2} \, n} \cos \left (2 \, n \arctan \left (-\frac{2 \, b x}{\pi } - \frac{2 \, a}{\pi } + \frac{\sqrt{4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}}}{\pi }\right )\right ) - 2 \,{\left (\pi b n x - 2 \, \pi a\right )}{\left (b^{2} x^{2} + 2 \, a b x + \frac{1}{4} \, \pi ^{2} + a^{2}\right )}^{\frac{1}{2} \, n} \sin \left (2 \, n \arctan \left (-\frac{2 \, b x}{\pi } - \frac{2 \, a}{\pi } + \frac{\sqrt{4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}}}{\pi }\right )\right )}{4 \,{\left (b^{2} n^{2} + 3 \, b^{2} n + 2 \, b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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 \operatorname{arcoth}\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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