Optimal. Leaf size=20 \[ -\frac{(a+b \coth (x))^{n+1}}{b (n+1)} \]
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Rubi [A] time = 0.0483491, antiderivative size = 20, 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 = {3506, 32} \[ -\frac{(a+b \coth (x))^{n+1}}{b (n+1)} \]
Antiderivative was successfully verified.
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Rule 3506
Rule 32
Rubi steps
\begin{align*} \int (a+b \coth (x))^n \text{csch}^2(x) \, dx &=-\frac{\operatorname{Subst}\left (\int (a+x)^n \, dx,x,b \coth (x)\right )}{b}\\ &=-\frac{(a+b \coth (x))^{1+n}}{b (1+n)}\\ \end{align*}
Mathematica [A] time = 0.196076, size = 19, normalized size = 0.95 \[ -\frac{(a+b \coth (x))^{n+1}}{b n+b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 21, normalized size = 1.1 \begin{align*} -{\frac{ \left ( a+b{\rm coth} \left (x\right ) \right ) ^{n+1}}{b \left ( n+1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.15537, size = 221, normalized size = 11.05 \begin{align*} -\frac{{\left (b \cosh \left (x\right ) + a \sinh \left (x\right )\right )} \cosh \left (n \log \left (\frac{b \cosh \left (x\right ) + a \sinh \left (x\right )}{\sinh \left (x\right )}\right )\right ) +{\left (b \cosh \left (x\right ) + a \sinh \left (x\right )\right )} \sinh \left (n \log \left (\frac{b \cosh \left (x\right ) + a \sinh \left (x\right )}{\sinh \left (x\right )}\right )\right )}{{\left (b n + b\right )} \sinh \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\left (b \coth \left (x\right ) + a\right )}^{n} \operatorname{csch}\left (x\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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