Optimal. Leaf size=20 \[ -\frac{\coth ^{n+1}(a+b x)}{b (n+1)} \]
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Rubi [A] time = 0.033743, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2607, 32} \[ -\frac{\coth ^{n+1}(a+b x)}{b (n+1)} \]
Antiderivative was successfully verified.
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Rule 2607
Rule 32
Rubi steps
\begin{align*} \int \coth ^n(a+b x) \text{csch}^2(a+b x) \, dx &=\frac{i \operatorname{Subst}\left (\int (-i x)^n \, dx,x,i \coth (a+b x)\right )}{b}\\ &=-\frac{\coth ^{1+n}(a+b x)}{b (1+n)}\\ \end{align*}
Mathematica [A] time = 0.022899, size = 20, normalized size = 1. \[ -\frac{\coth ^{n+1}(a+b x)}{b (n+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 21, normalized size = 1.1 \begin{align*} -{\frac{ \left ({\rm coth} \left (bx+a\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 = 2.01245, size = 192, normalized size = 9.6 \begin{align*} -\frac{\cosh \left (b x + a\right ) \cosh \left (n \log \left (\frac{\cosh \left (b x + a\right )}{\sinh \left (b x + a\right )}\right )\right ) + \cosh \left (b x + a\right ) \sinh \left (n \log \left (\frac{\cosh \left (b x + a\right )}{\sinh \left (b x + a\right )}\right )\right )}{{\left (b n + b\right )} \sinh \left (b x + a\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 \coth \left (b x + a\right )^{n} \operatorname{csch}\left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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