Optimal. Leaf size=16 \[ -\frac{\text{csch}^n(a+b x)}{b n} \]
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Rubi [A] time = 0.0279535, antiderivative size = 16, 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 = {2621, 30} \[ -\frac{\text{csch}^n(a+b x)}{b n} \]
Antiderivative was successfully verified.
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Rule 2621
Rule 30
Rubi steps
\begin{align*} \int \cosh (a+b x) \text{csch}^{1+n}(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int x^{-1+n} \, dx,x,\text{csch}(a+b x)\right )}{b}\\ &=-\frac{\text{csch}^n(a+b x)}{b n}\\ \end{align*}
Mathematica [A] time = 0.0198964, size = 16, normalized size = 1. \[ -\frac{\text{csch}^n(a+b x)}{b n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 17, normalized size = 1.1 \begin{align*} -{\frac{ \left ({\rm csch} \left (bx+a\right ) \right ) ^{n}}{bn}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.90992, size = 72, normalized size = 4.5 \begin{align*} -\frac{2^{n} e^{\left (-{\left (b x + a\right )} n - n \log \left (e^{\left (-b x - a\right )} + 1\right ) - n \log \left (-e^{\left (-b x - a\right )} + 1\right )\right )}}{b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.89532, size = 338, normalized size = 21.12 \begin{align*} -\frac{\cosh \left (n \log \left (\frac{2 \,{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1}\right )\right ) + \sinh \left (n \log \left (\frac{2 \,{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1}\right )\right )}{b n} \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 \coth{\left (a + b x \right )} \operatorname{csch}^{n}{\left (a + b x \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 \operatorname{csch}\left (b x + a\right )^{n} \coth \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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