Optimal. Leaf size=16 \[ -\frac{\text{sech}^n(a+b x)}{b n} \]
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Rubi [A] time = 0.0298977, 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 = {2622, 30} \[ -\frac{\text{sech}^n(a+b x)}{b n} \]
Antiderivative was successfully verified.
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Rule 2622
Rule 30
Rubi steps
\begin{align*} \int \text{sech}^{1+n}(a+b x) \sinh (a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int x^{-1+n} \, dx,x,\text{sech}(a+b x)\right )}{b}\\ &=-\frac{\text{sech}^n(a+b x)}{b n}\\ \end{align*}
Mathematica [A] time = 0.0375077, size = 16, normalized size = 1. \[ -\frac{\text{sech}^n(a+b x)}{b n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 17, normalized size = 1.1 \begin{align*} -{\frac{ \left ({\rm sech} \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 = 1.77211, size = 49, normalized size = 3.06 \begin{align*} -\frac{2^{n} e^{\left (-{\left (b x + a\right )} n - n \log \left (e^{\left (-2 \, b x - 2 \, 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 = 1.85129, 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 [A] time = 0.677922, size = 39, normalized size = 2.44 \begin{align*} \begin{cases} x \tanh{\left (a \right )} & \text{for}\: b = 0 \wedge n = 0 \\x \tanh{\left (a \right )} \operatorname{sech}^{n}{\left (a \right )} & \text{for}\: b = 0 \\x - \frac{\log{\left (\tanh{\left (a + b x \right )} + 1 \right )}}{b} & \text{for}\: n = 0 \\- \frac{\operatorname{sech}^{n}{\left (a + b x \right )}}{b n} & \text{otherwise} \end{cases} \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{sech}\left (b x + a\right )^{n} \tanh \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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