Optimal. Leaf size=11 \[ -\frac{\text{sech}(a+b x)}{b} \]
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Rubi [A] time = 0.0130824, antiderivative size = 11, 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 = {2606, 8} \[ -\frac{\text{sech}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2606
Rule 8
Rubi steps
\begin{align*} \int \text{sech}(a+b x) \tanh (a+b x) \, dx &=-\frac{\operatorname{Subst}(\int 1 \, dx,x,\text{sech}(a+b x))}{b}\\ &=-\frac{\text{sech}(a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0069194, size = 11, normalized size = 1. \[ -\frac{\text{sech}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 12, normalized size = 1.1 \begin{align*} -{\frac{{\rm sech} \left (bx+a\right )}{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.134, size = 31, normalized size = 2.82 \begin{align*} -\frac{2}{b{\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.69006, size = 154, normalized size = 14. \begin{align*} -\frac{2 \,{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2} + b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.320807, size = 17, normalized size = 1.55 \begin{align*} \begin{cases} - \frac{\operatorname{sech}{\left (a + b x \right )}}{b} & \text{for}\: b \neq 0 \\x \tanh{\left (a \right )} \operatorname{sech}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16443, size = 32, normalized size = 2.91 \begin{align*} -\frac{2 \, e^{\left (b x + a\right )}}{b{\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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