Optimal. Leaf size=34 \[ \frac{\tan ^{-1}(\sinh (a+b x))}{2 b}-\frac{\tanh (a+b x) \text{sech}(a+b x)}{2 b} \]
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Rubi [A] time = 0.0237051, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2611, 3770} \[ \frac{\tan ^{-1}(\sinh (a+b x))}{2 b}-\frac{\tanh (a+b x) \text{sech}(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 2611
Rule 3770
Rubi steps
\begin{align*} \int \text{sech}(a+b x) \tanh ^2(a+b x) \, dx &=-\frac{\text{sech}(a+b x) \tanh (a+b x)}{2 b}+\frac{1}{2} \int \text{sech}(a+b x) \, dx\\ &=\frac{\tan ^{-1}(\sinh (a+b x))}{2 b}-\frac{\text{sech}(a+b x) \tanh (a+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0158222, size = 34, normalized size = 1. \[ \frac{\tan ^{-1}(\sinh (a+b x))}{2 b}-\frac{\tanh (a+b x) \text{sech}(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 49, normalized size = 1.4 \begin{align*} -{\frac{\sinh \left ( bx+a \right ) }{b \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}}+{\frac{{\rm sech} \left (bx+a\right )\tanh \left ( bx+a \right ) }{2\,b}}+{\frac{\arctan \left ({{\rm e}^{bx+a}} \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.618, size = 89, normalized size = 2.62 \begin{align*} -\frac{\arctan \left (e^{\left (-b x - a\right )}\right )}{b} - \frac{e^{\left (-b x - a\right )} - e^{\left (-3 \, b x - 3 \, a\right )}}{b{\left (2 \, e^{\left (-2 \, b x - 2 \, a\right )} + e^{\left (-4 \, b x - 4 \, a\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.99816, size = 759, normalized size = 22.32 \begin{align*} -\frac{\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} -{\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \,{\left (3 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right )^{2} + 4 \,{\left (\cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \arctan \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) +{\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) - \cosh \left (b x + a\right )}{b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4} + 2 \, b \cosh \left (b x + a\right )^{2} + 2 \,{\left (3 \, b \cosh \left (b x + a\right )^{2} + b\right )} \sinh \left (b x + a\right )^{2} + 4 \,{\left (b \cosh \left (b x + a\right )^{3} + b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + b} \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 \sinh ^{2}{\left (a + b x \right )} \operatorname{sech}^{3}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16275, size = 108, normalized size = 3.18 \begin{align*} \frac{\pi + 2 \, \arctan \left (\frac{1}{2} \,{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-b x - a\right )}\right )}{4 \, b} - \frac{e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}}{{\left ({\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{2} + 4\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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