Optimal. Leaf size=23 \[ \frac{\text{sech}(a+b x)}{b}-\frac{\tanh ^{-1}(\cosh (a+b x))}{b} \]
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Rubi [A] time = 0.0257552, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2622, 321, 207} \[ \frac{\text{sech}(a+b x)}{b}-\frac{\tanh ^{-1}(\cosh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 2622
Rule 321
Rule 207
Rubi steps
\begin{align*} \int \text{csch}(a+b x) \text{sech}^2(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{-1+x^2} \, dx,x,\text{sech}(a+b x)\right )}{b}\\ &=\frac{\text{sech}(a+b x)}{b}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\text{sech}(a+b x)\right )}{b}\\ &=-\frac{\tanh ^{-1}(\cosh (a+b x))}{b}+\frac{\text{sech}(a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0307134, size = 26, normalized size = 1.13 \[ \frac{\text{sech}(a+b x)}{b}+\frac{\log \left (\tanh \left (\frac{1}{2} (a+b x)\right )\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 23, normalized size = 1. \begin{align*}{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{-1}-2\,{\it Artanh} \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 = 1.01946, size = 82, normalized size = 3.57 \begin{align*} -\frac{\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac{\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} + \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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Fricas [B] time = 2.08739, size = 462, normalized size = 20.09 \begin{align*} -\frac{{\left (\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 )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) -{\left (\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 )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) - 2 \, \cosh \left (b x + a\right ) - 2 \, \sinh \left (b x + a\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{csch}{\left (a + b x \right )} \operatorname{sech}^{2}{\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 = 1.16259, size = 95, normalized size = 4.13 \begin{align*} -\frac{\log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} + 2\right )}{2 \, b} + \frac{\log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} - 2\right )}{2 \, b} + \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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