Optimal. Leaf size=35 \[ \frac{\cosh (a-c) \tan ^{-1}(\sinh (b x+c))}{b}-\frac{\sinh (a-c) \text{sech}(b x+c)}{b} \]
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Rubi [A] time = 0.0308974, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {5627, 2606, 8, 3770} \[ \frac{\cosh (a-c) \tan ^{-1}(\sinh (b x+c))}{b}-\frac{\sinh (a-c) \text{sech}(b x+c)}{b} \]
Antiderivative was successfully verified.
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Rule 5627
Rule 2606
Rule 8
Rule 3770
Rubi steps
\begin{align*} \int \cosh (a+b x) \text{sech}^2(c+b x) \, dx &=\cosh (a-c) \int \text{sech}(c+b x) \, dx+\sinh (a-c) \int \text{sech}(c+b x) \tanh (c+b x) \, dx\\ &=\frac{\tan ^{-1}(\sinh (c+b x)) \cosh (a-c)}{b}-\frac{\sinh (a-c) \operatorname{Subst}(\int 1 \, dx,x,\text{sech}(c+b x))}{b}\\ &=\frac{\tan ^{-1}(\sinh (c+b x)) \cosh (a-c)}{b}-\frac{\text{sech}(c+b x) \sinh (a-c)}{b}\\ \end{align*}
Mathematica [B] time = 0.0822096, size = 83, normalized size = 2.37 \[ \frac{2 \cosh (a-c) \tan ^{-1}\left (\frac{(\cosh (c)-\sinh (c)) \left (\sinh (c) \cosh \left (\frac{b x}{2}\right )+\cosh (c) \sinh \left (\frac{b x}{2}\right )\right )}{\cosh (c) \cosh \left (\frac{b x}{2}\right )-\sinh (c) \cosh \left (\frac{b x}{2}\right )}\right )}{b}-\frac{\sinh (a-c) \text{sech}(b x+c)}{b} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.08, size = 183, normalized size = 5.2 \begin{align*} -{\frac{{{\rm e}^{bx+a}} \left ({{\rm e}^{2\,a}}-{{\rm e}^{2\,c}} \right ) }{b \left ({{\rm e}^{2\,bx+2\,a+2\,c}}+{{\rm e}^{2\,a}} \right ) }}+{\frac{{\frac{i}{2}}\ln \left ({{\rm e}^{bx+a}}+i{{\rm e}^{a-c}} \right ){{\rm e}^{-a-c}}{{\rm e}^{2\,a}}}{b}}+{\frac{{\frac{i}{2}}\ln \left ({{\rm e}^{bx+a}}+i{{\rm e}^{a-c}} \right ){{\rm e}^{-a-c}}{{\rm e}^{2\,c}}}{b}}-{\frac{{\frac{i}{2}}\ln \left ({{\rm e}^{bx+a}}-i{{\rm e}^{a-c}} \right ){{\rm e}^{-a-c}}{{\rm e}^{2\,a}}}{b}}-{\frac{{\frac{i}{2}}\ln \left ({{\rm e}^{bx+a}}-i{{\rm e}^{a-c}} \right ){{\rm e}^{-a-c}}{{\rm e}^{2\,c}}}{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.63424, size = 95, normalized size = 2.71 \begin{align*} -\frac{{\left (e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} \arctan \left (e^{\left (-b x - c\right )}\right ) e^{\left (-a - c\right )}}{b} - \frac{{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} e^{\left (-b x - a\right )}}{b{\left (e^{\left (-2 \, b x\right )} + e^{\left (2 \, c\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.83236, size = 1129, normalized size = 32.26 \begin{align*} \frac{2 \, \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) - \cosh \left (b x + c\right ) \sinh \left (-a + c\right )^{2} +{\left ({\left (\cosh \left (-a + c\right )^{2} + 1\right )} \cosh \left (b x + c\right )^{2} +{\left (\cosh \left (-a + c\right )^{2} - 2 \, \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + \sinh \left (-a + c\right )^{2} + 1\right )} \sinh \left (b x + c\right )^{2} +{\left (\cosh \left (b x + c\right )^{2} + 1\right )} \sinh \left (-a + c\right )^{2} + \cosh \left (-a + c\right )^{2} - 2 \,{\left (2 \, \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) - \cosh \left (b x + c\right ) \sinh \left (-a + c\right )^{2} -{\left (\cosh \left (-a + c\right )^{2} + 1\right )} \cosh \left (b x + c\right )\right )} \sinh \left (b x + c\right ) - 2 \,{\left (\cosh \left (b x + c\right )^{2} \cosh \left (-a + c\right ) + \cosh \left (-a + c\right )\right )} \sinh \left (-a + c\right ) + 1\right )} \arctan \left (\cosh \left (b x + c\right ) + \sinh \left (b x + c\right )\right ) -{\left (\cosh \left (-a + c\right )^{2} - 1\right )} \cosh \left (b x + c\right ) -{\left (\cosh \left (-a + c\right )^{2} - 2 \, \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + \sinh \left (-a + c\right )^{2} - 1\right )} \sinh \left (b x + c\right )}{b \cosh \left (b x + c\right )^{2} \cosh \left (-a + c\right ) +{\left (b \cosh \left (-a + c\right ) - b \sinh \left (-a + c\right )\right )} \sinh \left (b x + c\right )^{2} + b \cosh \left (-a + c\right ) + 2 \,{\left (b \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) - b \cosh \left (b x + c\right ) \sinh \left (-a + c\right )\right )} \sinh \left (b x + c\right ) -{\left (b \cosh \left (b x + c\right )^{2} + b\right )} \sinh \left (-a + c\right )} \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 \cosh{\left (a + b x \right )} \operatorname{sech}^{2}{\left (b x + c \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14912, size = 92, normalized size = 2.63 \begin{align*} \frac{{\left (e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} \arctan \left (e^{\left (b x + c\right )}\right ) e^{\left (-a - c\right )} - \frac{{\left (e^{\left (b x + 2 \, a\right )} - e^{\left (b x + 2 \, c\right )}\right )} e^{\left (-a\right )}}{e^{\left (2 \, b x + 2 \, c\right )} + 1}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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