Optimal. Leaf size=26 \[ \frac{\cosh (a-c) \log (\cosh (b x+c))}{b}+x \sinh (a-c) \]
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Rubi [A] time = 0.0158705, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {5624, 3475, 8} \[ \frac{\cosh (a-c) \log (\cosh (b x+c))}{b}+x \sinh (a-c) \]
Antiderivative was successfully verified.
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Rule 5624
Rule 3475
Rule 8
Rubi steps
\begin{align*} \int \text{sech}(c+b x) \sinh (a+b x) \, dx &=\cosh (a-c) \int \tanh (c+b x) \, dx+\sinh (a-c) \int 1 \, dx\\ &=\frac{\cosh (a-c) \log (\cosh (c+b x))}{b}+x \sinh (a-c)\\ \end{align*}
Mathematica [A] time = 0.11894, size = 26, normalized size = 1. \[ \frac{\cosh (a-c) \log (\cosh (b x+c))}{b}+x \sinh (a-c) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.036, size = 148, normalized size = 5.7 \begin{align*} x{{\rm e}^{a-c}}-{{\rm e}^{-a-c}}{{\rm e}^{2\,a}}x-{{\rm e}^{-a-c}}{{\rm e}^{2\,c}}x-{\frac{{{\rm e}^{-a-c}}{{\rm e}^{2\,a}}a}{b}}-{\frac{{{\rm e}^{-a-c}}{{\rm e}^{2\,c}}a}{b}}+{\frac{\ln \left ({{\rm e}^{2\,bx+2\,a}}+{{\rm e}^{2\,a-2\,c}} \right ){{\rm e}^{-a-c}}{{\rm e}^{2\,a}}}{2\,b}}+{\frac{\ln \left ({{\rm e}^{2\,bx+2\,a}}+{{\rm e}^{2\,a-2\,c}} \right ){{\rm e}^{-a-c}}{{\rm e}^{2\,c}}}{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.26344, size = 66, normalized size = 2.54 \begin{align*} \frac{{\left (e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} \log \left (e^{\left (-2 \, b x\right )} + e^{\left (2 \, c\right )}\right )}{2 \, b} + \frac{{\left (b x + a\right )} e^{\left (a - c\right )}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.15378, size = 231, normalized size = 8.88 \begin{align*} -\frac{2 \, b x -{\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 )} \log \left (\frac{2 \, \cosh \left (b x + c\right )}{\cosh \left (b x + c\right ) - \sinh \left (b x + c\right )}\right )}{2 \,{\left (b \cosh \left (-a + c\right ) - b \sinh \left (-a + c\right )\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 \sinh{\left (a + b x \right )} \operatorname{sech}{\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 = 1.14728, size = 66, normalized size = 2.54 \begin{align*} -\frac{2 \, b x e^{\left (-a + c\right )} -{\left (e^{\left (2 \, a + c\right )} + e^{\left (3 \, c\right )}\right )} e^{\left (-a - 2 \, c\right )} \log \left (e^{\left (2 \, b x + 2 \, c\right )} + 1\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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