Optimal. Leaf size=38 \[ \frac{\cosh (a-c) \tanh (b x+c)}{b}-\frac{\sinh (a-c) \text{sech}^2(b x+c)}{2 b} \]
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Rubi [A] time = 0.0418814, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5627, 2606, 30, 3767, 8} \[ \frac{\cosh (a-c) \tanh (b x+c)}{b}-\frac{\sinh (a-c) \text{sech}^2(b x+c)}{2 b} \]
Antiderivative was successfully verified.
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Rule 5627
Rule 2606
Rule 30
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \cosh (a+b x) \text{sech}^3(c+b x) \, dx &=\cosh (a-c) \int \text{sech}^2(c+b x) \, dx+\sinh (a-c) \int \text{sech}^2(c+b x) \tanh (c+b x) \, dx\\ &=\frac{(i \cosh (a-c)) \operatorname{Subst}(\int 1 \, dx,x,-i \tanh (c+b x))}{b}-\frac{\sinh (a-c) \operatorname{Subst}(\int x \, dx,x,\text{sech}(c+b x))}{b}\\ &=-\frac{\text{sech}^2(c+b x) \sinh (a-c)}{2 b}+\frac{\cosh (a-c) \tanh (c+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.167805, size = 35, normalized size = 0.92 \[ -\frac{\text{sech}(c) \text{sech}^2(b x+c) (\sinh (a)-\cosh (a-c) \sinh (2 b x+c))}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 56, normalized size = 1.5 \begin{align*} -{\frac{ \left ( 2\,{{\rm e}^{2\,bx+2\,a+2\,c}}+{{\rm e}^{2\,a}}+{{\rm e}^{2\,c}} \right ){{\rm e}^{3\,a-c}}}{b \left ({{\rm e}^{2\,bx+2\,a+2\,c}}+{{\rm e}^{2\,a}} \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.101, size = 161, normalized size = 4.24 \begin{align*} \frac{2 \, e^{\left (-2 \, b x + 3 \, c\right )}}{b{\left (2 \, e^{\left (-2 \, b x + a + 2 \, c\right )} + e^{\left (-4 \, b x + a\right )} + e^{\left (a + 4 \, c\right )}\right )}} + \frac{e^{\left (2 \, a + 3 \, c\right )}}{b{\left (2 \, e^{\left (-2 \, b x + a + 2 \, c\right )} + e^{\left (-4 \, b x + a\right )} + e^{\left (a + 4 \, c\right )}\right )}} + \frac{e^{\left (5 \, c\right )}}{b{\left (2 \, e^{\left (-2 \, b x + a + 2 \, c\right )} + e^{\left (-4 \, b x + a\right )} + e^{\left (a + 4 \, c\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.82322, size = 644, normalized size = 16.95 \begin{align*} -\frac{2 \,{\left (2 \, \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) - \cosh \left (b x + c\right ) \sinh \left (-a + c\right ) - \sinh \left (b x + c\right ) \sinh \left (-a + c\right )\right )}}{b \cosh \left (b x + c\right )^{3} \cosh \left (-a + c\right )^{2} + 3 \, b \cosh \left (b x + c\right ) \cosh \left (-a + c\right )^{2} +{\left (b \cosh \left (-a + c\right )^{2} - b \sinh \left (-a + c\right )^{2}\right )} \sinh \left (b x + c\right )^{3} + 3 \,{\left (b \cosh \left (b x + c\right ) \cosh \left (-a + c\right )^{2} - b \cosh \left (b x + c\right ) \sinh \left (-a + c\right )^{2}\right )} \sinh \left (b x + c\right )^{2} -{\left (b \cosh \left (b x + c\right )^{3} + 3 \, b \cosh \left (b x + c\right )\right )} \sinh \left (-a + c\right )^{2} +{\left (3 \, b \cosh \left (b x + c\right )^{2} \cosh \left (-a + c\right )^{2} + b \cosh \left (-a + c\right )^{2} -{\left (3 \, b \cosh \left (b x + c\right )^{2} + b\right )} \sinh \left (-a + c\right )^{2}\right )} \sinh \left (b x + 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}^{3}{\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 = 1.18868, size = 66, normalized size = 1.74 \begin{align*} -\frac{{\left (2 \, e^{\left (2 \, b x + 2 \, a + 2 \, c\right )} + e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )}}{b{\left (e^{\left (2 \, b x + 2 \, c\right )} + 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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