Optimal. Leaf size=43 \[ \frac{2 \cosh (a+b x) e^{n \cosh (a+b x)}}{b n}-\frac{2 e^{n \cosh (a+b x)}}{b n^2} \]
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Rubi [A] time = 0.0388002, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {12, 2176, 2194} \[ \frac{2 \cosh (a+b x) e^{n \cosh (a+b x)}}{b n}-\frac{2 e^{n \cosh (a+b x)}}{b n^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2176
Rule 2194
Rubi steps
\begin{align*} \int e^{n \cosh (a+b x)} \sinh (2 (a+b x)) \, dx &=\frac{i \operatorname{Subst}\left (\int -2 i e^{n x} x \, dx,x,\cosh (a+b x)\right )}{b}\\ &=\frac{2 \operatorname{Subst}\left (\int e^{n x} x \, dx,x,\cosh (a+b x)\right )}{b}\\ &=\frac{2 e^{n \cosh (a+b x)} \cosh (a+b x)}{b n}-\frac{2 \operatorname{Subst}\left (\int e^{n x} \, dx,x,\cosh (a+b x)\right )}{b n}\\ &=-\frac{2 e^{n \cosh (a+b x)}}{b n^2}+\frac{2 e^{n \cosh (a+b x)} \cosh (a+b x)}{b n}\\ \end{align*}
Mathematica [A] time = 0.0309611, size = 28, normalized size = 0.65 \[ \frac{2 e^{n \cosh (a+b x)} (n \cosh (a+b x)-1)}{b n^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 59, normalized size = 1.4 \begin{align*}{\frac{n{{\rm e}^{2\,bx+2\,a}}+n-2\,{{\rm e}^{bx+a}}}{b{n}^{2}}{{\rm e}^{-bx-a+{\frac{n{{\rm e}^{bx+a}}}{2}}+{\frac{n{{\rm e}^{-bx-a}}}{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.34352, size = 139, normalized size = 3.23 \begin{align*} \frac{e^{\left (b x + \frac{1}{2} \, n e^{\left (b x + a\right )} + \frac{1}{2} \, n e^{\left (-b x - a\right )} + a\right )}}{b n} + \frac{e^{\left (-b x + \frac{1}{2} \, n e^{\left (b x + a\right )} + \frac{1}{2} \, n e^{\left (-b x - a\right )} - a\right )}}{b n} - \frac{2 \, e^{\left (\frac{1}{2} \, n e^{\left (b x + a\right )} + \frac{1}{2} \, n e^{\left (-b x - a\right )}\right )}}{b n^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06515, size = 193, normalized size = 4.49 \begin{align*} \frac{2 \,{\left ({\left (n \cosh \left (b x + a\right ) - 1\right )} \cosh \left (n \cosh \left (b x + a\right )\right ) +{\left (n \cosh \left (b x + a\right ) - 1\right )} \sinh \left (n \cosh \left (b x + a\right )\right )\right )}}{b n^{2} \cosh \left (b x + a\right )^{2} - b n^{2} \sinh \left (b x + a\right )^{2}} \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 e^{n \cosh{\left (a + b x \right )}} \sinh{\left (2 a + 2 b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{\left (n \cosh \left (b x + a\right )\right )} \sinh \left (2 \, b x + 2 \, a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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