Optimal. Leaf size=64 \[ \frac{4 \cosh \left (\frac{a}{2}+\frac{b x}{2}\right ) e^{n \cosh \left (\frac{a}{2}+\frac{b x}{2}\right )}}{b n}-\frac{4 e^{n \cosh \left (\frac{a}{2}+\frac{b x}{2}\right )}}{b n^2} \]
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Rubi [A] time = 0.0487021, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {12, 2176, 2194} \[ \frac{4 \cosh \left (\frac{a}{2}+\frac{b x}{2}\right ) e^{n \cosh \left (\frac{a}{2}+\frac{b x}{2}\right )}}{b n}-\frac{4 e^{n \cosh \left (\frac{a}{2}+\frac{b x}{2}\right )}}{b n^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2176
Rule 2194
Rubi steps
\begin{align*} \int e^{n \cosh \left (\frac{a}{2}+\frac{b x}{2}\right )} \sinh (a+b x) \, dx &=\frac{(2 i) \operatorname{Subst}\left (\int -2 i e^{n x} x \, dx,x,\cosh \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b}\\ &=\frac{4 \operatorname{Subst}\left (\int e^{n x} x \, dx,x,\cosh \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b}\\ &=\frac{4 e^{n \cosh \left (\frac{a}{2}+\frac{b x}{2}\right )} \cosh \left (\frac{a}{2}+\frac{b x}{2}\right )}{b n}-\frac{4 \operatorname{Subst}\left (\int e^{n x} \, dx,x,\cosh \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b n}\\ &=-\frac{4 e^{n \cosh \left (\frac{a}{2}+\frac{b x}{2}\right )}}{b n^2}+\frac{4 e^{n \cosh \left (\frac{a}{2}+\frac{b x}{2}\right )} \cosh \left (\frac{a}{2}+\frac{b x}{2}\right )}{b n}\\ \end{align*}
Mathematica [A] time = 0.158314, size = 36, normalized size = 0.56 \[ \frac{4 e^{n \cosh \left (\frac{1}{2} (a+b x)\right )} \left (n \cosh \left (\frac{1}{2} (a+b x)\right )-1\right )}{b n^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.091, size = 63, normalized size = 1. \begin{align*} 2\,{\frac{ \left ( n{{\rm e}^{bx+a}}+n-2\,{{\rm e}^{1/2\,bx+a/2}} \right ){{\rm e}^{-1/2\,bx-a/2+1/2\,n{{\rm e}^{1/2\,bx+a/2}}+1/2\,n{{\rm e}^{-1/2\,bx-a/2}}}}}{{n}^{2}b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.33967, size = 158, normalized size = 2.47 \begin{align*} \frac{2 \, e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, n e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )} + \frac{1}{2} \, n e^{\left (-\frac{1}{2} \, b x - \frac{1}{2} \, a\right )} + \frac{1}{2} \, a\right )}}{b n} + \frac{2 \, e^{\left (-\frac{1}{2} \, b x + \frac{1}{2} \, n e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )} + \frac{1}{2} \, n e^{\left (-\frac{1}{2} \, b x - \frac{1}{2} \, a\right )} - \frac{1}{2} \, a\right )}}{b n} - \frac{4 \, e^{\left (\frac{1}{2} \, n e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )} + \frac{1}{2} \, n e^{\left (-\frac{1}{2} \, b x - \frac{1}{2} \, 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 = 2.02435, size = 258, normalized size = 4.03 \begin{align*} \frac{4 \,{\left ({\left (n \cosh \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right ) - 1\right )} \cosh \left (n \cosh \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )\right ) +{\left (n \cosh \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right ) - 1\right )} \sinh \left (n \cosh \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )\right )\right )}}{b n^{2} \cosh \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{2} - b n^{2} \sinh \left (\frac{1}{2} \, b x + \frac{1}{2} \, 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 (\frac{a}{2} + \frac{b x}{2} \right )}} \sinh{\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 = 1.18923, size = 343, normalized size = 5.36 \begin{align*} \frac{2 \,{\left (n e^{\left (b x + \frac{1}{4} \,{\left (2 \, b x e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )} + n e^{\left (b x + a\right )} + n\right )} e^{\left (-\frac{1}{2} \, b x - \frac{1}{2} \, a\right )} - \frac{1}{4} \,{\left (2 \, b x e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )} - n e^{\left (b x + a\right )} - n\right )} e^{\left (-\frac{1}{2} \, b x - \frac{1}{2} \, a\right )} + a\right )} + n e^{\left (\frac{1}{4} \,{\left (2 \, b x e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )} + n e^{\left (b x + a\right )} + n\right )} e^{\left (-\frac{1}{2} \, b x - \frac{1}{2} \, a\right )} - \frac{1}{4} \,{\left (2 \, b x e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )} - n e^{\left (b x + a\right )} - n\right )} e^{\left (-\frac{1}{2} \, b x - \frac{1}{2} \, a\right )}\right )} - 2 \, e^{\left (\frac{1}{2} \, b x + \frac{1}{4} \,{\left (2 \, b x e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )} + n e^{\left (b x + a\right )} + n\right )} e^{\left (-\frac{1}{2} \, b x - \frac{1}{2} \, a\right )} - \frac{1}{4} \,{\left (2 \, b x e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )} - n e^{\left (b x + a\right )} - n\right )} e^{\left (-\frac{1}{2} \, b x - \frac{1}{2} \, a\right )} + \frac{1}{2} \, a\right )}\right )} e^{\left (-\frac{1}{2} \, b x - \frac{1}{2} \, a\right )}}{b n^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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