Optimal. Leaf size=18 \[ \frac{\text{Ei}(n \sinh (c (a+b x)))}{b c} \]
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Rubi [A] time = 0.0229849, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {4340, 2178} \[ \frac{\text{Ei}(n \sinh (c (a+b x)))}{b c} \]
Antiderivative was successfully verified.
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Rule 4340
Rule 2178
Rubi steps
\begin{align*} \int e^{n \sinh (a c+b c x)} \coth (c (a+b x)) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{e^{n x}}{x} \, dx,x,\sinh (c (a+b x))\right )}{b c}\\ &=\frac{\text{Ei}(n \sinh (c (a+b x)))}{b c}\\ \end{align*}
Mathematica [A] time = 0.0619603, size = 18, normalized size = 1. \[ \frac{\text{Ei}(n \sinh (c (a+b x)))}{b c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 23, normalized size = 1.3 \begin{align*} -{\frac{{\it Ei} \left ( 1,-n\sinh \left ( bcx+ac \right ) \right ) }{cb}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \coth \left ({\left (b x + a\right )} c\right ) e^{\left (n \sinh \left (b c x + a c\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.13657, size = 42, normalized size = 2.33 \begin{align*} \frac{{\rm Ei}\left (n \sinh \left (b c x + a c\right )\right )}{b c} \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 \sinh{\left (a c + b c x \right )}} \coth{\left (a c + b c 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 \coth \left ({\left (b x + a\right )} c\right ) e^{\left (n \sinh \left (b c x + a c\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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