Optimal. Leaf size=28 \[ a \cosh (c) \text{Chi}(d x)+a \sinh (c) \text{Shi}(d x)+\frac{b \sinh (c+d x)}{d} \]
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Rubi [A] time = 0.148764, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6742, 2637, 3303, 3298, 3301} \[ a \cosh (c) \text{Chi}(d x)+a \sinh (c) \text{Shi}(d x)+\frac{b \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 6742
Rule 2637
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{(a+b x) \cosh (c+d x)}{x} \, dx &=\int \left (b \cosh (c+d x)+\frac{a \cosh (c+d x)}{x}\right ) \, dx\\ &=a \int \frac{\cosh (c+d x)}{x} \, dx+b \int \cosh (c+d x) \, dx\\ &=\frac{b \sinh (c+d x)}{d}+(a \cosh (c)) \int \frac{\cosh (d x)}{x} \, dx+(a \sinh (c)) \int \frac{\sinh (d x)}{x} \, dx\\ &=a \cosh (c) \text{Chi}(d x)+\frac{b \sinh (c+d x)}{d}+a \sinh (c) \text{Shi}(d x)\\ \end{align*}
Mathematica [A] time = 0.0279314, size = 39, normalized size = 1.39 \[ a \cosh (c) \text{Chi}(d x)+a \sinh (c) \text{Shi}(d x)+\frac{b \sinh (c) \cosh (d x)}{d}+\frac{b \cosh (c) \sinh (d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 52, normalized size = 1.9 \begin{align*} -{\frac{a{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2}}-{\frac{b{{\rm e}^{-dx-c}}}{2\,d}}-{\frac{a{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{2}}+{\frac{b{{\rm e}^{dx+c}}}{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.20109, size = 131, normalized size = 4.68 \begin{align*} -\frac{1}{2} \,{\left (b{\left (\frac{{\left (d x e^{c} - e^{c}\right )} e^{\left (d x\right )}}{d^{2}} + \frac{{\left (d x + 1\right )} e^{\left (-d x - c\right )}}{d^{2}}\right )} + \frac{2 \, a \cosh \left (d x + c\right ) \log \left (x\right )}{d} - \frac{{\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} +{\rm Ei}\left (d x\right ) e^{c}\right )} a}{d}\right )} d +{\left (b x + a \log \left (x\right )\right )} \cosh \left (d x + c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.96747, size = 142, normalized size = 5.07 \begin{align*} \frac{{\left (a d{\rm Ei}\left (d x\right ) + a d{\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) + 2 \, b \sinh \left (d x + c\right ) +{\left (a d{\rm Ei}\left (d x\right ) - a d{\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.54689, size = 34, normalized size = 1.21 \begin{align*} a \sinh{\left (c \right )} \operatorname{Shi}{\left (d x \right )} + a \cosh{\left (c \right )} \operatorname{Chi}\left (d x\right ) + b \left (\begin{cases} \frac{\sinh{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \cosh{\left (c \right )} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1822, size = 63, normalized size = 2.25 \begin{align*} \frac{a d{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a d{\rm Ei}\left (d x\right ) e^{c} + b e^{\left (d x + c\right )} - b e^{\left (-d x - c\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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