Optimal. Leaf size=73 \[ -\frac{\cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{a}-\frac{\sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{a}+\frac{\cosh (c) \text{Chi}(d x)}{a}+\frac{\sinh (c) \text{Shi}(d x)}{a} \]
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Rubi [A] time = 0.258935, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {6742, 3303, 3298, 3301} \[ -\frac{\cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{a}-\frac{\sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{a}+\frac{\cosh (c) \text{Chi}(d x)}{a}+\frac{\sinh (c) \text{Shi}(d x)}{a} \]
Antiderivative was successfully verified.
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Rule 6742
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{\cosh (c+d x)}{x (a+b x)} \, dx &=\int \left (\frac{\cosh (c+d x)}{a x}-\frac{b \cosh (c+d x)}{a (a+b x)}\right ) \, dx\\ &=\frac{\int \frac{\cosh (c+d x)}{x} \, dx}{a}-\frac{b \int \frac{\cosh (c+d x)}{a+b x} \, dx}{a}\\ &=\frac{\cosh (c) \int \frac{\cosh (d x)}{x} \, dx}{a}-\frac{\left (b \cosh \left (c-\frac{a d}{b}\right )\right ) \int \frac{\cosh \left (\frac{a d}{b}+d x\right )}{a+b x} \, dx}{a}+\frac{\sinh (c) \int \frac{\sinh (d x)}{x} \, dx}{a}-\frac{\left (b \sinh \left (c-\frac{a d}{b}\right )\right ) \int \frac{\sinh \left (\frac{a d}{b}+d x\right )}{a+b x} \, dx}{a}\\ &=\frac{\cosh (c) \text{Chi}(d x)}{a}-\frac{\cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{a d}{b}+d x\right )}{a}+\frac{\sinh (c) \text{Shi}(d x)}{a}-\frac{\sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.140752, size = 63, normalized size = 0.86 \[ \frac{-\cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (d \left (\frac{a}{b}+x\right )\right )-\sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right )+\cosh (c) \text{Chi}(d x)+\sinh (c) \text{Shi}(d x)}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 108, normalized size = 1.5 \begin{align*} -{\frac{{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2\,a}}+{\frac{1}{2\,a}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }-{\frac{{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{2\,a}}+{\frac{1}{2\,a}{{\rm e}^{-{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,-dx-c-{\frac{da-cb}{b}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.43734, size = 209, normalized size = 2.86 \begin{align*} \frac{1}{2} \, d{\left (\frac{b{\left (\frac{e^{\left (-c + \frac{a d}{b}\right )} E_{1}\left (\frac{{\left (b x + a\right )} d}{b}\right )}{b} + \frac{e^{\left (c - \frac{a d}{b}\right )} E_{1}\left (-\frac{{\left (b x + a\right )} d}{b}\right )}{b}\right )}}{a d} + \frac{2 \, \cosh \left (d x + c\right ) \log \left (b x + a\right )}{a d} - \frac{2 \, \cosh \left (d x + c\right ) \log \left (x\right )}{a d} + \frac{{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} +{\rm Ei}\left (d x\right ) e^{c}}{a d}\right )} -{\left (\frac{\log \left (b x + a\right )}{a} - \frac{\log \left (x\right )}{a}\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 = 2.06579, size = 277, normalized size = 3.79 \begin{align*} \frac{{\left ({\rm Ei}\left (d x\right ) +{\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) -{\left ({\rm Ei}\left (\frac{b d x + a d}{b}\right ) +{\rm Ei}\left (-\frac{b d x + a d}{b}\right )\right )} \cosh \left (-\frac{b c - a d}{b}\right ) +{\left ({\rm Ei}\left (d x\right ) -{\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right ) +{\left ({\rm Ei}\left (\frac{b d x + a d}{b}\right ) -{\rm Ei}\left (-\frac{b d x + a d}{b}\right )\right )} \sinh \left (-\frac{b c - a d}{b}\right )}{2 \, a} \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 \frac{\cosh{\left (c + d x \right )}}{x \left (a + b x\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15917, size = 101, normalized size = 1.38 \begin{align*} \frac{{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} -{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} +{\rm Ei}\left (d x\right ) e^{c} -{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )}}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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