How to fix a long multiplicative expression?

Question

Consider the following expression:

\documentclass[12pt, a4paper]{report}
\usepackage{amsfonts, graphicx, verbatim, mathtools,amssymb, amsthm, mathrsfs,amsmath}
\begin{document}
       \begin{align*}
    S\left(t\right)&\geq  S\left(t\right)\exp \left\lbrace {-\int_0^t \left[\beta_A A\left(\tau\right) +\beta_I I\left(\tau\right) \right]\;\mathrm{d}\tau-\mu t -\rho t} \right \rbrace\\[1ex]
    &+\exp \left\lbrace {-\int_0^t \left[\beta_A A\left(\tau\right) +\beta_I I\left(\tau\right) \right]\;\mathrm{d}\tau-\mu t -\rho t} \right \rbrace \left \lbrace\int_0^t b(1-\nu)  \exp\left \lbrace{\int_0^\tau \left[\beta_A A\left(\zeta\right) +\beta_I I\left(\zeta\right) \right]\;\mathrm{d}\zeta+\mu \tau +\rho \tau}\right \rbrace\;\mathrm{d}\tau\right \rbrace.
    \end{align*}
\end{document}

What would be an "ideal" way to write this?

Answer 1

The first thing you should do is to eliminate all \left/\right pairs, as they are doing nothing but introducing spurious spaces. I would also define a macro for the differential and give it an operator spacing (\, instead of \;). The brackets around the last term look superfluous to me.

\documentclass[12pt, a4paper]{report}

\usepackage{amsmath}

\newcommand*{\diff}{\mathop{}\!\mathrm{d}} % common definition

\begin{document}

\[
S(t) \geq
\begin{aligned}[t]
&S(t)\exp\biggl\lbrace {-\int_0^t [\beta_A A(\tau) +\beta_I I(\tau) ] \diff\tau-\mu t -\rho t} \biggr\rbrace\\[1ex]
&+\exp \biggl\lbrace {-\int_0^t [\beta_A A(\tau) +\beta_I I(\tau) ]\diff\tau-\mu t -\rho t} \biggr\rbrace \\
&\qquad\times \int_0^t b(1-\nu)  \exp\biggl\lbrace{\int_0^\tau [\beta_A A(\zeta) +\beta_I I(\zeta) ]\diff\zeta+\mu \tau +\rho \tau}\biggr\rbrace\diff\tau.
\end{aligned}
\]

\end{document}

Note that mathtools loads amsmath and graphicx, and amssymb loads amsfonts. Your

\usepackage{amsfonts, graphicx, verbatim, mathtools,amssymb, amsthm, mathrsfs,amsmath}

can be thus reduced to

\usepackage{verbatim,mathtools,amssymb, amsthm, mathrsfs}

Answer 2

Another option is to split it into multiple statements, such as

\documentclass[12pt, a4paper]{report}
\usepackage{verbatim, mathtools,amssymb, amsthm, mathrsfs}
\begin{document}
    \begin{equation*}
        S(t) \ge S(t) \exp[- J(t) ] + \exp[-J(t)] K(t)
    \end{equation*}
    where
    \begin{align*}
        J(t) &= \int_0^t [\beta_A A(\tau) + \beta_I I(\tau)] \; \mathrm{d}\tau - \mu t - \rho t\\
        K(t) &= \int_0^t b(1-\nu)  \exp\left \lbrace{\int_0^\tau [\beta_A A(\zeta) +\beta_I I(\zeta) ] \; \mathrm{d}\zeta+\mu \tau +\rho \tau}\right \rbrace\;\mathrm{d}\tau
    \end{align*}
\end{document}

giving the result

Answer 3

Does this layout with the multlined environment befit you?

Unrelated: you don't have to load amsfont when load amssymb as the latter does it for you. Similarly, needless to load amsmath if you load mathtools.

\documentclass[12pt, a4paper]{report}
\usepackage{graphicx, verbatim, mathtools,amssymb, amsthm, mathrsfs}

\begin{document}

       \begin{align*}
    S\left(t\right)&\geq S\left(t\right)\exp \left\lbrace {-\int_0^t \left[\beta_A A\left(\tau\right) +\beta_I I\left(\tau\right) \right]\;\mathrm{d}\tau-\mu t -\rho t} \right \rbrace\\[1ex]
    &\phantom{\geq}+\begin{multlined}[t]\exp \left\lbrace {-\int_0^t \left[\beta_A A\left(\tau\right) +\beta_I I\left(\tau\right) \right]\;\mathrm{d}\tau-\mu t -\rho t} \right \rbrace \times \\
    \left \lbrace\int_0^t b(1-\nu) \exp\left \lbrace{\int_0^\tau \left[\beta_A A\left(\zeta\right) +\beta_I I\left(\zeta\right) \right]\;\mathrm{d}\zeta+\mu \tau +\rho \tau}\right \rbrace\;\mathrm{d}\tau\right \rbrace.\end{multlined}
    \end{align*}

\end{document} 

Answer 4

You have already got several nice answers. Since the question is tagged typography and you ask for an ideal way of doing this, I feel that an approach different from the others could be of interest (perhaps not for you, right now, since you are using LaTeX and not ConTeXt, but maybe for other users).

I think that an ideal way of typesetting this kind of formula is probably one that is not based on math alignments, but on some paragraph type math environment. It also has the advantage that one can break it inside fences, even though that is not necessary here. (It could also potentially be better when it comes to tagging and voice reading support, but let us leave that aside.)

We define such an environment with

\defineformula[long][
    split=yes,
    textalign=middle,
    hang=yes,
    distance=1em,
    spaceinbetween=.5\lineheight,
]

Then we can typeset the formula, more or less as you did (I removed some of the \left( and \right) since they do not add anything at those places, and also added some formatting.)

\startlongformula
    S(t) 
    \alignhere 
        \geq 
        S(t) 
        \exp 
        \left\lbrace 
            -\int_{0}^{t} 
            \left[
                \beta_{A} A(\tau) +\beta_{I} I(\tau) 
            \right]
            \dd\tau
            -\mu t -\rho t
        \right\rbrace
    \breakhere
    \skiphere[1]
        +\exp
        \left\lbrace 
            -\int_{0}^{t} 
            \left[
                \beta_{A} A(\tau) + \beta_{I} I(\tau) 
            \right]
            \dd\tau
            -\mu t -\rho t 
        \right\rbrace 
    \breakhere
    \skiphere[2]
        \times
        \left(
            \int_{0}^{t} b(1-\nu)  
            \exp
            \left\lbrace
                \int_{0}^{\tau} 
                \left[
                    \beta_{A} A(\zeta) + \beta_{I} I(\zeta)
                \right]
                \dd\zeta 
                + \mu \tau +\rho \tau
            \right\rbrace
            \dd\tau
        \right).
\stoplongformula

It is worth to note the \alignhere that sets an align point (not really needed in this example, but if there would be another line it would be useful) the \breakhere that insert line breaks, and also the \skiphere[1] and \skiphere[2] that indent the second and third line. (The \dd gives a d set in the differential class, no fixing with spaces needed.)

The output looks like this: