# How to integrate a logsumexp function?

I am trying to integrate a logsumexp function which is a smooth function of another piecewise function.

Run

Integrate[ Log[Exp[s*a] + Exp[s*(a + v +a*t)]], {t, 0, (-v +vm)/a}, Assumptions -> {s > 0, a > 0, v >= 0, vm > 0}],

then, I got

(PolyLog[2, -e^(v * s)] - PolyLog[2, -e^(vm * s])] + a (-v + vm)*s^2)/(a*s).

But if I run

Integrate[ Log[Exp[s*a] + Exp[s*(a + v +a*t + (v +a*t)^2)]], {t, 0, (-v +vm)/a}, Assumptions -> {s > 0, a > 0, v >= 0, vm > 0}],

I cannot get any result.

• Are you aware that there might be integrals out there that cannot be expressed in closed form? – Henrik Schumacher Dec 11 '18 at 7:33
• @HenrikSchumacher Maybe you are right. If that is true, do you have any suggestion to solve this? numerical interation? – WZhao Dec 11 '18 at 7:37
• If you replace the free variables (s, am v, vm, ...) by numeric values then you can apply numeric integration by using NIntegrate instead of Integrate. – Henrik Schumacher Dec 11 '18 at 7:49
• Amplifying on Henrik's comment: See the tutorial on Integration, specificly the example Integrate[x^x, x] for which it is stated "This integral simply cannot be done in terms of standard mathematical functions. As a result, the Wolfram Language just leaves it undone. " – Bob Hanlon Dec 11 '18 at 17:24