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If I have for any piecewise continuous function $f$ $$ F(x) = \int f(x)\ dx $$ is it possible to calculate $$ \int f(ax+b)\ e^{cx}\ dx $$ If $f$ is $sin$ then a computer algebra system gives me $$ (e^{cx}\ (sin(ax + b) - a\ cos(ax + b)))/(a^2 + c^2) $$ However for a general $f$ the system says that there is no solution using standard mathematical functions.

I am not necessarily looking for an actual formula for it and argument for its existence would suffice (with not too onerous preconditions if required).

Mark Wassell
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    We know that $\int \frac{1}{x} , \mathrm{d}x = \log x$ (up to an additive constant), but it is proved that $\int \frac{\mathrm{e}^x}{x} , \mathrm{d}x$ has no elementary antiderivative. Even $f(x) = \sqrt{x}$ will give you yet another integral without elementary antiderivative. – Sangchul Lee May 18 '16 at 17:47
  • Thanks. Looking at your examples has made me realise I also have that $f$ is piecewise continuous. – Mark Wassell May 18 '16 at 17:59

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