Integral including a Bessel function of the first kind

Question

I tried to find the following integral using maple and mathematica but they would not do me the favour (only for $b=1$, but I am looking for generic real $b,a,c$).

$$\int_0^a x \sin(bx)\, J_1(cx)\,dx$$

Anybody an idea how to do it?

Answer 1

Maple does the case $b=c$, maybe that is the case you intended when you said $b=1$. $$ \int x \;\sin(cx) \;\mathrm{J}_1(cx)\;dx = {\frac {x \big( xc\sin \left( cx \right) {{\rm J}_1\left(cx\right)}+xc\cos \left( cx \right) {{\rm J}_0\left(cx\right)}-2\,\cos \left( cx \right) {{\rm J}_1\left(cx\right)} \big) }{3c}} $$