# Solving an indefinite integral with an unsual function

I am trying to solve this integral but do not get any point to start.I was thinking an U-substitution may help but do not know what to consider as U. Tried with mathematica but it does not provide any solution. Can anyone help? $$\int sin\theta *2\sin^{-1} [\frac{\cos \theta+\sin \gamma}{\sin \theta \tan \beta}] d\theta$$

• If Mma doesn't give an answer, it's fairly likely that there is no way to reduce it to standard functions. – The_Sympathizer Jul 4 at 7:55
• @The_Sympathizer I do not completely agree with you because I did one integration by hand where mathematica failed to solve. It might be my limited knowledge on mathematica as well. I always use basic commands there. – T. an Jul 4 at 8:02
• @Tanvir: cases where you can beat Mathematica are exceptional. – Yves Daoust Jul 4 at 8:03
• @Tanvir : Yup, but that's why I said merely "likely", not "certainly". – The_Sympathizer Jul 4 at 8:03
• @Yves Daoust: Though, to be fair, there are more published special functions than Mma has in its repertoire and one of them might be able to ork it. E.g. there are various exotic types of hypergeometric function Mma does not implement like Kampe de Feriet, which can be used to integrate $e^{\sin(x)}$, a famously Mma-resistant integrand. – The_Sympathizer Jul 4 at 8:04

Hint:

With $$t:=\cos\theta$$, renaming the constants and dropping the $$2$$,

$$I:=-\int \arcsin\frac{t+c}{b\sqrt{1-t^2}}dt.$$

By parts,

$$I:=-t\arcsin\frac{t+c}{b\sqrt{1-t^2}}+\int t\frac{t(t+c)+1-t^2}{b(1-t^2)^{3/2}\sqrt{1-\dfrac{(t+c)^2}{b^2(1-t^2)}}}dt \\=\cdots+\int t\frac{ct+1}{(1-t^2)\sqrt{b^2(1-t^2)-(t+c)^2}}dt.$$

This rationalizes the integral and Alpha is able to integrate. Strange.

Update:

Actually, Alpha does the integration by parts:

https://www.wolframalpha.com/input/?i=integrate+arcsin((t%2Bc)%2F(sqrt(1-t%5E2)+b))

• Sir, which alpha you are talking about? – T. an Jul 4 at 8:32
• @Tanvir: wolframalpha.com/input/…. – Yves Daoust Jul 4 at 8:33
• Yeah now it able to integrate but before it gives some result after quite long time and that was really awkward. – T. an Jul 4 at 8:35
• Now out of curiosity one question comes in my mind, why alpha was not able to use Usubstitution and by parts method? – T. an Jul 4 at 8:37
• @Tanvir: computation time exceeded. Try again with Mathematica. – Yves Daoust Jul 4 at 8:39