# Can this expression be calculated symbolically?

I want to do definite integration of the following expression

expr=(4*θ*Sqrt[1/(1 + (4*Cos[θ])/169)]*Cos[θ]*
Sqrt[(-4*Sqrt[1/(1 + (4*Cos[θ])/169)]*Sin[θ] +
(8/169)*(1/(1 + (4*Cos[θ])/169))^(3/2)*Cos[θ]*Sin[θ])^2 +
(4*Sqrt[1/(1 + (4*Cos[θ])/169)]*Cos[θ] +
(8/169)*(1/(1 + (4*Cos[θ])/169))^(3/2)*Sin[θ]^2)^2])/
(9/4 + (16*Cos[θ]^2)/(1 + (4*Cos[θ])/169) +
(16*Sin[θ]^2)/(1 + (4*Cos[θ])/169))^(3/2)


But when I do Integrate[expr,{θ,0,2*Pi}] it just goes on and on and dont stop in 5 mins. The expr is smooth on the whole interval of integration. Here is the plot.

Plot[expr, {θ, 0, 2*Pi}]


Please tell me what could be the reason of such behavior of Mathematica? How do I properly integrate this expression? Maybe the only way is to do NIntegrate[]? And how one knows if the expression can be integrated symbolically?

• It seems unlikely that this would have a simple symbolic result. Wouldn't a numerical one do? – Szabolcs Jun 23 '16 at 13:39
• @Szabolcs in Mapple it gives the symbolic result very fast. So there should be one. Maybe I need to use some options in Integrate[] function?NIntegrate[] gives the expected result, but I need the symbolic one. – Mr Bubble Hubble Jun 23 '16 at 13:41
• @Szabolcs sry, my bad. Maple doesnt give the symbolic result. Maybe u are right. I will edit my question now. – Mr Bubble Hubble Jun 23 '16 at 13:48
• I was just about to say, I've tried running the integration on python, and it doesn't do any better than Mathematica. – Feyre Jun 23 '16 at 14:01
• @Feyre ty, I think it can be calculated only numerically. – Mr Bubble Hubble Jun 23 '16 at 14:04

NIntegrate[expr,{\[Theta],0,2*Pi}]// Timing


{0.008, 0.0151049}

When you integrating it within a limit, it will always give you a number, not a function. Now, say you are interested in finding the integration as a function of the upper limit. In that case you can get an InterpolatingFunction which you can use in any further calculation.

data = Table[{x, NIntegrate[expr, {\[Theta], 0, x}]}, {x, 0., 4 Pi, Pi/50.}];
f[x_] = Interpolation[data][x];

Plot[f[x], {x, 0, 4 Pi}, Prolog -> Point[data]]


So your answer is f[x]. You can try to fit it with any trial function. Probably it is possible to get an analytic answer with proper Assumption, but I am not very sure about that. So wait for a better answer.

• Why it is a list and not one number? – Mr Bubble Hubble Jun 23 '16 at 13:51
• First one is the time and second is the result – Sumit Jun 23 '16 at 13:52
• @Sumit, you forgot the Timing ;) – Marius Ladegård Meyer Jun 23 '16 at 13:52
• @Sumit ok, I know that this can be calculated with NIntegrate[]. Can it be calculated symbolically? – Mr Bubble Hubble Jun 23 '16 at 13:53
• @Sumit, I need the expr to be integrate over \[Theta] not x – Mr Bubble Hubble Jun 23 '16 at 13:55