Integrating e^t sin(6t)/cos²(5t)

[BobbyBear]

Hello, I need help witht he following integral which has come up while trying to solve a differential equation.

$$<br /> \int \frac{e^t}{cos^2(5t)}sin(6t)dt<br />$$

I'm not sure it can be integrated analytically, I 've tried to think of possible trigonometric manipulations and integrating by parts ... but not leading anywhere, any help would be appreciated.

 

[Cyosis]

I am pretty sure a primitive of this function does not exist in closed form/elementary functions.

 

[Mute]

Gradshteyn & Rhyzik states that indefinite integrals of the forms exp[ax] divided by sin(bx), cos(bx), sin^2(bx) or cos^2(bx) aren't expressible as a finite combination of elementary functions, so your integral is similarly hopeless, as it's a linear combination of such integrals (for complex a).

You might be able to get an infinite series for it, though.

Or, if you had constant limits it might be a doable integral.

 

[BobbyBear]

Thank you, I've been breaking my head trying to find a closed form in terms of elementary functions, but wasn't able to come up with anything. I guess the only way then is to use Taylor expansions or the like and then integrate the terms of the resulting series.

No, I don't have constant limits: the integral came up in the process of trying to find the solution function of an ordinary differential equation.

Thank you for your help :)