Thinking about a kind of heavy integral

[yangyongkang]

Definite integral and indefinite integral can be said to be the core of calculus.I found that there is a kind of trigonometric integral, and XCAS can't do anything about it.

I will give the general form of it first.

Code:

int((a*sin(x)+b*cos(x))/(c+d*sin(2*x)+f*cos(2*x)), x)

Maple gives a general answer in the picture, and it can be seen that it is essentially necessary to solve a quadratic equation.

This type of indefinite integral can spawn many similar problems.

Code:

int(sin(x)/(2+sin(x)*cos(x)), x)

Answer from Wolfram Alpha

Code:

int(cos(x)/(2-sin(x)*cos(x)),x)

Answer from Wolfram Alpha

Code:

int((2sin(x)+3*cos(x))/(1+sin(x)*cos(x)),x)

Answer from Wolfram Alpha

Code:

int((sin(x)+cos(x))/(1-sin(x)*cos(x)),x)

Answer from Wolfram Alpha

[parisse]

Xcas does not try to match to a long table of integral types. It identifies a trig fraction, and runs the change of variable t=tan(x/2) (except if symmetries allows t=sin(x), t=cos(x) or t=tan(x)). This raises a 4th order equation, and it can not be solved with symbolic coefficients (well it can theoretically, but the roots would be much too complicated unless you are using rootofs). If the coefficients have specific values, it can be solved. I get a complex answer for int(sin(x)/(2+sin(x)*cos(x)), x) but without rootof, and you can evalc it to get back to real. Maple returns a simpler expression, but with rootofs, it does not csolve(900x^4+15x^2+1). I don't get anything with Wolfy.