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