1/(cosx(5+3cosx) ) 的不定积分

1/(cosx(5+3cosx) ) 的不定积分
1/(cosx(5+3cosx) ) 的不定积分

1/(cosx(5+3cosx) ) 的不定积分
int(1/(cos(x)*(5+3*cos(x))),x)
= int((-1-u^2)/(3*u^2+u^4-4),u) and int((-1-u^2)/(3*u^2+u^4-4),u)
= int(-(3/5)/(u^2+4)+1/(5*u+5)-1/(-5+5*u),u) and int(-(3/5)/(u^2+4)+1/(5*u+5)-1/(-5+5*u),u)
= int(-(3/5)/(u^2+4),u)+int(1/(5*u+5),u)+int(-1/(-5+5*u),u) and int(-(3/5)/(u^2+4),u)+int(1/(5*u+5),u)+int(-1/(-5+5*u),u)
= -(3/5)*(int(1/(u^2+4),u))+int(1/(5*u+5),u)+int(-1/(-5+5*u),u) and -(3/5)*(int(1/(u^2+4),u))+int(1/(5*u+5),u)+int(-1/(-5+5*u),u)
= -(3/5)*(int(1/2,u1))+int(1/(5*u+5),u)+int(-1/(-5+5*u),u) and -(3/5)*(int(1/2,u1))+int(1/(5*u+5),u)+int(-1/(-5+5*u),u)
= -(3/10)*u1+int(1/(5*u+5),u)+int(-1/(-5+5*u),u) and -(3/10)*u1+int(1/(5*u+5),u)+int(-1/(-5+5*u),u)
= -(3/10)*arctan((1/2)*u)+int(1/(5*u+5),u)+int(-1/(-5+5*u),u) and -(3/10)*arctan((1/2)*u)+int(1/(5*u+5),u)+int(-1/(-5+5*u),u)
= -(3/10)*arctan((1/2)*u)+(1/5)*(int(1/(u+1),u))+int(-1/(-5+5*u),u) and -(3/10)*arctan((1/2)*u)+(1/5)*(int(1/(u+1),u))+int(-1/(-5+5*u),u)
= -(3/10)*arctan((1/2)*u)+(1/5)*(int(1/u1,u1))+int(-1/(-5+5*u),u) and -(3/10)*arctan((1/2)*u)+(1/5)*(int(1/u1,u1))+int(-1/(-5+5*u),u)
= -(3/10)*arctan((1/2)*u)+(1/5)*ln(u1)+int(-1/(-5+5*u),u) and -(3/10)*arctan((1/2)*u)+(1/5)*ln(u1)+int(-1/(-5+5*u),u)
= -(3/10)*arctan((1/2)*u)+(1/5)*ln(u+1)+int(-1/(-5+5*u),u) and -(3/10)*arctan((1/2)*u)+(1/5)*ln(u+1)+int(-1/(-5+5*u),u)
= -(3/10)*arctan((1/2)*u)+(1/5)*ln(u+1)-(1/5)*(int(1/(-1+u),u)) and -(3/10)*arctan((1/2)*u)+(1/5)*ln(u+1)-(1/5)*(int(1/(-1+u),u))
= -(3/10)*arctan((1/2)*u)+(1/5)*ln(u+1)-(1/5)*(int(1/u1,u1)) and -(3/10)*arctan((1/2)*u)+(1/5)*ln(u+1)-(1/5)*(int(1/u1,u1))
= -(3/10)*arctan((1/2)*u)+(1/5)*ln(u+1)-(1/5)*ln(u1) and -(3/10)*arctan((1/2)*u)+(1/5)*ln(u+1)-(1/5)*ln(u1)
= -(3/10)*arctan((1/2)*u)+(1/5)*ln(u+1)-(1/5)*ln(-1+u) and -(3/10)*arctan((1/2)*u)+(1/5)*ln(u+1)-(1/5)*ln(-1+u)
= -(3/10)*arctan((1/2)*tan((1/2)*x))-(1/5)*ln(tan((1/2)*x)-1)+(1/5)*ln(tan((1/2)*x)+1)

用万能公式吧 楼上的过程太复杂了
这里我只提供思路，用一遍万能公式 设u=tan(x/2) 则cosx=(1-u^2)/(1+u^2)
代入化简后是一个真分式，再用待定系数法就可以解决了