The integrals below involve inverse trigonometric functions.
1) ∫
2) int x*sin^-1(x/a) dx = (x^2/2-a^2/4)sin^-1(x/a)+(xsqrt(a^2-x^2))/4
3) int x^2 sin^-1(x/a) dx = x^3/3sin^-1(x/a)+((x^2+2a^2)sqrt(a^2-x^2))/9
4) int (sin^-1(x/a))/x dx = x/a+(x/a)^3/(2*3*3)+(1*3(x/a)^5)/(2*4*5*5)+(1*3*5(x/a)^7)/(2*4*6*7*7)+...
5) int (sin^-1(x/a))/x^2 dx = -(sin^-1(x/a))/x-1/a ln((a+sqrt(a^2-x^2))/x)
6) int [sin^-1(x/a)]^2 dx = x[sin^-1(x/a)]^2-2x+2sqrt(a^2-x^2)*sin^-1(x/a)
7) int cos^-1(x/a) dx = x*cos^-1(x/a)-sqrt(a^2-x^2)
8) int x*cos^-1(x/a) dx = (x^2/2-a^2/4)cos^-1(x/a)-(xsqrt(a^2-x^2))/4
9) int x^2 cos^-1(x/a) dx = x^3/3 cos^-1(x/a)-((x^2+2a^2)sqrt(a^2-x^2))/9
10) int (cos^-1(x/a))/x dx = pi/2 lnx-int (sin^-1(x/a))/x dx
**[See integral #4 in this table]
11) int (cos^-1(x/a))/x^2 dx = -(cos^-1(x/a))/x+1/aln((a+sqrt(a^2-x^2))/x)
12) int [cos^-1(x/a)]^2 dx = x[cos^-1(x/a)]^2-2x-2sqrt(a^2-x^2) *cos^-1(x/a)
13) int tan^-1(x/a) dx = x*tan^-1(x/a)-a/2ln(x^2+a^2)
14) int x*tan^-1(x/a) dx = 1/2(x^2+a^2)tan^-1(x/a)-(ax)/2
15) int x^2*tan^-1(x/a) dx = x^3/3tan^-1(x/a)-(ax^2)/6+a^3/6ln(x^2+a^2)
16) int (tan^-1(x/a))/x dx = x/a-(x/a)^3/3^2+(x/a)^5/5^2-(x/a)^7/7^2+...
17) int (tan^-1(x/a))/x^2 dx = -1/xtan^-1(x/a)-1/(2a)ln((x^2+a^2)/x^2)
18) int cot^-1(x/a) dx = x*cot^-1(x/a)+a/2ln(x^2+a^2)
19) int x*cot^-1(x/a) dx = 1/2(x^2+a^2)cot^-1(x/a)+(ax)/2
20) int x^2 *cot^-1(x/a) dx = x^3/3cot^-1(x/a)+(ax^2)/6-a^3/6ln(x^2+a^2)
21) int (cot^-1(x/a))/x dx = pi/2lnx-int (tan^-1(x/a))/x dx
**[See integral #16 in this table]
22) int (cot^-1(x/a))/x^2 dx = -(cot^-1(x/a))/x+1/(2a)ln((x^2+a^2)/x^2)
23) int sec^-1(x/a) dx = x*sec^-1(x/a)-a ln(x+sqrt(x^2-a^2)).....For 0 < sec^-1(x/a) < pi/2
OR = x*sec^-1(x/a)+a ln(x+sqrt(x^2-a^2)).....For pi/2 < sec^-1(x/a) < pi
24) int x*sec^-1(x/a) dx = x^2/2sec^-1(x/a)-(asqrt(x^2-a^2))/2.....For 0 < sec^-1(x/a) < pi/2
OR = x^2/2sec^-1(x/a)+(asqrt(x^2-a^2))/2.....For pi/2 < sec^-1(x/a) < pi
25) int x^2*sec^-1(x/a) dx = x^3/3sec^-1(x/a)-(axsqrt(x^2-a^2))/6-a^3/6ln(x+sqrt(x^2-a^2)).....For 0 < sec^-1(x/a) < pi/2
OR = x^3/3sec^-1(x/a)+(axsqrt(x^2-a^2))/6+a^3/6ln(x+sqrt(x^2-a^2)).....For pi/2 < sec^-1(x/a) < pi
26) int (sec^-1(x/a))/x dx = pi/2ln x+a/x+(a/x)^3/(2*3*3)+(1*3(a/x)^5)/(2*4*5*5)+(1*3*5(a/x)^7)/(2*4*6*7*7)+...
27) int (sec^-1(x/a))/x^2 dx = -(sec^-1(x/a))/x+sqrt(x^2-a^2)/(ax).....For 0 < sec^-1(x/a) < pi/2
OR = -(sec^-1(x/a))/x-sqrt(x^2-a^2)/(ax).....For pi/2 < sec^-1(x/a) < pi
28) int csc^-1(x/a) dx = x*csc^-1(x/a)+a ln(x+sqrt(x^2-a^2)).....For 0 < csc^-1(x/a) < pi/2
OR = x*csc^-1(x/a)-a ln(x+sqrt(x^2-a^2)).....For -pi/2 < csc^-1(x/a) < 0
29) int x*csc^-1(x/a) dx = x^2/2 csc^-1(x/a)+(asqrt(x^2-a^2))/2.....For 0 < csc^-1(x/a) < pi/2
OR = x^2/2 csc^-1(x/a)-(asqrt(x^2-a^2))/2.....For -pi/2 < csc^-1(x/a) < 0
30) int x^2 csc^-1(x/a) dx = x^3/3csc^-1(x/a)+(axsqrt(x^2-a^2))/6+a^3/6ln(x+sqrt(x^2-a^2)).....For 0 < csc^-1(x/a) < pi/2
OR = x^3/3csc^-1(x/a)-(axsqrt(x^2-a^2))/6-a^3/6ln(x+sqrt(x^2-a^2)).....For -pi/2 < csc^-1(x/a) < 0
31) int (csc^-1(x/a))/x dx = -(a/x+(a/x)^3/(2*3*3)+(1*3(a/x)^5)/(2*4*5*5)+(1*3*5(a/x)^7)/(2*4*6*7*7)+...)
32) int (csc^-1(x/a))/x^2 dx = -(csc^-1(x/a))/x-sqrt(x^2-a^2)/(ax).....For 0 < csc^-1(x/a) < pi/2
OR = -(csc^-1(x/a))/x+sqrt(x^2-a^2)/(ax).....For -pi/2 < csc^-1(x/a) < 0
33) int x^m sin^-1(x/a) dx = x^(m+1)/(m+1)sin^-1(x/a)-1/(m+1) int x^(m+1)/sqrt(a^2-x^2) dx
34) int x^mcos^-1(x/a) dx = x^(m+1)/(m+1)cos^-1(x/a)+1/(m+1)int x^(m+1)/sqrt(a^2-x^2) dx
35) int x^mtan^-1(x/a) dx = x^(m+1)/(m+1)tan^-1(x/a)-a/(m+1)int x^(m+1)/(x^2+a^2) dx
36) int x^mcot^-1(x/a) dx = x^(m+1)/(m+1)cot^-1(x/a)+a/(m+1)int x^(m+1)/(x^2+a^2) dx
37) int x^msec^-1(x/a) dx = (x^(m+1)sec^-1(x/a))/(m+1)-a/(m+1)int x^m/sqrt(x^2-a^2) dx.....For 0 < sec^-1(x/a) < pi/2
OR = (x^(m+1)sec^-1(x/a))/(m+1)+a/(m+1)int x^m/sqrt(x^2-a^2) dx.....For pi/2 < sec^-1(x/a) < pi
38) int x^mcsc^-1(x/a) dx = (x^(m+1)csc^-1(x/a))/(m+1)+a/(m+1)int x^m/sqrt(x^2-a^2) dx.....For 0 < csc^-1(x/a) < pi/2
OR = (x^(m+1)csc^-1(x/a))/(m+1)-a/(m+1)int x^m/sqrt(x^2-a^2) dx.....For -pi/2 < csc^-1(x/a) < 0