The integrals below involve inverse hyperbolic functions.
1) `int sinh^-1(x/a) dx = x sinh^-1(x/a)-sqrt(x^2+a^2)`
2) `int x*sinh^-1(x/a) dx = (x^2/2+a^2/4)sinh^-1(x/a)-(xsqrt(x^2+a^2))/4`
3) `int x^2 sinh^-1(x/a) dx = x^3/3sinh^-1(x/a)+((2a^2-x^2)sqrt(x^2+a^2))/9`
4) `int (sinh^-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)+...` **[For `absx < a]`
OR `= (ln^2((2x)/a))/2-(a/x)^2/(2*2*2)+(1*3(a/x)^4)/(2*4*4*4)-(1*3*5(a/x)^6)/(2*4*6*6*6)+...` **[For `x > a`]
OR `=-(ln^2((-2x)/a))/2+(a/x)^2/(2*2*2)-(1*3(a/x)^4)/(2*4*4*4)+(1*3*5(a/x)^6)/(2*4*6*6*6)-...` **[For `x < -a`]
5) `int (sinh^-1(x/a))/x^2 dx = -(sinh^-1(x/a))/x-1/a ln((a+sqrt(x^2+a^2))/x)`
6) `int cosh^-1(x/a) dx = x*cosh^-1(x/a)-sqrt(x^2-a^2)` **[For `cosh^-1(x/a) > 0`]
OR `=x*cosh^-1(x/a)+sqrt(x^2-a^2)` **[For `cosh^-1(x/a) < 0`]
7) `int x*cosh^-1(x/a) dx = 1/4(2x^2-a^2) cosh^-1(x/a)-1/4 x sqrt(x^2-a^2)` **[For `cosh^-1(x/a) > 0`]
OR `=1/4(2x^2-a^2) cosh^-1(x/a)+1/4 x sqrt(x^2-a^2)` **[For `cosh^-1(x/a) < 0`]
8) `int x^2 cosh^-1(x/a) dx = 1/3x^3cosh^-1(x/a)-1/9(x^2+2a^2)sqrt(x^2-a^2)` **[For `cosh^-1(x/a) > 0`]
OR `= 1/3x^3cosh^-1(x/a)+1/9(x^2+2a^2)sqrt(x^2-a^2)` **[For `cosh^-1(x/a) < 0`]
9) `int (cosh^-1(x/a))/x dx = +-[1/2 ln^2((2x)/a)+(a/x)^2/(2*2*2)+(1*3(a/x)^4)/(2*4*4*4)+(1*3*5(a/x)^6)/(2*4*6*6*6)+...]`
**[`+` if `cosh^-1(x/a) > 0`; `-` if `cosh^-1(x/a) < 0`]
10) `int (cosh^-1(x/a))/x^2 dx = -(cosh^-1(x/a))/x+-1/a ln ((a+sqrt(x^2+a^2))/x)`
**[`+` if `cosh^-1(x/a) < 0`; `-` if `cosh^-1(x/a) > 0`]
11) `int tanh^-1(x/a) dx = x*tanh^-1(x/a)+a/2 ln (a^2-x^2)`
12) `int x*tanh^-1(x/a) dx = (ax)/2+1/2(x^2-a^2)tanh^-1(x/a)`
13) `int x^2tanh^-1(x/a) dx = (ax^2)/6+x^3/3 tanh^-1(x/a)+a^3/6 ln (a^2-x^2)`
14) `int (tanh^-1(x/a))/x dx = x/a+(x/a)^3/3^2+(x/a)^5/5^2+...`
15) `int (tanh^-1(x/a))/x^2 dx = -(tanh^-1(x/a))/x+1/(2a) ln (x^2/(a^2-x^2))`
16) `int coth^-1(x/a) dx = x*coth^-1x+a/2 ln(x^2-a^2)`
17) `int x*coth^-1(x/a) dx = (ax)/2+1/2(x^2-a^2)coth^-1(x/a)`
18) `int x^2coth^-1(x/a) dx = (ax^2)/6+x^3/3coth^-1(x/a)+a^3/6 ln (x^2-a^2)`
19) `int (coth^-1(x/a))/x dx = -(a/x+(a/x)^3/3^2+(a/x)^5/5^2+...)`
20) `int (coth^-1(x/a))/x^2 dx = -(coth^-1(x/a))/x+1/(2a) ln (x^2/(x^2-a^2))`
21) `int sech^-1(x/a) dx = x*sech^-1(x/a)+a sin^-1(x/a)` **[For `sech^-1(x/a) > 0`]
OR `= x*sech^-1(x/a)-a sin^-1(x/a)` **[For `sech^-1(x/a) < 0`]
22) `int x*sech^-1(x/a) dx = 1/2x^2sech^-1(x/a)-1/2asqrt(a^2-x^2)` **[For `sech^-1(x/a) > 0`]
OR `= 1/2x^2sech^-1(x/a)+1/2asqrt(a^2-x^2)` **[For `sech^-1(x/a) < 0`]
23) `int (sech^-1(x/a))/x dx = -1/2 ln(a/x) ln((4a)/x)-(x/a)^2/(2*2*2)-(1*3(x/a)^4)/(2*4*4*4)-...` **[For `sech^-1(x/a) > 0`]
OR `= 1/2 ln(a/x) ln((4a)/x)+(x/a)^2/(2*2*2)+(1*3(x/a)^4)/(2*4*4*4)+...` **[For `sech^-1(x/a) < 0`]
24) `int csch^-1(x/a) dx = x*csch^-1(x/a)+-a sinh^-1(x/a)` **[`+` if `x > 0`; `-` if `x < 0`]
25) `int x*csch^-1(x/a) dx = x^2/2csch^-1(x/a)+-(asqrt(x^2+a^2))/2` **[`+` if `x > 0`; `-` if `x < 0`]
26) `int (csch^-1(x/a))/x dx = 1/2 ln(x/a) ln((4a)/x)+(1(x/a)^2)/(2*2*2)-(1*3(x/a)^4)/(2*4*4*4)+...` **[For `0 < x < a`]
OR `= 1/2 ln(-x/a) ln(-x/(4a))-(x/a)^2/(2*2*2)+(1*3(x/a)^4)/(2*4*4*4)+...` **[For `-a < x < 0`]
OR `= -a/x+(a/x)^3/(2*3*3)-(1*3(a/x)^5)/(2*4*5*5)+...` **[For `absx > a`]
27) `int x^m sinh^-1(x/a) dx = x^(m+1)/(m+1)sinh^-1(x/a)-1/(m+1) int x^(m+1)/sqrt(x^2+a^2) dx`
28) `int x^m cosh^-1(x/a) dx = x^(m+1)/(m+1) cosh^-1(x/a)-1/(m+1) int x^(m+1)/sqrt(x^2-a^2) dx` **[For `cosh^-1(x/a) > 0`]
OR `= x^(m+1)/(m+1) cosh^-1(x/a)+1/(m+1) int x^(m+1)/sqrt(x^2-a^2) dx` **[For `cosh^-1(x/a) < 0`]
29) `int x^m tanh^-1(x/a) dx = x^(m+1)/(m+1) tanh^-1(x/a)-a/(m+1) int x^(m+1)/(a^2-x^2) dx`
30) `int x^m coth^-1(x/a) dx = x^(m+1)/(m+1) coth^-1(x/a)-a/(m+1) int x^(m+1)/(a^2-x^2) dx`
31) `int x^m sech^-1(x/a) dx = x^(m+1)/(m+1) sech^-1(x/a)+a/(m+1) int x^m/sqrt(a^2-x^2) dx` **[For `sech^-1(x/a) > 0`]
OR `= x^(m+1)/(m+1) sech^-1(x/a)-a/(m+1) int x^m/sqrt(a^2-x^2) dx` **[For `sech^-1(x/a) < 0`]
32) `int x^m csch^-1(x/a) dx = x^(m+1)/(m+1) csch^-1(x/a) +- a/(m+1) int x^m/sqrt(x^2+a^2) dx` **[`+` if `x > 0`; `-` if `x < 0`]