Table of Integrals - Forms Involving `sinh ax`

The integrals below involve `sinh ax`.

1) `int  sinh ax  dx = (cosh ax)/a`

2) `int  x sinh ax  dx = (x cosh ax)/a-(sinh ax)/a^2`

3) `int  x^2 sinh ax  dx = (x^2/a+2/a^3) cosh ax-(2x)/a^2 sinh ax`

4) `int  (sinh ax)/x  dx = ax+(ax)^3/(3*3!)+(ax)^5/(5*5!)+...`

5) `int  (sinh ax)/x^2  dx = -(sinh ax)/x+aint  (cosh ax)/x  dx`

                    **[See integral #4 in the table involving `cosh ax`]

6) `int  1/(sinh ax)  dx = 1/a ln tanh ((ax)/2)`

7) `int x/(sinh ax)  dx =   1/a^2{ax-(ax)^3/18+(7(ax)^5)/1800-...+(2(-1)^n(2^(2n)-1)B_n(ax)^(2n+1))/((2n+1)!)+...}`

8) `int  sinh^2 ax  dx = (sinh ax cosh ax)/(2a)-x/2`

9) `int  x*sinh^2 ax  dx = (x sinh 2ax)/(4a)-(cosh 2ax)/(8a^2)-x^2/4`

10) `int  1/(sinh^2 ax)  dx = -(coth ax)/a`

11) `int  sinh ax*sinhpx  dx = (sinh(a+p)x)/(2(a+p))-(sinh(a-p)x)/(2(a-p))`

                    **[For `a=+-p`, see integral #8 in this table]

12) `int  sinh ax*sin px  dx = (a cosh ax*sin px-p sinh ax *cos px)/(a^2+p^2)`

13) `int  sinh ax*cos px  dx = (a cosh ax*cos px+p sinh ax *sin px)/(a^2+p^2)`

14) `int  1/(p+q sinh ax)  dx = 1/(asqrt(p^2+q^2))ln((qe^(ax)+p-sqrt(p^2+q^2))/(qe^(ax)+p+sqrt(p^2+q^2)))`

15) `int  1/(p+q sinh ax)^2  dx = (-q cosh ax)/(a(p^2+q^2)(p+q sinh ax))+p/(p^2+q^2) int  1/(p+q sinh ax)  dx`

16) `int  1/(p^2+q^2 sinh^2 ax)  dx = 1/(apsqrt(q^2-p^2)) tan^-1((sqrt(q^2-p^2)*tanh ax)/p)`

                                  OR `= 1/(2apsqrt(p^2-q^2)) ln ((p+sqrt(p^2-q^2)*tanh ax)/(p-sqrt(p^2-q^2)*tanh ax))`

17) `int  1/(p^2-q^2 sinh^2 ax)  dx = 1/(2apsqrt(p^2+q^2)) ln ((p+sqrt(p^2+q^2)*tanh ax)/(p-sqrt(p^2+q^2)*tanh ax))`

18) `int  x^m sinh ax  dx = (x^m cosh ax)/a-m/a int  x^(m-1) cosh ax   dx`

                    **[see integral #24 in the table involving `cosh ax`]

19) `int  sinh^n ax  dx = (sinh^(n-1) ax cosh ax)/(an)-(n-1)/n int  sinh^(n-2) ax  dx`

20) `int  (sinh ax)/x^n  dx = (-sinh ax)/((n-1)x^(n-1))+a/(n-1) int (cosh ax)/x^(n-1)  dx`

                    **[see integral #26 int the table involving `cosh ax`]

21) `int  1/(sinh^n ax)  dx = (-cosh ax)/(a(n-1)sinh^(n-1) ax)-(n-2)/(n-1) int 1/(sinh^(n-2) ax)  dx`

22) `int  x/(sinh^n ax)  dx = (-x cosh ax)/(a(n-1)sinh^(n-1)ax)-1/(a^2(n-1)(n-2)sinh^(n-2)ax)-(n-2)/(n-1) int  x/(sinh^(n-2) ax)  dx`