∫ dx
sinh^2 axcoshax=−^1 atan−^1 (sinhax)−a^1 cschax+C∫ dx
sinh^2 axcosh^2 ax=−2
acothax+C∫ sinh (^2) ax
coshaxdx=
1
asinhax−
1
atan
− (^1) (sinhax) +C
652.
∫ cosh (^2) ax
sinhaxdx=
1
acoshax+
1
aln|tanh
ax
2 |+C
653.
∫ dx
coshax(1 + sinhax)=
1
2 aln
∣∣
∣∣1 + sinhax
coshax
∣∣
∣∣+^1
atan
− (^1) eax+C
654.
∫ dx
sinhax(coshax+ 1)=
1
2 aln|tanh
ax
2 |+
1
2 a(coshax+ 1)+C
655.
∫ dx
sinhax(coshax−1)=−
1
2 aln|tanh
ax
2 |−
1
2 a(coshax−1)+C
656.
∫ dx
α+βcoshsinhaxax
=α 2 αx−β 2 −a(α 2 β−β (^2) )ln|βsinhax+αcoshax|+C
657.
∫ dx
α+βcoshsinhaxax
=α 2 αx−β 2 +a(α 2 β−β (^2) )ln|αsinhax+βcoshax|+C
658.
∫ dx
bcoshax+csinhax=
1
a
√
b^2 −c^2
sec−^1
[bcoshax+csinhax
√
b^2 −c^2
]
+C, b^2 > c^2
− 1
a
√
c^2 −b^2
csch−^1
[bcoshax+csinhax
√
c^2 −b^2
]
+C, b^2 < c^2
Integrals containing the hyperbolic functions tanhax, cothax, sechax, cschax
Express integrals in terms ofsinhaxandcoshaxand see previous listings.
Integrals containing inverse hyperbolic functions
659.
∫
sinh−^1 xadx=xsinh−^1 xa−
√
x^2 +a^2 +C
660.
∫
cosh−^1 xadx=
{
xcosh−^1 (x/a)−
√
x^2 −a^2 , cosh−^1 (x/a)> 0
xcosh−^1 (x/a) +
√
x^2 −a^2 , cosh−^1 (x/a)< 0
661.
∫
tanh−^1 xadx=xtanh−^1 xa+a 2 ln|a^2 −x^2 |+C
662.
∫
coth−^1 xadx=xcoth−^1 xa+a 2 ln|x^2 −a^2 |+C
663.
∫
sech−^1 xadx=
xsech−^1 xa+asin−^1 xa+C, sech−^1 (x/a)> 0
xsech−^1 xa−asin−^1 xa+C, sech−^1 (x/a)< 0
Appendix C