∫ dx

cosh^2 ax

=^1 atanhax+C

∫ x dx

cosh^2 ax=

1

axtanhax−

1

a^2 ln|coshax|+C

∫ dx

coshnax=

1

(n−1)a

xsinhax

coshn−^1 ax+

n− 2

n− 1

∫ dx

coshn−^2 ax, n > 1

∫

coshaxcoshbx dx=2(a^1 −b)sinh(a−b)x+2(a^1 +b)sinh(a+b)x+C

∫

coshaxsinbx dx=a (^2) +^1 b 2 [asinhaxsinbx−bcoshaxcosbx] +C
622.
∫
coshaxcosbx dx=a (^2) +^1 b 2 [asinhaxcosbx+bcoshaxsinbx] +C
623.
∫ dx
α+βcoshax=





√^2
β^2 −α^2
tan−^1 βe
ax+α
√
β^2 −α^2
+C, β^2 > α^2
1
a
√
α^2 −β^2
ln
∣∣
∣∣
∣
βeax+α−
√
α^2 −β^2
βeax+α+
√
α^2 −β^2
∣∣
∣∣
∣+C, β
(^2) < α 2
624.
∫ dx
1 + coshax=
1
atanhax+C
625.
∫ x dx
1 + coshax=
x
atanh
ax
2 −
2
a^2 ln|cosh
ax
2 |+C
626.
∫ dx
−1 + coshax=−
1
acoth
ax
2 +C
627.
∫ dx
(α+βcoshax)^2 =
βsinhax
a(β^2 −α^2 )(α+βcoshax)−
α
β^2 −α^2
∫ dx
α+βcoshax
628.
∫ dx
α^2 −β^2 cosh^2 ax






1
2 aα
√
α^2 −β^2
ln
∣∣
∣∣
∣
αtanhax+
√
α^2 −β^2
αtanhax−
√
α^2 −β^2
∣∣
∣∣
∣+C, α
(^2) > β 2
− 1
aα
√
β^2 −α^2
tan−^1 √αtanhax
β^2 −α^2
+C, α^2 < β^2
629.
∫ dx
α^2 +β^2 cosh^2 ax
=^1
aα
√
α^2 +β^2
tanh−^1
(
√αtanhax
α^2 +β^2
)
+C
Integrals containing the hyperbolic functions sinhaxand coshax
630.
∫
sinhaxcoshax dx= 21 asinh^2 ax+C
631.
∫
sinhaxcoshbx dx=2(a^1 +b)cosh(a+b)x+2(a^1 −b)cosh(a−b)x+C
Appendix C