The following is a list of integrals (anti-derivative functions) of hyperbolic functions. For a complete list of integral functions, see list of integrals.
In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration.
∫
sinh
a
x
d
x
=
1
a
cosh
a
x
+
C
∫
sinh
2
a
x
d
x
=
1
4
a
sinh
2
a
x
−
x
2
+
C
∫
sinh
n
a
x
d
x
=
1
a
n
sinh
n
−
1
a
x
cosh
a
x
−
n
−
1
n
∫
sinh
n
−
2
a
x
d
x
(for
n
>
0
)
also:
∫
sinh
n
a
x
d
x
=
1
a
(
n
+
1
)
sinh
n
+
1
a
x
cosh
a
x
−
n
+
2
n
+
1
∫
sinh
n
+
2
a
x
d
x
(for
n
<
0
,
n
≠
−
1
)
∫
d
x
sinh
a
x
=
1
a
ln
|
tanh
a
x
2
|
+
C
also:
∫
d
x
sinh
a
x
=
1
a
ln
|
cosh
a
x
−
1
sinh
a
x
|
+
C
∫
d
x
sinh
a
x
=
1
a
ln
|
sinh
a
x
cosh
a
x
+
1
|
+
C
∫
d
x
sinh
a
x
=
1
2
a
ln
|
cosh
a
x
−
1
cosh
a
x
+
1
|
+
C
∫
d
x
sinh
n
a
x
=
−
cosh
a
x
a
(
n
−
1
)
sinh
n
−
1
a
x
−
n
−
2
n
−
1
∫
d
x
sinh
n
−
2
a
x
(for
n
≠
1
)
∫
x
sinh
a
x
d
x
=
1
a
x
cosh
a
x
−
1
a
2
sinh
a
x
+
C
∫
sinh
a
x
sinh
b
x
d
x
=
1
a
2
−
b
2
(
a
sinh
b
x
cosh
a
x
−
b
cosh
b
x
sinh
a
x
)
+
C
(for
a
2
≠
b
2
)
∫
cosh
a
x
d
x
=
1
a
sinh
a
x
+
C
∫
cosh
2
a
x
d
x
=
1
4
a
sinh
2
a
x
+
x
2
+
C
∫
cosh
n
a
x
d
x
=
1
a
n
sinh
a
x
cosh
n
−
1
a
x
+
n
−
1
n
∫
cosh
n
−
2
a
x
d
x
(for
n
>
0
)
also:
∫
cosh
n
a
x
d
x
=
−
1
a
(
n
+
1
)
sinh
a
x
cosh
n
+
1
a
x
+
n
+
2
n
+
1
∫
cosh
n
+
2
a
x
d
x
(for
n
<
0
,
n
≠
−
1
)
∫
d
x
cosh
a
x
=
2
a
arctan
e
a
x
+
C
also:
∫
d
x
cosh
a
x
=
1
a
arctan
(
sinh
a
x
)
+
C
∫
d
x
cosh
n
a
x
=
sinh
a
x
a
(
n
−
1
)
cosh
n
−
1
a
x
+
n
−
2
n
−
1
∫
d
x
cosh
n
−
2
a
x
(for
n
≠
1
)
∫
x
cosh
a
x
d
x
=
1
a
x
sinh
a
x
−
1
a
2
cosh
a
x
+
C
∫
x
2
cosh
a
x
d
x
=
−
2
x
cosh
a
x
a
2
+
(
x
2
a
+
2
a
3
)
sinh
a
x
+
C
∫
cosh
a
x
cosh
b
x
d
x
=
1
a
2
−
b
2
(
a
sinh
a
x
cosh
b
x
−
b
sinh
b
x
cosh
a
x
)
+
C
(for
a
2
≠
b
2
)
∫
tanh
x
d
x
=
ln
cosh
x
+
C
∫
tanh
2
a
x
d
x
=
x
−
tanh
a
x
a
+
C
∫
tanh
n
a
x
d
x
=
−
1
a
(
n
−
1
)
tanh
n
−
1
a
x
+
∫
tanh
n
−
2
a
x
d
x
(for
n
≠
1
)
∫
coth
x
d
x
=
ln
|
sinh
x
|
+
C
,
for
x
≠
0
∫
coth
n
a
x
d
x
=
−
1
a
(
n
−
1
)
coth
n
−
1
a
x
+
∫
coth
n
−
2
a
x
d
x
(for
n
≠
1
)
∫
sech
x
d
x
=
arctan
(
sinh
x
)
+
C
∫
csch
x
d
x
=
ln
|
tanh
x
2
|
+
C
,
for
x
≠
0
Integrals involving hyperbolic sine and cosine functions
∫
cosh
a
x
sinh
b
x
d
x
=
1
a
2
−
b
2
(
a
sinh
a
x
sinh
b
x
−
b
cosh
a
x
cosh
b
x
)
+
C
(for
a
2
≠
b
2
)
∫
cosh
n
a
x
sinh
m
a
x
d
x
=
cosh
n
−
1
a
x
a
(
n
−
m
)
sinh
m
−
1
a
x
+
n
−
1
n
−
m
∫
cosh
n
−
2
a
x
sinh
m
a
x
d
x
(for
m
≠
n
)
also:
∫
cosh
n
a
x
sinh
m
a
x
d
x
=
−
cosh
n
+
1
a
x
a
(
m
−
1
)
sinh
m
−
1
a
x
+
n
−
m
+
2
m
−
1
∫
cosh
n
a
x
sinh
m
−
2
a
x
d
x
(for
m
≠
1
)
∫
cosh
n
a
x
sinh
m
a
x
d
x
=
−
cosh
n
−
1
a
x
a
(
m
−
1
)
sinh
m
−
1
a
x
+
n
−
1
m
−
1
∫
cosh
n
−
2
a
x
sinh
m
−
2
a
x
d
x
(for
m
≠
1
)
∫
sinh
m
a
x
cosh
n
a
x
d
x
=
sinh
m
−
1
a
x
a
(
m
−
n
)
cosh
n
−
1
a
x
+
m
−
1
n
−
m
∫
sinh
m
−
2
a
x
cosh
n
a
x
d
x
(for
m
≠
n
)
∫
sinh
m
a
x
cosh
n
a
x
d
x
=
sinh
m
+
1
a
x
a
(
n
−
1
)
cosh
n
−
1
a
x
+
m
−
n
+
2
n
−
1
∫
sinh
m
a
x
cosh
n
−
2
a
x
d
x
(for
n
≠
1
)
∫
sinh
m
a
x
cosh
n
a
x
d
x
=
−
sinh
m
−
1
a
x
a
(
n
−
1
)
cosh
n
−
1
a
x
+
m
−
1
n
−
1
∫
sinh
m
−
2
a
x
cosh
n
−
2
a
x
d
x
(for
n
≠
1
)
Integrals involving hyperbolic and trigonometric functions
∫
sinh
(
a
x
+
b
)
sin
(
c
x
+
d
)
d
x
=
a
a
2
+
c
2
cosh
(
a
x
+
b
)
sin
(
c
x
+
d
)
−
c
a
2
+
c
2
sinh
(
a
x
+
b
)
cos
(
c
x
+
d
)
+
C
∫
sinh
(
a
x
+
b
)
cos
(
c
x
+
d
)
d
x
=
a
a
2
+
c
2
cosh
(
a
x
+
b
)
cos
(
c
x
+
d
)
+
c
a
2
+
c
2
sinh
(
a
x
+
b
)
sin
(
c
x
+
d
)
+
C
∫
cosh
(
a
x
+
b
)
sin
(
c
x
+
d
)
d
x
=
a
a
2
+
c
2
sinh
(
a
x
+
b
)
sin
(
c
x
+
d
)
−
c
a
2
+
c
2
cosh
(
a
x
+
b
)
cos
(
c
x
+
d
)
+
C
∫
cosh
(
a
x
+
b
)
cos
(
c
x
+
d
)
d
x
=
a
a
2
+
c
2
sinh
(
a
x
+
b
)
cos
(
c
x
+
d
)
+
c
a
2
+
c
2
cosh
(
a
x
+
b
)
sin
(
c
x
+
d
)
+
C
derivatives of hyperbolic functions(http://math.info/Calculus/Derivatives_Hyp_InvHyp/)
