∫
sinh^2 axcosh^2 ax dx= 321 asinh 4ax−^18 x+C∫
sinhnaxcoshax dx=(n+ 1)^1 asinhn+1ax+C, n 6 =− 1∫
coshnaxsinhax dx=(n+ 1)^1 acoshn+1ax+C, n 6 =− 1∫ sinhax
coshaxdx=1
aln|coshax|+C∫ coshax
sinhaxdx=1
aln|sinhax|+C∫ dx
sinhaxcoshax=1
aln|tanhax|+C∫ xsinhax
coshax dx=1
a^2[a (^3) x 3
3 −
a^5 x^5
15 +···+ (−1)
n^22 n(2^2 n−1)Bna^2 n+1x^2 n+1
(2n+ 1)! +···
]
+C
639.
∫ xcoshax
sinhax dx=
1
a^2
[
ax+a
(^3) x 3
9 −
a^5 x^5
225 +···+ (−1)
n− 122 nBna^2 n+1x^2 n+1
(2n+ 1)! +···
]
+C
640.
∫ sinh (^2) ax
cosh^2 ax
dx=x−a^1 tanhax+C
641.
∫ cosh (^2) ax
sinh^2 ax
dx=x−a^1 cothax+C
642.
∫ xsinh (^2) ax
cosh^2 ax
dx=^12 x^2 −a^1 xtanhax+a^12 ln|coshax|+C
643.
∫ xcosh (^2) ax
sinh^2 ax
dx=^12 x^2 −a^1 xcothax+a^12 ln|sinhax|+C
644.
∫ sinhax
xcoshaxdx=ax−
a^3 x^3
9 +···+ (−1)
n− 122 n(2^2 n−1)Bna^2 n−^1 x^2 n−^1
(2n−1)(2n)! +···+C
645.
∫ coshax
xsinhaxdx=−
1
ax+
ax
3 −
a^3 x^3
135 +···+ (−1)
n^22 nBna^2 n−^1 x^2 n−^1
(2n−1)(2n)! +···+C
646.
∫ sinh (^3) ax
cosh^3 axdx=
1
aln|coshax|−
1
2 atanh
(^2) ax+C
647.
∫ cosh (^3) ax
sinh^3 ax
dx=^1 aln|sinhax|− 21 acoth^2 ax+C
648.
∫ dx
sinhaxcosh^2 ax
=^1 asechax+a^1 ln tanhax 2 |+C
Appendix C