List of integrals of inverse hyperbolic functions

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The following is a list of integrals (antiderivative 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.

sinhaxdx=1acoshax+C
coshaxdx=1asinhax+C
sinh2axdx=14asinh2axx2+C
cosh2axdx=14asinh2ax+x2+C
tanh2axdx=xtanhaxa+C
sinhnaxdx=1ansinhn1axcoshaxn1nsinhn2axdx(for n>0)
also: sinhnaxdx=1a(n+1)sinhn+1axcoshaxn+2n+1sinhn+2axdx(for n<0n1)
coshnaxdx=1ansinhaxcoshn1ax+n1ncoshn2axdx(for n>0)
also: coshnaxdx=1a(n+1)sinhaxcoshn+1ax+n+2n+1coshn+2axdx(for n<0n1)
dxsinhax=1aln|tanhax2|+C
also: dxsinhax=1aln|coshax1sinhax|+C
also: dxsinhax=1aln|sinhaxcoshax+1|+C
also: dxsinhax=12aln|coshax1coshax+1|+C
dxcoshax=2aarctaneax+C
also: dxcoshax=1aarctan(sinhax)+C
dxsinhnax=coshaxa(n1)sinhn1axn2n1dxsinhn2ax(for n1)
dxcoshnax=sinhaxa(n1)coshn1ax+n2n1dxcoshn2ax(for n1)
coshnaxsinhmaxdx=coshn1axa(nm)sinhm1ax+n1nmcoshn2axsinhmaxdx(for mn)
also: coshnaxsinhmaxdx=coshn+1axa(m1)sinhm1ax+nm+2m1coshnaxsinhm2axdx(for m1)
also: coshnaxsinhmaxdx=coshn1axa(m1)sinhm1ax+n1m1coshn2axsinhm2axdx(for m1)
sinhmaxcoshnaxdx=sinhm1axa(mn)coshn1ax+m1nmsinhm2axcoshnaxdx(for mn)
also: sinhmaxcoshnaxdx=sinhm+1axa(n1)coshn1ax+mn+2n1sinhmaxcoshn2axdx(for n1)
also: sinhmaxcoshnaxdx=sinhm1axa(n1)coshn1ax+m1n1sinhm2axcoshn2axdx(for n1)
xsinhaxdx=1axcoshax1a2sinhax+C
xcoshaxdx=1axsinhax1a2coshax+C
x2coshaxdx=2xcoshaxa2+(x2a+2a3)sinhax+C
tanhaxdx=1alncoshax+C
cothaxdx=1aln|sinhax|+C
tanhnaxdx=1a(n1)tanhn1ax+tanhn2axdx(for n1)
cothnaxdx=1a(n1)cothn1ax+cothn2axdx(for n1)
sinhaxsinhbxdx=1a2b2(asinhbxcoshaxbcoshbxsinhax)+C(for a2b2)
coshaxcoshbxdx=1a2b2(asinhaxcoshbxbsinhbxcoshax)+C(for a2b2)
coshaxsinhbxdx=1a2b2(asinhaxsinhbxbcoshaxcoshbx)+C(for a2b2)
sinh(ax+b)sin(cx+d)dx=aa2+c2cosh(ax+b)sin(cx+d)ca2+c2sinh(ax+b)cos(cx+d)+C
sinh(ax+b)cos(cx+d)dx=aa2+c2cosh(ax+b)cos(cx+d)+ca2+c2sinh(ax+b)sin(cx+d)+C
cosh(ax+b)sin(cx+d)dx=aa2+c2sinh(ax+b)sin(cx+d)ca2+c2cosh(ax+b)cos(cx+d)+C
cosh(ax+b)cos(cx+d)dx=aa2+c2sinh(ax+b)cos(cx+d)+ca2+c2cosh(ax+b)sin(cx+d)+C

Template:Lists of integrals

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