We need
#cosh^2x-sinh^2x=1#
#arcsinhx=ln(x+sqrt(x^2+1))#
I prefer to use the substitution
#x=sinhu#, #=>#, #dx=coshudu#
Therefore,
#int(x^3dx)/sqrt(1+x^2)=int(sinh^3ucoshudu)/(coshu)#
#=intsinh^3udu#
#=intsinh^2usinhudu#
#=int(cosh^2u-1)sinhudu#
Let #v=coshu#, #=>#, #dv=sinhudu#
Therefore,
#int(cosh^2u-1)sinhudu=int(v^2-1)dv#
#=1/3v^3-v#
#=1/3cosh^3u-coshu#
#=1/3cosh ^3(arcsinhx)-cosharcsinhx+C#
#cosh(arcsinh(x))=1/2(e^(arcsinhx)+e^(-arcsinhx))#
#=1/2(e^(ln(x+sqrt(x^2+1)))+e^(-ln(x+sqrt(x^2+1))))#
#=1/2((x+sqrt(x^2+1))+1/(x+sqrt(x^2+1)))#
#=1/2((x+sqrt(x^2+1))+(x-sqrt(x^2+1))/(x+sqrt(x^2+1)(x-sqrt(x^2+1))))#
#=1/2((x+sqrt(x^2+1))+(x-sqrt(x^2-1)/(x^2-x^2-1)))#
#=sqrt(x^2+1)#
And
#cosh^3(arcsinhx)=(sqrt(x^2+1))^3#
#=(x^2+1)sqrt(x^2+1)#
Finally,
#int(x^3dx)/sqrt(1+x^2)=1/3(x^2+1)sqrt(x^2+1)-sqrt(x^2+1)+C#
#=1/3(x^2-2)sqrt(x^2+1)+C#