In[]:=
α[a_,b_,t_]:={a*Exp[b*t]*Cos[t],a*Exp[b*t]*Sin[t]}​​derivα[a_,b_,t_]:=D[α[a,b,t],t]​​s1[a_,b_,tt_]:=​​Integrate[​​Sqrt[derivα[a,b,t].derivα[a,b,t]],{t,0,tt}]​​s2[a_,b_,tt_]:=​​Integrate[​​Sqrt[derivα[a,b,t].derivα[a,b,t]],{t,0,tt}]//​​PowerExpand//Simplify
In[]:=
s1[a,b,t]
Out[]=
-
2
a
(1+
2
b
)
+
2
a
(1+
2
b
)
2bt

b
In[]:=
s2[a,b,t]
Out[]=
a
1+
2
b
(-1+
bt

)
b
In[]:=
s1inv:=InverseFunction[s1,3,3]​​s2inv:=InverseFunction[s2,3,3]
In[]:=
s1[1,-0.1,π]
Out[]=
2.70942
In[]:=
s2[1,-0.1,π]
Out[]=
2.70942
In[]:=
s1inv[1,-0.1,2.709419422750013`]
InverseFunction
:Inverse functions are being used. Values may be lost for multivalued inverses.
Out[]=
3.14159-31.4159
In[]:=
s2inv[1,-0.1,2.709419422750013`]
InverseFunction
:Inverse functions are being used. Values may be lost for multivalued inverses.
Out[]=
3.14159