Questionable are these large imaginary parts in the results for a real integrand:
In[39]:= With[{x = 1/2},
Integrate[Sqrt[1 - (x - \[Tau])^2] Sqrt[1 - \[Tau]^2], {\[Tau], x - 1, 1}, Assumptions -> x >= 0]
]
Out[39]= 1/48 (150 I Sqrt[2] + 51 Sqrt[3] +
34 Sqrt[15] EllipticE[16/15] -
68 EllipticE[ArcSin[2 Sqrt[2/5]], 15/16] -
68 EllipticE[ArcSin[Sqrt[6/5]], 15/16] +
8 (EllipticF[ArcSin[2 Sqrt[2/5]], 15/16] +
EllipticF[ArcSin[Sqrt[6/5]], 15/16]) -
2 Sqrt[15] EllipticK[16/15])
In[40]:= %39 // N
Out[40]= 1.05223 + 4.41942 I
In[52]:= With[{x = 1/3},
Plot[Sqrt[1 - (x - \[Tau])^2] Sqrt[1 - \[Tau]^2], {\[Tau], x - 1, 1}]
]
as well as
In[41]:= With[{x = -1/2},
Integrate[
Sqrt[1 - (x - \[Tau])^2] Sqrt[1 - \[Tau]^2], {\[Tau], -1, 1 + x},
Assumptions -> x <= 0]
]
Out[41]= 1/48 (150 I Sqrt[2] + 51 Sqrt[3] +
34 Sqrt[15] EllipticE[16/15] -
68 EllipticE[ArcSin[2 Sqrt[2/5]], 15/16] -
68 EllipticE[ArcSin[Sqrt[6/5]], 15/16] +
8 (EllipticF[ArcSin[2 Sqrt[2/5]], 15/16] +
EllipticF[ArcSin[Sqrt[6/5]], 15/16]) -
2 Sqrt[15] EllipticK[16/15])
In[42]:= %41 // N
Out[42]= 1.05223 + 4.41942 I
In[47]:= With[{x = -1/2},
Plot[Sqrt[1 - (x - \[Tau])^2] Sqrt[1 - \[Tau]^2], {\[Tau], -1, 1 + x}]
]
whereas
In[35]:= With[{x = 3/2},
Integrate[
Sqrt[1 - (x - \[Tau])^2] Sqrt[1 - \[Tau]^2], {\[Tau], x - 1, 1},
Assumptions -> x >= 0]
]
Out[35]= 1/168 (-175 I Sqrt[7] EllipticE[-(9/7)] -
350 EllipticE[7/16] + 175 Sqrt[7] EllipticE[16/7] +
400 I Sqrt[7] EllipticK[-(9/7)] - 252 I EllipticK[9/16] +
81 Sqrt[7] EllipticK[16/7])
In[36]:= With[{x = -3/2},
Integrate[
Sqrt[1 - (x - \[Tau])^2] Sqrt[1 - \[Tau]^2], {\[Tau], -1, 1 + x},
Assumptions -> x <= 0]
]
Out[36]= 1/168 (-175 I Sqrt[7] EllipticE[-(9/7)] -
350 EllipticE[7/16] + 175 Sqrt[7] EllipticE[16/7] +
400 I Sqrt[7] EllipticK[-(9/7)] - 252 I EllipticK[9/16] +
81 Sqrt[7] EllipticK[16/7])
In[37]:= N[%36, 39]
Out[37]= 0.171366442761431135877826465734053857640 + 0.*10^-40 I
In[38]:= N[%35, 39]
Out[38]= 0.171366442761431135877826465734053857640 + 0.*10^-40 I
looks good.