<s>[INST] I am not able to solve the integral $I=\int { \frac { \sqrt { \sqrt { x } +\sqrt { x-1 }  }  }{ 1+\sqrt { x }  }  }$ $J=\int { \frac { \sqrt { \sqrt { x } -\sqrt { x-1 }  }  }{ 1+\sqrt { x }  }  } $ then I-J
I substituted $x=\sec ^{ 2 }{ \theta  } $.
$I-J=\frac { \sqrt { \sec { \theta  } +\tan { \theta  }  } -\sqrt { \sec { \theta  } -\tan { \theta  }  }  }{ 1+\sec { \theta  }  } \times 2\sec ^{ 2 }{ \theta  } \tan { \theta d\theta  } \\ I-J=\frac { \sqrt { \sec { \theta  } +\tan { \theta  }  } -\frac { 1 }{ \sqrt { \sec { \theta  } +\tan { \theta  }  }  }  }{ 1+\sec { \theta  }  } \times 2\sec ^{ 2 }{ \theta  } \tan { \theta d\theta  } \\ I-J=\frac { \frac { \sec { \theta  } +\tan { \theta  } -1 }{ \sqrt { \sec { \theta  } +\tan { \theta  }  }  }  }{ 1+\sec { \theta  }  } \times 2\sec ^{ 2 }{ \theta  } \tan { \theta d\theta  } $
I am not able to solve further Please help
 [/INST] There is some simplification available here as
\begin{align}
\left(\sqrt{\sqrt{x}+\sqrt{x-1}}\right)\left(\sqrt{\sqrt{x}-\sqrt{x-1}}\right) &= \sqrt{1}
\end{align}
Since both $I$ and $J$ have these terms in, multiplication of these terms should give you something to work with once you have simplified the integrands using high school methods.
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