<s>[INST] How to evaluate $\int \frac{x dx}{x^4 + 6x^2 + 5}$? $$\int \frac{x dx}{x^4 + 6x^2 + 5}$$
How to evaluate this integral ?
 [/INST] $$\int  \frac { xdx }{ x^{ 4 }+6x^{ 2 }+5 } =\frac { 1 }{ 2 } \int { \frac { d{ x }^{ 2 } }{ { \left( { x }^{ 2 }+3 \right)  }^{ 2 }-4 }  } =\frac { 1 }{ 2 } \left[ \int { \frac { d{ x }^{ 2 } }{ \left( { x }^{ 2 }+1 \right) \left( { x }^{ 2 }+5 \right)  }  }  \right] =\frac { 1 }{ 8 } \left[ \int { \frac { d{ x }^{ 2 } }{ \left( { x }^{ 2 }+1 \right)  } -\int { \frac { d{ x }^{ 2 } }{ \left( { x }^{ 2 }+5 \right)  }  }  }  \right] =\\ =\frac { 1 }{ 8 } \left[ \int { \frac { d\left( { x }^{ 2 }+1 \right)  }{ \left( { x }^{ 2 }+1 \right)  } -\int { \frac { d\left( { x }^{ 2 }+5 \right)  }{ \left( { x }^{ 2 }+5 \right)  }  }  }  \right] =\frac { 1 }{ 8 } \ln { \left| \frac { { x }^{ 2 }+1 }{ { x }^{ 2 }+5 }  \right|  } +C$$
</s>