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∫dx/(x^4+1)
= (1/2)∫[(x²+1)-(x²-1)]/(x^4+1)
= (1/2)∫(x²+1)/(x^4+1) dx - (1/2)∫(x²-1)/(x^4+1)
= (1/2)∫(1+1/x²)/(x²+1/x²) dx - (1/2)∫(1-1/x²)/(x²+1/x²) dx
= (1/2)d(x-1/x)/[(x-1/x)²+2] - (1/2)∫d(x+1/x)/[(x+1/x)²-2]
= (1/2)(1/√2)arctan[(x-1/x)/√2] - (1/2)*[1/(2√2)]ln|[(x+1/x+√2)/(x+1/x-√2)| + C
= arctan[x/√2 - 1/√2x] / (2√2) - ln|(x²-√2x+1)/(x²+√2x+1)| / (4√2) + C
= (1/2)∫[(x²+1)-(x²-1)]/(x^4+1)
= (1/2)∫(x²+1)/(x^4+1) dx - (1/2)∫(x²-1)/(x^4+1)
= (1/2)∫(1+1/x²)/(x²+1/x²) dx - (1/2)∫(1-1/x²)/(x²+1/x²) dx
= (1/2)d(x-1/x)/[(x-1/x)²+2] - (1/2)∫d(x+1/x)/[(x+1/x)²-2]
= (1/2)(1/√2)arctan[(x-1/x)/√2] - (1/2)*[1/(2√2)]ln|[(x+1/x+√2)/(x+1/x-√2)| + C
= arctan[x/√2 - 1/√2x] / (2√2) - ln|(x²-√2x+1)/(x²+√2x+1)| / (4√2) + C
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