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原式=(1/2)*∫(x^2+1)/(1+x^4)dx-(1/2)*∫(x^2-1)/(1+x^4)dx
=(1/2)*∫(1+1/x^2)/(x^2+1/x^2)dx-(1/2)*∫(1-1/x^2)/(x^2+1/x^2)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√2)*arctan[(x-1/x)/√2]-(1/4√2)*∫[1/(x+1/x-√2)-1/(x+1/x+√2)]d(x+1/x)
=(√2/4)*arctan[(x-1/x)/√2]-(√2/8)*[ln|x+1/x-√2|-ln|x+1/x+√2|]+C，其中C是任意常数
=(1/2)*∫(1+1/x^2)/(x^2+1/x^2)dx-(1/2)*∫(1-1/x^2)/(x^2+1/x^2)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√2)*arctan[(x-1/x)/√2]-(1/4√2)*∫[1/(x+1/x-√2)-1/(x+1/x+√2)]d(x+1/x)
=(√2/4)*arctan[(x-1/x)/√2]-(√2/8)*[ln|x+1/x-√2|-ln|x+1/x+√2|]+C，其中C是任意常数
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