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Math Help - For major players only: Integrate ∫ 1/√(1+x^4) dx...?

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    For major players only: Integrate ∫ 1/√(1+x^4) dx...?

    For major players only: Integrate ∫ 1/√(1+x^4) dx

    I broke this down using trig substitution (triangle: 1, x^2, √(x^4+1) to
    ∫ 1/√(2sin2) d

    If you could integrate in terms of or x it would be amazing.
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    Quote Originally Posted by cdrappi2552 View Post
    For major players only: Integrate ∫ 1/√(1+x^4) dx

    I broke this down using trig substitution (triangle: 1, x^2, √(x^4+1) to
    ∫ 1/√(2sin2) d

    If you could integrate in terms of or x it would be amazing.
    The answer is a special function, called the "inverse hyperbolic lemniscatic sine"... The inverse lemniscatic sine equals \int_0^x \frac{dt}{\sqrt{1-t^4}} (a generalization of the arcsinus, relating to the lemniscate curve like the sinus relates to the circle...), and its hyperbolic counterpart is the function you need: \int_0^x\frac{dt}{\sqrt{1+t^4}}. It is tightly connected to elliptic functions.

    In a word: there is no expression for this anti-derivative in terms of "elementary functions"...
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