# ∫sqrt (1+sec^4(x)) dx

by skeem
Tags: ∫sqrt, sec4x
 Emeritus Sci Advisor PF Gold P: 4,500 Maple gives a closed form solution $$-\sqrt {{\frac {1+\cos \left( x \right) }{\cos \left( x \right) }}} \sqrt {-2\,{\frac {\cos \left( x \right) }{1+\cos \left( x \right) }}} \sqrt {2}{\it arctanh} \left( 1/2\,\sqrt {2}\sin \left( x \right) \sqrt {-2\,{\frac {\cos \left( x \right) }{1+\cos \left( x \right) }}} \left( \cos \left( x \right) \right) ^{-1} \right)$$ Thankfully, it lets you convert automatically to latex :) It should be noted that when I take the integral of that expression in maple, it just doesn't do it, instead giving me the "symbolic" answer of an integral sign followed up by the expression. Simplify doesn't seem to simplify it very much either
 Emeritus Sci Advisor PF Gold P: 4,500 I apologize.... I went back to have maple numerically solve from 0 to pi/4, and realized earlier I had mistyped, and only evaluated the integral of $$[1+sec(x)]^{1/2}$$ Maple doesn't actually do the integral you're told in a closed form solution, but the numerical approximation is 1.277978059