Evaluation of $\displaystyle \int\frac{1}{\sqrt{\sin x\cos 7x}}dx$

$\bf{My\; Try::}$ Let $$I = \int\frac{1}{\sqrt{\sin x\cos 7x}}dx$$

Now Put $\sin x\cos 7x = t^2\;,$ Then $(-7\sin x\sin 7x+\cos 7x\cos x)dx=2tdt$

Now How can I solve after that, Help Required, Thanks

  • 1
    $\begingroup$ Are sure that it isn't $$\cos^7x?$$ Then we can use math.stackexchange.com/questions/1883963/… $\endgroup$ – lab bhattacharjee Aug 27 '16 at 7:25
  • 2
    $\begingroup$ I agree with lab bhattacharjee's idea. Otherwise, I really do not see any approach for the posted problem (even using special functions). $\endgroup$ – Claude Leibovici Aug 27 '16 at 7:55
  • $\begingroup$ By virtue of Mathematica, the integral can only be expressed in terms of the Incomplete Elliptic Integral of the Third Kind, making a closed form extremely unlikely. $\endgroup$ – Jack Lam Aug 27 '16 at 11:18

The function inside the integral (as currently given) has multiple discontinuities where it approaches infinity then becomes purely imaginary then repeats. As such it is highly unlikely that a closed form exists.

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