A large discrepancy evaluating $\int_{5.566}^{5.568}\frac{(\sin 6x)^{14}}{(\cos 11x)^{8}}dx$ with Hiper Scientific Calculator vs WolframAlpha

Question

So, in a nutshell, dealing with Hiper Scientific Calculator and Wolframalpha Integral Calculator . Only this integral

$$\int_{5.566}^{5.568}\frac{(\sin 6x)^{14}}{(\cos 11x)^{8}}dx$$

is being strangely computed by the respective calculators $28,066,763,000$ for hiper and $5.61409 \times 10^{10}$ for WA. What could've gone wrong? Why is that happening anyway ?

Kindly look into the attached screenshots for the specifics. Hoping to find a way out of this.

Pic1

Pic2

Answer 1

Maple agrees with Wolfram Alpha. Using 100 digits of precision, both numerical integration and the (rather unpleasant) closed-form formula give results of approximately $$ 5.614093676901550871856281466889638286126491248415545\times 10^{10}$$

Answer 2

If you set HiperCalc to radians, you will get (almost) the same answer as Wolfram Alpha (I got $5.6134 \cdot 10^{10}$, which is an error of about $1.3\%$). Numerical methods will get you closer.

Answer 3

There is an explicit antiderivative and then we have the exact value of the definite integral.

To explain the huge numbers : there is a denominator

$$(2 \cos (2 x)-2 \cos (4 x)+2 \cos (6 x)-2 \cos (8 x)+2 \cos (10 x)-1)^7$$

Using $\cos(2t)=x$ $$32 t^5-16 t^4-32 t^3+12 t^2+6 t-1$$ shows five real roots and one of them corresponds to $x=5.5691869768182698318\cdots$

So, at the lower bound, the denominator is $-4.70495\times 10^{-10}$ and $-4.6264\times 10^{-13}$ at the upper bound.