I have learned integration by parts, integration by u-substitution, partial fractions, power rule, integrals of $e^x$ and $\ln(x)$, the derivatives of the standard trigonometric functions and inverse trigonometric functions.
But I have no idea how to solve this question, Please let me know if I need to learn more information to solve this question such as learning special non-elementary integrals (by the way, I have no idea how to tell whether or not an integral is elementary except that all integrals of rational polynomial functions are elementary).
Thanks in advance.
I am skeptical about a possible closed form (even using special functions.
Assuminf that you want to compute $$I=\int_0^a \frac{\sec ^2(x)}{(x+5)^2(x-1)}\,dx$$ with $a<1$, just use Taylor series built at $x=0$ to get $$\frac{\sec ^2(x)}{(x+5)^2(x-1)}=-\frac{1}{25}-\frac{3 x}{125}-\frac{43 x^2}{625}-\frac{161 x^3}{3125}-\frac{781 x^4}{9375}-\frac{5569 x^5}{78125}-\frac{316168 x^6}{3515625}+O\left(x^7\right)$$ and integrate termwise to get $$I=-\frac{a}{25}-\frac{3 a^2}{250}-\frac{43 a^3}{1875}-\frac{161 a^4}{12500}-\frac{781 a^5}{46875}-\frac{5569 a^6}{468750}-\frac{316168 a^7}{24609375}+O\left(a^{8}\right)$$
The table below reports values of the above expression and the result of the numerical integration $$\left( \begin{array}{ccc} a & \text{approximation} & \text{exact} \\ 0.05 & -0.0020330 & -0.0020330 \\ 0.10 & -0.0041444 & -0.0041444 \\ 0.15 & -0.0063553 & -0.0063553 \\ 0.20 & -0.0086903 & -0.0086904 \\ 0.25 & -0.0111786 & -0.0111788 \\ 0.30 & -0.0138555 & -0.0138564 \\ 0.35 & -0.0167642 & -0.0167677 \\ 0.40 & -0.0199578 & -0.0199688 \\ 0.45 & -0.0235021 & -0.0235326 \\ 0.50 & -0.0274783 & -0.0275554 \\ 0.55 & -0.0319871 & -0.0321682 \\ 0.60 & -0.0371524 & -0.0375539 \\ 0.65 & -0.0431263 & -0.0439777 \\ 0.70 & -0.0500947 & -0.0518410 \\ 0.75 & -0.0582835 & -0.0617866 \end{array} \right)$$ It is clear that the approximation starts to be bad when $a$ increases. But you could add more terms.