Evaluate the indefinite integral.
\[
\begin{array}{l}
\int\left(-5 \sin (t)+2 \cos (t)-4 \sec ^{2}(t)-5 e^{t}+\frac{1}{\sqrt{1-t^{2}}}+\frac{5}{1+t^{2}}\right) d t= \\
\square+C
\end{array}
\]

Step 1 ：Break the integral up into several smaller integrals: \(\int -5 \sin (t) dt, \int 2 \cos (t) dt, \int -4 \sec ^{2}(t) dt, \int -5 e^{t} dt, \int \frac{1}{\sqrt{1-t^{2}}} dt, \int \frac{5}{1+t^{2}} dt\)

Step 2 ：Find the integral of each function individually using standard integral formulas: \(-5 \cos (t), 2 \sin (t), -4 \tan (t), -5 e^{t}, \sin ^{-1}(t), 5 \arctan (t)\)

Step 3 ：Add the results of the integrals together to get the final answer: \(-5 \cos (t) + 2 \sin (t) - 4 \tan (t) -5 e^{t} + \sin ^{-1}(t) + 5 \arctan (t)\)

Step 4 ：Final Answer: \(\boxed{-5 e^{t}+2 \sin (t)-4 \tan (t)+5 \arctan (t)+\sin ^{-1}(t)+C}\)