Difficult integrals?

Main Question or Discussion Point

I'm looking for some tricky/difficult integrals within the scope of calc I and II that I can play around with. Most of the integrals in my books (Stewart and Spivak) are fairly straight forward, and the only real practice I get is in "rigor". I can't really make up my own problems either, because I always come up with something unsolvable (without a CAS et al).

What are some good integrals??

What are some good integrals??
You may hit the SEARCH of this forum with 'integrals'.

https://www.physicsforums.com/showpost.php?p=3433157&postcount=272
$$\int \sin(\ln x) + \cos(\ln x)dx$$
$$\int \frac{x^2}{x^2 +4x + 8} dx$$
$$\int \frac{1}{\sqrt{5x-3}+\sqrt{5x+2}} dx$$
$$\int \left( x^2 + 1\right) e^{x^2}dx$$
$$\int \frac{1}{\sqrt[3]{x} + x} dx$$
The integral below is tricky, BUT it can be solved using only simple substitutions.
Show that

$$I_4 \, = \, \int_{0}^{\infty} \dfrac{x^{29}}{(5x^2+49)^{17}} \, dx \,=\, \dfrac{14!}{2\cdot 49^2 \cdot 5^{15 }\cdot 16!}$$

What I like about these integrals, is that most of them have simple, clever solutions.

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Here's one which had me stumped for a while:

$$\int\frac{4x^5-1}{(x^5+x+1)^2}dx$$

Once you see the solution of this one, you immediately get it. But without seeing the solution, it can be quite hard.

I'd suggest getting Apostol's calculus book. It is filled with hard integrals.