$$ \int_0^\infty x^{s-1}f(x)dx=\Gamma(s)\phi(-s). $$
$\bf{2.}$ Glasser's Master Theorem: $$ \int_{-\infty}^\infty f\left(x-\sum_{n}\frac{|a_n|}{x-b_n}\right)dx=\int_{-\infty}^\infty f\left(x\right)dx. $$ See also George Boole, "On the Comparison of Transcendents, with certain applications to the Theory of Definite Integrals", Philosophical Transactions of the Royal Society, Vol. 147, 745-804 (1857), where one can find more general formulas.
$\bf{3.}$ This result is given in American Mathematical Monthly Problem 11234 - 06 by J. Brennan and R. Ehrenborg:
Let the $2n-1$ real numbers satisfy $a_1 < b_1< a_2 <...< a_{n-1} < b_{n-1} < a_n$, then $$ \int_{-\infty}^\infty f\left(\frac{(x-a_1)...(x-a_n)}{(x-b_1)...(x-b_{n-1})}\right)dx=\int_{-\infty}^\infty f\left(x\right)dx. $$
$\bf{4.}$ This result is due to M.L. Glasser (see the arxiv preprint): Let $F(p)=\int_0^\infty e^{-pt}f(t)dt$ be Laplace transform of a function and $a$ any real number, then $$ \int_{-\infty}^\infty \frac{F(x^2+i\pi x)\cosh x}{\cosh(2x)+\cosh(2a)}dx=\frac{\pi F[(\pi/2)^2+a^2]}{2\cosh a}. $$
$\bf{5.}$ B. Berndt gives the following formula in Ramanujan's Notebooks, part IV, page 318, with the conditions for its validity: $$ \int_{-\infty}^\infty \frac{F(iu/x)}{1+x^2}dx=F(u). $$
