Can science and religion be integrated? What comes to mind immediately is that religions themselves cannot agree with one another whereas science is basically monolithic. How can there even be trade between the two, let alone integration?
First, it is only a perception that religions are pluralistic and science is not. Science is monolithic only so far as science of matter–physics and chemistry–is concerned. Psychology, the science of the psyche, has three different paradigms–behavioral-cognitive consisting of hard science orientation, depth psychology consisting of Freudian psychoanalysis and Jungian analytical psychology and their derivatives with psychotherapy orientation, and humanistic-transpersonal-yoga psychology with positive mental health orientation. Both the later paradigms of psychology acknowledge downward causation and subtle bodies in some form or other. Medicine has the conventional allopathic medicine and also alternative medicine practices that complement it. A prominent part of alternative medicine is Eastern medicine that emphasizes subtle energies called variously as prana, chi, and ki. And biology is in transition right now. The materialist biology is highly developed but with some unsolved (maybe unsolvable) problems. Alternative biology is biology that sees life as the handiwork of a purposive designer with the power of downward causation; but at present it is so poorly developed that hardly anyone can call it a genuine alternative biology.