∫sec⁡xatan⁡x+bdx is equal to

Let J=∫sec⁡xatan⁡x+bdx=∫1asin⁡x+bcos⁡xdx=∫1a2+b2aa2+b2sin⁡x+ba2+b2cos⁡xdx Choose sin⁡α=aa2+b2,cos⁡α=ba2+b2,tan⁡α=ab⇒α=tan−1⁡abJ=1a2+b2∫1sin⁡αsin⁡x+cos⁡αcos⁡xdxJ=1a2+b2∫1cos⁡(x−α)dx=1a2+b2∫sec⁡(x−α)dxJ=1a2+b2log⁡tan⁡π4+x−α2+CJ=1a2+b2log⁡tan⁡π4+x2−12tan−1⁡ab+K