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Find $$\int\frac{e^{2x}-e^x+1}{(e^x\sin x+\cos x)(e^x\cos x-\sin x)}dx$$ I observed that $(e^x\sin x+\cos x)'=e^x\cos x+e^x\sin x-\sin x$ and I tried to denote this by $t$ to change the variable but I failed. Any help?

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  • $\begingroup$ where did you get this? I am not sure there is an elementary antiderivative. $\endgroup$ – Vasya Mar 7 at 15:48
  • $\begingroup$ It`s from a magazine of mathematics. Any help is welcomed. $\endgroup$ – user651692 Mar 7 at 15:54
  • $\begingroup$ Have you tried expanding out the denominator? $\endgroup$ – Alex Nelson Mar 7 at 15:55
  • $\begingroup$ Also, I don't think your observation $(e^x\sin x+\cos x)=e^x\cos x-\sin x$ is true, what happens at $x=5\pi$? $\endgroup$ – Alex Nelson Mar 7 at 16:03
  • $\begingroup$ It`s $(e^x\sin x+\cos x)'=e^x\cos x-\sin x$. Sorry for this. I edit it. $\endgroup$ – user651692 Mar 7 at 16:24
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let $f(x)=e^x\cos x-\sin x$ and

$g(x)=e^x\sin x+\cos x$

and $e^{2x}-e^x+1=(e^x\cos x-\sin x)(e^x\sin x+\cos x)'-(e^x\sin x+\cos x)(e^x\cos x-\sin x)'$

integration is $\displaystyle \int\frac{f(x)g'(x)-g(x)f'(x)}{f(x)g(x)}dx$

$$\int\frac{g'(x)}{g(x)}dx-\int\frac{f'(x)}{f(x)}dx=\ln\bigg|\frac{g(x)}{f(x)}\bigg|+C$$

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