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Answer
$$ \displaystyle\int \cos\left(x\right){\cdot}\ln\left(x\right)\, \mathrm d x = {{2\,\ln x\,\sin x+i\,\Gamma\left(0 , i\,x\right)-i\,\Gamma\left(0 , -i\,x\right)}\over{2}} $$
Explanation
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