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Question Number 129173 by bramlexs22 last updated on 13/Jan/21

$$\mathrm{J}=\int \frac{\mathrm{d}\theta }{\tan \theta +\cot \theta +\sec \theta +\csc \theta }?$$

Answered by liberty last updated on 13/Jan/21

$$\mathrm{J}=\int \frac{\mathrm{d}\theta }{\frac{\sin \theta +1}{\cos \theta }+\frac{\cos \theta +1}{\sin \theta }}=\int \frac{\cos \theta \sin \theta \mathrm{d}\theta }{\sin {}^2\theta +\sin \theta +\cos {}^2\theta +\cos \theta }$$$$\mathrm{J}=\int \frac{\sin \theta \cos \theta \mathrm{d}\theta }{1+\sin \theta +\cos \theta }=\int \frac{2\sin \left(\frac{\theta }{2}\right)\cos \left(\frac{\theta }{2}\right)\cos \theta }{2\sin \left(\frac{\theta }{2}\right)\cos \left(\frac{\theta }{2}\right)+2\cos {}^2\left(\frac{\theta }{2}\right)}\mathrm{d}\theta$$$$\mathrm{J}=\int \frac{\sin \left(\frac{\theta }{2}\right)\cos \theta }{\sin \left(\frac{\theta }{2}\right)+\cos \left(\frac{\theta }{2}\right)}\mathrm{d}\theta$$$$\mathrm{J}=\int \sin \left(\frac{\theta }{2}\right)(\cos \left(\frac{\theta }{2}\right)-\sin \left(\frac{\theta }{2}\right))\mathrm{d}\theta$$$$\mathrm{J}=\int \frac{1}{2}\sin \theta -(\frac{1}{2}-\frac{1}{2}\cos \theta )\mathrm{d}\theta$$$$\underset{―}{\mathbf{J}=-\frac{1}{2}\mathbf{cos}\mathit{\theta }-\frac{1}{2}\theta +\frac{1}{2}\mathbf{sin}\mathit{\theta }+\mathbf{C}}$$$$$$

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