integration of dx/(((cosx)^2 +4sinx-5)cosx)

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on Jan 25, 2025 at 9:55 pm



You asked:

Evaluate the integral: ∫1(cos⁡(x)2+4sin⁡(x)−5)⋅cos⁡(x)dx\int{\frac{1}{\left({\cos\left( x \right)}^{2} + 4 \sin\left( x \right) - 5\right) \cdot \cos\left( x \right)}}dx

MathBot Answer:

Antiderivative

∫1(cos⁡(x)2+4sin⁡(x)−5)⋅cos⁡(x)dx=−8log⁡(1+tan⁡2(x2)−tan⁡(x2))18−18tan⁡(x2)+18tan⁡2(x2)−2log⁡(1+tan⁡(x2))18−18tan⁡(x2)+18tan⁡2(x2)+3tan⁡(x2)18−18tan⁡(x2)+18tan⁡2(x2)+18log⁡(−1+tan⁡(x2))18−18tan⁡(x2)+18tan⁡2(x2)−18log⁡(−1+tan⁡(x2))tan⁡(x2)18−18tan⁡(x2)+18tan⁡2(x2)−8tan⁡2(x2)log⁡(1+tan⁡2(x2)−tan⁡(x2))18−18tan⁡(x2)+18tan⁡2(x2)−2tan⁡2(x2)log⁡(1+tan⁡(x2))18−18tan⁡(x2)+18tan⁡2(x2)+2log⁡(1+tan⁡(x2))tan⁡(x2)18−18tan⁡(x2)+18tan⁡2(x2)+8log⁡(1+tan⁡2(x2)−tan⁡(x2))tan⁡(x2)18−18tan⁡(x2)+18tan⁡2(x2)+18tan⁡2(x2)log⁡(−1+tan⁡(x2))18−18tan⁡(x2)+18tan⁡2(x2)+C\int{\frac{1}{\left({\cos\left( x \right)}^{2} + 4 \sin\left( x \right) - 5\right) \cdot \cos\left( x \right)}}dx = - \frac{8 \log{\left(1 + \tan^{2}{\left(\frac{x}{2} \right)} - \tan{\left(\frac{x}{2} \right)} \right)}}{18 - 18 \tan{\left(\frac{x}{2} \right)} + 18 \tan^{2}{\left(\frac{x}{2} \right)}} - \frac{2 \log{\left(1 + \tan{\left(\frac{x}{2} \right)} \right)}}{18 - 18 \tan{\left(\frac{x}{2} \right)} + 18 \tan^{2}{\left(\frac{x}{2} \right)}} + \frac{3 \tan{\left(\frac{x}{2} \right)}}{18 - 18 \tan{\left(\frac{x}{2} \right)} + 18 \tan^{2}{\left(\frac{x}{2} \right)}} + \frac{18 \log{\left(-1 + \tan{\left(\frac{x}{2} \right)} \right)}}{18 - 18 \tan{\left(\frac{x}{2} \right)} + 18 \tan^{2}{\left(\frac{x}{2} \right)}} - \frac{18 \log{\left(-1 + \tan{\left(\frac{x}{2} \right)} \right)} \tan{\left(\frac{x}{2} \right)}}{18 - 18 \tan{\left(\frac{x}{2} \right)} + 18 \tan^{2}{\left(\frac{x}{2} \right)}} - \frac{8 \tan^{2}{\left(\frac{x}{2} \right)} \log{\left(1 + \tan^{2}{\left(\frac{x}{2} \right)} - \tan{\left(\frac{x}{2} \right)} \right)}}{18 - 18 \tan{\left(\frac{x}{2} \right)} + 18 \tan^{2}{\left(\frac{x}{2} \right)}} - \frac{2 \tan^{2}{\left(\frac{x}{2} \right)} \log{\left(1 + \tan{\left(\frac{x}{2} \right)} \right)}}{18 - 18 \tan{\left(\frac{x}{2} \right)} + 18 \tan^{2}{\left(\frac{x}{2} \right)}} + \frac{2 \log{\left(1 + \tan{\left(\frac{x}{2} \right)} \right)} \tan{\left(\frac{x}{2} \right)}}{18 - 18 \tan{\left(\frac{x}{2} \right)} + 18 \tan^{2}{\left(\frac{x}{2} \right)}} + \frac{8 \log{\left(1 + \tan^{2}{\left(\frac{x}{2} \right)} - \tan{\left(\frac{x}{2} \right)} \right)} \tan{\left(\frac{x}{2} \right)}}{18 - 18 \tan{\left(\frac{x}{2} \right)} + 18 \tan^{2}{\left(\frac{x}{2} \right)}} + \frac{18 \tan^{2}{\left(\frac{x}{2} \right)} \log{\left(-1 + \tan{\left(\frac{x}{2} \right)} \right)}}{18 - 18 \tan{\left(\frac{x}{2} \right)} + 18 \tan^{2}{\left(\frac{x}{2} \right)}} + C

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