Difference in my and wolfram's integration.

There's no error in your solution. Both results (yours and Wolfram's) are equal up to a constant (which for an indefinite integral is arbitrary). You can check that using the equalities $$ \cos (5\alpha) = \cos^5 \alpha - 10\cos^3\alpha \sin\alpha + 5\cos\alpha \sin^4\alpha = 16\cos^5\alpha -20\cos^3\alpha + 5\cos\alpha$$ $$ \cos (3\alpha) = \cos^3\alpha - 3\cos\alpha\sin^2\alpha = 4\cos^3\alpha -3\cos\alpha$$ $$ \frac{1}{\tan^2\alpha +1} = \cos^2\alpha$$ $$ \sec\alpha = \frac{1}{\cos\alpha}$$ Just write both results using only $\cos(\frac{x}{2})$ and compare.