Integral of x/cos^2x^2 dx
The solution
The answer (Indefinite)
[src]
/ 4/x\ / /x\\ / /x\\ 2/x\ / 2/x\\ 5/x\ /x\ 2/x\ / 2/x\\ 4/x\ / /x\\ 4/x\ / /x\\ 6/x\ / 2/x\\ 6/x\ / /x\\ 6/x\ / /x\\ 3/x\ 2/x\ / /x\\ 2/x\ / /x\\ 4/x\ / 2/x\\
| 2*tan |-| 2*log|1 + tan|-|| 2*log|-1 + tan|-|| 2*tan |-| 2*log|1 + tan |-|| 6*x*tan |-| 6*x*tan|-| 6*tan |-|*log|1 + tan |-|| 6*tan |-|*log|1 + tan|-|| 6*tan |-|*log|-1 + tan|-|| 2*tan |-|*log|1 + tan |-|| 2*tan |-|*log|1 + tan|-|| 2*tan |-|*log|-1 + tan|-|| 4*x*tan |-| 6*tan |-|*log|1 + tan|-|| 6*tan |-|*log|-1 + tan|-|| 6*tan |-|*log|1 + tan |-||
| x \2/ \ \2// \ \2// \2/ \ \2// \2/ \2/ \2/ \ \2// \2/ \ \2// \2/ \ \2// \2/ \ \2// \2/ \ \2// \2/ \ \2// \2/ \2/ \ \2// \2/ \ \2// \2/ \ \2//
| ------- dx = C - -------------------------------------- - -------------------------------------- - -------------------------------------- + -------------------------------------- + -------------------------------------- - -------------------------------------- - -------------------------------------- - -------------------------------------- - -------------------------------------- - -------------------------------------- - -------------------------------------- + -------------------------------------- + -------------------------------------- + -------------------------------------- + -------------------------------------- + -------------------------------------- + --------------------------------------
| 4 4/x\ 6/x\ 2/x\ 4/x\ 6/x\ 2/x\ 4/x\ 6/x\ 2/x\ 4/x\ 6/x\ 2/x\ 4/x\ 6/x\ 2/x\ 4/x\ 6/x\ 2/x\ 4/x\ 6/x\ 2/x\ 4/x\ 6/x\ 2/x\ 4/x\ 6/x\ 2/x\ 4/x\ 6/x\ 2/x\ 4/x\ 6/x\ 2/x\ 4/x\ 6/x\ 2/x\ 4/x\ 6/x\ 2/x\ 4/x\ 6/x\ 2/x\ 4/x\ 6/x\ 2/x\ 4/x\ 6/x\ 2/x\ 4/x\ 6/x\ 2/x\
| cos (x) -3 - 9*tan |-| + 3*tan |-| + 9*tan |-| -3 - 9*tan |-| + 3*tan |-| + 9*tan |-| -3 - 9*tan |-| + 3*tan |-| + 9*tan |-| -3 - 9*tan |-| + 3*tan |-| + 9*tan |-| -3 - 9*tan |-| + 3*tan |-| + 9*tan |-| -3 - 9*tan |-| + 3*tan |-| + 9*tan |-| -3 - 9*tan |-| + 3*tan |-| + 9*tan |-| -3 - 9*tan |-| + 3*tan |-| + 9*tan |-| -3 - 9*tan |-| + 3*tan |-| + 9*tan |-| -3 - 9*tan |-| + 3*tan |-| + 9*tan |-| -3 - 9*tan |-| + 3*tan |-| + 9*tan |-| -3 - 9*tan |-| + 3*tan |-| + 9*tan |-| -3 - 9*tan |-| + 3*tan |-| + 9*tan |-| -3 - 9*tan |-| + 3*tan |-| + 9*tan |-| -3 - 9*tan |-| + 3*tan |-| + 9*tan |-| -3 - 9*tan |-| + 3*tan |-| + 9*tan |-| -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|
| \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/
/
$$\int \frac{x}{\cos^{4}{\left(x \right)}}\, dx = C - \frac{6 x \tan^{5}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} + \frac{4 x \tan^{3}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} - \frac{6 x \tan{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} + \frac{2 \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{6}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} - \frac{6 \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{4}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} + \frac{6 \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} - \frac{2 \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} + \frac{2 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{6}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} - \frac{6 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} + \frac{6 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} - \frac{2 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} - \frac{2 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{6}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} + \frac{6 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} - \frac{6 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} + \frac{2 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} - \frac{2 \tan^{4}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} + \frac{2 \tan^{2}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3}$$
Use the examples entering the upper and lower limits of integration.