What is the integral of x^2sec^2(x) ?

The integral of x^2sec^2(x) is x^2tan(x)-i\Li_{2}(-e^{2ix})-ix^2+2xln(1+cos(2x)+isin(2x))+C

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integral of x^2sec^2(x)

\int x^{2} sec^{2} (x) d x

x^{2} tan (x) - i Li_{2} (- e^{2 i x}) - i x^{2} + 2 x ln (1 + cos (2 x) + i sin (2 x)) + C

Solution steps

\int x^{2} sec^{2} (x) d x

Simplify

x^{2} : xx

= \int xx sec^{2} (x) d x

Apply Integration By Parts

: x^{2} tan (x) - \int 2 x tan (x) d x

= x^{2} tan (x) - \int 2 x tan (x) d x

\int 2 x tan (x) d x = i Li_{2} (- e^{2 i x}) + i x^{2} - 2 x ln (1 + cos (2 x) + i sin (2 x))

= x^{2} tan (x) - (i Li_{2} (- e^{2 i x}) + i x^{2} - 2 x ln (1 + cos (2 x) + i sin (2 x)))

Simplify

= x^{2} tan (x) - i Li_{2} (- e^{2 i x}) - i x^{2} + 2 x ln (1 + cos (2 x) + i sin (2 x))

Add a constant to the solution

= x^{2} tan (x) - i Li_{2} (- e^{2 i x}) - i x^{2} + 2 x ln (1 + cos (2 x) + i sin (2 x)) + C

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What is the integral of x^2sec^2(x) ?
The integral of x^2sec^2(x) is x^2tan(x)-i\Li_{2}(-e^{2ix})-ix^2+2xln(1+cos(2x)+isin(2x))+C