What is the integral of θtan(θ) ?

The integral of θtan(θ) is i 1/2 \Li_{2}(-e^{2iθ})+i 1/2 θ^2-θln(1+cos(2θ)+isin(2θ))+C

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integral of θtan(θ)

\int θ tan (θ) d θ

i \frac{1}{2} Li_{2} (- e^{2 i θ}) + i \frac{1}{2} θ^{2} - θ ln (1 + cos (2 θ) + i sin (2 θ)) + C

Solution steps

\int θ tan (θ) d θ

Use the nonelementary integral:

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

= i \frac{1}{2} Li_{2} (- e^{2 i θ}) + i \frac{1}{2} θ^{2} - θ ln (1 + e^{2 i θ})

Apply imaginary number rule:

e^{ia} = cos (a) + i sin (a)

= i \frac{1}{2} Li_{2} (- e^{2 i θ}) + i \frac{1}{2} θ^{2} - θ ln (1 + cos (2 θ) + i sin (2 θ))

Add a constant to the solution

= i \frac{1}{2} Li_{2} (- e^{2 i θ}) + i \frac{1}{2} θ^{2} - θ ln (1 + cos (2 θ) + i sin (2 θ)) + C

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What is the integral of θtan(θ) ?
The integral of θtan(θ) is i 1/2 \Li_{2}(-e^{2iθ})+i 1/2 θ^2-θln(1+cos(2θ)+isin(2θ))+C