Integrate $\int\frac{1}{\sin x+\cos x+\tan x+\cot x+\csc x+\sec x}dx$

Solution 1:

You want

$\begin{align} \int\frac{dx}{\sin x+\cos x+\tan x+\cot x+\csc x+\sec x} &= \int\frac{dx}{\sin x+\cos x+\frac{\sin x}{\cos x} +\frac{\cos x}{\sin x}+\frac1{\sin x}+\frac1{\cos x}}\\ &= \int\frac{\sin x \cos x\ dx}{\sin^2 x \cos x+\cos^2 x \sin x+\sin^2 x +\cos^2 x+\cos x+\sin x}\\ &= \int\frac{\sin x \cos x\ dx}{\sin x \cos x(\sin x+ \cos x)+1+\cos x+\sin x}\\ &= \int\frac{\sin x \cos x\ dx}{(\sin x \cos x+1)(\sin x+ \cos x)+1}\\ \end{align} $.

Applying substitutions $\sin x = \frac{2t}{1+t^2}, \cos x = \frac{1-t^2}{1+t^2}, dx = \frac{2 dt}{1+t^2}$, this becomes

$\begin{align} \int\frac{\sin x \cos x\ dx}{(\sin x \cos x+1)(\sin x+ \cos x)+1} &=\int\frac{\frac{(2t)(1-t^2)(2dt)}{(1+t^2)^3}} {(\frac{(2t)(1-t^2)}{(1+t^2)^2}+1)(\frac{2t}{1+t^2}+ \frac{1-t^2}{1+t^2})+1}\\ &=\int\frac{(2t)(1-t^2)(2dt)} {((2t)(1-t^2)+(1+t^2)^2)(1+2t-t^2)+(1+t^2)^3}\\ &=\int\frac{4t(1-t^2)dt} {((2t-2t^3)+1+2t^2+t^4)(1+2t-t^2)+1+3t^2+3t^4+t^6}\\ &=\int\frac{4t(1-t^2)dt} {(1+2t+2t^2-2t^3+t^4)(1+2t-t^2)+1+3t^2+3t^4+t^6}\\ &=\int\frac{4t(1-t^2)dt} {2 (t+1) (2 t^4-3 t^3+3 t^2+t+1)} \quad \text{ (according to Wolfram Alpha)}\\ &=\int\frac{2t(1-t)dt} {2 t^4-3 t^3+3 t^2+t+1}\\ \end{align} $.

According to Wolfram Alpha, that quartic can be factored as the product of two quadratics, but the coefficients look irrational.

I'll leave it at this.

Solution 2:

If you can't see anything simpler: when you have a rational function of sin and cos, you can always use the tangent half-angle substitution $t = \tan (x/2)$. Then $$\sin x = \frac{2t}{1+t^2}, \cos x = \frac{1-t^2}{1+t^2}, dx = \frac{2 dt}{1+t^2}.$$ Substituting these converts your integral into the integral of a rational function of $t$, which can then be integrated by partial fractions.

Solution 3:

You can use Euler's identities and expand it in terms of complex exponents like (after some simplifications): $$\frac{1}{\sin x+\cos x+\tan x+\cot x+\csc x+\sec x}=\frac{e^{\mathrm i x} \left((1+\mathrm i) e^{2 \mathrm i x}-2 \mathrm i e^{\mathrm i x}+(\mathrm i-1)\right)}{e^{4 \mathrm i x}-(1+\mathrm i) e^{3 \mathrm i x}+6 \mathrm i e^{2 \mathrm i x}+(1-\mathrm i) e^{\mathrm i x}-1}$$ So $$ \begin{eqnarray} \int\!\!\frac{1}{\sin x\!+\!\cos x\!+\!\tan x\!+\!\cot x\!+\!\csc x\!+\!\sec x}\!\mathrm dx&=&\frac{1}{\mathrm i}\!\!\int \!\! \frac{(1+\mathrm i) e^{2 \mathrm i x}-2 \mathrm i e^{\mathrm i x}+(\mathrm i-1)}{e^{4 \mathrm i x}-(1+\mathrm i) e^{3 \mathrm i x}+6 \mathrm i e^{2 \mathrm i x}+(1-\mathrm i) e^{\mathrm i x}-1}\!\!\mathrm de^{\mathrm i x}\!\!=\\ &=&\frac{1}{\mathrm i}\!\!\int \!\! \frac{(1+\mathrm i) t^{2}-2 \mathrm i t+(\mathrm i-1)}{t^{4}-(1+\mathrm i) t^{3}+6 \mathrm i t^{2}+(1-\mathrm i) t^{}-1} \!\!\mathrm dt&=&\\ &=&\frac{1}{\mathrm i}\!\!\int \!\! \frac{(t-t_{11})(t-t_{12})}{(t-t_{21})(t-t_{22})(t-t_{32})(t-t_{42})} \!\!\mathrm dt \end{eqnarray} $$ After that you can use partial fraction expansion to simplify the answer (Mathematica gives 4 distinct roots for denominator).