Indefinite Integrals of $trig^5(x)$
Index
\[\begin{aligned}
& \int \sin^5(x) \,\mathrm{d}x = -\cos(x) + \dfrac{2\cos^3(x)}{3} - \dfrac{\cos^5(x)}{5} + C \\
& \int \cos^5(x) \,\mathrm{d}x = \sin(x) - \dfrac{2\sin^3(x)}{3} + \dfrac{\sin^5(x)}{5} + C \\
& \int \tan^5(x) \,\mathrm{d}x = \dfrac{\tan^4(x)}{4} - \dfrac{\tan^2(x)}{2} - \ln(\vert \cos(x) \vert) + C = \dfrac{\tan^4(x)}{4} - \dfrac{\tan^2(x)}{2} + \ln(\vert \sec(x) \vert) + C \\
& \int \cot^5(x) \,\mathrm{d}x = -\dfrac{\cot^4(x)}{4} + \dfrac{\cot^2(x)}{2} + \ln(\vert \sin(x) \vert) + C = -\dfrac{\cot^4(x)}{4} + \dfrac{\cot^2(x)}{2} - \ln(\vert \csc(x) \vert) + C \\
& \int \sec^5(x) \,\mathrm{d}x = \dfrac{\sec^3(x)\tan(x)}{4} + \dfrac{3\sec(x)\tan(x)}{8} + \dfrac{3\ln(\vert \sec(x)+\tan(x) \vert)}{8} + C
= \dfrac{\sec^3(x)\tan(x)}{4} + \dfrac{3\sec(x)\tan(x)}{8} - \dfrac{3\ln(\vert \sec(x)-\tan(x) \vert)}{8} + C \\
& \int \csc^5(x) \,\mathrm{d}x = -\dfrac{\csc^3(x)\cot(x)}{4} - \dfrac{3\csc(x)\cot(x)}{8} - \dfrac{3\ln(\vert \csc(x)+\cot(x) \vert)}{8} + C
= -\dfrac{\csc^3(x)\cot(x)}{4} - \dfrac{3\csc(x)\cot(x)}{8} + \dfrac{3\ln(\vert \csc(x)-\cot(x) \vert)}{8} + C
\end{aligned}\]
sin
\[\begin{aligned}
\int \sin^5(x) \,\mathrm{d}x &= \int \sin^4(x)\sin(x) \,\mathrm{d}x \\
&= -\int \sin^4(x) \,\mathrm{d}\cos(x) \\
&= -\int (1 - \cos^2(x))^2 \,\mathrm{d}\cos(x) \\
&= -\int (1 - 2\cos^2(x) + \cos^4(x)) \,\mathrm{d}\cos(x) \\
&= -\left(\cos(x) - \dfrac{2\cos^3(x)}{3} + \dfrac{\cos^5(x)}{5}\right) + C \\
&= -\cos(x) + \dfrac{2\cos^3(x)}{3} - \dfrac{\cos^5(x)}{5} + C
\end{aligned}\]
cos
\[\begin{aligned}
\int \cos^5(x) \,\mathrm{d}x &= \int \cos^4(x)\cos(x) \,\mathrm{d}x \\
&= \int \cos^4(x) \,\mathrm{d}\sin(x) \\
&= \int (1 - \sin^2(x))^2 \,\mathrm{d}\sin(x) \\
&= \int (1 - 2\sin^2(x) + \sin^4(x)) \,\mathrm{d}\sin(x) \\
&= \sin(x) - \dfrac{2\sin^3(x)}{3} + \dfrac{\sin^5(x)}{5} + C
\end{aligned}\]
tan
\[\begin{aligned}
\int \tan^3(x) \,\mathrm{d}x &= \dfrac{\tan^2(x)}{2} + \ln(\vert \cos(x) \vert) + C = \dfrac{\tan^2(x)}{2} - \ln(\vert \sec(x) \vert) + C\\
\\
\therefore
\int \tan^5(x) \,\mathrm{d}x &= \int \tan^3(x)\tan^2(x) \,\mathrm{d}x \\
&= \int \tan^3(x)(\sec^2(x) - 1)\,\mathrm{d}x \\
&= \int (\tan^3(x)\sec^2(x) - \tan^3(x))\,\mathrm{d}x \\
&= \int \tan^3(x)\sec^2(x) \,\mathrm{d}x - \int \tan^3(x)\,\mathrm{d}x \\
&= \int \tan^3(x) \,\mathrm{d}\tan(x) - \int \tan^3(x)\,\mathrm{d}x \\
&= \dfrac{\tan^4(x)}{4} - \left(\dfrac{\tan^2(x)}{2} + \ln(\vert \cos(x) \vert)\right) + C \\
&= \dfrac{\tan^4(x)}{4} - \dfrac{\tan^2(x)}{2} - \ln(\vert \cos(x) \vert) + C \\
&\overset{\text{or}}{=} \dfrac{\tan^4(x)}{4} - \dfrac{\tan^2(x)}{2} + \ln(\vert \sec(x) \vert) + C
\end{aligned}\]
cot
\[\begin{aligned}
\int \cot^3(x) \,\mathrm{d}x &= -\dfrac{\cot^2(x)}{2} - \ln(\vert \sin(x) \vert) + C \\
&= -\dfrac{\cot^2(x)}{2} + \ln(\vert \csc(x) \vert) + C \\
\\
\therefore
\int \cot^5(x) \,\mathrm{d}x &= \int \cot^3(x)\cot^2(x) \,\mathrm{d}x \\
&= \int \cot^3(x)(\csc^2(x) - 1)\,\mathrm{d}x \\
&= \int (\cot^3(x)\csc^2(x) - \cot^3(x))\,\mathrm{d}x \\
&= \int \cot^3(x)\csc^2(x) \,\mathrm{d}x - \int \cot^3(x)\,\mathrm{d}x \\
&= -\int \cot^3(x) \,\mathrm{d}\cot(x) - \int \cot^3(x)\,\mathrm{d}x \\
&= -\dfrac{\cot^4(x)}{4} - \left(-\dfrac{\cot^2(x)}{2} - \ln(\vert \sin(x) \vert)\right) + C \\
&= -\dfrac{\cot^4(x)}{4} + \dfrac{\cot^2(x)}{2} + \ln(\vert \sin(x) \vert) + C \\
&\overset{\text{or}}{=} -\dfrac{\cot^4(x)}{4} + \dfrac{\cot^2(x)}{2} - \ln(\vert \csc(x) \vert) + C
\end{aligned}\]
sec
\[\begin{aligned}
\dfrac{\mathrm{d}}{\mathrm{d}x}\sec(x) &= \sec(x)\tan(x) \\
\int \sec^3(x) \,\mathrm{d}x &= \dfrac{1}{2}\sec(x)\tan(x) + \dfrac{1}{2}\ln(\vert \sec(x)+\tan(x) \vert) + C \\
&= \dfrac{1}{2}\sec(x)\tan(x) - \dfrac{1}{2}\ln(\vert \sec(x)-\tan(x) \vert) + C \\
\\
\therefore
\int \sec^5(x) \,\mathrm{d}x &= \int \sec^3(x)\sec^2(x) \,\mathrm{d}x \\
&= \int \sec^3(x)\,\mathrm{d}\tan(x) \\
&= \sec^3(x)\tan(x) - \int\tan(x)\,\mathrm{d}\sec^3(x) \\
&= \sec^3(x)\tan(x) - \int\tan(x)\cdot 3\sec^2(x) \cdot \sec(x)\tan(x)\,\mathrm{d}x \\
&= \sec^3(x)\tan(x) - 3\int\tan^2(x)\sec^3(x)\,\mathrm{d}x \\
&= \sec^3(x)\tan(x) - 3\int(\sec^2(x)-1)\sec^3(x)\,\mathrm{d}x \\
&= \sec^3(x)\tan(x) - 3\int\sec^5(x)\,\mathrm{d}x + 3\int\sec^3(x)\,\mathrm{d}x \\
\\
\therefore
\int \sec^5(x) \,\mathrm{d}x &= \dfrac{1}{4}\left(\sec^3(x)\tan(x) + 3\int\sec^3(x)\,\mathrm{d}x \right) \\
&= \dfrac{\sec^3(x)\tan(x)}{4} + \dfrac{3}{4}\left({\dfrac{1}{2}\sec(x)\tan(x) + \dfrac{1}{2}\ln(\vert \sec(x)+\tan(x) \vert)}\right) + C \\
&= \dfrac{\sec^3(x)\tan(x)}{4} + \dfrac{3\sec(x)\tan(x)}{8} + \dfrac{3\ln(\vert \sec(x)+\tan(x) \vert)}{8} + C \\
&\overset{\text{or}}{=} \dfrac{\sec^3(x)\tan(x)}{4} + \dfrac{3\sec(x)\tan(x)}{8} - \dfrac{3\ln(\vert \sec(x)-\tan(x) \vert)}{8} + C
\end{aligned}\]
csc
\[\begin{aligned}
\dfrac{\mathrm{d}}{\mathrm{d}x}\csc(x) &= -\csc(x)\cot(x) \\
\int \csc^3(x) \,\mathrm{d}x &= -\dfrac{1}{2}\csc(x)\cot(x) - \dfrac{1}{2}\ln(\vert \csc(x)+\cot(x) \vert) + C \\
&= -\dfrac{1}{2}\csc(x)\cot(x) + \dfrac{1}{2}\ln(\vert \csc(x)-\cot(x) \vert) + C \\
\\
\therefore
\int \csc^5(x) \,\mathrm{d}x &= \int \csc^3(x)\csc^2(x) \,\mathrm{d}x \\
&= -\int \csc^3(x)\,\mathrm{d}\cot(x) \\
&= -\csc^3(x)\cot(x) + \int\cot(x)\,\mathrm{d}\csc^3(x) \\
&= -\csc^3(x)\cot(x) - \int\cot(x)\cdot 3\csc^2(x) \cdot \csc(x)\cot(x)\,\mathrm{d}x \\
&= -\csc^3(x)\cot(x) - 3\int\cot^2(x)\csc^3(x)\,\mathrm{d}x \\
&= -\csc^3(x)\cot(x) - 3\int(\csc^2(x)-1)\csc^3(x)\,\mathrm{d}x \\
&= -\csc^3(x)\cot(x) - 3\int\csc^5(x)\,\mathrm{d}x + 3\int\csc^3(x)\,\mathrm{d}x \\
\\
\therefore
\int \csc^5(x) \,\mathrm{d}x &= \dfrac{1}{4}\left(-\csc^3(x)\cot(x) + 3\int\csc^3(x)\,\mathrm{d}x \right) \\
&= -\dfrac{\csc^3(x)\cot(x)}{4} + \dfrac{3}{4}\left({-\dfrac{1}{2}\csc(x)\cot(x) - \dfrac{1}{2}\ln(\vert \csc(x)+\cot(x) \vert)}\right) + C \\
&= -\dfrac{\csc^3(x)\cot(x)}{4} - \dfrac{3\csc(x)\cot(x)}{8} - \dfrac{3\ln(\vert \csc(x)+\cot(x) \vert)}{8} + C \\
&\overset{\text{or}}{=} -\dfrac{\csc^3(x)\cot(x)}{4} - \dfrac{3\csc(x)\cot(x)}{8} + \dfrac{3\ln(\vert \csc(x)-\cot(x) \vert)}{8} + C
\end{aligned}\]