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