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The following is a list of integrals (antiderivative functions) of trigonometric functions. For antiderivatives involving both exponential and trigonometric functions, see List of integrals of exponential functions. For a complete list of antiderivative functions, see Lists of integrals. For the special antiderivatives involving trigonometric functions, see Trigonometric integral.

Generally, if the function $\sin x$ is any trigonometric function, and $\cos x$ is its derivative,

\int a\cos nx\,dx={\frac {a}{n))\sin nx+C

In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration.

Integrands involving only sine

$\int \sin ax\,dx=-{\frac {1}{a))\cos ax+C$
$\int \sin ^{2}{ax}\,dx={\frac {x}{2))-{\frac {1}{4a))\sin 2ax+C={\frac {x}{2))-{\frac {1}{2a))\sin ax\cos ax+C$
$\int \sin ^{3}{ax}\,dx={\frac {\cos 3ax}{12a))-{\frac {3\cos ax}{4a))+C$
$\int x\sin ^{2}{ax}\,dx={\frac {x^{2)){4))-{\frac {x}{4a))\sin 2ax-{\frac {1}{8a^{2))}\cos 2ax+C$
$\int x^{2}\sin ^{2}{ax}\,dx={\frac {x^{3)){6))-\left({\frac {x^{2)){4a))-{\frac {1}{8a^{3))}\right)\sin 2ax-{\frac {x}{4a^{2))}\cos 2ax+C$
$\int x\sin ax\,dx={\frac {\sin ax}{a^{2))}-{\frac {x\cos ax}{a))+C$
$\int (\sin b_{1}x)(\sin b_{2}x)\,dx={\frac {\sin((b_{2}-b_{1})x)}{2(b_{2}-b_{1})))-{\frac {\sin((b_{1}+b_{2})x)}{2(b_{1}+b_{2})))+C\qquad {\mbox{(for ))|b_{1}|\neq |b_{2}|{\mbox{)))$
$\int \sin ^{n}{ax}\,dx=-{\frac {\sin ^{n-1}ax\cos ax}{na))+{\frac {n-1}{n))\int \sin ^{n-2}ax\,dx\qquad {\mbox{(for ))n>0{\mbox{)))$
$\int {\frac {dx}{\sin ax))=-{\frac {1}{a))\ln {\left|\csc {ax}+\cot {ax}\right|}+C$
$\int {\frac {dx}{\sin ^{n}ax))={\frac {\cos ax}{a(1-n)\sin ^{n-1}ax))+{\frac {n-2}{n-1))\int {\frac {dx}{\sin ^{n-2}ax))\qquad {\mbox{(for ))n>1{\mbox{)))$
${\begin{aligned}\int x^{n}\sin ax\,dx&=-{\frac {x^{n)){a))\cos ax+{\frac {n}{a))\int x^{n-1}\cos ax\,dx\\&=\sum _{k=0}^{2k\leq n}(-1)^{k+1}{\frac {x^{n-2k)){a^{1+2k))}{\frac {n!}{(n-2k)!))\cos ax+\sum _{k=0}^{2k+1\leq n}(-1)^{k}{\frac {x^{n-1-2k)){a^{2+2k))}{\frac {n!}{(n-2k-1)!))\sin ax\\&=-\sum _{k=0}^{n}{\frac {x^{n-k)){a^{1+k))}{\frac {n!}{(n-k)!))\cos \left(ax+k{\frac {\pi }{2))\right)\qquad {\mbox{(for ))n>0{\mbox{)))\end{aligned))$
$\int {\frac {\sin ax}{x))\,dx=\sum _{n=0}^{\infty }(-1)^{n}{\frac {(ax)^{2n+1)){(2n+1)\cdot (2n+1)!))+C$
$\int {\frac {\sin ax}{x^{n))}\,dx=-{\frac {\sin ax}{(n-1)x^{n-1))}+{\frac {a}{n-1))\int {\frac {\cos ax}{x^{n-1))}\,dx$
${\displaystyle \int {\sin {\mathrm {(} }{ax}^{2}\mathrm {+} {bx}\mathrm {+} {c}{\mathrm {)} }{dx))\mathrm {=} \left\((\begin{aligned}&((\sqrt {a)){\sqrt {\frac {\mathit {\pi )){2))}\cos \left({\frac ((b}^{2}\mathrm {-} {4}{ac)){4a))\right){S}\left({\frac ((2}{ax}\mathrm {+} {b)){\sqrt ((2}{a}{\mathit {\pi ))))}\right)\mathrm {+} {\sqrt {a)){\sqrt {\frac {\mathit {\pi )){2))}\sin \left({\frac ((b}^{2}\mathrm {-} {4}{ac)){4a))\right){C}\left({\frac ((2}{ax}\mathrm {+} {b)){\sqrt ((2}{a}{\mathit {\pi ))))}\right)\;{to}\;{b}^{2}\mathrm {-} {4}{ac}\;{\mathrm {>} }\;{0))\\&((\sqrt {a)){\sqrt {\frac {\mathit {\pi )){2))}\cos \left({\frac ((b}^{2}\mathrm {-} {4}{ac)){4a))\right){S}\left({\frac ((2}{ax}\mathrm {+} {b)){\sqrt ((2}{a}{\mathit {\pi ))))}\right)\mathrm {-} {\sqrt {a)){\sqrt {\frac {\mathit {\pi )){2))}\sin \left({\frac ((b}^{2}\mathrm {-} {4}{ac)){4a))\right){C}\left({\frac ((2}{ax}\mathrm {+} {b)){\sqrt ((2}{a}{\mathit {\pi ))))}\right)\;{to}\;{b}^{2}\mathrm {-} {4}{ac}\;{\mathrm {<} }\;{0))\end{aligned))\right.\;\;{for}\;{a}\diagup \!\!\!\!{\mathrm {=} }{0}{\mathrm {,} }\;{a}{\mathrm {>} }{0))$
$\int {\frac {dx}{1\pm \sin ax))={\frac {1}{a))\tan \left({\frac {ax}{2))\mp {\frac {\pi }{4))\right)+C$
$\int {\frac {x\,dx}{1+\sin ax))={\frac {x}{a))\tan \left({\frac {ax}{2))-{\frac {\pi }{4))\right)+{\frac {2}{a^{2))}\ln \left|\cos \left({\frac {ax}{2))-{\frac {\pi }{4))\right)\right|+C$
$\int {\frac {x\,dx}{1-\sin ax))={\frac {x}{a))\cot \left({\frac {\pi }{4))-{\frac {ax}{2))\right)+{\frac {2}{a^{2))}\ln \left|\sin \left({\frac {\pi }{4))-{\frac {ax}{2))\right)\right|+C$
$\int {\frac {\sin ax\,dx}{1\pm \sin ax))=\pm x+{\frac {1}{a))\tan \left({\frac {\pi }{4))\mp {\frac {ax}{2))\right)+C$

Integrands involving only cosine

$\int \cos ax\,dx={\frac {1}{a))\sin ax+C$
$\int \cos ^{2}{ax}\,dx={\frac {x}{2))+{\frac {1}{4a))\sin 2ax+C={\frac {x}{2))+{\frac {1}{2a))\sin ax\cos ax+C$
$\int \cos ^{n}ax\,dx={\frac {\cos ^{n-1}ax\sin ax}{na))+{\frac {n-1}{n))\int \cos ^{n-2}ax\,dx\qquad {\mbox{(for ))n>0{\mbox{)))$
$\int x\cos ax\,dx={\frac {\cos ax}{a^{2))}+{\frac {x\sin ax}{a))+C$
$\int x^{2}\cos ^{2}{ax}\,dx={\frac {x^{3)){6))+\left({\frac {x^{2)){4a))-{\frac {1}{8a^{3))}\right)\sin 2ax+{\frac {x}{4a^{2))}\cos 2ax+C$
${\begin{aligned}\int x^{n}\cos ax\,dx&={\frac {x^{n}\sin ax}{a))-{\frac {n}{a))\int x^{n-1}\sin ax\,dx\\&=\sum _{k=0}^{2k+1\leq n}(-1)^{k}{\frac {x^{n-2k-1)){a^{2+2k))}{\frac {n!}{(n-2k-1)!))\cos ax+\sum _{k=0}^{2k\leq n}(-1)^{k}{\frac {x^{n-2k)){a^{1+2k))}{\frac {n!}{(n-2k)!))\sin ax\\&=\sum _{k=0}^{n}(-1)^{\lfloor k/2\rfloor }{\frac {x^{n-k)){a^{1+k))}{\frac {n!}{(n-k)!))\cos \left(ax-{\frac {(-1)^{k}+1}{2)){\frac {\pi }{2))\right)\\&=\sum _{k=0}^{n}{\frac {x^{n-k)){a^{1+k))}{\frac {n!}{(n-k)!))\sin \left(ax+k{\frac {\pi }{2))\right)\qquad {\mbox{(for ))n>0{\mbox{)))\end{aligned))$
$\int {\frac {\cos ax}{x))\,dx=\ln |ax|+\sum _{k=1}^{\infty }(-1)^{k}{\frac {(ax)^{2k)){2k\cdot (2k)!))+C$
$\int {\frac {\cos ax}{x^{n))}\,dx=-{\frac {\cos ax}{(n-1)x^{n-1))}-{\frac {a}{n-1))\int {\frac {\sin ax}{x^{n-1))}\,dx\qquad {\mbox{(for ))n\neq 1{\mbox{)))$
$\int {\frac {dx}{\cos ax))={\frac {1}{a))\ln \left|\tan \left({\frac {ax}{2))+{\frac {\pi }{4))\right)\right|+C$
$\int {\frac {dx}{\cos ^{n}ax))={\frac {\sin ax}{a(n-1)\cos ^{n-1}ax))+{\frac {n-2}{n-1))\int {\frac {dx}{\cos ^{n-2}ax))\qquad {\mbox{(for ))n>1{\mbox{)))$
$\int {\frac {dx}{1+\cos ax))={\frac {1}{a))\tan {\frac {ax}{2))+C$
$\int {\frac {dx}{1-\cos ax))=-{\frac {1}{a))\cot {\frac {ax}{2))+C$
$\int {\frac {x\,dx}{1+\cos ax))={\frac {x}{a))\tan {\frac {ax}{2))+{\frac {2}{a^{2))}\ln \left|\cos {\frac {ax}{2))\right|+C$
$\int {\frac {x\,dx}{1-\cos ax))=-{\frac {x}{a))\cot {\frac {ax}{2))+{\frac {2}{a^{2))}\ln \left|\sin {\frac {ax}{2))\right|+C$
$\int {\frac {\cos ax\,dx}{1+\cos ax))=x-{\frac {1}{a))\tan {\frac {ax}{2))+C$
$\int {\frac {\cos ax\,dx}{1-\cos ax))=-x-{\frac {1}{a))\cot {\frac {ax}{2))+C$
$\int (\cos a_{1}x)(\cos a_{2}x)\,dx={\frac {\sin((a_{2}-a_{1})x)}{2(a_{2}-a_{1})))+{\frac {\sin((a_{2}+a_{1})x)}{2(a_{2}+a_{1})))+C\qquad {\mbox{(for ))|a_{1}|\neq |a_{2}|{\mbox{)))$

Integrands involving only tangent

$\int \tan ax\,dx=-{\frac {1}{a))\ln |\cos ax|+C={\frac {1}{a))\ln |\sec ax|+C$
$\int \tan ^{2}{x}\,dx=\tan {x}-x+C$
$\int \tan ^{n}ax\,dx={\frac {1}{a(n-1)))\tan ^{n-1}ax-\int \tan ^{n-2}ax\,dx\qquad {\mbox{(for ))n\neq 1{\mbox{)))$
$\int {\frac {dx}{q\tan ax+p))={\frac {1}{p^{2}+q^{2))}(px+{\frac {q}{a))\ln |q\sin ax+p\cos ax|)+C\qquad {\mbox{(for ))p^{2}+q^{2}\neq 0{\mbox{)))$
$\int {\frac {dx}{\tan ax\pm 1))=\pm {\frac {x}{2))+{\frac {1}{2a))\ln |\sin ax\pm \cos ax|+C$
$\int {\frac {\tan ax\,dx}{\tan ax\pm 1))={\frac {x}{2))\mp {\frac {1}{2a))\ln |\sin ax\pm \cos ax|+C$

Integrands involving only secant

Further information: Integral of the secant function

$\int \sec {ax}\,dx={\frac {1}{a))\ln {\left|\sec {ax}+\tan {ax}\right|}+C={\frac {1}{a))\ln {\left|\tan {\left({\frac {ax}{2))+{\frac {\pi }{4))\right)}\right|}+C={\frac {1}{a))\operatorname {artanh} {\left(\sin {ax}\right)}+C$
$\int \sec ^{2}{x}\,dx=\tan {x}+C$
$\int \sec ^{3}{x}\,dx={\frac {1}{2))\sec x\tan x+{\frac {1}{2))\ln |\sec x+\tan x|+C.$
$\int \sec ^{n}{ax}\,dx={\frac {\sec ^{n-2}{ax}\tan {ax)){a(n-1)))\,+\,{\frac {n-2}{n-1))\int \sec ^{n-2}{ax}\,dx\qquad {\mbox{ (for ))n\neq 1{\mbox{)))$
$\int {\frac {dx}{\sec {x}+1))=x-\tan {\frac {x}{2))+C$
$\int {\frac {dx}{\sec {x}-1))=-x-\cot {\frac {x}{2))+C$

Integrands involving only cosecant

$\int \csc {ax}\,dx=-{\frac {1}{a))\ln {\left|\csc {ax}+\cot {ax}\right|}+C={\frac {1}{a))\ln {\left|\csc {ax}-\cot {ax}\right|}+C={\frac {1}{a))\ln {\left|\tan {\left({\frac {ax}{2))\right)}\right|}+C$
$\int \csc ^{2}{x}\,dx=-\cot {x}+C$
$\int \csc ^{3}{x}\,dx=-{\frac {1}{2))\csc x\cot x-{\frac {1}{2))\ln |\csc x+\cot x|+C=-{\frac {1}{2))\csc x\cot x+{\frac {1}{2))\ln |\csc x-\cot x|+C$
$\int \csc ^{n}{ax}\,dx=-{\frac {\csc ^{n-2}{ax}\cot {ax)){a(n-1)))\,+\,{\frac {n-2}{n-1))\int \csc ^{n-2}{ax}\,dx\qquad {\mbox{ (for ))n\neq 1{\mbox{)))$
$\int {\frac {dx}{\csc {x}+1))=x-{\frac {2}{\cot {\frac {x}{2))+1))+C$
$\int {\frac {dx}{\csc {x}-1))=-x+{\frac {2}{\cot {\frac {x}{2))-1))+C$

Integrands involving only cotangent

$\int \cot ax\,dx={\frac {1}{a))\ln |\sin ax|+C$
$\int \cot ^{2}{x}\,dx=-\cot {x}-x+C$
$\int \cot ^{n}ax\,dx=-{\frac {1}{a(n-1)))\cot ^{n-1}ax-\int \cot ^{n-2}ax\,dx\qquad {\mbox{(for ))n\neq 1{\mbox{)))$
$\int {\frac {dx}{1+\cot ax))=\int {\frac {\tan ax\,dx}{\tan ax+1))={\frac {x}{2))-{\frac {1}{2a))\ln |\sin ax+\cos ax|+C$
$\int {\frac {dx}{1-\cot ax))=\int {\frac {\tan ax\,dx}{\tan ax-1))={\frac {x}{2))+{\frac {1}{2a))\ln |\sin ax-\cos ax|+C$

Integrands involving both sine and cosine

An integral that is a rational function of the sine and cosine can be evaluated using Bioche's rules.

$\int {\frac {dx}{\cos ax\pm \sin ax))={\frac {1}{a{\sqrt {2))))\ln \left|\tan \left({\frac {ax}{2))\pm {\frac {\pi }{8))\right)\right|+C$
$\int {\frac {dx}{(\cos ax\pm \sin ax)^{2))}={\frac {1}{2a))\tan \left(ax\mp {\frac {\pi }{4))\right)+C$
$\int {\frac {dx}{(\cos x+\sin x)^{n))}={\frac {1}{2(n-1)))\left({\frac {\sin x-\cos x}{(\cos x+\sin x)^{n-1))}+(n-2)\int {\frac {dx}{(\cos x+\sin x)^{n-2))}\right)$
$\int {\frac {\cos ax\,dx}{\cos ax+\sin ax))={\frac {x}{2))+{\frac {1}{2a))\ln \left|\sin ax+\cos ax\right|+C$
$\int {\frac {\cos ax\,dx}{\cos ax-\sin ax))={\frac {x}{2))-{\frac {1}{2a))\ln \left|\sin ax-\cos ax\right|+C$
$\int {\frac {\sin ax\,dx}{\cos ax+\sin ax))={\frac {x}{2))-{\frac {1}{2a))\ln \left|\sin ax+\cos ax\right|+C$
$\int {\frac {\sin ax\,dx}{\cos ax-\sin ax))=-{\frac {x}{2))-{\frac {1}{2a))\ln \left|\sin ax-\cos ax\right|+C$
$\int {\frac {\cos ax\,dx}{(\sin ax)(1+\cos ax)))=-{\frac {1}{4a))\tan ^{2}{\frac {ax}{2))+{\frac {1}{2a))\ln \left|\tan {\frac {ax}{2))\right|+C$
$\int {\frac {\cos ax\,dx}{(\sin ax)(1-\cos ax)))=-{\frac {1}{4a))\cot ^{2}{\frac {ax}{2))-{\frac {1}{2a))\ln \left|\tan {\frac {ax}{2))\right|+C$
$\int {\frac {\sin ax\,dx}{(\cos ax)(1+\sin ax)))={\frac {1}{4a))\cot ^{2}\left({\frac {ax}{2))+{\frac {\pi }{4))\right)+{\frac {1}{2a))\ln \left|\tan \left({\frac {ax}{2))+{\frac {\pi }{4))\right)\right|+C$
$\int {\frac {\sin ax\,dx}{(\cos ax)(1-\sin ax)))={\frac {1}{4a))\tan ^{2}\left({\frac {ax}{2))+{\frac {\pi }{4))\right)-{\frac {1}{2a))\ln \left|\tan \left({\frac {ax}{2))+{\frac {\pi }{4))\right)\right|+C$
$\int (\sin ax)(\cos ax)\,dx={\frac {1}{2a))\sin ^{2}ax+C$
$\int (\sin a_{1}x)(\cos a_{2}x)\,dx=-{\frac {\cos((a_{1}-a_{2})x)}{2(a_{1}-a_{2})))-{\frac {\cos((a_{1}+a_{2})x)}{2(a_{1}+a_{2})))+C\qquad {\mbox{(for ))|a_{1}|\neq |a_{2}|{\mbox{)))$
$\int (\sin ^{n}ax)(\cos ax)\,dx={\frac {1}{a(n+1)))\sin ^{n+1}ax+C\qquad {\mbox{(for ))n\neq -1{\mbox{)))$
$\int (\sin ax)(\cos ^{n}ax)\,dx=-{\frac {1}{a(n+1)))\cos ^{n+1}ax+C\qquad {\mbox{(for ))n\neq -1{\mbox{)))$
${\begin{aligned}\int (\sin ^{n}ax)(\cos ^{m}ax)\,dx&=-{\frac {(\sin ^{n-1}ax)(\cos ^{m+1}ax)}{a(n+m)))+{\frac {n-1}{n+m))\int (\sin ^{n-2}ax)(\cos ^{m}ax)\,dx\qquad {\mbox{(for ))m,n>0{\mbox{)))\\&={\frac {(\sin ^{n+1}ax)(\cos ^{m-1}ax)}{a(n+m)))+{\frac {m-1}{n+m))\int (\sin ^{n}ax)(\cos ^{m-2}ax)\,dx\qquad {\mbox{(for ))m,n>0{\mbox{)))\end{aligned))$
$\int {\frac {dx}{(\sin ax)(\cos ax)))={\frac {1}{a))\ln \left|\tan ax\right|+C$
$\int {\frac {dx}{(\sin ax)(\cos ^{n}ax)))={\frac {1}{a(n-1)\cos ^{n-1}ax))+\int {\frac {dx}{(\sin ax)(\cos ^{n-2}ax)))\qquad {\mbox{(for ))n\neq 1{\mbox{)))$
$\int {\frac {dx}{(\sin ^{n}ax)(\cos ax)))=-{\frac {1}{a(n-1)\sin ^{n-1}ax))+\int {\frac {dx}{(\sin ^{n-2}ax)(\cos ax)))\qquad {\mbox{(for ))n\neq 1{\mbox{)))$
$\int {\frac {\sin ax\,dx}{\cos ^{n}ax))={\frac {1}{a(n-1)\cos ^{n-1}ax))+C\qquad {\mbox{(for ))n\neq 1{\mbox{)))$
$\int {\frac {\sin ^{2}ax\,dx}{\cos ax))=-{\frac {1}{a))\sin ax+{\frac {1}{a))\ln \left|\tan \left({\frac {\pi }{4))+{\frac {ax}{2))\right)\right|+C$
$\int {\frac {\sin ^{2}ax\,dx}{\cos ^{n}ax))={\frac {\sin ax}{a(n-1)\cos ^{n-1}ax))-{\frac {1}{n-1))\int {\frac {dx}{\cos ^{n-2}ax))\qquad {\mbox{(for ))n\neq 1{\mbox{)))$
${\begin{aligned}\int {\frac {\sin ^{2}x}{1+\cos ^{2}x))\,dx&={\sqrt {2))\operatorname {arctangant} \left({\frac {\tan x}{\sqrt {2))}\right)-x\qquad {\mbox{(for x in))]-{\frac {\pi }{2));+{\frac {\pi }{2))[{\mbox{)))\\&={\sqrt {2))\operatorname {arctangant} \left({\frac {\tan x}{\sqrt {2))}\right)-\operatorname {arctangant} \left(\tan x\right)\qquad {\mbox{(this time x being any real number )){\mbox{)))\end{aligned))$
$\int {\frac {\sin ^{n}ax\,dx}{\cos ax))=-{\frac {\sin ^{n-1}ax}{a(n-1)))+\int {\frac {\sin ^{n-2}ax\,dx}{\cos ax))\qquad {\mbox{(for ))n\neq 1{\mbox{)))$
$\int {\frac {\sin ^{n}ax\,dx}{\cos ^{m}ax))={\begin{cases}{\frac {\sin ^{n+1}ax}{a(m-1)\cos ^{m-1}ax))-{\frac {n-m+2}{m-1))\int {\frac {\sin ^{n}ax\,dx}{\cos ^{m-2}ax))&{\mbox{(for ))m\neq 1{\mbox{)))\\{\frac {\sin ^{n-1}ax}{a(m-1)\cos ^{m-1}ax))-{\frac {n-1}{m-1))\int {\frac {\sin ^{n-2}ax\,dx}{\cos ^{m-2}ax))&{\mbox{(for ))m\neq 1{\mbox{)))\\-{\frac {\sin ^{n-1}ax}{a(n-m)\cos ^{m-1}ax))+{\frac {n-1}{n-m))\int {\frac {\sin ^{n-2}ax\,dx}{\cos ^{m}ax))&{\mbox{(for ))m\neq n{\mbox{)))\end{cases))$
$\int {\frac {\cos ax\,dx}{\sin ^{n}ax))=-{\frac {1}{a(n-1)\sin ^{n-1}ax))+C\qquad {\mbox{(for ))n\neq 1{\mbox{)))$
$\int {\frac {\cos ^{2}ax\,dx}{\sin ax))={\frac {1}{a))\left(\cos ax+\ln \left|\tan {\frac {ax}{2))\right|\right)+C$
$\int {\frac {\cos ^{2}ax\,dx}{\sin ^{n}ax))=-{\frac {1}{n-1))\left({\frac {\cos ax}{a\sin ^{n-1}ax))+\int {\frac {dx}{\sin ^{n-2}ax))\right)\qquad {\mbox{(for ))n\neq 1{\mbox{)))$
$\int {\frac {\cos ^{n}ax\,dx}{\sin ^{m}ax))={\begin{cases}-{\frac {\cos ^{n+1}ax}{a(m-1)\sin ^{m-1}ax))-{\frac {n-m+2}{m-1))\int {\frac {\cos ^{n}ax\,dx}{\sin ^{m-2}ax))&{\mbox{(for ))n\neq 1{\mbox{)))\\-{\frac {\cos ^{n-1}ax}{a(m-1)\sin ^{m-1}ax))-{\frac {n-1}{m-1))\int {\frac {\cos ^{n-2}ax\,dx}{\sin ^{m-2}ax))&{\mbox{(for ))m\neq 1{\mbox{)))\\{\frac {\cos ^{n-1}ax}{a(n-m)\sin ^{m-1}ax))+{\frac {n-1}{n-m))\int {\frac {\cos ^{n-2}ax\,dx}{\sin ^{m}ax))&{\mbox{(for ))m\neq n{\mbox{)))\end{cases))$

Integrands involving both sine and tangent

$\int (\sin ax)(\tan ax)\,dx={\frac {1}{a))(\ln |\sec ax+\tan ax|-\sin ax)+C$
$\int {\frac {\tan ^{n}ax\,dx}{\sin ^{2}ax))={\frac {1}{a(n-1)))\tan ^{n-1}(ax)+C\qquad {\mbox{(for ))n\neq 1{\mbox{)))$

Integrand involving both cosine and tangent

$\int {\frac {\tan ^{n}ax\,dx}{\cos ^{2}ax))={\frac {1}{a(n+1)))\tan ^{n+1}ax+C\qquad {\mbox{(for ))n\neq -1{\mbox{)))$

Integrand involving both sine and cotangent

$\int {\frac {\cot ^{n}ax\,dx}{\sin ^{2}ax))=-{\frac {1}{a(n+1)))\cot ^{n+1}ax+C\qquad {\mbox{(for ))n\neq -1{\mbox{)))$

Integrand involving both cosine and cotangent

$\int {\frac {\cot ^{n}ax\,dx}{\cos ^{2}ax))={\frac {1}{a(1-n)))\tan ^{1-n}ax+C\qquad {\mbox{(for ))n\neq 1{\mbox{)))$

Integrand involving both secant and tangent

$\int (\sec x)(\tan x)\,dx=\sec x+C$

Integrand involving both cosecant and cotangent

$\int (\csc x)(\cot x)\,dx=-\csc x+C$

Integrals in a quarter period

Using the beta function $B(a,b)$ one can write

$\int _{0}^{\frac {\pi }{2))\sin ^{n}x\,dx=\int _{0}^{\frac {\pi }{2))\cos ^{n}x\,dx={\frac {1}{2))B\left({\frac {n+1}{2)),{\frac {1}{2))\right)={\begin{cases}{\frac {n-1}{n))\cdot {\frac {n-3}{n-2))\cdots {\frac {3}{4))\cdot {\frac {1}{2))\cdot {\frac {\pi }{2)),&{\text{if ))n{\text{ is even))\\{\frac {n-1}{n))\cdot {\frac {n-3}{n-2))\cdots {\frac {4}{5))\cdot {\frac {2}{3)),&{\text{if ))n{\text{ is odd and more than 1))\\1,&{\text{if ))n=1\end{cases))$

Integrals with symmetric limits

$\int _{-c}^{c}\sin {x}\,dx=0$
${\displaystyle \int _{-c}^{c}\cos {x}\,dx=2\int _{0}^{c}\cos {x}\,dx=2\int _{-c}^{0}\cos {x}\,dx=2\sin {c))$
$\int _{-c}^{c}\tan {x}\,dx=0$
$\int _{-{\frac {a}{2))}^{\frac {a}{2))x^{2}\cos ^{2}{\frac {n\pi x}{a))\,dx={\frac {a^{3}(n^{2}\pi ^{2}-6)}{24n^{2}\pi ^{2))}\qquad {\mbox{(for ))n=1,3,5...{\mbox{)))$
$\int _{\frac {-a}{2))^{\frac {a}{2))x^{2}\sin ^{2}{\frac {n\pi x}{a))\,dx={\frac {a^{3}(n^{2}\pi ^{2}-6(-1)^{n})}{24n^{2}\pi ^{2))}={\frac {a^{3)){24))(1-6{\frac {(-1)^{n)){n^{2}\pi ^{2))})\qquad {\mbox{(for ))n=1,2,3,...{\mbox{)))$