Integration of Transcendental Functions

$\int \sin x \sin 3 x d x$

$\int \sin 2 x \cos 4 x d x$

$\int \cos x \cos 4 x d x$

$\int \sin (3 x+2) \cos (x-1) d x$

$\int \sin x \sin 2 x \sin 3 x d x$

$\int \cos x \cos 3 x \cos 5 x d x$

$\int \sin^{2} x \cos (3 x+1) d x$;

$\int \cos^{2} 2 x \cos^{2} 3 x d x$.

$\int \operatorname {sh} x \operatorname {sh} 7 x d x$;

$\int \operatorname {sh}^{3} x d x$

$\int \operatorname {ch} x \operatorname {ch} 2 x \operatorname {ch} 3 x d x$;

$\int \frac{d x}{\operatorname {sh}^{2} x+\operatorname {ch}^{2} x}$.

$\int \frac{\sin 2 x}{\cos^{3} x} d x$;

$\int \frac{\sin 3 x}{\cos x} d x$

$\int \frac{\cos 3 x}{\sin^{5} x} d x$;

$\int \frac{\cos^{2} x}{\sin 4 x} d x$.

$\int \cos^{3} x d x$;

$\int \sin^{3} x \cos^{4} x d x$;

$\int \cos^{3} x \sin^{8} x d x$

$\int \cos^{5} 2 x \sin^{7} 2 x d x$;

$\int \cos^{3} x \cos 2 x d x$;

$\int \cos^{2} 3 x \sin x d x$.

$\int \operatorname {ch}^{3} x \operatorname {sh} x d x$;

$\int \operatorname {sh}^{2} x \operatorname {ch}^{3} x d x$;

$\int \operatorname {sh}^{4} x \operatorname {ch} \frac{x}{2} d x$;

$\int \operatorname {sh}^{3} x \operatorname {ch} 2 x d x$.

$\int \sin^{2} x \cos^{2} x d x$;

$\int \sin^{4} 4 x d x$;

$\int \cos^{6} 3 x d x$;

$\int \sin^{4} x \cos^{6} x d x$.

$\int \operatorname {sh}^{2} 2 x \operatorname {ch}^{2} 2 x d x$;

$\int \operatorname {ch}^{4} x d x$

$\int \operatorname {sh}^{2} x \operatorname {ch}^{4} x d x$;

$\int \operatorname {sh}^{4} \frac{x}{2} \operatorname {ch}^{2} \frac{x}{2} d x$.

$\int \frac{d x}{\cos^{3} x}$;

$\int \frac{\sin^{4} x}{\cos x} d x$;

$\int \frac{d x}{\sin^{2} x \cos^{3} x}$;

$\int \frac{\cos^{4} x}{\sin^{3} x} d x$.

$\int \frac{d x}{\operatorname {sh} x \operatorname {ch}^{2} x}$;

$\int \frac{d x}{\operatorname {sh}^{3} x \operatorname {ch}^{2} x}$;

$\int \frac{d x}{\operatorname {ch}^{5} x}$;

$\int \frac{\operatorname {sh}^{4} x}{\operatorname {ch}^{3} x} d x$.

$\int \frac{\cos^{3} x}{\sin^{5} x} d x$;

$\int \frac{\cos^{5} x}{\sin^{3} x} d x$;

$\int \frac{d x}{\sin x \cos^{3} x}$;

$\int \frac{d x}{\sin^{2} x \cos^{4} x}$;

$\int \frac{d x}{\sin^{4} x}$;

$\int \frac{\sin^{4} x}{\cos^{10} x} d x$;

$\int \operatorname {tg}^{5} x d x$;

$\int \operatorname {tg}^{6} x d x$.

$\int \frac{\operatorname {sh}^{4} x}{\operatorname {ch}^{6} x} d x$;

$\int \frac{\operatorname {ch}^{5} x}{\operatorname {sh} x} d x$;

$\int \frac{d x}{(1+\operatorname {ch} x)^{2}}$;

$\int \frac{d x}{\operatorname {sh}^{4} x \operatorname {ch}^{2} x}$;

$\int \operatorname {th}^{3} x d x$;

$\int \operatorname {th}^{4} x d x$.

$\int \frac{\sin x d x}{(3 \cos x-1)^{3}}$;

$\int \frac{d x}{\sin x(1+\cos x)}$

$\int \frac{\sin x+\sin^{3} x}{\cos 2 x} d x$;

$\int \frac{2 \sin^{3} x+\cos^{2} x \sin 2 x}{\sin^{4} x+3 \cos^{2} x} d x$;

$\int \frac{\operatorname {sh} 2 x+4 \operatorname {sh} x}{\operatorname {ch}^{2} x-3 \operatorname {ch} x} d x$;

$\int \frac{\operatorname {sh} 2 x+3 \operatorname {sh} x}{\operatorname {ch}^{2} x+2 \operatorname {ch}^{2}(x / 2)} d x$.

$\int \frac{\cos x d x}{\sin^{2} x-6 \sin x+5}$;

$\int \frac{\sin 2 x d x}{3+4 \sin^{2} x}$;

$\int \frac{\cos^{5} x+\cos^{3} x}{\sin^{4} x+\sin^{2} x} d x$;

$\int \frac{\cos x-\cos 3 x}{1-\sin^{4} x} d x$;

$\int \frac{d x}{\operatorname {ch}^{3} x+3 \operatorname {ch} x}$;

$\int \frac{\operatorname {sh} 2 x d x}{(\operatorname {sh} x+1)\left(\operatorname {ch}^{2} x-\operatorname {sh} x\right)}$.

$\int \frac{\operatorname {tg} x d x}{\operatorname {tg} x-3}$;

$\int \frac{\cos x-\sin x}{\cos x+\sin x} d x$;

$\int \frac{\sin \frac{x-a}{2}}{\sin \frac{x+a}{2}} d x$;

$\int \frac{a_{1} \cos x+b_{1} \sin x}{a \cos x+b \sin x} d x, a^{2}+b^{2} \neq 0$;

$\int \frac{4 \operatorname {ch} x-3 \operatorname {sh} x}{2 \operatorname {ch} x-\operatorname {sh} x} d x$;

$\int \frac{d x}{1-\operatorname {th} x}$;

$\int \frac{a_{1} \operatorname {ch} x+b_{1} \operatorname {sh} x}{a \operatorname {ch} x+b \operatorname {sh} x} d x, a^{2}+b^{2} \neq 0$.

$\int \frac{d x}{2 \cos^{2} x+\sin x \cos x+\sin^{2} x}$;

$\int \frac{d x}{\cos 2 x-\sin 2 x}$

$\int \frac{d x}{4 \cos^{2} x-2 \sin 2 x+\sin^{2} x}$;

$\int \frac{d x}{5+\cos^{2} x},\left|x\right| < \frac{\pi }{2}$;

$\int \frac{d x}{2+3 \sin 2 x-4 \cos^{2} x}$;

$\int \frac{d x}{a \cos^{2} x+b \sin 2 x+c \sin^{2} x}, c > 0$;

$\int \frac{d x}{3 \operatorname {sh}^{2} x-7 \operatorname {sh} x \operatorname {ch} x+3 \operatorname {ch}^{2} x}$;

$\int \frac{d x}{10 \operatorname {ch}^{2} x-2 \operatorname {sh} 2 x-1}$;

$\int \frac{d x}{4+3 \operatorname {sh}^{2} x}$;

$\int \frac{d x}{1-6 \operatorname {sh} 2 x-37 \operatorname {ch} x}$.

$\int \frac{d x}{\operatorname {tg}^{2} x+4 \operatorname {tg} x}$;

$\int \frac{\operatorname {th} x d x}{(\operatorname {th} x+2)^{2}}$.

$\int \frac{1+\operatorname {tg} x}{\sin 2 x} d x$;

$\int \frac{\cos x d x}{\sin^{3} x+\cos^{3} x}$;

$\int \frac{\sin 2 x d x}{\sin^{4} x+\cos^{4} x}$;

$\int \frac{d x}{\sin^{4} x+\cos^{4} x}$;

$\int \frac{\cos^{2} x d x}{\left(a^{2} \sin^{2} x+b^{2} \cos^{2} x\right)^{2}}, a^{2}+b^{2} \neq 0$;

$\int \frac{d x}{\sin^{6} x+\cos^{6} x}$;

$\int \frac{\operatorname {sh}^{2} x d x}{1-\operatorname {sh}^{2} x}$;

$\int \frac{\operatorname {ch} 2 x d x}{\operatorname {sh}^{4} x+\operatorname {ch}^{4} x}$;

$\int \frac{\operatorname {sh} 2 x d x}{1+\operatorname {sh}^{4} x}$;

$\int \frac{d x}{\left(\operatorname {ch} 2 x+\operatorname {ch}^{2} x\right)^{2}}$.

$\int \frac{d x}{\sin x+\cos x}$;

$\int \frac{d x}{\sqrt{3} \cos x+\sin x}$;

$\int \frac{d x}{\operatorname {sh} x+2 \operatorname {ch} x}$;

$\int \frac{d x}{2 \operatorname {sh} x-\operatorname {ch} x}$;

$\int \frac{d x}{a \operatorname {ch} x+b \operatorname {sh} x}, a > 0$;

$\int \frac{d x}{a \cos x+b \sin x}, a^{2}+b^{2} \neq 0$.

$\int \frac{d x}{1+4 \cos x}$;

$\int \frac{d x}{4+\cos x}$;

$\int \frac{d x}{4-\sin x}$;

$\int \frac{d x}{6-5 \sin x+\sin^{2} x}$;

$\int \frac{d x}{1+10 \operatorname {ch} x}$

$\int \frac{d x}{2 \operatorname {sh}^{2} x+5 \operatorname {sh} x+2}$;

$\int \frac{d x}{b \operatorname {sh} x+c}, c > 0$;

$\int \frac{d x}{a \cos x+c}, a > 0$.

$\int \frac{d x}{(1+\varepsilon \cos x)^{2}}, 0 < \varepsilon < 1$;

$\int \frac{d x}{(1+\varepsilon \cos x)^{3}}, \varepsilon > 1$.

$\int \frac{d x}{\cos x+\sin x+1}$;

$\int \frac{d x}{4 \cos x-3 \sin x-5}$;

$\int \frac{d x}{3 \cos x+\sin x+5}$

$\int \frac{d x}{7 \cos x-4 \sin x+8}$

$\int \frac{d x}{\operatorname {ch} x+\operatorname {sh} x+2}$;

$\int \frac{d x}{3 \operatorname {ch} x+5 \operatorname {sh} x+3}$

$\int \frac{d x}{a \operatorname {ch} x+b \operatorname {sh} x+c}, c^{2} > a^{2}-b^{2}$;

$\int \frac{d x}{a \cos x+b \sin x+c}, c > \sqrt{a^{2}+b^{2}} > 0$.

$\int \frac{1-\sin x+\cos x}{1+\sin x-\cos x} d x$;

$\int \frac{\sin x d x}{\cos x+\sin x+\sqrt{2}}$

$\int \frac{\cos x+2 \sin x}{4 \cos x+3 \sin x-2} d x$;

$\int \frac{2 \cos x+\sin x-3}{2 \cos x-\sin x-3} d x$;

$\int \frac{1-\cos (x-a)}{1-\cos (x+a)} d x$;

$\int \frac{\operatorname {sh} x+2 \operatorname {ch} x}{2 \operatorname {sh} x-\operatorname {ch} x-1} d x$;

$\int \frac{\operatorname {ch} x+2 \operatorname {sh} x+3}{4 \operatorname {ch} x+5 \operatorname {sh} x+6} d x$.

$\int \frac{\sin^{2} x}{2 \cos x+\sin x} d x$;

$\int \frac{1-2 \sin 2 x+2 \cos^{2} x}{\sin x+\cos x} d x$;

$\int \frac{2 \cos^{2} x-\sin x \cos x+\sin^{2} x}{\sin x+2 \cos x} d x$;

$\int \frac{a_{1} \cos^{2} x+b_{1} \cos x \sin x+c_{1} \sin^{2} x}{a \cos x+b \sin x} d x, a^{2}+b^{2} \neq 0$;

$\int \frac{\operatorname {sh} 2 x}{5 \operatorname {sh} x+3 \operatorname {ch} x} d x$;

$\int \frac{\operatorname {ch} 2 x}{3 \operatorname {sh} x+5 \operatorname {ch} x} d x$.

$\int \frac{\cos x-2 \sin x}{3 \sin^{2} x+4 \cos^{2} x} d x$;

$\int \frac{2 \sin x-\cos x}{(3 \sin x+4 \cos x)^{2}} d x$;

$\int \frac{a_{1} \cos x+b_{1} \sin x}{a \cos^{2} x+b \sin^{2} x} d x, b > a > 0$;

$\int \frac{a_{1} \cos x+b_{1} \sin x}{(a \cos x+b \sin x)^{2}} d x, a^{2}+b^{2} \neq 0$;

$\int \frac{2 \operatorname {sh} x-\operatorname {ch} x}{3 \operatorname {sh}^{2} x+4 \operatorname {ch}^{2} x} d x$;

$\int \frac{2 \operatorname {sh} x+\operatorname {ch} x}{(3 \operatorname {sh} x+4 \operatorname {ch} x)^{2}} d x$.

$\int \frac{(\cos x+\sin x) d x}{5 \cos^{2} x-2 \sin 2 x+2 \sin^{2} x}$;

$\int \frac{\sin x-2 \cos x}{1+2 \sin 2 x} d x$.

$\int \frac{(1+\sin x) d x}{\sin 2 x+2 \sin x}$;

$\int \frac{d x}{\sin 2 x+4 \sin x-4 \sin^{2} x}$;

$\int \frac{\sin x d x}{(1-\cos x+\sin x)^{2}}$;

$\int \frac{2 \sin x-\sin 2 x}{\sin^{3} x+(1-\cos x)^{3}} d x$;

$\int \frac{\operatorname {ch} x+2 \operatorname {sh} x-1}{\operatorname {sh} x(\operatorname {ch} x-3 \operatorname {sh} x-1)} d x$;

$\int \frac{\operatorname {sh} 2 x-2 \operatorname {sh} x}{\operatorname {sh}^{6}(x / 2)-\operatorname {sh}^{3} x} d x$.

$\int \sin^{5} x \sqrt[3]{\cos x} d x$;

$\int \frac{\cos^{3} x}{\sqrt[5]{\sin x}} d x$

$\int \frac{\sin^{3} x}{\cos x \sqrt[3]{\cos x}} d x$;

$\int \frac{d x}{\cos x \sqrt[3]{\sin^{2} x}} d x$;

$\int \operatorname {ch}^{3} x \sqrt[3]{\operatorname {sh}^{2} x} d x$;

$\int \frac{\operatorname {sh}^{3} x}{\sqrt[3]{\operatorname {ch}^{2} x}} d x$.

$\int \frac{d x}{\cos^{3} x \sqrt{\sin 2 x}}$;

$\int \frac{d x}{\cos x \sqrt{\sin^{3} 2 x}}$;

$\int \sqrt{\operatorname {tg} x} d x$;

$\int \sqrt{\operatorname {tg}^{3} x} d x$;

$\int \frac{\sqrt[3]{\operatorname {th}^{2} x}}{\operatorname {ch}^{4} x} d x$;

$\int \sqrt{\operatorname {th} x} d x$.

$\int \frac{(\cos ((x+\varphi ) / 2))^{n-1}}{(\sin ((x-\varphi ) / 2))^{n+1}} d x, n \in \mathbb {N}, \varphi \neq \frac{2 k+1}{2} \pi , k \in \mathbb {Z}$;

$\int \frac{(\operatorname {ch}((x-\varphi ) / 2))^{n-1}}{(\operatorname {sh}((x+\varphi ) / 2))^{n+1}} d x, n \in \mathbb {N}$.

$\int \frac{e^{x}+e^{3 x}}{1-e^{2 x}+e^{4 x}} d x$;

$\int \frac{d x}{2-e^{x}-e^{2 x}}$;

$\int \frac{d x}{1+e^{x}+e^{2 x}+e^{3 x}}$

$\int \frac{d x}{\left(e^{x}-1\right)^{4}}$;

$\int \frac{d x}{\left(e^{x-1}+1\right)^{2}-\left(e^{x+1}+1\right)^{2}}$

$\int \frac{d x}{\sqrt{1+e^{x}+e^{2 x}}}$;

$\int \sqrt{e^{2 x}+4 e^{x}-1} d x$;

$\int \frac{d x}{\sqrt{1+e^{x}}+\sqrt{1-e^{x}}}$.

$\int \left(x^{3}+x\right) e^{-x^{2}} d x$;

$\int \frac{x+\operatorname {sh} x}{\operatorname {ch} x-\operatorname {sh} x} d x$;

$\int \frac{x d x}{\sin^{2} x}$;

$\int \frac{x d x}{1+\sin x}$;

$\int x \sin x \cos 2 x d x$

$\int (\sin x-x)^{3} d x$

$\int x^{2} e^{x} \sin x d x$;

$\int e^{a x} \sin^{3} b x d x, a^{2}+b^{2} \neq 0$.

$\int \frac{\ln \left(1-x+x^{2}\right)}{x^{2}} d x$;

$\int e^{x} \ln \left(1+e^{-x}\right) d x$;

$\int \ln (\sqrt{x+1}+\sqrt{x-1}) d x$;

$\int \frac{(x \ln x)^{2}}{\sqrt{x}} d x$.

$\int \operatorname {arctg} \frac{1}{x-1} d x$;

$\int x \operatorname {arctg}(x+1) d x$;

$\int x \arccos (5 x-2) d x$;

$\int x^{7} \operatorname {arctg} x d x$

$\int \arcsin \sqrt{x} d x$;

$\int \sqrt{x} \operatorname {arctg} \sqrt{x} d x$;

$\int x \sqrt{1-x^{2}} \arccos x d x$;

$\int \frac{x \operatorname {arctg} x}{\sqrt{1+x^{2}}} d x$.

$\int e^{-x} \arcsin e^{x} d x$;

$\int e^{\arcsin x} d x ;$

$\int (2 x+1) e^{\operatorname {arctg} x} d x$;