سری اول : *

\[ \int {\sinh } axdx = {1 \over a}\cosh ax + C \] \[ \int {\cosh } axdx = {1 \over a}\sinh ax + C \] \[ \int {{{\sinh }^2}} axdx = {{\sinh 2ax} \over {4a}} - {x \over 2} + C \] \[ \int {{{\cosh }^2}} axdx = {{\sinh 2ax} \over {4a}} + {x \over 2} + C \] \[ \int {{{\sinh }^n}} axdx = {{{{\sinh }^{n - 1}}ax\cosh ax} \over {na}} - {{n - 1} \over n}\int {{{\sinh }^{n - 2}}} axdx \ \ \ \ ,n \ne 0 \] \[ \int {{{\cosh }^n}} axdx = {{{{\cosh }^{n - 1}}ax\sinh ax} \over {na}} + {{n - 1} \over n}\int {{{\cosh }^{n - 2}}} axdx \ \ \ \ ,n \ne 0 \] \[ \int x \sinh axdx = {x \over a}\cosh ax - {1 \over {{a^2}}}\sinh ax + C \] \[ \int x \cosh axdx = {x \over a}\sinh ax - {1 \over {{a^2}}}\cosh ax + C \] \[ \int {{x^n}} \sinh axdx = {{{x^n}} \over a}\cosh ax - {n \over a}\int {{x^{n - 1}}} \cosh axdx \] \[ \int {{x^n}} \cosh axdx = {{{x^n}} \over a}\sinh ax - {n \over a}\int {{x^{n - 1}}} \sinh axdx \] \[ \int {\tanh } axdx = {1 \over a}\ln (\cosh ax) + C \] \[ \int {\coth } axdx = {1 \over a}\ln |\sinh ax| + C \] \[ \int {{{\tanh }^2}} axdx = x - {1 \over a}\tanh ax + C \] \[ \int {{{\coth }^2}} axdx = x - {1 \over a}\coth ax + C \] \[ \int {{{\tanh }^n}} axdx = - {{{{\tanh }^{n - 1}}ax} \over {(n - 1)a}} + \int {{{\tanh }^{n - 2}}} axdx \ \ \ \ ,n \ne 1 \] \[ \int {{{\coth }^n}} axdx = - {{{{\coth }^{n - 1}}ax} \over {(n - 1)a}} + \int {{{\coth }^{n - 2}}} axdx \ \ \ \ ,n \ne 1 \] \[ \int {sechaxdx} = {1 \over a}{\sin ^{ - 1}}(\tanh ax) + C \] \[ \int {cschaxdx} = {1 \over a}\ln \left| {\tanh {{ax} \over 2}} \right| + C \] \[ \int {\rm{s}} {\rm{ec}}{{\rm{h}}^2}axdx = {1 \over a}\tanh ax + C \] \[ \int {\rm{c}} {\rm{sc}}{{\rm{h}}^2}axdx = - {1 \over a}\coth ax + C \] \[ \int {\rm{s}} {\rm{ec}}{{\rm{h}}^n}axdx = {{{\rm{sec}}{{\rm{h}}^{n - 2}}ax\tanh ax} \over {(n - 1)a}} + {{n - 2} \over {n - 1}}\int {\rm{s}} {\rm{ec}}{{\rm{h}}^{n - 2}}axdx \ \ \ \ ,n \ne 1 \] \[ \int {\rm{c}} {\rm{sc}}{{\rm{h}}^n}axdx = - {{{\rm{csc}}{{\rm{h}}^{n - 2}}ax\coth ax} \over {(n - 1)a}} - {{n - 2} \over {n - 1}}\int {\rm{c}} {\rm{sc}}{{\rm{h}}^{n - 2}}axdx \ \ \ \ ,n \ne 1 \] \[ \int {\rm{s}} {\rm{ec}}{{\rm{h}}^n}ax\tanh axdx = - {{{\rm{sec}}{{\rm{h}}^n}ax} \over {na}} + C \ \ \ \ ,n \ne 0 \] \[ \int {\rm{c}} {\rm{sc}}{{\rm{h}}^n}ax\coth axdx = - {{{\rm{csc}}{{\rm{h}}^n}ax} \over {na}} + C \ \ \ \ ,n \ne 0 \] \[ \int {{e^{ax}}} \sinh bxdx = {{{e^{ax}}} \over 2}\left[ {{{{e^{bx}}} \over {a + b}} - {{{e^{ - bx}}} \over {a - b}}} \right] + C \ \ \ \ ,{a^2} \ne {b^2} \] \[ \int {{e^{ax}}} \cosh bxdx = {{{e^{ax}}} \over 2}\left[ {{{{e^{bx}}} \over {a + b}} + {{{e^{ - bx}}} \over {a - b}}} \right] + C \ \ \ \ ,{a^2} \ne {b^2} \]

---

سری دوم : *

\[ \int \cosh ax\ dx =\frac{1}{a} \sinh ax \] \[ \int e^{ax} \cosh bx \ dx = \begin{cases} \displaystyle{\frac{e^{ax}}{a^2-b^2} }[ a \cosh bx - b \sinh bx ] & a\ne b \\ \displaystyle{\frac{e^{2ax}}{4a} + \frac{x}{2}} & a = b \end{cases} \] \[ \int \sinh ax\ dx = \frac{1}{a} \cosh ax \] \[ \int e^{ax} \sinh bx \ dx = \begin{cases} \displaystyle{\frac{e^{ax}}{a^2-b^2} }[ -b \cosh bx + a \sinh bx ] & a\ne b \\ \displaystyle{\frac{e^{2ax}}{4a} - \frac{x}{2}} & a = b \end{cases} \] \[ \int \tanh ax\hspace{1.5pt} dx =\frac{1}{a} \ln \cosh ax \] \[ \int e^{ax} \tanh bx\ dx = \begin{cases} \displaystyle{ \frac{ e^{(a+2b)x}}{(a+2b)} {_2F_1}\left[ 1+\frac{a}{2b},1,2+\frac{a}{2b}, -e^{2bx}\right] }& \\ \displaystyle{ \hspace{1cm}-\frac{1}{a}e^{ax}{_2F_1}\left[ 1, \frac{a}{2b},1+\frac{a}{2b}, -e^{2bx}\right] } & a\ne b \\ \displaystyle{\frac{e^{ax}-2\tan^{-1}[e^{ax}]}{a} } & a = b \end{cases} \] \[ \int \cos ax \cosh bx\ dx = \frac{1}{a^2 + b^2} \left[ a \sin ax \cosh bx + b \cos ax \sinh bx \right] \] \[ \int \cos ax \sinh bx\ dx = \frac{1}{a^2 + b^2} \left[ b \cos ax \cosh bx + a \sin ax \sinh bx \right] \] \[ \int \sin ax \cosh bx \ dx = \frac{1}{a^2 + b^2} \left[ -a \cos ax \cosh bx + b \sin ax \sinh bx \right] \] \[ \int \sin ax \sinh bx \ dx = \frac{1}{a^2 + b^2} \left[ b \cosh bx \sin ax - a \cos ax \sinh bx \right] \] \[ \int \sinh ax \cosh ax dx= \frac{1}{4a}\left[ -2ax + \sinh 2ax \right] \] \[ \int \sinh ax \cosh bx \ dx = \frac{1}{b^2-a^2}\left[ b \cosh bx \sinh ax - a \cosh ax \sinh bx \right] \]