La siguiente es una lista de integrales ( funciones antiderivadas ) de funciones hiperbólicas . Para obtener una lista completa de funciones integrales, consulte lista de integrales .
En todas las fórmulas se supone que la constante a es distinta de cero y C
denota la constante de integración .
Integrales que involucran solo funciones seno hiperbólicas
![{\displaystyle \int \sinh ax\,dx={\frac {1}{a}}\cosh ax+C}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int \sinh ^{2}ax\,dx={\frac {1}{4a}}\sinh 2ax-{\frac {x}{2}}+C}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int \sinh ^{n}ax\,dx={\begin{cases}{\frac {1}{an}}(\sinh ^{n-1}ax)(\cosh ax)-{ \frac {n-1}{n}}\displaystyle \int \sinh ^{n-2}ax\,dx,&n>0\\{\frac {1}{a(n+1)}}(\ sinh ^{n+1}ax)(\cosh ax)-{\frac {n+2}{n+1}}\displaystyle \int \sinh ^{n+2}ax\,dx,&n<0, n\neq -1\end{casos}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle {\begin{aligned}\int {\frac {dx}{\sinh ax}}&={\frac {1}{a}}\ln \left|\tanh {\frac {ax}{2 }}\right|+C\\&={\frac {1}{a}}\ln \left|{\frac {\cosh ax+1}{\sinh ax}}\right|+C\\& ={\frac {1}{a}}\ln \left|{\frac {\sinh ax}{\cosh ax+1}}\right|+C\\&={\frac {1}{2a} }\ln \left|{\frac {\cosh ax-1}{\cosh ax+1}}\right|+C\end{aligned}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {dx}{\sinh ^{n}ax}}=-{\frac {\cosh hacha}{a(n-1)\sinh ^{n-1}ax}}- {\frac {n-2}{n-1}}\int {\frac {dx}{\sinh ^{n-2}ax}}\qquad {\mbox{(para }}n\neq 1{\ mbox{)}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int x\sinh ax\,dx={\frac {1}{a}}x\cosh ax-{\frac {1}{a^{2}}}\sinh ax+C}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int (\sinh ax)(\sinh bx)\,dx={\frac {1}{a^{2}-b^{2}}}{\big (}a(\sinh bx) (\cosh ax)-b(\cosh bx)(\sinh ax){\big )}+C\qquad {\mbox{(para }}a^{2}\neq b^{2}{\mbox{ )}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Integrales que involucran solo funciones cosenos hiperbólicas
![{\displaystyle \int \cosh ax\,dx={\frac {1}{a}}\sinh ax+C}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int \cosh ^{2}ax\,dx={\frac {1}{4a}}\sinh 2ax+{\frac {x}{2}}+C}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int \cosh ^{n}ax\,dx={\begin{casos}{\frac {1}{an}}(\sinh ax)(\cosh ^{n-1}ax)+{ \frac {n-1}{n}}\displaystyle \int \cosh ^{n-2}ax\,dx,&n>0\\-{\frac {1}{a(n+1)}}( \sinh ax)(\cosh ^{n+1}ax)+{\frac {n+2}{n+1}}\displaystyle \int \cosh ^{n+2}ax\,dx,&n<0 ,n\neq -1\end{casos}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle {\begin{aligned}\int {\frac {dx}{\cosh ax}}&={\frac {2}{a}}\arctan e^{ax}+C\\&={\ frac {1}{a}}\arctan(\sinh ax)+C\end{aligned}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {dx}{\cosh ^{n}ax}}={\frac {\sinh hacha}{a(n-1)\cosh ^{n-1}ax}}+{ \frac {n-2}{n-1}}\int {\frac {dx}{\cosh ^{n-2}ax}}\qquad {\mbox{(para }}n\neq 1{\mbox {)}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int x\cosh ax\,dx={\frac {1}{a}}x\sinh ax-{\frac {1}{a^{2}}}\cosh ax+C}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int x^{2}\cosh hacha\,dx=-{\frac {2x\cosh hacha}{a^{2}}}+\left({\frac {x^{2}}{ a}}+{\frac {2}{a^{3}}}\right)\sinh ax+C}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int (\cosh ax)(\cosh bx)\,dx={\frac {1}{a^{2}-b^{2}}}{\big (}a(\sinh ax) (\cosh bx)-b(\sinh bx)(\cosh ax){\big )}+C\qquad {\mbox{(para }}a^{2}\neq b^{2}{\mbox{ )}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
o veces La Función Logística![{\displaystyle {\frac {2}{a}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Otras integrales
Integrales de funciones hiperbólicas tangente, cotangente, secante y cosecante
![{\displaystyle \int \tanh x\,dx=\ln \cosh x+C}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int \tanh ^{2}ax\,dx=x-{\frac {\tanh ax}{a}}+C}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int \tanh ^{n}ax\,dx=-{\frac {1}{a(n-1)}}\tanh ^{n-1}ax+\int \tanh ^{n-2 }ax\,dx\qquad {\mbox{(para }}n\neq 1{\mbox{)}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int \coth x\,dx=\ln |\sinh x|+C,{\text{ para }}x\neq 0}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int \coth ^{n}ax\,dx=-{\frac {1}{a(n-1)}}\coth ^{n-1}ax+\int \coth ^{n-2 }ax\,dx\qquad {\mbox{(para }}n\neq 1{\mbox{)}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int \operatorname {sech} \,x\,dx=\arctan \,(\sinh x)+C}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int \operatorname {csch} \,x\,dx=\ln \left|\tanh {x \over 2}\right|+C=\ln \left|\coth {x}-\operatorname { csch} {x}\right|+C,{\text{ para }}x\neq 0}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Integrales que involucran funciones hiperbólicas seno y coseno
![{\displaystyle \int (\cosh ax)(\sinh bx)\,dx={\frac {1}{a^{2}-b^{2}}}{\big (}a(\sinh ax) (\sinh bx)-b(\cosh ax)(\cosh bx){\big )}+C\qquad {\mbox{(para }}a^{2}\neq b^{2}{\mbox{ )}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle {\begin{aligned}\int {\frac {\cosh ^{n}ax}{\sinh ^{m}ax}}\,dx&={\frac {\cosh ^{n-1}ax }{a(nm)\sinh ^{m-1}ax}}+{\frac {n-1}{nm}}\int {\frac {\cosh ^{n-2}ax}{\sinh ^ {m}ax}}\,dx\qquad {\mbox{(para }}m\neq n{\mbox{)}}\\&=-{\frac {\cosh ^{n+1}ax}{ a(m-1)\sinh ^{m-1}ax}}+{\frac {n-m+2}{m-1}}\int {\frac {\cosh ^{n}ax}{\ sinh ^{m-2}ax}}\,dx\qquad {\mbox{(para }}m\neq 1{\mbox{)}}\\&=-{\frac {\cosh ^{n-1 }ax}{a(m-1)\sinh ^{m-1}ax}}+{\frac {n-1}{m-1}}\int {\frac {\cosh ^{n-2} hacha}{\sinh ^{m-2}ax}}\,dx\qquad {\mbox{(para }}m\neq 1{\mbox{)}}\end{aligned}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle {\begin{aligned}\int {\frac {\sinh ^{m}ax}{\cosh ^{n}ax}}\,dx&={\frac {\sinh ^{m-1}ax }{a(mn)\cosh ^{n-1}ax}}+{\frac {m-1}{nm}}\int {\frac {\sinh ^{m-2}ax}{\cosh ^ {n}ax}}\,dx\qquad {\mbox{(para }}m\neq n{\mbox{)}}\\&={\frac {\sinh ^{m+1}ax}{a (n-1)\cosh ^{n-1}ax}}+{\frac {m-n+2}{n-1}}\int {\frac {\sinh ^{m}ax}{\cosh ^{n-2}ax}}\,dx\qquad {\mbox{(para }}n\neq 1{\mbox{)}}\\&=-{\frac {\sinh ^{m-1} hacha}{a(n-1)\cosh ^{n-1}ax}}+{\frac {m-1}{n-1}}\int {\frac {\sinh ^{m-2}hacha }{\cosh ^{n-2}ax}}\,dx\qquad {\mbox{(para }}n\neq 1{\mbox{)}}\end{aligned}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Integrales que involucran funciones hiperbólicas y trigonométricas.
![{\displaystyle \int \sinh(ax+b)\sin(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\cosh(ax+b) \sin(cx+d)-{\frac {c}{a^{2}+c^{2}}}\sinh(ax+b)\cos(cx+d)+C}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int \sinh(ax+b)\cos(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\cosh(ax+b) \cos(cx+d)+{\frac {c}{a^{2}+c^{2}}}\sinh(ax+b)\sin(cx+d)+C}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int \cosh(ax+b)\sin(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\sinh(ax+b) \sin(cx+d)-{\frac {c}{a^{2}+c^{2}}}\cosh(ax+b)\cos(cx+d)+C}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int \cosh(ax+b)\cos(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\sinh(ax+b) \cos(cx+d)+{\frac {c}{a^{2}+c^{2}}}\cosh(ax+b)\sin(cx+d)+C}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)