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Questa pagina contiene una tavola di integrali indefiniti di funzioni iperboliche. Per altri integrali vedi Integrale § Tavole di integrali.
![{\displaystyle \int \sinh cx\,\mathrm {d} x={\frac {1}{c}}\cosh cx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/058a32f5d1b90276001a52a82183b35db193039d)
![{\displaystyle \int \cosh cx\,\mathrm {d} x={\frac {1}{c}}\sinh cx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c708b0c1820e344d27f2ec9a674e83ef381770)
![{\displaystyle \int \sinh ^{2}cx\,\mathrm {d} x={\frac {1}{4c}}\sinh 2cx-{\frac {x}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7fec2edd6f55b655de004ee6d75dd675dcb1fbd)
![{\displaystyle \int \cosh ^{2}cx\,\mathrm {d} x={\frac {1}{4c}}\sinh 2cx+{\frac {x}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68e2d806b5a2d683270cd11682e0a29a90a43061)
![{\displaystyle \int \sinh ^{n}cx\,\mathrm {d} x={\frac {1}{cn}}\sinh ^{n-1}cx\cosh cx-{\frac {n-1}{n}}\int \sinh ^{n-2}cx\,\mathrm {d} x\qquad {\mbox{(per }}n>0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a202bbeb7e5541dd06c087bfd74c1b9c67c542a)
- anche:
![{\displaystyle \int \sinh ^{n}cx\,\mathrm {d} x={\frac {1}{c(n+1)}}\sinh ^{n+1}cx\cosh cx-{\frac {n+2}{n+1}}\int \sinh ^{n+2}cx\,\mathrm {d} x\qquad {\mbox{(per }}n<0{\mbox{, }}n\neq -1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59fc58128db8ecfafc881bc6b8edf142beaa8964)
![{\displaystyle \int \cosh ^{n}cx\,\mathrm {d} x={\frac {1}{cn}}\sinh cx\cosh ^{n-1}cx+{\frac {n-1}{n}}\int \cosh ^{n-2}cx\,\mathrm {d} x\qquad {\mbox{(per }}n>0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ee9c5013257458a64de3e3d79480c05dbe4d2a9)
- anche:
![{\displaystyle \int \cosh ^{n}cx\,\mathrm {d} x=-{\frac {1}{c(n+1)}}\sinh cx\cosh ^{n+1}cx-{\frac {n+2}{n+1}}\int \cosh ^{n+2}cx\,\mathrm {d} x\qquad {\mbox{(per }}n<0{\mbox{, }}n\neq -1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e74fa99e6764e0dd5589dfaeabaeaa525dbf4c1)
![{\displaystyle \int {\frac {\mathrm {d} x}{\sinh cx}}={\frac {1}{c}}\log \left|\tanh {\frac {cx}{2}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73b3fc6ca48f06d122a0a7cf8e20d19cea429dd2)
- anche:
![{\displaystyle \int {\frac {\mathrm {d} x}{\sinh cx}}={\frac {1}{c}}\log \left|{\frac {\cosh cx-1}{\sinh cx}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70966f1b52177ee5a8e92baae59684b3d7acd835)
- anche:
![{\displaystyle \int {\frac {\mathrm {d} x}{\sinh cx}}={\frac {1}{c}}\log \left|{\frac {\sinh cx}{\cosh cx+1}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8280a544638f88c463dcfe4a966638cabdfbb67)
- anche:
![{\displaystyle \int {\frac {\mathrm {d} x}{\sinh cx}}={\frac {1}{c}}\log \left|{\frac {\cosh cx-1}{\cosh cx+1}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a23478b010e3d619a1040535c8a0e16c08b50f9)
![{\displaystyle \int {\frac {\mathrm {d} x}{\cosh cx}}={\frac {2}{c}}\arctan e^{cx}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/535a6a11312b0bbc20321a12afa84892b36a6868)
![{\displaystyle \int {\frac {\mathrm {d} x}{\sinh ^{n}cx}}={\frac {\cosh cx}{c(n-1)\sinh ^{n-1}cx}}-{\frac {n-2}{n-1}}\int {\frac {\mathrm {d} x}{\sinh ^{n-2}cx}}\qquad {\mbox{(per }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c91918d6852dd966c41964df2ee9ba72727fcf74)
![{\displaystyle \int {\frac {\mathrm {d} x}{\cosh ^{n}cx}}={\frac {\sinh cx}{c(n-1)\cosh ^{n-1}cx}}+{\frac {n-2}{n-1}}\int {\frac {\mathrm {d} x}{\cosh ^{n-2}cx}}\qquad {\mbox{(per }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d9a26c32475841490dc443eb521fa4d020021fb)
![{\displaystyle \int {\frac {\cosh ^{n}cx}{\sinh ^{m}cx}}\mathrm {d} x={\frac {\cosh ^{n-1}cx}{c(n-m)\sinh ^{m-1}cx}}+{\frac {n-1}{n-m}}\int {\frac {\cosh ^{n-2}cx}{\sinh ^{m}cx}}\mathrm {d} x\qquad {\mbox{(per }}m\neq n{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fe5aacc2096f8c3c0e101539354593f53563f07)
- anche:
![{\displaystyle \int {\frac {\cosh ^{n}cx}{\sinh ^{m}cx}}\mathrm {d} x=-{\frac {\cosh ^{n+1}cx}{c(m-1)\sinh ^{m-1}cx}}+{\frac {n-m+2}{m-1}}\int {\frac {\cosh ^{n}cx}{\sinh ^{m-2}cx}}\mathrm {d} x\qquad {\mbox{(per }}m\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92af88a6572908e17b43276476078078581d5dae)
- anche:
![{\displaystyle \int {\frac {\cosh ^{n}cx}{\sinh ^{m}cx}}\mathrm {d} x=-{\frac {\cosh ^{n-1}cx}{c(m-1)\sinh ^{m-1}cx}}+{\frac {n-1}{m-1}}\int {\frac {\cosh ^{n-2}cx}{\sinh ^{m-2}cx}}\mathrm {d} x\qquad {\mbox{(per }}m\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1622987bad01151d4fd76b84acf1c31bf7194de3)
![{\displaystyle \int {\frac {\sinh ^{m}cx}{\cosh ^{n}cx}}\mathrm {d} x={\frac {\sinh ^{m-1}cx}{c(m-n)\cosh ^{n-1}cx}}+{\frac {m-1}{m-n}}\int {\frac {\sinh ^{m-2}cx}{\cosh ^{n}cx}}\mathrm {d} x\qquad {\mbox{(per }}m\neq n{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43e9847935a9b268096a2aa3f91eb23093ecc9bf)
- anche:
![{\displaystyle \int {\frac {\sinh ^{m}cx}{\cosh ^{n}cx}}\mathrm {d} x={\frac {\sinh ^{m+1}cx}{c(n-1)\cosh ^{n-1}cx}}+{\frac {m-n+2}{n-1}}\int {\frac {\sinh ^{m}cx}{\cosh ^{n-2}cx}}\mathrm {d} x\qquad {\mbox{(per }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f73b9b9cb9f29939f09de514094f2e8f7b595aea)
- anche:
![{\displaystyle \int {\frac {\sinh ^{m}cx}{\cosh ^{n}cx}}\mathrm {d} x=-{\frac {\sinh ^{m-1}cx}{c(n-1)\cosh ^{n-1}cx}}+{\frac {m-1}{n-1}}\int {\frac {\sinh ^{m-2}cx}{\cosh ^{n-2}cx}}\mathrm {d} x\qquad {\mbox{(per }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/081129f947da295b89b7bb8772d6d4e8105df2d1)
![{\displaystyle \int x\sinh cx\,\mathrm {d} x={\frac {1}{c}}x\cosh cx-{\frac {1}{c^{2}}}\sinh cx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9605a33f4f5d2fe4f7c75aa77f0e7db3fcda0899)
![{\displaystyle \int x\cosh cx\,\mathrm {d} x={\frac {1}{c}}x\sinh cx-{\frac {1}{c^{2}}}\cosh cx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b939eb8815ed66adbd9c00386ab36d1c1e8184ed)
![{\displaystyle \int \tanh cx\,\mathrm {d} x={\frac {1}{c}}\log |\cosh cx|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5895481828f6b61bebf4a9c2829929b1947a57cb)
![{\displaystyle \int \coth cx\,\mathrm {d} x={\frac {1}{c}}\log |\sinh cx|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b295a715fffa8cd6362ac6a76377e68f9e9ad65b)
![{\displaystyle \int \tanh ^{n}cx\,\mathrm {d} x=-{\frac {1}{c(n-1)}}\tanh ^{n-1}cx+\int \tanh ^{n-2}cx\,\mathrm {d} x\qquad {\mbox{(per }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a546682de27abcb1219d183f97f8b834e8695b71)
![{\displaystyle \int \coth ^{n}cx\,\mathrm {d} x=-{\frac {1}{c(n-1)}}\coth ^{n-1}cx+\int \coth ^{n-2}cx\,\mathrm {d} x\qquad {\mbox{(per }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/236866496693e572337a26a3c9e1d5034684d44c)
![{\displaystyle \int \sinh bx\sinh cx\,\mathrm {d} x={\frac {1}{b^{2}-c^{2}}}(b\sinh cx\cosh bx-c\cosh cx\sinh bx)\qquad {\mbox{(per }}b^{2}\neq c^{2}{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73798016730dced8aa274a7889fba16e10edc087)
![{\displaystyle \int \cosh bx\cosh cx\,\mathrm {d} x={\frac {1}{b^{2}-c^{2}}}(b\sinh bx\cosh cx-c\sinh cx\cosh bx)\qquad {\mbox{(per }}b^{2}\neq c^{2}{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af8c4024c7bfc8c4dd2d435ee0b810774bebe4cc)
![{\displaystyle \int \cosh bx\sinh cx\,\mathrm {d} x={\frac {1}{b^{2}-c^{2}}}(b\sinh bx\sinh cx-c\cosh bx\cosh cx)\qquad {\mbox{(per }}b^{2}\neq c^{2}{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82b1b9906b791ca963f7252154cba002c9dc4100)
![{\displaystyle \int \sinh(ax+b)\sin(cx+d)\,\mathrm {d} x={\frac {a}{a^{2}+c^{2}}}\cosh(ax+b)\sin(cx+d)-{\frac {c}{a^{2}+c^{2}}}\sinh(ax+b)\cos(cx+d)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20d1208d652ba8ff28eeed1ea73fb47c170b44dd)
![{\displaystyle \int \sinh(ax+b)\cos(cx+d)\,\mathrm {d} x={\frac {a}{a^{2}+c^{2}}}\cosh(ax+b)\cos(cx+d)+{\frac {c}{a^{2}+c^{2}}}\sinh(ax+b)\sin(cx+d)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36731f5d94b4fa76ebe265efe55e20d70e9c3dfe)
![{\displaystyle \int \cosh(ax+b)\sin(cx+d)\,\mathrm {d} x={\frac {a}{a^{2}+c^{2}}}\sinh(ax+b)\sin(cx+d)-{\frac {c}{a^{2}+c^{2}}}\cosh(ax+b)\cos(cx+d)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd3f1a241a16d6b41074ea4f9de3f0585f9008ae)
![{\displaystyle \int \cosh(ax+b)\cos(cx+d)\,\mathrm {d} x={\frac {a}{a^{2}+c^{2}}}\sinh(ax+b)\cos(cx+d)+{\frac {c}{a^{2}+c^{2}}}\cosh(ax+b)\sin(cx+d)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9375b67236bec43c5dc350867fa052aad216f906)
- Murray R. Spiegel, Manuale di matematica, Etas Libri, 1974, pp. 86-92.