{"version":9,"randomSeed":"dc0d2658aaeed7efd83465fef13c92fc","graph":{"viewport":{"xmin":-10.447568884322465,"ymin":-20.031254504586116,"xmax":10.802431115677543,"ymax":27.16187415944868}},"expressions":{"list":[{"type":"expression","id":"1","color":"#c74440","latex":"\\int_{ }^{ }\\operatorname{csch}^{2}udu=\\int_{ }^{ }\\frac{1}{\\sinh^{2}u}du=\\int_{ }^{ }\\frac{1}{\\left(\\frac{e^{u}-e^{-u}}{2}\\right)^{2}}du=\\int_{ }^{ }\\frac{4}{\\left(e^{u}-e^{-u}\\right)^{2}}du"},{"type":"expression","id":"21","color":"#c74440","latex":"w=e^{u},\\ \\int_{ }^{ }\\frac{4}{\\left(e^{u}-e^{-u}\\right)^{2}}du=4\\int_{ }^{ }\\frac{1}{\\left(w-\\frac{1}{w}\\right)^{2}}du=4\\int_{ }^{ }\\frac{w^{2}}{\\left(w^{2}-1\\right)^{2}}du"},{"type":"expression","id":"22","color":"#2d70b3","latex":"p=w^{2}=e^{2u},\\ \\ \\int_{ }^{ }\\frac{w^{2}}{\\left(w^{2}-1\\right)^{2}}du=\\int_{ }^{ }\\frac{p}{\\left(p-1\\right)^{2}}\\cdot\\frac{dp}{\\frac{dp}{du}}=\\int_{ }^{ }\\frac{p}{\\left(p-1\\right)^{2}}\\cdot\\frac{dp}{2e^{2u}}=\\int_{ }^{ }\\frac{p}{\\left(p-1\\right)^{2}}\\cdot\\frac{dp}{2p}=\\frac{1}{2}\\int_{ }^{ }\\frac{1}{\\left(p-1\\right)^{2}}dp","labelSize":"medium"},{"type":"expression","id":"23","color":"#388c46","latex":"s=p-1,\\ \\ \\ \\int_{ }^{ }\\frac{1}{\\left(p-1\\right)^{2}}dp=\\int_{ }^{ }\\frac{1}{s^{2}}ds=-\\frac{1}{s}+C=-\\frac{1}{p-1}+C=-\\frac{1}{e^{2u}-1}+C"},{"type":"expression","id":"24","color":"#6042a6","latex":"\\int_{ }^{ }\\operatorname{csch}^{2}udu=4\\int_{ }^{ }\\frac{1}{\\left(w-\\frac{1}{w}\\right)^{2}}du=4\\cdot\\frac{1}{2}\\int_{ }^{ }\\frac{1}{\\left(p-1\\right)^{2}}dp=4\\cdot\\frac{1}{2}\\cdot\\left(-\\frac{1}{e^{2u}-1}+C\\right)=-\\frac{2}{e^{2u}-1}+C"},{"type":"expression","id":"29","color":"#6042a6","latex":"-\\coth x=-\\frac{\\cosh x}{\\sinh x}=-\\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}=-\\frac{e^{2x}+1}{e^{2x}-1}=-\\frac{2}{e^{2x}-1}-1"}]}}