List of integrals of hyperbolic functions explained

The following is a list of integrals (anti-derivative functions) of hyperbolic functions. For a complete list of integral functions, see list of integrals.

In all formulas the constant a is assumed to be nonzero, and Cdenotes the constant of integration.

Integrals involving only hyperbolic sine functions

\int\sinh axdx = (1)/(a) \cosh ax + C \int\sinh ^{2} axdx = (1)/(4 a) \sinh 2 ax - (x)/(2) + C \int\sinh ^{n} axdx = \begin{cases} (1)/(an) (\sinh ^{n-1} ax)(\cosh ax) - (n -1)/(n) \displaystyle\int\sinh ^{n-2} axdx, & n> 0 \\ (1)/(a(n +1 )) (\sinh ^{n+1} ax)(\cosh ax) - (n +2)/(n +1) \displaystyle\int\sinh ^{n+2} axdx, & n< 0 , n ≠ -1 \end{cases} \begin{align} \int (dx)/(\sinh ax) & = (1)/(ln \left|\tanh a) (ax)/(2) \right| + C \\ & = (1)/(ln \left| a) (\cosh ax +1)/(\sinh ax) \right| + C \\ & = (1)/(ln \left| a) (\sinh ax)/(\cosh ax +1) \right| + C \\ & = (1)/(ln \left| 2 a) (\cosh ax -1)/(\cosh ax +1) \right| + C \end{align} \int (dx)/(\sinh ^{n} ax) =- (\cosh ax)/(- a(n -1 )\sinh ^{n-1} ax) (n -2)/(\int n -1) (dx)/(\sinh ^{n-2} ax) (for n ≠ 1 ) \int x\sinh axdx = (1)/(a) x\cosh ax - (1)/(a ^{2}) \sinh ax + C \int (\sinh ax)(\sinh bx)dx = (1)/(a ^{2-b} ^{2}) ( a(\sinh bx)(\cosh ax) - b(\cosh bx)(\sinh ax) ) + C (for a ^{2 ≠} b ^{2)}

Integrals involving only hyperbolic cosine functions

\int\cosh axdx = (1)/(a) \sinh ax + C \int\cosh ^{2} axdx = (1)/(4 a) \sinh 2 ax + (x)/(2) + C \int\cosh ^{n} axdx = \begin{cases} (1)/(an) (\sinh ax)(\cosh ^{n-1} ax) + (n -1)/(n) \displaystyle\int\cosh ^{n-2} axdx, & n> 0 \\ - (1)/(a(n +1 )) (\sinh ax)(\cosh ^{n+1} ax) + (n +2)/(n +1) \displaystyle\int\cosh ^{n+2} axdx, & n< 0 , n ≠ -1 \end{cases} \begin{align} \int (dx)/(\cosh ax) & = (2)/(a) \arctan e ^{ax} + C \\ & = (1)/(a) \arctan (\sinh ax) + C \end{align} \int (dx)/(\cosh ^{n} ax) = (\sinh ax)/(+ a(n -1 )\cosh ^{n-1} ax) (n -2)/(\int n -1) (dx)/(\cosh ^{n-2} ax) (for n ≠ 1 ) \int x\cosh axdx = (1)/(a) x\sinh ax - (1)/(a ^{2}) \cosh ax + C \int x ^{2} \cosh axdx =- (2 x \cosh ax)/(a ^{2}) + \left( (x ^{2})/(+ a) (2)/(a ^{3}) \right) \sinh ax + C \int (\cosh ax)(\cosh bx)dx = (1)/(a ^{2-b} ^{2}) ( a(\sinh ax)(\cosh bx) - b(\sinh bx)(\cosh ax) ) + C (for a ^{2 ≠} b ^{2)} \int (dx)/(1+ \cosh(ax)) = (2)/(a) (1)/(1+ e ^{-ax}) + C or (2)/(a) times The Logistic Function

Other integrals

Integrals of hyperbolic tangent, cotangent, secant, cosecant functions

\int \tanh x dx = ln \cosh x + C \int\tanh ^{2} axdx = x - (\tanh ax)/(a) + C \int \tanh ^{n} axdx =- (1)/(a(n -1 )) \tanh ^{n-1} ax + \int\tanh ^{n-2} axdx (for n ≠ 1 ) \int \coth x dx = ln | \sinh x | + C , for x ≠ 0 \int \coth ^{n} axdx =- (1)/(a(n -1 )) \coth ^{n-1} ax + \int\coth ^{n-2} axdx (for n ≠ 1 ) \int \operatorname{sech}x dx = \arctan(\sinh x) + C \int \operatorname{csch}x dx = ln \left| \tanh {x \over 2 }\right| + C = ln \left|\coth{x} - \operatorname{csch}{x}\right| + C, for x ≠ 0

Integrals involving hyperbolic sine and cosine functions

\int (\cosh ax)(\sinh bx)dx = (1)/(a ^{2-b} ^{2}) ( a(\sinh ax)(\sinh bx) - b(\cosh ax)(\cosh bx) ) + C (for a ^{2 ≠} b ^{2)} \begin{align} \int (\cosh ^{n} ax)/(\sinh ^{m} ax) dx & = (\cosh ^{n-1} ax)/(a(n - m)\sinh ^{m-1} ax) + (n -1)/(\int n - m) (\cosh ^{n-2} ax)/(\sinh ^{m} ax) dx (for m ≠ n ) \\ & =- (\cosh ^{n+1} ax)/(a(m -1 )\sinh ^{m-1} ax) + (n - m +2)/(\int m -1) (\cosh ^{n} ax)/(\sinh ^{m-2} ax) dx (for m ≠ 1 ) \\ & =- (\cosh ^{n-1} ax)/(a(m -1 )\sinh ^{m-1} ax) + (n -1)/(\int m -1) (\cosh ^{n-2} ax)/(\sinh ^{m-2} ax) dx (for m ≠ 1 ) \end{align} \begin{align} \int (\sinh ^{m} ax)/(\cosh ^{n} ax) dx & = (\sinh ^{m-1} ax)/(a(m - n)\cosh ^{n-1} ax) + (m -1)/(\int n - m) (\sinh ^{m-2} ax)/(\cosh ^{n} ax) dx (for m ≠ n ) \\ & = (\sinh ^{m+1} ax)/(a(n -1 )\cosh ^{n-1} ax) + (m - n +2)/(\int n -1) (\sinh ^{m} ax)/(\cosh ^{n-2} ax) dx (for n ≠ 1 ) \\ & =- (\sinh ^{m-1} ax)/(a(n -1 )\cosh ^{n-1} ax) + (m -1)/(\int n -1) (\sinh ^{m} ax)/(\cosh ^{n-2} ax) dx (for n ≠ 1 ) \end{align}

Integrals involving hyperbolic and trigonometric functions

\int \sinh (ax + b)\sin (cx + d)dx = (a)/(\cosh(ax + b)\sin(cx + d) - a ^{2+c} ^{2}) (c)/(a ^{2+c} ^{2}) \sinh(ax + b)\cos(cx + d) + C \int \sinh (ax + b)\cos (cx + d)dx = (a)/(\cosh(ax + b)\cos(cx + d) + a ^{2+c} ^{2}) (c)/(a ^{2+c} ^{2}) \sinh(ax + b)\sin(cx + d) + C \int \cosh (ax + b)\sin (cx + d)dx = (a)/(\sinh(ax + b)\sin(cx + d) - a ^{2+c} ^{2}) (c)/(a ^{2+c} ^{2}) \cosh(ax + b)\cos(cx + d) + C \int \cosh (ax + b)\cos (cx + d)dx = (a)/(\sinh(ax + b)\cos(cx + d) + a ^{2+c} ^{2}) (c)/(a ^{2+c} ^{2}) \cosh(ax + b)\sin(cx + d) + C