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Solving integrals/ 1/ch(x)^3

Integral of 1/ch(x)^3 dx

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The solution

You have entered [src] 
 oo            
  /            
 |             
 |     1       
 |  -------- dx
 |      3      
 |  cosh (x)   
 |             
/              
0
$$\int\limits_{0}^{\infty} \frac{1}{\cosh^{3}{\left(x \right)}}\, dx$$ 
Integral(1/(cosh(x)^3), (x, 0, oo))
The answer (Indefinite) [src] 
  /                            /    /x\\                      /x\                         3/x\                 4/x\     /    /x\\           2/x\     /    /x\\
 |                         atan|tanh|-||                  tanh|-|                     tanh |-|             tanh |-|*atan|tanh|-||     2*tanh |-|*atan|tanh|-||
 |    1                        \    \2//                      \2/                          \2/                  \2/     \    \2//            \2/     \    \2//
 | -------- dx = C + ------------------------- + ------------------------- - ------------------------- + ------------------------- + -------------------------
 |     3                     4/x\         2/x\           4/x\         2/x\           4/x\         2/x\           4/x\         2/x\           4/x\         2/x\
 | cosh (x)          1 + tanh |-| + 2*tanh |-|   1 + tanh |-| + 2*tanh |-|   1 + tanh |-| + 2*tanh |-|   1 + tanh |-| + 2*tanh |-|   1 + tanh |-| + 2*tanh |-|
 |                            \2/          \2/            \2/          \2/            \2/          \2/            \2/          \2/            \2/          \2/
/
$$\int \frac{1}{\cosh^{3}{\left(x \right)}}\, dx = C + \frac{\tanh^{4}{\left(\frac{x}{2} \right)} \operatorname{atan}{\left(\tanh{\left(\frac{x}{2} \right)} \right)}}{\tanh^{4}{\left(\frac{x}{2} \right)} + 2 \tanh^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{\tanh^{3}{\left(\frac{x}{2} \right)}}{\tanh^{4}{\left(\frac{x}{2} \right)} + 2 \tanh^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \tanh^{2}{\left(\frac{x}{2} \right)} \operatorname{atan}{\left(\tanh{\left(\frac{x}{2} \right)} \right)}}{\tanh^{4}{\left(\frac{x}{2} \right)} + 2 \tanh^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{\tanh{\left(\frac{x}{2} \right)}}{\tanh^{4}{\left(\frac{x}{2} \right)} + 2 \tanh^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{\operatorname{atan}{\left(\tanh{\left(\frac{x}{2} \right)} \right)}}{\tanh^{4}{\left(\frac{x}{2} \right)} + 2 \tanh^{2}{\left(\frac{x}{2} \right)} + 1}$$
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The graph
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The answer [src] 
pi
--
4
$$\frac{\pi}{4}$$ 
=
= 
pi
--
4
$$\frac{\pi}{4}$$ 
pi/4

    Use the examples entering the upper and lower limits of integration.

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