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Integral of 1/((chx)^2+(shx)^2) dx

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The solution

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  1                       
  /                       
 |                        
 |           1            
 |  ------------------- dx
 |      2          2      
 |  cosh (x) + sinh (x)   
 |                        
/                         
0                         
$$\int\limits_{0}^{1} \frac{1}{\sinh^{2}{\left(x \right)} + \cosh^{2}{\left(x \right)}}\, dx$$
Integral(1/(cosh(x)^2 + sinh(x)^2), (x, 0, 1))
The answer (Indefinite) [src]
                                                 /        /x\     \                                 /        /x\     \                                                  /        /x\     \                                                 /        /x\     \
                                                 |    tanh|-|     |                                 |    tanh|-|     |               _____________    _____________     |    tanh|-|     |              _____________    _____________     |    tanh|-|     |
                                                 |        \2/     |                         ___     |        \2/     |              /         ___    /         ___      |        \2/     |       ___   /         ___    /         ___      |        \2/     |
                                           3*atan|----------------|                     2*\/ 2 *atan|----------------|          3*\/  3 - 2*\/ 2  *\/  3 + 2*\/ 2  *atan|----------------|   2*\/ 2 *\/  3 - 2*\/ 2  *\/  3 + 2*\/ 2  *atan|----------------|
  /                                              |   _____________|                                 |   _____________|                                                  |   _____________|                                                 |   _____________|
 |                                               |  /         ___ |                                 |  /         ___ |                                                  |  /         ___ |                                                 |  /         ___ |
 |          1                                    \\/  3 - 2*\/ 2  /                                 \\/  3 - 2*\/ 2  /                                                  \\/  3 + 2*\/ 2  /                                                 \\/  3 + 2*\/ 2  /
 | ------------------- dx = C + --------------------------------------------- + --------------------------------------------- - ---------------------------------------------------------- - ----------------------------------------------------------------
 |     2          2                  _____________              _____________        _____________              _____________              _____________              _____________                        _____________              _____________          
 | cosh (x) + sinh (x)              /         ___        ___   /         ___        /         ___        ___   /         ___              /         ___        ___   /         ___                        /         ___        ___   /         ___           
 |                              7*\/  3 - 2*\/ 2   + 5*\/ 2 *\/  3 - 2*\/ 2     7*\/  3 - 2*\/ 2   + 5*\/ 2 *\/  3 - 2*\/ 2           7*\/  3 - 2*\/ 2   + 5*\/ 2 *\/  3 - 2*\/ 2                     7*\/  3 - 2*\/ 2   + 5*\/ 2 *\/  3 - 2*\/ 2            
/                                                                                                                                                                                                                                                            
$$\int \frac{1}{\sinh^{2}{\left(x \right)} + \cosh^{2}{\left(x \right)}}\, dx = C + \frac{2 \sqrt{2} \operatorname{atan}{\left(\frac{\tanh{\left(\frac{x}{2} \right)}}{\sqrt{3 - 2 \sqrt{2}}} \right)}}{7 \sqrt{3 - 2 \sqrt{2}} + 5 \sqrt{2} \sqrt{3 - 2 \sqrt{2}}} + \frac{3 \operatorname{atan}{\left(\frac{\tanh{\left(\frac{x}{2} \right)}}{\sqrt{3 - 2 \sqrt{2}}} \right)}}{7 \sqrt{3 - 2 \sqrt{2}} + 5 \sqrt{2} \sqrt{3 - 2 \sqrt{2}}} - \frac{3 \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} \operatorname{atan}{\left(\frac{\tanh{\left(\frac{x}{2} \right)}}{\sqrt{2 \sqrt{2} + 3}} \right)}}{7 \sqrt{3 - 2 \sqrt{2}} + 5 \sqrt{2} \sqrt{3 - 2 \sqrt{2}}} - \frac{2 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} \operatorname{atan}{\left(\frac{\tanh{\left(\frac{x}{2} \right)}}{\sqrt{2 \sqrt{2} + 3}} \right)}}{7 \sqrt{3 - 2 \sqrt{2}} + 5 \sqrt{2} \sqrt{3 - 2 \sqrt{2}}}$$
The graph
The answer [src]
                                                                                                     _____________    _____________                                     _____________    _____________                       
                 /   tanh(1/2)    \                         ___     /   tanh(1/2)    \              /         ___    /         ___      /   tanh(1/2)    \       ___   /         ___    /         ___      /   tanh(1/2)    \
           3*atan|----------------|                     2*\/ 2 *atan|----------------|          3*\/  3 - 2*\/ 2  *\/  3 + 2*\/ 2  *atan|----------------|   2*\/ 2 *\/  3 - 2*\/ 2  *\/  3 + 2*\/ 2  *atan|----------------|
                 |   _____________|                                 |   _____________|                                                  |   _____________|                                                 |   _____________|
                 |  /         ___ |                                 |  /         ___ |                                                  |  /         ___ |                                                 |  /         ___ |
                 \\/  3 - 2*\/ 2  /                                 \\/  3 - 2*\/ 2  /                                                  \\/  3 + 2*\/ 2  /                                                 \\/  3 + 2*\/ 2  /
--------------------------------------------- + --------------------------------------------- - ---------------------------------------------------------- - ----------------------------------------------------------------
     _____________              _____________        _____________              _____________              _____________              _____________                        _____________              _____________          
    /         ___        ___   /         ___        /         ___        ___   /         ___              /         ___        ___   /         ___                        /         ___        ___   /         ___           
7*\/  3 - 2*\/ 2   + 5*\/ 2 *\/  3 - 2*\/ 2     7*\/  3 - 2*\/ 2   + 5*\/ 2 *\/  3 - 2*\/ 2           7*\/  3 - 2*\/ 2   + 5*\/ 2 *\/  3 - 2*\/ 2                     7*\/  3 - 2*\/ 2   + 5*\/ 2 *\/  3 - 2*\/ 2            
$$- \frac{3 \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} \operatorname{atan}{\left(\frac{\tanh{\left(\frac{1}{2} \right)}}{\sqrt{2 \sqrt{2} + 3}} \right)}}{7 \sqrt{3 - 2 \sqrt{2}} + 5 \sqrt{2} \sqrt{3 - 2 \sqrt{2}}} - \frac{2 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} \operatorname{atan}{\left(\frac{\tanh{\left(\frac{1}{2} \right)}}{\sqrt{2 \sqrt{2} + 3}} \right)}}{7 \sqrt{3 - 2 \sqrt{2}} + 5 \sqrt{2} \sqrt{3 - 2 \sqrt{2}}} + \frac{2 \sqrt{2} \operatorname{atan}{\left(\frac{\tanh{\left(\frac{1}{2} \right)}}{\sqrt{3 - 2 \sqrt{2}}} \right)}}{7 \sqrt{3 - 2 \sqrt{2}} + 5 \sqrt{2} \sqrt{3 - 2 \sqrt{2}}} + \frac{3 \operatorname{atan}{\left(\frac{\tanh{\left(\frac{1}{2} \right)}}{\sqrt{3 - 2 \sqrt{2}}} \right)}}{7 \sqrt{3 - 2 \sqrt{2}} + 5 \sqrt{2} \sqrt{3 - 2 \sqrt{2}}}$$
=
=
                                                                                                     _____________    _____________                                     _____________    _____________                       
                 /   tanh(1/2)    \                         ___     /   tanh(1/2)    \              /         ___    /         ___      /   tanh(1/2)    \       ___   /         ___    /         ___      /   tanh(1/2)    \
           3*atan|----------------|                     2*\/ 2 *atan|----------------|          3*\/  3 - 2*\/ 2  *\/  3 + 2*\/ 2  *atan|----------------|   2*\/ 2 *\/  3 - 2*\/ 2  *\/  3 + 2*\/ 2  *atan|----------------|
                 |   _____________|                                 |   _____________|                                                  |   _____________|                                                 |   _____________|
                 |  /         ___ |                                 |  /         ___ |                                                  |  /         ___ |                                                 |  /         ___ |
                 \\/  3 - 2*\/ 2  /                                 \\/  3 - 2*\/ 2  /                                                  \\/  3 + 2*\/ 2  /                                                 \\/  3 + 2*\/ 2  /
--------------------------------------------- + --------------------------------------------- - ---------------------------------------------------------- - ----------------------------------------------------------------
     _____________              _____________        _____________              _____________              _____________              _____________                        _____________              _____________          
    /         ___        ___   /         ___        /         ___        ___   /         ___              /         ___        ___   /         ___                        /         ___        ___   /         ___           
7*\/  3 - 2*\/ 2   + 5*\/ 2 *\/  3 - 2*\/ 2     7*\/  3 - 2*\/ 2   + 5*\/ 2 *\/  3 - 2*\/ 2           7*\/  3 - 2*\/ 2   + 5*\/ 2 *\/  3 - 2*\/ 2                     7*\/  3 - 2*\/ 2   + 5*\/ 2 *\/  3 - 2*\/ 2            
$$- \frac{3 \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} \operatorname{atan}{\left(\frac{\tanh{\left(\frac{1}{2} \right)}}{\sqrt{2 \sqrt{2} + 3}} \right)}}{7 \sqrt{3 - 2 \sqrt{2}} + 5 \sqrt{2} \sqrt{3 - 2 \sqrt{2}}} - \frac{2 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} \operatorname{atan}{\left(\frac{\tanh{\left(\frac{1}{2} \right)}}{\sqrt{2 \sqrt{2} + 3}} \right)}}{7 \sqrt{3 - 2 \sqrt{2}} + 5 \sqrt{2} \sqrt{3 - 2 \sqrt{2}}} + \frac{2 \sqrt{2} \operatorname{atan}{\left(\frac{\tanh{\left(\frac{1}{2} \right)}}{\sqrt{3 - 2 \sqrt{2}}} \right)}}{7 \sqrt{3 - 2 \sqrt{2}} + 5 \sqrt{2} \sqrt{3 - 2 \sqrt{2}}} + \frac{3 \operatorname{atan}{\left(\frac{\tanh{\left(\frac{1}{2} \right)}}{\sqrt{3 - 2 \sqrt{2}}} \right)}}{7 \sqrt{3 - 2 \sqrt{2}} + 5 \sqrt{2} \sqrt{3 - 2 \sqrt{2}}}$$
3*atan(tanh(1/2)/sqrt(3 - 2*sqrt(2)))/(7*sqrt(3 - 2*sqrt(2)) + 5*sqrt(2)*sqrt(3 - 2*sqrt(2))) + 2*sqrt(2)*atan(tanh(1/2)/sqrt(3 - 2*sqrt(2)))/(7*sqrt(3 - 2*sqrt(2)) + 5*sqrt(2)*sqrt(3 - 2*sqrt(2))) - 3*sqrt(3 - 2*sqrt(2))*sqrt(3 + 2*sqrt(2))*atan(tanh(1/2)/sqrt(3 + 2*sqrt(2)))/(7*sqrt(3 - 2*sqrt(2)) + 5*sqrt(2)*sqrt(3 - 2*sqrt(2))) - 2*sqrt(2)*sqrt(3 - 2*sqrt(2))*sqrt(3 + 2*sqrt(2))*atan(tanh(1/2)/sqrt(3 + 2*sqrt(2)))/(7*sqrt(3 - 2*sqrt(2)) + 5*sqrt(2)*sqrt(3 - 2*sqrt(2)))
Numerical answer [src]
0.650880168023008
0.650880168023008

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