Mister Exam
Expression simplification/
Fraction Decomposition into the simple/
-tanh(3/2+5*x^2/2)/(-5+5*tanh(3/2+5*x^2/2)^2)
How do you -tanh(3/2+5*x^2/2)/(-5+5*tanh(3/2+5*x^2/2)^2) in partial fractions?
The solution
You have entered
[src]
/ 2\
|3 5*x |
-tanh|- + ----|
\2 2 /
----------------------
/ 2\
2|3 5*x |
-5 + 5*tanh |- + ----|
\2 2 /
$$\frac{\left(-1\right) \tanh{\left(\frac{5 x^{2}}{2} + \frac{3}{2} \right)}}{5 \tanh^{2}{\left(\frac{5 x^{2}}{2} + \frac{3}{2} \right)} - 5}$$
(-tanh(3/2 + (5*x^2)/2))/(-5 + 5*tanh(3/2 + (5*x^2)/2)^2)
Fraction decomposition
[src]
-1/(10*(1 + tanh(3/2 + 5*x^2/2))) - 1/(10*(-1 + tanh(3/2 + 5*x^2/2)))
$$- \frac{1}{10 \left(\tanh{\left(\frac{5 x^{2}}{2} + \frac{3}{2} \right)} + 1\right)} - \frac{1}{10 \left(\tanh{\left(\frac{5 x^{2}}{2} + \frac{3}{2} \right)} - 1\right)}$$
1 1
- ----------------------- - ------------------------
/ / 2\\ / / 2\\
| |3 5*x || | |3 5*x ||
10*|1 + tanh|- + ----|| 10*|-1 + tanh|- + ----||
\ \2 2 // \ \2 2 //
General simplification
[src]
/ 2\
sinh\3 + 5*x /
--------------
10
$$\frac{\sinh{\left(5 x^{2} + 3 \right)}}{10}$$
sinh(3 + 5*x^2)/10
Numerical answer
[src]
-tanh(3/2 + (5*x^2)/2)/(-5.0 + 5.0*tanh(3/2 + (5*x^2)/2)^2)
-tanh(3/2 + (5*x^2)/2)/(-5.0 + 5.0*tanh(3/2 + (5*x^2)/2)^2)
Rational denominator
[src]
/ 2\
|3 + 5*x |
-tanh|--------|
\ 2 /
----------------------
/ 2\
2|3 5*x |
-5 + 5*tanh |- + ----|
\2 2 /
$$- \frac{\tanh{\left(\frac{5 x^{2} + 3}{2} \right)}}{5 \tanh^{2}{\left(\frac{5 x^{2}}{2} + \frac{3}{2} \right)} - 5}$$
-tanh((3 + 5*x^2)/2)/(-5 + 5*tanh(3/2 + (5*x^2)/2)^2)
Common denominator
[src]
/ 2\
|3 5*x |
-tanh|- + ----|
\2 2 /
----------------------
/ 2\
2|3 5*x |
-5 + 5*tanh |- + ----|
\2 2 /
$$- \frac{\tanh{\left(\frac{5 x^{2}}{2} + \frac{3}{2} \right)}}{5 \tanh^{2}{\left(\frac{5 x^{2}}{2} + \frac{3}{2} \right)} - 5}$$
-tanh(3/2 + 5*x^2/2)/(-5 + 5*tanh(3/2 + 5*x^2/2)^2)
Powers
[src]
/ 2\
|3 5*x |
-tanh|- + ----|
\2 2 /
----------------------
/ 2\
2|3 5*x |
-5 + 5*tanh |- + ----|
\2 2 /
$$- \frac{\tanh{\left(\frac{5 x^{2}}{2} + \frac{3}{2} \right)}}{5 \tanh^{2}{\left(\frac{5 x^{2}}{2} + \frac{3}{2} \right)} - 5}$$
-tanh(3/2 + 5*x^2/2)/(-5 + 5*tanh(3/2 + 5*x^2/2)^2)
Assemble expression
[src]
/ 2\
|3 5*x |
-tanh|- + ----|
\2 2 /
----------------------
/ 2\
2|3 5*x |
-5 + 5*tanh |- + ----|
\2 2 /
$$- \frac{\tanh{\left(\frac{5 x^{2}}{2} + \frac{3}{2} \right)}}{5 \tanh^{2}{\left(\frac{5 x^{2}}{2} + \frac{3}{2} \right)} - 5}$$
-tanh(3/2 + (5*x^2)/2)/(-5 + 5*tanh(3/2 + (5*x^2)/2)^2)
Combinatorics
[src]
/ 2\
|3 5*x |
-tanh|- + ----|
\2 2 /
--------------------------------------------
/ / 2\\ / / 2\\
| |3 5*x || | |3 5*x ||
5*|1 + tanh|- + ----||*|-1 + tanh|- + ----||
\ \2 2 // \ \2 2 //
$$- \frac{\tanh{\left(\frac{5 x^{2}}{2} + \frac{3}{2} \right)}}{5 \left(\tanh{\left(\frac{5 x^{2}}{2} + \frac{3}{2} \right)} - 1\right) \left(\tanh{\left(\frac{5 x^{2}}{2} + \frac{3}{2} \right)} + 1\right)}$$
-tanh(3/2 + 5*x^2/2)/(5*(1 + tanh(3/2 + 5*x^2/2))*(-1 + tanh(3/2 + 5*x^2/2)))
Combining rational expressions
[src]
/ 2\
|3 + 5*x |
-tanh|--------|
\ 2 /
------------------------
/ / 2\\
| 2|3 + 5*x ||
5*|-1 + tanh |--------||
\ \ 2 //
$$- \frac{\tanh{\left(\frac{5 x^{2} + 3}{2} \right)}}{5 \left(\tanh^{2}{\left(\frac{5 x^{2} + 3}{2} \right)} - 1\right)}$$
-tanh((3 + 5*x^2)/2)/(5*(-1 + tanh((3 + 5*x^2)/2)^2))
Trigonometric part
[src]
/ 2\
|3 5*x |
-tanh|- + ----|
\2 2 /
----------------------
/ 2\
2|3 5*x |
-5 + 5*tanh |- + ----|
\2 2 /
$$- \frac{\tanh{\left(\frac{5 x^{2}}{2} + \frac{3}{2} \right)}}{5 \tanh^{2}{\left(\frac{5 x^{2}}{2} + \frac{3}{2} \right)} - 5}$$
-tanh(3/2 + 5*x^2/2)/(-5 + 5*tanh(3/2 + 5*x^2/2)^2)