Mister Exam

# Least common denominator sqrt(4/x+1)*x+2*log(sqrt(4/x+1)+1)-2*log(sqrt(4/x+1)-1)

An expression to simplify:

### The solution

You have entered [src]
    _______          /    _______    \        /    _______    \
/ 4               |   / 4         |        |   / 4         |
/  - + 1 *x + 2*log|  /  - + 1  + 1| - 2*log|  /  - + 1  - 1|
\/   x               \\/   x         /        \\/   x         /
$$\left(x \sqrt{1 + \frac{4}{x}} + 2 \log{\left(\sqrt{1 + \frac{4}{x}} + 1 \right)}\right) - 2 \log{\left(\sqrt{1 + \frac{4}{x}} - 1 \right)}$$
sqrt(4/x + 1)*x + 2*log(sqrt(4/x + 1) + 1) - 2*log(sqrt(4/x + 1) - 1)
General simplification [src]
       /         _______\        /        _______\         _______
|        / 4 + x |        |       / 4 + x |        / 4 + x
- 2*log|-1 +   /  ----- | + 2*log|1 +   /  ----- | + x*  /  -----
\     \/     x   /        \    \/     x   /     \/     x   
$$x \sqrt{\frac{x + 4}{x}} - 2 \log{\left(\sqrt{\frac{x + 4}{x}} - 1 \right)} + 2 \log{\left(\sqrt{\frac{x + 4}{x}} + 1 \right)}$$
-2*log(-1 + sqrt((4 + x)/x)) + 2*log(1 + sqrt((4 + x)/x)) + x*sqrt((4 + x)/x)
Common denominator [src]
       /         _______\        /        _______\         _______
|        /     4 |        |       /     4 |        /     4
- 2*log|-1 +   /  1 + - | + 2*log|1 +   /  1 + - | + x*  /  1 + -
\     \/       x /        \    \/       x /     \/       x 
$$x \sqrt{1 + \frac{4}{x}} - 2 \log{\left(\sqrt{1 + \frac{4}{x}} - 1 \right)} + 2 \log{\left(\sqrt{1 + \frac{4}{x}} + 1 \right)}$$
-2*log(-1 + sqrt(1 + 4/x)) + 2*log(1 + sqrt(1 + 4/x)) + x*sqrt(1 + 4/x)
Rational denominator [src]
       /         _______\        /        _______\         _______
|        /     4 |        |       /     4 |        /     4
- 2*log|-1 +   /  1 + - | + 2*log|1 +   /  1 + - | + x*  /  1 + -
\     \/       x /        \    \/       x /     \/       x 
$$x \sqrt{1 + \frac{4}{x}} - 2 \log{\left(\sqrt{1 + \frac{4}{x}} - 1 \right)} + 2 \log{\left(\sqrt{1 + \frac{4}{x}} + 1 \right)}$$
-2*log(-1 + sqrt(1 + 4/x)) + 2*log(1 + sqrt(1 + 4/x)) + x*sqrt(1 + 4/x)
Trigonometric part [src]
       /         _______\        /        _______\         _______
|        /     4 |        |       /     4 |        /     4
- 2*log|-1 +   /  1 + - | + 2*log|1 +   /  1 + - | + x*  /  1 + -
\     \/       x /        \    \/       x /     \/       x 
$$x \sqrt{1 + \frac{4}{x}} - 2 \log{\left(\sqrt{1 + \frac{4}{x}} - 1 \right)} + 2 \log{\left(\sqrt{1 + \frac{4}{x}} + 1 \right)}$$
-2*log(-1 + sqrt(1 + 4/x)) + 2*log(1 + sqrt(1 + 4/x)) + x*sqrt(1 + 4/x)
Combining rational expressions [src]
       /         _______\        /        _______\         _______
|        / 4 + x |        |       / 4 + x |        / 4 + x
- 2*log|-1 +   /  ----- | + 2*log|1 +   /  ----- | + x*  /  -----
\     \/     x   /        \    \/     x   /     \/     x   
$$x \sqrt{\frac{x + 4}{x}} - 2 \log{\left(\sqrt{\frac{x + 4}{x}} - 1 \right)} + 2 \log{\left(\sqrt{\frac{x + 4}{x}} + 1 \right)}$$
-2*log(-1 + sqrt((4 + x)/x)) + 2*log(1 + sqrt((4 + x)/x)) + x*sqrt((4 + x)/x)
Combinatorics [src]
       /         _______\        /        _______\         _______
|        /     4 |        |       /     4 |        /     4
- 2*log|-1 +   /  1 + - | + 2*log|1 +   /  1 + - | + x*  /  1 + -
\     \/       x /        \    \/       x /     \/       x 
$$x \sqrt{1 + \frac{4}{x}} - 2 \log{\left(\sqrt{1 + \frac{4}{x}} - 1 \right)} + 2 \log{\left(\sqrt{1 + \frac{4}{x}} + 1 \right)}$$
-2*log(-1 + sqrt(1 + 4/x)) + 2*log(1 + sqrt(1 + 4/x)) + x*sqrt(1 + 4/x)
2.0*log(sqrt(4/x + 1) + 1) - 2.0*log(sqrt(4/x + 1) - 1) + 2.0*x*(0.25 + 1/x)^0.5
2.0*log(sqrt(4/x + 1) + 1) - 2.0*log(sqrt(4/x + 1) - 1) + 2.0*x*(0.25 + 1/x)^0.5
Powers [src]
       /         _______\        /        _______\         _______
|        /     4 |        |       /     4 |        /     4
- 2*log|-1 +   /  1 + - | + 2*log|1 +   /  1 + - | + x*  /  1 + -
\     \/       x /        \    \/       x /     \/       x 
$$x \sqrt{1 + \frac{4}{x}} - 2 \log{\left(\sqrt{1 + \frac{4}{x}} - 1 \right)} + 2 \log{\left(\sqrt{1 + \frac{4}{x}} + 1 \right)}$$
-2*log(-1 + sqrt(1 + 4/x)) + 2*log(1 + sqrt(1 + 4/x)) + x*sqrt(1 + 4/x)
Assemble expression [src]
       /    _______    \        /    _______    \         _______
|   / 4         |        |   / 4         |        /     4
- 2*log|  /  - + 1  - 1| + 2*log|  /  - + 1  + 1| + x*  /  1 + -
\\/   x         /        \\/   x         /     \/       x 
$$x \sqrt{1 + \frac{4}{x}} - 2 \log{\left(\sqrt{1 + \frac{4}{x}} - 1 \right)} + 2 \log{\left(\sqrt{1 + \frac{4}{x}} + 1 \right)}$$
-2*log(sqrt(4/x + 1) - 1) + 2*log(sqrt(4/x + 1) + 1) + x*sqrt(1 + 4/x)