Solve 1+x-y/x+y/1-frac{x-y{x+y}}+1+frac{x-y/x^{2+y^2}}{1-x^2-y/x^2+y^2}=

$$\frac { 1 + \frac { x - y } { x + y } } { 1 - \frac { x - y } { x + y } } + \frac { 1 + \frac { x - y } { x ^ { 2 } + y ^ { 2 } } } { 1 - \frac { x ^ { 2 } - y } { x ^ { 2 } + y ^ { 2 } } } =$$

Evaluate

$$\frac{x^{2}+xy+2x+y^{2}-y}{y\left(y+1\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x+y}{x+y}.

$$\frac{\frac{x+y}{x+y}+\frac{x-y}{x+y}}{1-\frac{x-y}{x+y}}+\frac{1+\frac{x-y}{x^{2}+y^{2}}}{1-\frac{x^{2}-y}{x^{2}+y^{2}}}$$

Since \frac{x+y}{x+y} and \frac{x-y}{x+y} have the same denominator, add them by adding their numerators.

$$\frac{\frac{x+y+x-y}{x+y}}{1-\frac{x-y}{x+y}}+\frac{1+\frac{x-y}{x^{2}+y^{2}}}{1-\frac{x^{2}-y}{x^{2}+y^{2}}}$$

Combine like terms in x+y+x-y.

$$\frac{\frac{2x}{x+y}}{1-\frac{x-y}{x+y}}+\frac{1+\frac{x-y}{x^{2}+y^{2}}}{1-\frac{x^{2}-y}{x^{2}+y^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x+y}{x+y}.

$$\frac{\frac{2x}{x+y}}{\frac{x+y}{x+y}-\frac{x-y}{x+y}}+\frac{1+\frac{x-y}{x^{2}+y^{2}}}{1-\frac{x^{2}-y}{x^{2}+y^{2}}}$$

Since \frac{x+y}{x+y} and \frac{x-y}{x+y} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{2x}{x+y}}{\frac{x+y-\left(x-y\right)}{x+y}}+\frac{1+\frac{x-y}{x^{2}+y^{2}}}{1-\frac{x^{2}-y}{x^{2}+y^{2}}}$$

Do the multiplications in x+y-\left(x-y\right).

$$\frac{\frac{2x}{x+y}}{\frac{x+y-x+y}{x+y}}+\frac{1+\frac{x-y}{x^{2}+y^{2}}}{1-\frac{x^{2}-y}{x^{2}+y^{2}}}$$

Combine like terms in x+y-x+y.

$$\frac{\frac{2x}{x+y}}{\frac{2y}{x+y}}+\frac{1+\frac{x-y}{x^{2}+y^{2}}}{1-\frac{x^{2}-y}{x^{2}+y^{2}}}$$

Divide \frac{2x}{x+y} by \frac{2y}{x+y} by multiplying \frac{2x}{x+y} by the reciprocal of \frac{2y}{x+y}.

$$\frac{2x\left(x+y\right)}{\left(x+y\right)\times 2y}+\frac{1+\frac{x-y}{x^{2}+y^{2}}}{1-\frac{x^{2}-y}{x^{2}+y^{2}}}$$

Cancel out 2\left(x+y\right) in both numerator and denominator.

$$\frac{x}{y}+\frac{1+\frac{x-y}{x^{2}+y^{2}}}{1-\frac{x^{2}-y}{x^{2}+y^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x^{2}+y^{2}}{x^{2}+y^{2}}.

$$\frac{x}{y}+\frac{\frac{x^{2}+y^{2}}{x^{2}+y^{2}}+\frac{x-y}{x^{2}+y^{2}}}{1-\frac{x^{2}-y}{x^{2}+y^{2}}}$$

Since \frac{x^{2}+y^{2}}{x^{2}+y^{2}} and \frac{x-y}{x^{2}+y^{2}} have the same denominator, add them by adding their numerators.

$$\frac{x}{y}+\frac{\frac{x^{2}+y^{2}+x-y}{x^{2}+y^{2}}}{1-\frac{x^{2}-y}{x^{2}+y^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x^{2}+y^{2}}{x^{2}+y^{2}}.

$$\frac{x}{y}+\frac{\frac{x^{2}+y^{2}+x-y}{x^{2}+y^{2}}}{\frac{x^{2}+y^{2}}{x^{2}+y^{2}}-\frac{x^{2}-y}{x^{2}+y^{2}}}$$

Since \frac{x^{2}+y^{2}}{x^{2}+y^{2}} and \frac{x^{2}-y}{x^{2}+y^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x}{y}+\frac{\frac{x^{2}+y^{2}+x-y}{x^{2}+y^{2}}}{\frac{x^{2}+y^{2}-\left(x^{2}-y\right)}{x^{2}+y^{2}}}$$

Do the multiplications in x^{2}+y^{2}-\left(x^{2}-y\right).

$$\frac{x}{y}+\frac{\frac{x^{2}+y^{2}+x-y}{x^{2}+y^{2}}}{\frac{x^{2}+y^{2}-x^{2}+y}{x^{2}+y^{2}}}$$

Combine like terms in x^{2}+y^{2}-x^{2}+y.

$$\frac{x}{y}+\frac{\frac{x^{2}+y^{2}+x-y}{x^{2}+y^{2}}}{\frac{y^{2}+y}{x^{2}+y^{2}}}$$

Divide \frac{x^{2}+y^{2}+x-y}{x^{2}+y^{2}} by \frac{y^{2}+y}{x^{2}+y^{2}} by multiplying \frac{x^{2}+y^{2}+x-y}{x^{2}+y^{2}} by the reciprocal of \frac{y^{2}+y}{x^{2}+y^{2}}.

$$\frac{x}{y}+\frac{\left(x^{2}+y^{2}+x-y\right)\left(x^{2}+y^{2}\right)}{\left(x^{2}+y^{2}\right)\left(y^{2}+y\right)}$$

Cancel out x^{2}+y^{2} in both numerator and denominator.

$$\frac{x}{y}+\frac{x^{2}+x+y^{2}-y}{y^{2}+y}$$

Factor y^{2}+y.

$$\frac{x}{y}+\frac{x^{2}+x+y^{2}-y}{y\left(y+1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y and y\left(y+1\right) is y\left(y+1\right). Multiply \frac{x}{y} times \frac{y+1}{y+1}.

$$\frac{x\left(y+1\right)}{y\left(y+1\right)}+\frac{x^{2}+x+y^{2}-y}{y\left(y+1\right)}$$

Since \frac{x\left(y+1\right)}{y\left(y+1\right)} and \frac{x^{2}+x+y^{2}-y}{y\left(y+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{x\left(y+1\right)+x^{2}+x+y^{2}-y}{y\left(y+1\right)}$$

Do the multiplications in x\left(y+1\right)+x^{2}+x+y^{2}-y.

$$\frac{xy+x+x^{2}+x+y^{2}-y}{y\left(y+1\right)}$$

Combine like terms in xy+x+x^{2}+x+y^{2}-y.

$$\frac{y^{2}+xy+2x+x^{2}-y}{y\left(y+1\right)}$$

Expand y\left(y+1\right).

$$\frac{y^{2}+xy+2x+x^{2}-y}{y^{2}+y}$$

Expand

$$\frac{x^{2}+xy+2x+y^{2}-y}{y\left(y+1\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x+y}{x+y}.

$$\frac{\frac{x+y}{x+y}+\frac{x-y}{x+y}}{1-\frac{x-y}{x+y}}+\frac{1+\frac{x-y}{x^{2}+y^{2}}}{1-\frac{x^{2}-y}{x^{2}+y^{2}}}$$

Since \frac{x+y}{x+y} and \frac{x-y}{x+y} have the same denominator, add them by adding their numerators.

$$\frac{\frac{x+y+x-y}{x+y}}{1-\frac{x-y}{x+y}}+\frac{1+\frac{x-y}{x^{2}+y^{2}}}{1-\frac{x^{2}-y}{x^{2}+y^{2}}}$$

Combine like terms in x+y+x-y.

$$\frac{\frac{2x}{x+y}}{1-\frac{x-y}{x+y}}+\frac{1+\frac{x-y}{x^{2}+y^{2}}}{1-\frac{x^{2}-y}{x^{2}+y^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x+y}{x+y}.

$$\frac{\frac{2x}{x+y}}{\frac{x+y}{x+y}-\frac{x-y}{x+y}}+\frac{1+\frac{x-y}{x^{2}+y^{2}}}{1-\frac{x^{2}-y}{x^{2}+y^{2}}}$$

Since \frac{x+y}{x+y} and \frac{x-y}{x+y} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{2x}{x+y}}{\frac{x+y-\left(x-y\right)}{x+y}}+\frac{1+\frac{x-y}{x^{2}+y^{2}}}{1-\frac{x^{2}-y}{x^{2}+y^{2}}}$$

Do the multiplications in x+y-\left(x-y\right).

$$\frac{\frac{2x}{x+y}}{\frac{x+y-x+y}{x+y}}+\frac{1+\frac{x-y}{x^{2}+y^{2}}}{1-\frac{x^{2}-y}{x^{2}+y^{2}}}$$

Combine like terms in x+y-x+y.

$$\frac{\frac{2x}{x+y}}{\frac{2y}{x+y}}+\frac{1+\frac{x-y}{x^{2}+y^{2}}}{1-\frac{x^{2}-y}{x^{2}+y^{2}}}$$

Divide \frac{2x}{x+y} by \frac{2y}{x+y} by multiplying \frac{2x}{x+y} by the reciprocal of \frac{2y}{x+y}.

$$\frac{2x\left(x+y\right)}{\left(x+y\right)\times 2y}+\frac{1+\frac{x-y}{x^{2}+y^{2}}}{1-\frac{x^{2}-y}{x^{2}+y^{2}}}$$

Cancel out 2\left(x+y\right) in both numerator and denominator.

$$\frac{x}{y}+\frac{1+\frac{x-y}{x^{2}+y^{2}}}{1-\frac{x^{2}-y}{x^{2}+y^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x^{2}+y^{2}}{x^{2}+y^{2}}.

$$\frac{x}{y}+\frac{\frac{x^{2}+y^{2}}{x^{2}+y^{2}}+\frac{x-y}{x^{2}+y^{2}}}{1-\frac{x^{2}-y}{x^{2}+y^{2}}}$$

Since \frac{x^{2}+y^{2}}{x^{2}+y^{2}} and \frac{x-y}{x^{2}+y^{2}} have the same denominator, add them by adding their numerators.

$$\frac{x}{y}+\frac{\frac{x^{2}+y^{2}+x-y}{x^{2}+y^{2}}}{1-\frac{x^{2}-y}{x^{2}+y^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x^{2}+y^{2}}{x^{2}+y^{2}}.

$$\frac{x}{y}+\frac{\frac{x^{2}+y^{2}+x-y}{x^{2}+y^{2}}}{\frac{x^{2}+y^{2}}{x^{2}+y^{2}}-\frac{x^{2}-y}{x^{2}+y^{2}}}$$

Since \frac{x^{2}+y^{2}}{x^{2}+y^{2}} and \frac{x^{2}-y}{x^{2}+y^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x}{y}+\frac{\frac{x^{2}+y^{2}+x-y}{x^{2}+y^{2}}}{\frac{x^{2}+y^{2}-\left(x^{2}-y\right)}{x^{2}+y^{2}}}$$

Do the multiplications in x^{2}+y^{2}-\left(x^{2}-y\right).

$$\frac{x}{y}+\frac{\frac{x^{2}+y^{2}+x-y}{x^{2}+y^{2}}}{\frac{x^{2}+y^{2}-x^{2}+y}{x^{2}+y^{2}}}$$

Combine like terms in x^{2}+y^{2}-x^{2}+y.

$$\frac{x}{y}+\frac{\frac{x^{2}+y^{2}+x-y}{x^{2}+y^{2}}}{\frac{y^{2}+y}{x^{2}+y^{2}}}$$

Divide \frac{x^{2}+y^{2}+x-y}{x^{2}+y^{2}} by \frac{y^{2}+y}{x^{2}+y^{2}} by multiplying \frac{x^{2}+y^{2}+x-y}{x^{2}+y^{2}} by the reciprocal of \frac{y^{2}+y}{x^{2}+y^{2}}.

$$\frac{x}{y}+\frac{\left(x^{2}+y^{2}+x-y\right)\left(x^{2}+y^{2}\right)}{\left(x^{2}+y^{2}\right)\left(y^{2}+y\right)}$$

Cancel out x^{2}+y^{2} in both numerator and denominator.

$$\frac{x}{y}+\frac{x^{2}+x+y^{2}-y}{y^{2}+y}$$

Factor y^{2}+y.

$$\frac{x}{y}+\frac{x^{2}+x+y^{2}-y}{y\left(y+1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y and y\left(y+1\right) is y\left(y+1\right). Multiply \frac{x}{y} times \frac{y+1}{y+1}.

$$\frac{x\left(y+1\right)}{y\left(y+1\right)}+\frac{x^{2}+x+y^{2}-y}{y\left(y+1\right)}$$

Since \frac{x\left(y+1\right)}{y\left(y+1\right)} and \frac{x^{2}+x+y^{2}-y}{y\left(y+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{x\left(y+1\right)+x^{2}+x+y^{2}-y}{y\left(y+1\right)}$$

Do the multiplications in x\left(y+1\right)+x^{2}+x+y^{2}-y.

$$\frac{xy+x+x^{2}+x+y^{2}-y}{y\left(y+1\right)}$$

Combine like terms in xy+x+x^{2}+x+y^{2}-y.

$$\frac{y^{2}+xy+2x+x^{2}-y}{y\left(y+1\right)}$$

Expand y\left(y+1\right).

$$\frac{y^{2}+xy+2x+x^{2}-y}{y^{2}+y}$$