Solve -1/1-x+1/left(1-xright)^2+-1/1+x+1/left(1+xright)^2

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

Evaluate

$$4\times \left(\frac{x}{1-x^{2}}\right)^{2}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 1-x and \left(1-x\right)^{2} is \left(-x+1\right)\left(-x+1\right)^{2}. Multiply \frac{-1}{1-x} times \frac{\left(-x+1\right)^{2}}{\left(-x+1\right)^{2}}. Multiply \frac{1}{\left(1-x\right)^{2}} times \frac{-x+1}{-x+1}.

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

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

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

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

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

Combine like terms in -x^{2}+2x-1-x+1.

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

Factor the expressions that are not already factored in \frac{-x^{2}+x}{\left(-x+1\right)\left(-x+1\right)^{2}}.

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

Cancel out -x+1 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(-x+1\right)^{2} and 1+x is \left(x+1\right)\left(-x+1\right)^{2}. Multiply \frac{x}{\left(-x+1\right)^{2}} times \frac{x+1}{x+1}. Multiply \frac{-1}{1+x} times \frac{\left(-x+1\right)^{2}}{\left(-x+1\right)^{2}}.

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

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

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

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

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

Combine like terms in x^{2}+x-x^{2}+2x-1.

$$\frac{3x-1}{\left(x+1\right)\left(-x+1\right)^{2}}+\frac{1}{\left(1+x\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+1\right)\left(-x+1\right)^{2} and \left(1+x\right)^{2} is \left(x-1\right)^{2}\left(x+1\right)^{2}. Multiply \frac{3x-1}{\left(x+1\right)\left(-x+1\right)^{2}} times \frac{x+1}{x+1}. Multiply \frac{1}{\left(1+x\right)^{2}} times \frac{\left(x-1\right)^{2}}{\left(x-1\right)^{2}}.

$$\frac{\left(3x-1\right)\left(x+1\right)}{\left(x-1\right)^{2}\left(x+1\right)^{2}}+\frac{\left(x-1\right)^{2}}{\left(x-1\right)^{2}\left(x+1\right)^{2}}$$

Since \frac{\left(3x-1\right)\left(x+1\right)}{\left(x-1\right)^{2}\left(x+1\right)^{2}} and \frac{\left(x-1\right)^{2}}{\left(x-1\right)^{2}\left(x+1\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(3x-1\right)\left(x+1\right)+\left(x-1\right)^{2}}{\left(x-1\right)^{2}\left(x+1\right)^{2}}$$

Do the multiplications in \left(3x-1\right)\left(x+1\right)+\left(x-1\right)^{2}.

$$\frac{3x^{2}+3x-x-1+x^{2}-2x+1}{\left(x-1\right)^{2}\left(x+1\right)^{2}}$$

Combine like terms in 3x^{2}+3x-x-1+x^{2}-2x+1.

$$\frac{4x^{2}}{\left(x-1\right)^{2}\left(x+1\right)^{2}}$$

Expand \left(x-1\right)^{2}\left(x+1\right)^{2}.

$$\frac{4x^{2}}{x^{4}-2x^{2}+1}$$

Expand

$$4\times \left(\frac{x}{x^{2}-1}\right)^{2}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 1-x and \left(1-x\right)^{2} is \left(-x+1\right)\left(-x+1\right)^{2}. Multiply \frac{-1}{1-x} times \frac{\left(-x+1\right)^{2}}{\left(-x+1\right)^{2}}. Multiply \frac{1}{\left(1-x\right)^{2}} times \frac{-x+1}{-x+1}.

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

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

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

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

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

Combine like terms in -x^{2}+2x-1-x+1.

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

Factor the expressions that are not already factored in \frac{-x^{2}+x}{\left(-x+1\right)\left(-x+1\right)^{2}}.

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

Cancel out -x+1 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(-x+1\right)^{2} and 1+x is \left(x+1\right)\left(-x+1\right)^{2}. Multiply \frac{x}{\left(-x+1\right)^{2}} times \frac{x+1}{x+1}. Multiply \frac{-1}{1+x} times \frac{\left(-x+1\right)^{2}}{\left(-x+1\right)^{2}}.

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

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

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

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

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

Combine like terms in x^{2}+x-x^{2}+2x-1.

$$\frac{3x-1}{\left(x+1\right)\left(-x+1\right)^{2}}+\frac{1}{\left(1+x\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+1\right)\left(-x+1\right)^{2} and \left(1+x\right)^{2} is \left(x-1\right)^{2}\left(x+1\right)^{2}. Multiply \frac{3x-1}{\left(x+1\right)\left(-x+1\right)^{2}} times \frac{x+1}{x+1}. Multiply \frac{1}{\left(1+x\right)^{2}} times \frac{\left(x-1\right)^{2}}{\left(x-1\right)^{2}}.

$$\frac{\left(3x-1\right)\left(x+1\right)}{\left(x-1\right)^{2}\left(x+1\right)^{2}}+\frac{\left(x-1\right)^{2}}{\left(x-1\right)^{2}\left(x+1\right)^{2}}$$

Since \frac{\left(3x-1\right)\left(x+1\right)}{\left(x-1\right)^{2}\left(x+1\right)^{2}} and \frac{\left(x-1\right)^{2}}{\left(x-1\right)^{2}\left(x+1\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(3x-1\right)\left(x+1\right)+\left(x-1\right)^{2}}{\left(x-1\right)^{2}\left(x+1\right)^{2}}$$

Do the multiplications in \left(3x-1\right)\left(x+1\right)+\left(x-1\right)^{2}.

$$\frac{3x^{2}+3x-x-1+x^{2}-2x+1}{\left(x-1\right)^{2}\left(x+1\right)^{2}}$$

Combine like terms in 3x^{2}+3x-x-1+x^{2}-2x+1.

$$\frac{4x^{2}}{\left(x-1\right)^{2}\left(x+1\right)^{2}}$$

Expand \left(x-1\right)^{2}\left(x+1\right)^{2}.

$$\frac{4x^{2}}{x^{4}-2x^{2}+1}$$