Solve 1/x-1+1/x+1+2x/x^2+1+4x^2/x^4+1

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

Evaluate

$$\frac{4x^{2}\left(x^{5}+x^{4}+x-1\right)}{x^{8}-1}$$

Short Solution Steps

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

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

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

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

Combine like terms in x+1+x-1.

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

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) and x^{2}+1 is \left(x-1\right)\left(x+1\right)\left(x^{2}+1\right). Multiply \frac{2x}{\left(x-1\right)\left(x+1\right)} times \frac{x^{2}+1}{x^{2}+1}. Multiply \frac{2x}{x^{2}+1} times \frac{\left(x-1\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}.

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

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

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

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

$$\frac{2x^{3}+2x+2x^{3}+2x^{2}-2x^{2}-2x}{\left(x-1\right)\left(x+1\right)\left(x^{2}+1\right)}+\frac{4x^{2}}{x^{4}+1}$$

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

$$\frac{4x^{3}}{\left(x-1\right)\left(x+1\right)\left(x^{2}+1\right)}+\frac{4x^{2}}{x^{4}+1}$$

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)\left(x^{2}+1\right) and x^{4}+1 is \left(x-1\right)\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right). Multiply \frac{4x^{3}}{\left(x-1\right)\left(x+1\right)\left(x^{2}+1\right)} times \frac{x^{4}+1}{x^{4}+1}. Multiply \frac{4x^{2}}{x^{4}+1} times \frac{\left(x-1\right)\left(x+1\right)\left(x^{2}+1\right)}{\left(x-1\right)\left(x+1\right)\left(x^{2}+1\right)}.

$$\frac{4x^{3}\left(x^{4}+1\right)}{\left(x-1\right)\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)}+\frac{4x^{2}\left(x-1\right)\left(x+1\right)\left(x^{2}+1\right)}{\left(x-1\right)\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)}$$

Since \frac{4x^{3}\left(x^{4}+1\right)}{\left(x-1\right)\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)} and \frac{4x^{2}\left(x-1\right)\left(x+1\right)\left(x^{2}+1\right)}{\left(x-1\right)\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{4x^{3}\left(x^{4}+1\right)+4x^{2}\left(x-1\right)\left(x+1\right)\left(x^{2}+1\right)}{\left(x-1\right)\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)}$$

Do the multiplications in 4x^{3}\left(x^{4}+1\right)+4x^{2}\left(x-1\right)\left(x+1\right)\left(x^{2}+1\right).

$$\frac{4x^{7}+4x^{3}+4x^{6}+4x^{4}+4x^{5}+4x^{3}-4x^{5}-4x^{3}-4x^{4}-4x^{2}}{\left(x-1\right)\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)}$$

Combine like terms in 4x^{7}+4x^{3}+4x^{6}+4x^{4}+4x^{5}+4x^{3}-4x^{5}-4x^{3}-4x^{4}-4x^{2}.

$$\frac{4x^{7}+4x^{3}+4x^{6}-4x^{2}}{\left(x-1\right)\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)}$$

Expand \left(x-1\right)\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right).

$$\frac{4x^{7}+4x^{3}+4x^{6}-4x^{2}}{x^{8}-1}$$

Differentiate w.r.t. x

$$\frac{4x\left(2-3x-6x^{4}-7x^{5}+6x^{8}-5x^{9}-2x^{12}-x^{13}\right)}{\left(x^{8}-1\right)^{2}}$$