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

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

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 1+x^{4} and x+1 is \left(x+1\right)\left(x^{4}+1\right). Multiply \frac{4}{1+x^{4}} times \frac{x+1}{x+1}. Multiply \frac{1}{x+1} times \frac{x^{4}+1}{x^{4}+1}.

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

Since \frac{4\left(x+1\right)}{\left(x+1\right)\left(x^{4}+1\right)} and \frac{x^{4}+1}{\left(x+1\right)\left(x^{4}+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

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

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

Combine like terms in 4x+4-x^{4}-1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+1\right)\left(x^{4}+1\right) and x^{2}+1 is \left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right). Multiply \frac{4x+3-x^{4}}{\left(x+1\right)\left(x^{4}+1\right)} times \frac{x^{2}+1}{x^{2}+1}. Multiply \frac{1}{x^{2}+1} times \frac{\left(x+1\right)\left(x^{4}+1\right)}{\left(x+1\right)\left(x^{4}+1\right)}.

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

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

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

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

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

Combine like terms in 4x^{3}+4x+3x^{2}+3-x^{6}-x^{4}-x^{5}-x-x^{4}-1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right) and x-1 is \left(x-1\right)\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right). Multiply \frac{4x^{3}-x^{6}+3x+3x^{2}+2-2x^{4}-x^{5}}{\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)} times \frac{x-1}{x-1}. Multiply \frac{1}{x-1} times \frac{\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)}{\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)}.

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

Since \frac{\left(4x^{3}-x^{6}+3x+3x^{2}+2-2x^{4}-x^{5}\right)\left(x-1\right)}{\left(x-1\right)\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)} and \frac{\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+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.

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

Do the multiplications in \left(4x^{3}-x^{6}+3x+3x^{2}+2-2x^{4}-x^{5}\right)\left(x-1\right)+\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right).

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

Combine like terms in 4x^{4}-4x^{3}-x^{7}+x^{6}+3x^{2}-3x+3x^{3}-3x^{2}+2x-2-2x^{5}+2x^{4}-x^{6}+x^{5}+x^{7}+x^{3}+x^{5}+x+x^{6}+x^{2}+x^{4}+1.

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

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

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

Differentiate w.r.t. x

$$\frac{2x\left(-x^{12}-14x^{10}-3x^{8}+4x^{6}-3x^{4}-14x^{2}-1\right)}{\left(x^{8}-1\right)^{2}}$$