Solve 1/1-a+1/1+a+2/1+a^2+4/1+a^4-frac{8}{1-a^2}

$$\frac { 1 } { 1 - a } + \frac { 1 } { 1 + a } + \frac { 2 } { 1 + a ^ { 2 } } + \frac { 4 } { 1 + a ^ { 4 } } - \frac { 8 } { 1 - a ^ { 2 } }$$

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

Combine like terms in 2a^{2}+2-2a^{2}+2a-2a+2.

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

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

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

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

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

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

$$\frac{4a^{4}+4-4a^{4}-4a^{2}+4a^{3}+4a-4a^{3}-4a+4a^{2}+4}{\left(a+1\right)\left(-a+1\right)\left(a^{2}+1\right)\left(a^{4}+1\right)}-\frac{8}{1-a^{2}}$$

Combine like terms in 4a^{4}+4-4a^{4}-4a^{2}+4a^{3}+4a-4a^{3}-4a+4a^{2}+4.

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

Factor 1-a^{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a+1\right)\left(-a+1\right)\left(a^{2}+1\right)\left(a^{4}+1\right) and \left(a-1\right)\left(-a-1\right) is \left(a-1\right)\left(a+1\right)\left(a^{2}+1\right)\left(a^{4}+1\right). Multiply \frac{8}{\left(a+1\right)\left(-a+1\right)\left(a^{2}+1\right)\left(a^{4}+1\right)} times \frac{-1}{-1}. Multiply \frac{8}{\left(a-1\right)\left(-a-1\right)} times \frac{-\left(a^{2}+1\right)\left(a^{4}+1\right)}{-\left(a^{2}+1\right)\left(a^{4}+1\right)}.

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

Since \frac{8\left(-1\right)}{\left(a-1\right)\left(a+1\right)\left(a^{2}+1\right)\left(a^{4}+1\right)} and \frac{8\left(-1\right)\left(a^{2}+1\right)\left(a^{4}+1\right)}{\left(a-1\right)\left(a+1\right)\left(a^{2}+1\right)\left(a^{4}+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 8\left(-1\right)-8\left(-1\right)\left(a^{2}+1\right)\left(a^{4}+1\right).

$$\frac{-8+8a^{6}+8a^{2}+8a^{4}+8}{\left(a-1\right)\left(a+1\right)\left(a^{2}+1\right)\left(a^{4}+1\right)}$$

Combine like terms in -8+8a^{6}+8a^{2}+8a^{4}+8.

$$\frac{8a^{6}+8a^{2}+8a^{4}}{\left(a-1\right)\left(a+1\right)\left(a^{2}+1\right)\left(a^{4}+1\right)}$$

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

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

Differentiate w.r.t. a

$$-\frac{16a\left(a^{12}+2a^{10}+3a^{8}+3a^{4}+2a^{2}+1\right)}{\left(a^{8}-1\right)^{2}}$$