9. $$ \left.\frac{\left(a+\frac{a b}{a-b}\right)}{\left(a^{2}-\frac{2 a^{2} b^{2}}{a^{2}+b^{2}}\right)}\right)\left(\frac{\left(\frac{1}{a^{2}}-\frac{1}{b^{2}}\right)}{\left(\frac{1}{a}-\frac{1}{b}\right)}\right)= $$

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Calculate or simplify the expression $$ \frac{(a+\frac{a b}{a-b})}{(a^{2}-\frac{2 a^{2} b^{2}}{a^{2}+b^{2}})}	imes \frac{(\frac{1}{a^{2}}-\frac{1}{b^{2}})}{(\frac{1}{a}-\frac{1}{b})} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\left(a+\frac{ab}{a-b}ight)}{\left(a^{2}-\frac{2a^{2}b^{2}}{a^{2}+b^{2}}ight)}	imes \frac{\left(\frac{1}{a^{2}}-\frac{1}{b^{2}}ight)}{\left(\frac{1}{a}-\frac{1}{b}ight)}$$
- step1: Remove the parentheses:
  $$\frac{a+\frac{ab}{a-b}}{a^{2}-\frac{2a^{2}b^{2}}{a^{2}+b^{2}}}	imes \frac{\frac{1}{a^{2}}-\frac{1}{b^{2}}}{\frac{1}{a}-\frac{1}{b}}$$
- step2: Add the terms:
  $$\frac{\frac{a^{2}}{a-b}}{a^{2}-\frac{2a^{2}b^{2}}{a^{2}+b^{2}}}	imes \frac{\frac{1}{a^{2}}-\frac{1}{b^{2}}}{\frac{1}{a}-\frac{1}{b}}$$
- step3: Subtract the terms:
  $$\frac{\frac{a^{2}}{a-b}}{\frac{a^{4}-a^{2}b^{2}}{a^{2}+b^{2}}}	imes \frac{\frac{1}{a^{2}}-\frac{1}{b^{2}}}{\frac{1}{a}-\frac{1}{b}}$$
- step4: Subtract the terms:
  $$\frac{\frac{a^{2}}{a-b}}{\frac{a^{4}-a^{2}b^{2}}{a^{2}+b^{2}}}	imes \frac{\frac{b^{2}-a^{2}}{a^{2}b^{2}}}{\frac{1}{a}-\frac{1}{b}}$$
- step5: Subtract the terms:
  $$\frac{\frac{a^{2}}{a-b}}{\frac{a^{4}-a^{2}b^{2}}{a^{2}+b^{2}}}	imes \frac{\frac{b^{2}-a^{2}}{a^{2}b^{2}}}{\frac{b-a}{ab}}$$
- step6: Divide the terms:
  $$\frac{a^{2}+b^{2}}{\left(a-bight)\left(a^{2}-b^{2}ight)}	imes \frac{\frac{b^{2}-a^{2}}{a^{2}b^{2}}}{\frac{b-a}{ab}}$$
- step7: Divide the terms:
  $$\frac{a^{2}+b^{2}}{\left(a-bight)\left(a^{2}-b^{2}ight)}	imes \frac{b+a}{ab}$$
- step8: Rewrite the expression:
  $$\frac{a^{2}+b^{2}}{\left(a-bight)\left(a+bight)\left(a-bight)}	imes \frac{a+b}{ab}$$
- step9: Reduce the fraction:
  $$\frac{a^{2}+b^{2}}{\left(a-bight)\left(a-bight)}	imes \frac{1}{ab}$$
- step10: Multiply the terms:
  $$\frac{a^{2}+b^{2}}{\left(a-bight)^{2}ab}$$
- step11: Expand the expression:
  $$\frac{a^{2}+b^{2}}{a^{3}b-2a^{2}b^{2}+b^{3}a}$$
The simplified expression is $$ \frac{a^{2}+b^{2}}{a^{3}b-2a^{2}b^{2}+b^{3}a} $$.
The simplified expression is $$ \frac{a^{2}+b^{2}}{a^{3}b-2a^{2}b^{2}+b^{3}a} $$.

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