Why do Wolfram Alpha and Mathematica give different answers? [closed]

I've just got my hands on Wolfram Mathematica and tried to see if I can figure out how to use it by myself. I've plugged in an integral $\int \frac{dx}{(1-x^2)^{2/3}}$, knowing that the result is $\frac{x}{\sqrt{1-x^2}}$. But Mathematica gave me something different:

In[10]:= Integrate[1/((1-x^2)^(2/3)), x]
Out[10]:= x Hypergeometric2F1[1/2,2/3,3/2,x^2]


What does this mean and what am I doing wrong?

This is more a long comment than an answer.

If you calculate:

D[x Hypergeometric2F1[1/2, 2/3, 3/2, x^2], x]
FullSimplify@D[x/Sqrt[1 - x^2], x]


You find respectively:

1/(1 - x^2)^(2/3)


and

1/(1 - x^2)^(3/2)


Mathematica gives the correct answer: Check the exponents!!

• Umm. Jeez. I'm so stupid. – Akiiino Jan 31 '16 at 8:38
• No. Your mistake is common. I've seen it made by plenty of people ranging from highschool students to engineering/science professors at good universities. You, like me, were probably taught calculus by people who had little experience with it in the real world. You were taught to expect that there is a "correct" answer to the integral. Instead you should have investigated whether the solution had the properties of a correct solution. – Searke Jan 31 '16 at 23:45