calculus and analysis - Mathematica 9 can't integrate this function but earlier versions could

Integrate[ ArcTan[x]/(1 + x) Log[(1 + x^2)/2], {x, -1, 1}]

I used Mathematica 9.0.1 on Windows7 32bit, Mathematica 9 cannot compute this, but Mathematica 8 gives Pi^3/96, is this a bug?

Answer

This issue reminds many similar problems with Integrate. We have in Mathematica 8.0.4:

Integrate[ ArcTan[x]/(1 + x) Log[(1 + x^2)/2], {x, -1, 1}]

Pi^3/96

However in Mathematica 9.0.1 it takes quite a long time yielding a different result:

Integrate[ ArcTan[x]/(1 + x) Log[(1 + x^2)/2], {x, -1, 1}]

Infinity

This is a bug, we can compare it with the NIntegrate result which yields the number numerically the same as in ver. 8.

Of course the result should be the same if we substitute ArcTan by its equivalent:

TrigToExp[ ArcTan[x] ]

1/2 I Log[1 - I x] - 1/2 I Log[1 + I x]

but now the result is the same as in ver. 8:

Integrate[ TrigToExp[ ArcTan[x]]/(1 + x) Log[(1 + x^2)/2], {x, -1, 1}]

Pi^3/96

Comments

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