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Topic: The Charlwood Fifty
Replies: 52   Last Post: Jun 24, 2013 10:24 PM

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 clicliclic@freenet.de Posts: 1,245 Registered: 4/26/08
Re: The Charlwood Fifty
Posted: Jun 8, 2013 5:12 PM

Albert Rich schrieb:
>
> [...] Presumably only a finite number of improvements are possible...
>

Are you sure? Here are shorter versions of the antiderivatives #41, #42
and #44 from Charlwood's appendix:

INT(LN(SIN(x))*SQRT(1 + SIN(x)), x) =
2*COS(x)*(2 - LN(SIN(x)))/SQRT(1 + SIN(x))
- 4*COS(x)/SQRT(COS(x)^2)*ATANH(SQRT(1 - SIN(x)))

INT(SEC(x)/SQRT(SEC(x)^4 - 1), x) =
- 1/SQRT(2)*ATANH(SQRT(SEC(x)^4 - 1)/(SQRT(2)*SEC(x)*TAN(x)))

INT(SIN(x)/SQRT(1 - SIN(x)^6), x) =
SQRT(3)/6*ATANH(SQRT(3)*COS(x)*(1 + SIN(x)^2)/(2*SQRT(1 - SIN(x)^6)))

These were arrived at by piecewise-constant acrobatics.

Martin.