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Topic: first anniversary of the IITS
Replies: 21   Last Post: Jan 10, 2015 4:01 PM

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 clicliclic@freenet.de Posts: 1,245 Registered: 4/26/08
Re: first anniversary of the IITS
Posted: Mar 8, 2014 9:22 AM

Albert Rich schrieb:
>
> On Sunday, March 2, 2014 12:29:28 AM UTC-10, clicl...@freenet.de wrote:
>

> > I have done some extra work and arrived at the following alternative
> > evaluations for the Examples 62, 64, 66 (p. 268) and 118 (p. 309) from
> > Chapter 5:
> >
> > INT(SQRT(TAN(x)),x)=-1/SQRT(2)*ATANH((1+TAN(x))/(SQRT(2)*SQRT(TA~
> > N(x))))+1/SQRT(2)*ATAN(1+SQRT(2)*SQRT(TAN(x)))-1/SQRT(2)*ATAN(1-~
> > SQRT(2)*SQRT(TAN(x)))=-1/SQRT(2)*LN((1+TAN(x)+SQRT(2)*SQRT(TAN(x~
> > )))/SEC(x))+1/SQRT(2)*ATAN(1+SQRT(2)*SQRT(TAN(x)))-1/SQRT(2)*ATA~
> > N(1-SQRT(2)*SQRT(TAN(x)))
> >
> > [...]

>
> For example 62, you combined the two logs into an inverse hyperbolic
> tangent. The two inverse tangents can also be combined to yield the
> elegant antiderivative for sqrt(tan(x)):
>
> -ArcTan[(1-Tan[x])/(Sqrt[2]*Sqrt[Tan[x]])]/Sqrt[2] -
> ArcTanh[(1+Tan[x])/(Sqrt[2]*Sqrt[Tan[x]])]/Sqrt[2]
>
> Did you not do this because of continuity problems?
>

My interest in Example 44 (like the similar but simpler 62, 64, 66, and
118) was mostly in the asymmetric LN evaluation. Please fuse the ATAN's
in numbers 62, 64, 66 as you see fit; I find no continuity problems:

ATAN(1 + SQRT(2)*SQRT(TAN(x))) - ATAN(1 - SQRT(2)*SQRT(TAN(x)))
= ATAN((TAN(x) - 1)/(SQRT(2)*SQRT(TAN(x)))) + pi/2

ATAN(3 + SQRT(2)*SQRT(4 + 3*TAN(2*x)))
- ATAN(3 - SQRT(2)*SQRT(4 + 3*TAN(2*x)))
= ATAN((3*TAN(2*x) - 1)/(SQRT(2)*SQRT(3*TAN(2*x) + 4))) + pi/2

My original proposals for optimization usually excluded evaluations
merely differing in paired LN's rewritten as ATANH, or by the fusing or
splitting of ATAN and ATANH. However, in my optimized evaluations I
freely made such changes, always keeping an eye on the continuity
properties - this is why ACOTH appears in Example 81 (warning: this is
Derive's ACOTH, which differs from Mathematica's ArcCoth, I believe).
Discontinuities at integrand poles never worry me much; I propose that
definite integrals be split automatically where such poles reside in the
integration range (unless there are too many such poles, perhaps even an
infinite number).

I followed Timofeev in Examples 70-71 and 80 where a fused double ATAN
can be written as a simpler ASIN:

ATAN(1 + 2*COS(x)/SQRT(SIN(2*x))) - ATAN(1 - 2*COS(x)/SQRT(SIN(2*x)))
+- pi/2 = ATAN((COS(x) - SIN(x))/SQRT(SIN(2*x)))
= ASIN(COS(x) - SIN(x))

I similarly fused ATAN in Examples 77 and 81, but without using ASIN
because of the awkward piecewise constants.

ATAN was also fused in number 109, but not in 112 and 118 (no longer
mere square roots here, but continuity problems do not arise); we have:

ATAN((2*(1 - 3*SEC(x)^2)^(1/6) - 1)/SQRT(3))
- ATAN((2*(1 - 3*SEC(x)^2)^(1/6) + 1)/SQRT(3))
= ATAN((2*(1 - 3*SEC(x)^2)^(1/3) + 1)/SQRT(3)) - pi/2

ATAN((1 - 2*(1 + 2*COS(x)^9)^(1/6))/SQRT(3))
- ATAN((1 + 2*(1 + 2*COS(x)^9)^(1/6))/SQRT(3))
= ATAN((1 - (1 + 2*COS(x)^9)^(1/3))/(SQRT(3)*(1 + 2*COS(x)^9)^(1/6)))
- pi/2

ATAN(1 - SQRT(2)*COS(2*x)^(1/4)) - ATAN(1 + SQRT(2)*COS(2*x)^(1/4))
= ATAN((1 - SQRT(COS(2*x)))/(SQRT(2)*COS(2*x)^(1/4)))

I thought Examples 90-91 (p. 289) didn't merit attention, yet the ATAN's
in these two actually simplify on recombination or fusing:

INT((SIN(x)*COS(2*x)-2*(SIN(x)-1)*COS(x)^3)/(SIN(x)^2*SQRT(SIN(x~
)^2-5)),x)=(2+2/5*CSC(x))*SQRT(SIN(x)^2-5)-2*ATANH(SIN(x)/SQRT(S~
IN(x)^2-5))+2*ATAN(COS(x)/SQRT(SIN(x)^2-5))-2/SQRT(5)*ATAN(SQRT(~
SIN(x)^2-5)/SQRT(5))-1/SQRT(5)*ATAN(SQRT(5)*COS(x)/SQRT(SIN(x)^2~
-5))=(2+2/5*CSC(x))*SQRT(SIN(x)^2-5)-2*LN(SIN(x)+SQRT(SIN(x)^2-5~
))+2*ATAN(COS(x)/SQRT(SIN(x)^2-5))-2/SQRT(5)*ATAN(SQRT(SIN(x)^2-~
5)/SQRT(5))-1/SQRT(5)*ATAN(SQRT(5)*COS(x)/SQRT(SIN(x)^2-5))

INT(COS(3*x)/(SQRT(3*COS(x)^2-SIN(x)^2)-SQRT(8*COS(x)^2-1)),x)=-~
1/2*SIN(x)*SQRT(4*COS(x)^2-1)-1/2*SIN(x)*SQRT(8*COS(x)^2-1)+3/4*~
ASIN(2*SIN(x)/SQRT(3))+5*SQRT(2)/8*ASIN(4*SIN(x)/SQRT(14))-3/4*A~
TAN(SIN(x)/SQRT(4*COS(x)^2-1))-3/4*ATAN(SIN(x)/SQRT(8*COS(x)^2-1~
))

Sorry for having been a bit unsystematic.

Martin.