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Do some definite integral calculation.

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Hongyi Zhao

unread,
May 4, 2009, 11:08:50 PM5/4/09
to
Hi all,

I've the following equations:


\[\sin \left( \theta \right) = a{x^{^2}} + bx + c\]

and

\[y = \int_0^x {\tan } \left( \theta \right)dx\]

I want to obtain the expression of y as the function of x. How should
I write the code within maple?

Thanks in advance.

--
.: Hongyi Zhao [ hongyi.zhao AT gmail.com ] Free as in Freedom :.

Nasser Abbasi

unread,
May 5, 2009, 1:27:24 AM5/5/09
to

"Hongyi Zhao" <hongy...@gmail.com> wrote in message
news:ncbvv49qvf3up5il8...@4ax.com...

Please see my answer to the same question you posted on sci.math.symbolic
(also you posted the same question on matlab newsgroup).

http://groups.google.com/group/sci.math.symbolic/browse_thread/thread/4d59d6b6fe17c577#

Fyi, It is considered not to be a good idea to post the same question
separately to many news groups. If you want to post the same question to
many newsgroup, you should do it all at once. This way, less people waste
their time answering the same question.

--Nasser


Hongyi Zhao

unread,
May 5, 2009, 4:30:12 AM5/5/09
to
On Mon, 4 May 2009 21:18:43 -0700, "Nasser Abbasi" <n...@12000.org>
wrote:

>> Hi all,
>>
>> I've the following equations:
>>
>> \[\sin \left( \theta \right) = a{x^{^2}} + bx + c\]
>>
>> and
>>
>> \[y = \int_0^x {\tan } \left( \theta \right)dx\]
>>
>> I want to obtain the expression of y as the function of x. How should

>> I write the code with appropriate tools?


>>
>> Thanks in advance.
>>
>> --
>> .: Hongyi Zhao [ hongyi.zhao AT gmail.com ] Free as in Freedom :.
>

>The second equation above is wrong.
>
>You can't have the limit of integration be x, and you are integrating with
>respect to x as well.
>
>(It will also help if you post you equation not in latex form, some might
>have hard time reading it if they are not familiar with latex)

Sorry for my LaTeX form of this issue, now, I give it in the following
ones:

theta=arcsin(a*x^2+b*x+c)

and

y=integrate(tan(theta),x,0,x').

Best regards,

Alan Weiss

unread,
May 5, 2009, 7:32:32 AM5/5/09
to

You can sort of do this with the Symbolic Math Toolbox.

syms a b c x t
mtheta = asin(a*x^2 + b*x + c);
y = int(tan(mtheta),0,t)

The result is only partially satisfactory:

Warning: Explicit integral could not be found.

y =

int((a*x^2 + b*x + c)/(1 - (a*x^2 + b*x + c)^2)^(1/2), x = 0..t)

MATLAB simplified the tan(arcsin) expression, but then did not integrate
what remained.

Alan Weiss
MATLAB mathematical toolbox documentation

Nasser Abbasi

unread,
May 5, 2009, 3:05:27 PM5/5/09
to

"Alan Weiss" <awe...@mathworks.com> wrote in message
news:gtp84g$5ll$1...@fred.mathworks.com...
> Hongyi Zhao wrote:

>> theta=arcsin(a*x^2+b*x+c)
>>
>> and
>>
>> y=integrate(tan(theta),x,0,x').
>>


>
> You can sort of do this with the Symbolic Math Toolbox.
>
> syms a b c x t
> mtheta = asin(a*x^2 + b*x + c);
> y = int(tan(mtheta),0,t)
>
> The result is only partially satisfactory:
>
> Warning: Explicit integral could not be found.
>
> y =
>
> int((a*x^2 + b*x + c)/(1 - (a*x^2 + b*x + c)^2)^(1/2), x = 0..t)
>
> MATLAB simplified the tan(arcsin) expression, but then did not integrate
> what remained.
>
> Alan Weiss
> MATLAB mathematical toolbox documentation

I tired this in Mathematica 7.0, under the assumptions that (a,b,c) are
real, and the upper limit of integration (which I called z) is also real.

theta = ArcSin[a*x^2 + b*x + c]
Assuming[Element[{a, b, c, z}, Reals], Integrate[Tan[theta], {x, 0, z}]]

If[(NotElement[(b + Sqrt[b^2 - 4*a*(1 + c)])/(a*z), Reals] ||
(Element[(b + Sqrt[b^2 - 4*a*(1 + c)])/(a*z), Reals] &&
(Re[(b + Sqrt[b^2 - 4*a*(1 + c)])/(a*z)] <= -2 ||
Re[(b + Sqrt[b^2 - 4*a*(1 + c)])/(a*z)] >= 0))) &&
(NotElement[(b + Sqrt[4*a + b^2 - 4*a*c])/(a*z), Reals] ||
Re[(b + Sqrt[4*a + b^2 - 4*a*c])/(a*z)] <= -2 ||
Re[(b + Sqrt[4*a + b^2 - 4*a*c])/(a*z)] >= 0) &&
(NotElement[(b + Sqrt[b^2 - 4*a*(1 + c)])/(a*z), Reals] ||
Re[(b + Sqrt[b^2 - 4*a*(1 + c)])/(a*z)] <= -2 ||
Re[(b + Sqrt[b^2 - 4*a*(1 + c)])/(a*z)] >= 0) &&
(NotElement[(-b + Sqrt[b^2 - 4*a*(-1 + c)])/(a*z), Reals] ||
Re[(-b + Sqrt[b^2 - 4*a*(-1 + c)])/(a*z)] == 0 ||
Re[(-b + Sqrt[4*a + b^2 - 4*a*c])/(a*z)] >= 2 ||
Re[(b - Sqrt[4*a + b^2 - 4*a*c])/(a*z)] >= 0) &&
(NotElement[(-b + Sqrt[b^2 - 4*a*(1 + c)])/(a*z), Reals] ||
Re[(-b + Sqrt[b^2 - 4*a*(1 + c)])/(a*z)] == 0 ||
Re[(-b + Sqrt[b^2 - 4*a*(1 + c)])/(a*z)] >= 2 ||
Re[(b - Sqrt[b^2 - 4*a*(1 + c)])/(a*z)] >= 0) &&
(NotElement[(b - Sqrt[4*a + b^2 - 4*a*c])/(a*z), Reals] ||
NotElement[(-b + Sqrt[4*a + b^2 - 4*a*c])/(a*z), Reals] ||
Re[(b - Sqrt[4*a + b^2 - 4*a*c])/(a*z)] <= -2 ||
Re[(b - Sqrt[4*a + b^2 - 4*a*c])/(a*z)] >= 0 ||
Re[(-b + Sqrt[4*a + b^2 - 4*a*c])/(a*z)] == 0) &&
(NotElement[(b - Sqrt[b^2 - 4*a*(1 + c)])/(a*z), Reals] ||
NotElement[(-b + Sqrt[b^2 - 4*a*(1 + c)])/(a*z), Reals] ||
Re[(b - Sqrt[b^2 - 4*a*(1 + c)])/(a*z)] <= -2 ||
Re[(b - Sqrt[b^2 - 4*a*(1 + c)])/(a*z)] >= 0 ||
Re[(-b + Sqrt[b^2 - 4*a*(1 + c)])/(a*z)] == 0) &&
(NotElement[-(b/(2*a*z)) - Sqrt[-4*a + b^2 - 4*a*c]/(2*a*z), Reals] ||
Re[-(b/(2*a*z)) - Sqrt[-4*a + b^2 - 4*a*c]/(2*a*z)] == 0 ||
Re[-(b/(2*a*z)) - Sqrt[-4*a + b^2 - 4*a*c]/(2*a*z)] <= 0 ||
NotElement[b/(2*a*z) + Sqrt[-4*a + b^2 - 4*a*c]/(2*a*z), Reals] ||
Re[b/(2*a*z) + Sqrt[-4*a + b^2 - 4*a*c]/(2*a*z)] == -1 ||
Re[b/(2*a*z) + Sqrt[-4*a + b^2 - 4*a*c]/(2*a*z)] <= -1),
-((c*(-b + Sqrt[-4*a + b^2 - 4*a*c])^2*Sqrt[((-b - Sqrt[4*a + b^2 -
4*a*c])*
(-((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + (-b + Sqrt[-4*a + b^2 -
4*a*c])/
(2*a)))/((-b + Sqrt[-4*a + b^2 - 4*a*c])*
(-((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + (-b - Sqrt[4*a + b^2 -
4*a*c])/
(2*a)))]*((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a) - (-b + Sqrt[4*a
+ b^2 - 4*a*c])/
(2*a))*Sqrt[((-b + Sqrt[4*a + b^2 - 4*a*c])*
(-((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + (-b + Sqrt[-4*a + b^2 -
4*a*c])/
(2*a)))/((-b + Sqrt[-4*a + b^2 - 4*a*c])*
(-((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + (-b + Sqrt[4*a + b^2 -
4*a*c])/
(2*a)))]*Sqrt[((-b - Sqrt[-4*a + b^2 - 4*a*c])*
(-((-b + Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + (-b + Sqrt[4*a + b^2 -
4*a*c])/(2*a)))/
((-b + Sqrt[-4*a + b^2 - 4*a*c])*(-((-b - Sqrt[-4*a + b^2 -
4*a*c])/(2*a)) +
(-b + Sqrt[4*a + b^2 - 4*a*c])/(2*a)))]*
EllipticF[ArcSin[Sqrt[((-b - Sqrt[-4*a + b^2 - 4*a*c])*
(-((-b + Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + (-b + Sqrt[4*a +
b^2 - 4*a*c])/
(2*a)))/((-b + Sqrt[-4*a + b^2 - 4*a*c])*
(-((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + (-b + Sqrt[4*a +
b^2 - 4*a*c])/
(2*a)))]], (((-b + Sqrt[-4*a + b^2 - 4*a*c])/(2*a) -
(-b - Sqrt[4*a + b^2 - 4*a*c])/(2*a))*((-b - Sqrt[-4*a + b^2 -
4*a*c])/(2*a) -
(-b + Sqrt[4*a + b^2 - 4*a*c])/(2*a)))/(((-b - Sqrt[-4*a + b^2 -
4*a*c])/(2*a) -
(-b - Sqrt[4*a + b^2 - 4*a*c])/(2*a))*((-b + Sqrt[-4*a + b^2 -
4*a*c])/(2*a) -
(-b + Sqrt[4*a + b^2 - 4*a*c])/(2*a)))])/(2*a^2*Sqrt[1 - c^2]*
(-((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + (-b + Sqrt[-4*a + b^2 -
4*a*c])/(2*a))*
(-((-b + Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + (-b + Sqrt[4*a + b^2 -
4*a*c])/(2*a)))) +
(2*c*((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a) - (-b + Sqrt[4*a + b^2 -
4*a*c])/(2*a))*
Sqrt[((-((-b + Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + (-b + Sqrt[4*a +
b^2 - 4*a*c])/
(2*a))*(-((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + z))/
((-((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + (-b + Sqrt[4*a + b^2 -
4*a*c])/(2*a))*
(-((-b + Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + z))]*
(-((-b + Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + z)^2*
Sqrt[((-((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + (-b + Sqrt[-4*a +
b^2 - 4*a*c])/
(2*a))*(-((-b - Sqrt[4*a + b^2 - 4*a*c])/(2*a)) + z))/
((-((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + (-b - Sqrt[4*a + b^2 -
4*a*c])/(2*a))*
(-((-b + Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + z))]*
Sqrt[((-((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + (-b + Sqrt[-4*a +
b^2 - 4*a*c])/
(2*a))*(-((-b + Sqrt[4*a + b^2 - 4*a*c])/(2*a)) + z))/
((-((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + (-b + Sqrt[4*a + b^2 -
4*a*c])/(2*a))*
(-((-b + Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + z))]*
EllipticF[ArcSin[Sqrt[((-((-b + Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) +
(-b + Sqrt[4*a + b^2 - 4*a*c])/(2*a))*(-((-b - Sqrt[-4*a + b^2 -
4*a*c])/
(2*a)) + z))/((-((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) +
(-b + Sqrt[4*a + b^2 - 4*a*c])/(2*a))*(-((-b + Sqrt[-4*a + b^2 -
4*a*c])/
(2*a)) + z))]], (((-b + Sqrt[-4*a + b^2 - 4*a*c])/(2*a) -
(-b - Sqrt[4*a + b^2 - 4*a*c])/(2*a))*((-b - Sqrt[-4*a + b^2 -
4*a*c])/(2*a) -
(-b + Sqrt[4*a + b^2 - 4*a*c])/(2*a)))/(((-b - Sqrt[-4*a + b^2 -
4*a*c])/(2*a) -
(-b - Sqrt[4*a + b^2 - 4*a*c])/(2*a))*((-b + Sqrt[-4*a + b^2 -
4*a*c])/(2*a) -
(-b + Sqrt[4*a + b^2 - 4*a*c])/(2*a)))])/
((-((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + (-b + Sqrt[-4*a + b^2 -
4*a*c])/(2*a))*
(-((-b + Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + (-b + Sqrt[4*a + b^2 -
4*a*c])/(2*a))*
Sqrt[1 - (c + z*(b + a*z))^2]) - (b*(-b + Sqrt[-4*a + b^2 - 4*a*c])^2*
Sqrt[((b + Sqrt[-4*a + b^2 - 4*a*c])*(Sqrt[-4*a + b^2 - 4*a*c] -
Sqrt[4*a + b^2 - 4*a*c]))/((-b + Sqrt[-4*a + b^2 - 4*a*c])*
(Sqrt[-4*a + b^2 - 4*a*c] + Sqrt[4*a + b^2 - 4*a*c]))]*
Sqrt[(Sqrt[-4*a + b^2 - 4*a*c]*(-b - Sqrt[4*a + b^2 - 4*a*c]))/
(a*(-b + Sqrt[-4*a + b^2 - 4*a*c])*(-((-b - Sqrt[-4*a + b^2 -
4*a*c])/(2*a)) +
(-b - Sqrt[4*a + b^2 - 4*a*c])/(2*a)))]*((-b - Sqrt[-4*a + b^2 -
4*a*c])/(2*a) -
(-b + Sqrt[4*a + b^2 - 4*a*c])/(2*a))*
Sqrt[(Sqrt[-4*a + b^2 - 4*a*c]*(-b + Sqrt[4*a + b^2 - 4*a*c]))/
(a*(-b + Sqrt[-4*a + b^2 - 4*a*c])*(-((-b - Sqrt[-4*a + b^2 -
4*a*c])/(2*a)) +
(-b + Sqrt[4*a + b^2 - 4*a*c])/(2*a)))]*
(-((1/(2*a))*((-b + Sqrt[-4*a + b^2 - 4*a*c])*EllipticF[
ArcSin[Sqrt[((b + Sqrt[-4*a + b^2 - 4*a*c])*(Sqrt[-4*a + b^2 -
4*a*c] - Sqrt[
4*a + b^2 - 4*a*c]))/((-b + Sqrt[-4*a + b^2 - 4*a*c])*
(Sqrt[-4*a + b^2 - 4*a*c] + Sqrt[4*a + b^2 - 4*a*c]))]],
(Sqrt[-4*a + b^2 - 4*a*c] + Sqrt[4*a + b^2 - 4*a*c])^2/
(Sqrt[-4*a + b^2 - 4*a*c] - Sqrt[4*a + b^2 - 4*a*c])^2])) +
(1/a)*(Sqrt[-4*a + b^2 - 4*a*c]*EllipticPi[
(-((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + (-b + Sqrt[4*a + b^2 -
4*a*c])/(2*a))/
(-((-b + Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + (-b + Sqrt[4*a + b^2 -
4*a*c])/
(2*a)), ArcSin[Sqrt[((b + Sqrt[-4*a + b^2 - 4*a*c])*(Sqrt[-4*a +
b^2 - 4*a*c] -
Sqrt[4*a + b^2 - 4*a*c]))/((-b + Sqrt[-4*a + b^2 - 4*a*c])*
(Sqrt[-4*a + b^2 - 4*a*c] + Sqrt[4*a + b^2 - 4*a*c]))]],
(Sqrt[-4*a + b^2 - 4*a*c] + Sqrt[4*a + b^2 - 4*a*c])^2/
(Sqrt[-4*a + b^2 - 4*a*c] - Sqrt[4*a + b^2 - 4*a*c])^2])))/
(2*a^2*Sqrt[1 - c^2]*(-((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) +
(-b + Sqrt[-4*a + b^2 - 4*a*c])/(2*a))*((-b + Sqrt[-4*a + b^2 -
4*a*c])/(2*a) -
(-b + Sqrt[4*a + b^2 - 4*a*c])/(2*a))) - (1/Sqrt[1 - c^2])*
(a*(-(((-b - Sqrt[-4*a + b^2 - 4*a*c])*(-b - Sqrt[4*a + b^2 - 4*a*c])*
(-b + Sqrt[4*a + b^2 - 4*a*c]))/(8*a^3)) +
(1/(4*a^2))*((-b + Sqrt[-4*a + b^2 - 4*a*c])^2*
Sqrt[((b + Sqrt[-4*a + b^2 - 4*a*c])*(Sqrt[-4*a + b^2 - 4*a*c] -
Sqrt[4*a + b^2 - 4*a*c]))/((-b + Sqrt[-4*a + b^2 - 4*a*c])*
(Sqrt[-4*a + b^2 - 4*a*c] + Sqrt[4*a + b^2 - 4*a*c]))]*
Sqrt[(Sqrt[-4*a + b^2 - 4*a*c]*(-b - Sqrt[4*a + b^2 - 4*a*c]))/
(a*(-b + Sqrt[-4*a + b^2 - 4*a*c])*(-((-b - Sqrt[-4*a + b^2 -
4*a*c])/(2*a)) +
(-b - Sqrt[4*a + b^2 - 4*a*c])/(2*a)))]*
Sqrt[(Sqrt[-4*a + b^2 - 4*a*c]*(-b + Sqrt[4*a + b^2 - 4*a*c]))/
(a*(-b + Sqrt[-4*a + b^2 - 4*a*c])*(-((-b - Sqrt[-4*a + b^2 -
4*a*c])/(2*a)) +
(-b + Sqrt[4*a + b^2 - 4*a*c])/(2*a)))]*
(-((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + (-b + Sqrt[4*a + b^2 -
4*a*c])/(2*a))*
((a*(-((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + (-b - Sqrt[4*a +
b^2 - 4*a*c])/
(2*a))*EllipticE[ArcSin[Sqrt[((b + Sqrt[-4*a + b^2 - 4*a*c])*
(Sqrt[-4*a + b^2 - 4*a*c] - Sqrt[4*a + b^2 - 4*a*c]))/(
(-b + Sqrt[-4*a + b^2 - 4*a*c])*(Sqrt[-4*a + b^2 - 4*a*c] +
Sqrt[4*a + b^2 - 4*a*c]))]], (Sqrt[-4*a + b^2 - 4*a*c] +
Sqrt[
4*a + b^2 - 4*a*c])^2/(Sqrt[-4*a + b^2 - 4*a*c] - Sqrt[4*a +
b^2 - 4*a*c])^
2])/Sqrt[-4*a + b^2 - 4*a*c] +
(a*(((-b + Sqrt[-4*a + b^2 - 4*a*c])*(-((-b + Sqrt[-4*a + b^2 -
4*a*c])/(2*a)) -
(-b + Sqrt[4*a + b^2 - 4*a*c])/(2*a)))/(2*a) -
((-b - Sqrt[-4*a + b^2 - 4*a*c])*((-b + Sqrt[-4*a + b^2 -
4*a*c])/(2*a) -
(-b + Sqrt[4*a + b^2 - 4*a*c])/(2*a)))/(2*a))*
EllipticF[ArcSin[Sqrt[((b + Sqrt[-4*a + b^2 - 4*a*c])*(Sqrt[-4*a
+ b^2 -
4*a*c] - Sqrt[4*a + b^2 - 4*a*c]))/((-b + Sqrt[-4*a +
b^2 - 4*a*c])*
(Sqrt[-4*a + b^2 - 4*a*c] + Sqrt[4*a + b^2 - 4*a*c]))]],
(Sqrt[-4*a + b^2 - 4*a*c] + Sqrt[4*a + b^2 - 4*a*c])^2/
(Sqrt[-4*a + b^2 - 4*a*c] - Sqrt[4*a + b^2 - 4*a*c])^2])/
(Sqrt[-4*a + b^2 - 4*a*c]*(-((-b + Sqrt[-4*a + b^2 -
4*a*c])/(2*a)) +
(-b + Sqrt[4*a + b^2 - 4*a*c])/(2*a))) -
((-((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) - (-b + Sqrt[-4*a +
b^2 - 4*a*c])/
(2*a) - (-b - Sqrt[4*a + b^2 - 4*a*c])/(2*a) - (-b + Sqrt[4*a +
b^2 - 4*a*c])/
(2*a))*EllipticPi[(-((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) +
(-b + Sqrt[4*a + b^2 - 4*a*c])/(2*a))/(-((-b + Sqrt[-4*a +
b^2 - 4*a*c])/
(2*a)) + (-b + Sqrt[4*a + b^2 - 4*a*c])/(2*a)),
ArcSin[Sqrt[((b + Sqrt[-4*a + b^2 - 4*a*c])*(Sqrt[-4*a + b^2 -
4*a*c] -
Sqrt[4*a + b^2 - 4*a*c]))/((-b + Sqrt[-4*a + b^2 - 4*a*c])*
(Sqrt[-4*a + b^2 - 4*a*c] + Sqrt[4*a + b^2 - 4*a*c]))]],
(Sqrt[-4*a + b^2 - 4*a*c] + Sqrt[4*a + b^2 - 4*a*c])^2/
(Sqrt[-4*a + b^2 - 4*a*c] - Sqrt[4*a + b^2 - 4*a*c])^2])/
(-((-b + Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + (-b + Sqrt[4*a + b^2 -
4*a*c])/
(2*a)))))) + (2*b*((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a) -
(-b + Sqrt[4*a + b^2 - 4*a*c])/(2*a))*(-((-b + Sqrt[-4*a + b^2 -
4*a*c])/(2*a)) + z)^
2*Sqrt[(Sqrt[-4*a + b^2 - 4*a*c]*(-((-b - Sqrt[4*a + b^2 -
4*a*c])/(2*a)) + z))/
(a*(-((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + (-b - Sqrt[4*a + b^2 -
4*a*c])/(2*a))*
(-((-b + Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + z))]*
Sqrt[(Sqrt[-4*a + b^2 - 4*a*c]*(-((-b + Sqrt[4*a + b^2 - 4*a*c])/(2*a))
+ z))/
(a*(-((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + (-b + Sqrt[4*a + b^2 -
4*a*c])/(2*a))*
(-((-b + Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + z))]*
Sqrt[((Sqrt[-4*a + b^2 - 4*a*c] - Sqrt[4*a + b^2 - 4*a*c])*
(b + Sqrt[-4*a + b^2 - 4*a*c] + 2*a*z))/((Sqrt[-4*a + b^2 - 4*a*c] +
Sqrt[4*a + b^2 - 4*a*c])*(-b + Sqrt[-4*a + b^2 - 4*a*c] - 2*a*z))]*
(-((1/(2*a))*((-b + Sqrt[-4*a + b^2 - 4*a*c])*EllipticF[
ArcSin[Sqrt[((Sqrt[-4*a + b^2 - 4*a*c] - Sqrt[4*a + b^2 - 4*a*c])*
(b + Sqrt[-4*a + b^2 - 4*a*c] + 2*a*z))/((Sqrt[-4*a + b^2 -
4*a*c] + Sqrt[
4*a + b^2 - 4*a*c])*(-b + Sqrt[-4*a + b^2 - 4*a*c] -
2*a*z))]],
(Sqrt[-4*a + b^2 - 4*a*c] + Sqrt[4*a + b^2 - 4*a*c])^2/
(Sqrt[-4*a + b^2 - 4*a*c] - Sqrt[4*a + b^2 - 4*a*c])^2])) +
(1/a)*(Sqrt[-4*a + b^2 - 4*a*c]*EllipticPi[
(-((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + (-b + Sqrt[4*a + b^2 -
4*a*c])/(2*a))/
(-((-b + Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + (-b + Sqrt[4*a + b^2 -
4*a*c])/
(2*a)), ArcSin[Sqrt[((Sqrt[-4*a + b^2 - 4*a*c] - Sqrt[4*a +
b^2 - 4*a*c])*
(b + Sqrt[-4*a + b^2 - 4*a*c] + 2*a*z))/((Sqrt[-4*a + b^2 -
4*a*c] +
Sqrt[4*a + b^2 - 4*a*c])*(-b + Sqrt[-4*a + b^2 - 4*a*c] -
2*a*z))]],
(Sqrt[-4*a + b^2 - 4*a*c] + Sqrt[4*a + b^2 - 4*a*c])^2/
(Sqrt[-4*a + b^2 - 4*a*c] - Sqrt[4*a + b^2 - 4*a*c])^2])))/
((-((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + (-b + Sqrt[-4*a + b^2 -
4*a*c])/(2*a))*
((-b + Sqrt[-4*a + b^2 - 4*a*c])/(2*a) - (-b + Sqrt[4*a + b^2 -
4*a*c])/(2*a))*
Sqrt[1 - (c + z*(b + a*z))^2]) +
(a*((-((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + z)*
(-((-b - Sqrt[4*a + b^2 - 4*a*c])/(2*a)) + z)*
(-((-b + Sqrt[4*a + b^2 - 4*a*c])/(2*a)) + z) +
(-((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + (-b + Sqrt[4*a + b^2 -
4*a*c])/(2*a))*
(-((-b + Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + z)^2*
Sqrt[(Sqrt[-4*a + b^2 - 4*a*c]*(-((-b - Sqrt[4*a + b^2 -
4*a*c])/(2*a)) + z))/
(a*(-((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + (-b - Sqrt[4*a +
b^2 - 4*a*c])/
(2*a))*(-((-b + Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + z))]*
Sqrt[(Sqrt[-4*a + b^2 - 4*a*c]*(-((-b + Sqrt[4*a + b^2 -
4*a*c])/(2*a)) + z))/
(a*(-((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + (-b + Sqrt[4*a +
b^2 - 4*a*c])/
(2*a))*(-((-b + Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + z))]*
Sqrt[((Sqrt[-4*a + b^2 - 4*a*c] - Sqrt[4*a + b^2 - 4*a*c])*
(b + Sqrt[-4*a + b^2 - 4*a*c] + 2*a*z))/((Sqrt[-4*a + b^2 - 4*a*c]
+
Sqrt[4*a + b^2 - 4*a*c])*(-b + Sqrt[-4*a + b^2 - 4*a*c] -
2*a*z))]*
((a*(-((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + (-b - Sqrt[4*a +
b^2 - 4*a*c])/
(2*a))*EllipticE[ArcSin[Sqrt[((Sqrt[-4*a + b^2 - 4*a*c] -
Sqrt[4*a + b^2 -
4*a*c])*(b + Sqrt[-4*a + b^2 - 4*a*c] + 2*a*z))/
((Sqrt[-4*a + b^2 - 4*a*c] + Sqrt[4*a + b^2 - 4*a*c])*(-b +
Sqrt[-4*a + b^2 - 4*a*c] - 2*a*z))]],
(Sqrt[-4*a + b^2 - 4*a*c] + Sqrt[4*a + b^2 - 4*a*c])^2/
(Sqrt[-4*a + b^2 - 4*a*c] - Sqrt[4*a + b^2 - 4*a*c])^2])/
Sqrt[-4*a + b^2 - 4*a*c] +
(a*(((-b + Sqrt[-4*a + b^2 - 4*a*c])*(-((-b + Sqrt[-4*a + b^2 -
4*a*c])/(2*a)) -
(-b + Sqrt[4*a + b^2 - 4*a*c])/(2*a)))/(2*a) -
((-b - Sqrt[-4*a + b^2 - 4*a*c])*((-b + Sqrt[-4*a + b^2 -
4*a*c])/(2*a) -
(-b + Sqrt[4*a + b^2 - 4*a*c])/(2*a)))/(2*a))*
EllipticF[ArcSin[Sqrt[((Sqrt[-4*a + b^2 - 4*a*c] - Sqrt[4*a +
b^2 - 4*a*c])*(b +
Sqrt[-4*a + b^2 - 4*a*c] + 2*a*z))/((Sqrt[-4*a + b^2 -
4*a*c] +
Sqrt[4*a + b^2 - 4*a*c])*(-b + Sqrt[-4*a + b^2 - 4*a*c] -
2*a*z))]],
(Sqrt[-4*a + b^2 - 4*a*c] + Sqrt[4*a + b^2 - 4*a*c])^2/
(Sqrt[-4*a + b^2 - 4*a*c] - Sqrt[4*a + b^2 - 4*a*c])^2])/
(Sqrt[-4*a + b^2 - 4*a*c]*(-((-b + Sqrt[-4*a + b^2 - 4*a*c])/(2*a))
+
(-b + Sqrt[4*a + b^2 - 4*a*c])/(2*a))) -
((-((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) - (-b + Sqrt[-4*a + b^2 -
4*a*c])/
(2*a) - (-b - Sqrt[4*a + b^2 - 4*a*c])/(2*a) - (-b + Sqrt[4*a +
b^2 - 4*a*c])/
(2*a))*EllipticPi[(-((-b - Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) +
(-b + Sqrt[4*a + b^2 - 4*a*c])/(2*a))/(-((-b + Sqrt[-4*a +
b^2 - 4*a*c])/(2*
a)) + (-b + Sqrt[4*a + b^2 - 4*a*c])/(2*a)),
ArcSin[Sqrt[((Sqrt[-4*a + b^2 - 4*a*c] - Sqrt[4*a + b^2 -
4*a*c])*(b +
Sqrt[-4*a + b^2 - 4*a*c] + 2*a*z))/((Sqrt[-4*a + b^2 -
4*a*c] +
Sqrt[4*a + b^2 - 4*a*c])*(-b + Sqrt[-4*a + b^2 - 4*a*c] -
2*a*z))]],
(Sqrt[-4*a + b^2 - 4*a*c] + Sqrt[4*a + b^2 - 4*a*c])^2/
(Sqrt[-4*a + b^2 - 4*a*c] - Sqrt[4*a + b^2 - 4*a*c])^2])/
(-((-b + Sqrt[-4*a + b^2 - 4*a*c])/(2*a)) + (-b + Sqrt[4*a + b^2 -
4*a*c])/
(2*a)))))/Sqrt[1 - (c + z*(b + a*z))^2],
Integrate[(c + x*(b + a*x))/Sqrt[1 - (c + x*(b + a*x))^2], {x, 0, z},
Assumptions -> Element[a | b | c | z, Reals] &&
!((NotElement[(b + Sqrt[b^2 - 4*a*(1 + c)])/(a*z), Reals] ||
(Element[(b + Sqrt[b^2 - 4*a*(1 + c)])/(a*z), Reals] &&
(Re[(b + Sqrt[b^2 - 4*a*(1 + c)])/(a*z)] <= -2 ||
Re[(b + Sqrt[b^2 - 4*a*(1 + c)])/(a*z)] >= 0))) &&
(NotElement[(b + Sqrt[4*a + b^2 - 4*a*c])/(a*z), Reals] ||
Re[(b + Sqrt[4*a + b^2 - 4*a*c])/(a*z)] <= -2 ||
Re[(b + Sqrt[4*a + b^2 - 4*a*c])/(a*z)] >= 0) &&
(NotElement[(b + Sqrt[b^2 - 4*a*(1 + c)])/(a*z), Reals] ||
Re[(b + Sqrt[b^2 - 4*a*(1 + c)])/(a*z)] <= -2 ||
Re[(b + Sqrt[b^2 - 4*a*(1 + c)])/(a*z)] >= 0) &&
(NotElement[(-b + Sqrt[b^2 - 4*a*(-1 + c)])/(a*z), Reals] ||
Re[(-b + Sqrt[b^2 - 4*a*(-1 + c)])/(a*z)] == 0 ||
Re[(-b + Sqrt[4*a + b^2 - 4*a*c])/(a*z)] >= 2 ||
Re[(b - Sqrt[4*a + b^2 - 4*a*c])/(a*z)] >= 0) &&
(NotElement[(-b + Sqrt[b^2 - 4*a*(1 + c)])/(a*z), Reals] ||
Re[(-b + Sqrt[b^2 - 4*a*(1 + c)])/(a*z)] == 0 ||
Re[(-b + Sqrt[b^2 - 4*a*(1 + c)])/(a*z)] >= 2 ||
Re[(b - Sqrt[b^2 - 4*a*(1 + c)])/(a*z)] >= 0) &&
(NotElement[(b - Sqrt[4*a + b^2 - 4*a*c])/(a*z), Reals] ||
NotElement[(-b + Sqrt[4*a + b^2 - 4*a*c])/(a*z), Reals] ||
Re[(b - Sqrt[4*a + b^2 - 4*a*c])/(a*z)] <= -2 ||
Re[(b - Sqrt[4*a + b^2 - 4*a*c])/(a*z)] >= 0 ||
Re[(-b + Sqrt[4*a + b^2 - 4*a*c])/(a*z)] == 0) &&
(NotElement[(b - Sqrt[b^2 - 4*a*(1 + c)])/(a*z), Reals] ||
NotElement[(-b + Sqrt[b^2 - 4*a*(1 + c)])/(a*z), Reals] ||
Re[(b - Sqrt[b^2 - 4*a*(1 + c)])/(a*z)] <= -2 ||
Re[(b - Sqrt[b^2 - 4*a*(1 + c)])/(a*z)] >= 0 ||
Re[(-b + Sqrt[b^2 - 4*a*(1 + c)])/(a*z)] == 0) &&
(NotElement[-(b/(2*a*z)) - Sqrt[-4*a + b^2 - 4*a*c]/(2*a*z), Reals]
||
Re[-(b/(2*a*z)) - Sqrt[-4*a + b^2 - 4*a*c]/(2*a*z)] == 0 ||
Re[-(b/(2*a*z)) - Sqrt[-4*a + b^2 - 4*a*c]/(2*a*z)] <= 0 ||
NotElement[b/(2*a*z) + Sqrt[-4*a + b^2 - 4*a*c]/(2*a*z), Reals] ||
Re[b/(2*a*z) + Sqrt[-4*a + b^2 - 4*a*c]/(2*a*z)] == -1 ||
Re[b/(2*a*z) + Sqrt[-4*a + b^2 - 4*a*c]/(2*a*z)] <= -1))]]


--Nasser


clicl...@freenet.de

unread,
May 5, 2009, 4:45:53 PM5/5/09
to

Nasser Abbasi schrieb:

> news:gtp84g$5ll$1...@fred.mathworks.com...
> > Hongyi Zhao wrote:
>
> >> theta=arcsin(a*x^2+b*x+c)
> >>
> >> and
> >>
> >> y=integrate(tan(theta),x,0,x').
> >>
>

This is a beauty, isn't it - and it doesn't contain the imaginary unit
anymore! But I suspect a less impressive expression also not involving
"ImaginaryI" can be obtained by a suitable transformation of Andreas'
result.

Martin.

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