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Editor: pabjr Time: 2020/04/14 22:03:00 GMT+0
Note: Exp Integral

added:

From pabjr Tue Apr 14 22:03:00 +0000 2020
From: pabjr
Date: Tue, 14 Apr 2020 22:03:00 +0000
Subject: Exp Integral
Message-ID: <20200414220300+0000@fricas-wiki.math.uni.wroc.pl>

integrate((1 - x/a)*%e^(-(b (x - a))/(x + d*a)),x)

Integration

Let's do some integration examples:

fricas

integrate(%e^x, x)

$\label{eq1}{e}^{x}$

(1)

Type: Union(Expression(Integer),...)

int(cos(x),x,0,pi);

reduce

$\displaylines{\qdd 0 \cr}$

fricas

integrate(x^2/sqrt(4-x^2),x)

$\label{eq2}{\left( \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle -{{32}\ {\sqrt{-{{x}^{2}}+ 4}}}- \ \ \displaystyle {8 \ {{x}^{2}}}+{64}$

(2)

Type: Union(Expression(Integer),...)

Below FriCAS? gives up because sign of a is unknown:

fricas

integrate(exp(-a*x^2),x=0..%plusInfinity)

$\label{eq3}\mbox{\tt "failed"}$

(3)

Type: Union(fail: failed,...)

The following won't "work", see CommonMistakes?:

fricas

integrate(exp(-a::PositiveInteger*x^2),x=0..%plusInfinity)

   Cannot convert from type Variable(a) to PositiveInteger for value
   a

fricas

integrate((x^3+x^2+2)/(x*(x^2-1)^2), x)

$\label{eq4}{\left( \begin{array}{@{}l} \displaystyle {{\left(-{5 \ {{x}^{2}}}+ 5 \right)}\ {\log \left({x + 1}\right)}}+{{\left({8 \ {{x}^{2}}}- 8 \right)}\ {\log \left({x}\right)}}+ \ \ \displaystyle {{\left(-{3 \ {{x}^{2}}}+ 3 \right)}\ {\log \left({x - 1}\right)}}-{2 \ x}- 6$

(4)

Type: Union(Expression(Integer),...)

fricas

integrate(2*x/sin(x)^2,x)

$\label{eq5}{\left( \begin{array}{@{}l} \displaystyle {2 \ {\sin \left({x}\right)}\ {\log \left({{\sin \left({x}\right)}\over{{\cos \left({x}\right)}+ 1}}\right)}}-{2 \ {\sin \left({x}\right)}\ {\log \left({2 \over{{\cos \left({x}\right)}+ 1}}\right)}}- \ \ \displaystyle {2 \ x \ {\cos \left({x}\right)}}$

(5)

Type: Union(Expression(Integer),...)

Comparing Axiom and Reduce:

fricas

integrate(sin(1/x),x)

$\label{eq6}\int^{ \displaystyle x}{{\sin \left({1 \over \%A}\right)}\ {d \%A}}$

(6)

Type: Union(Expression(Integer),...)

int(sin(1/x),x);

reduce

$\displaylines{\qdd \int {\sin \(\frac{1}{ x}$

load_package algint;

int(sin(1/x),x);

reduce

$\displaylines{\qdd \int {\sin \(\frac{1}{ x}$

A different problem, where Axiom has to give up:

fricas

integrate(sqrt(sin(1/x)),x)

   >> Error detected within library code:
   integrate: implementation incomplete (constant residues)

In Reduce:

load_package algint;

int(sqrt(sin(1/x)),x);

reduce

$\displaylines{\qdd \frac{2\cdot \sqrt{\sin \(\frac{1}{ x}$

fricas

integrate(exp(-x^2),x)

$\label{eq7}{{\erf \left({x}\right)}\ {\sqrt{\pi}}}\over 2$

(7)

Type: Union(Expression(Integer),...)

fricas

integrate(sin(x)/x,x)

$\label{eq8}Si \left({x}\right)$

(8)

Type: Union(Expression(Integer),...)

fricas

differentiate(%,x)

$\label{eq9}{\sin \left({x}\right)}\over x$

(9)

Type: Expression(Integer)

fricas

integrate(sin(1/x),x=%minusInfinity..%plusInfinity,"noPole")

   >> Error detected within library code:
   integrate: pole in path of integration

fricas

integrate(2*x/sin(x)^2,x=1/2..1)

$\label{eq10}potentialPole$

(10)

Type: Union(pole: potentialPole,...)

fricas

integrate(sin(x),x=0..%pi/2)

$\label{eq11}1$

(11)

Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

fricas

integrate(atan(x/a)/x,x)

$\label{eq12}\int^{ \displaystyle x}{{{\arctan \left({\%A \over a}\right)}\over \%A}\ {d \%A}}$

(12)

Type: Union(Expression(Integer),...)

fricas

integrate(1/(a+z^3), z=0..1,"noPole")

$\label{eq13}{\left( \begin{array}{@{}l} \displaystyle -{{\sqrt{3}}\ {\log{\left({{3 \ {{a}^{2}}\ {{\root{3}\of{{a}^{2}}}^{2}}}+{{\left(-{2 \ {{a}^{3}}}+{{a}^{2}}\right)}\ {\root{3}\of{{a}^{2}}}}+{{a}^{4}}-{2 \ {{a}^{3}}}}\right)}}}+{2 \ {\sqrt{3}}\ {\log \left({{{\root{3}\of{{a}^{2}}}^{2}}+{2 \ a \ {\root{3}\of{{a}^{2}}}}+{{a}^{2}}}\right)}}+ \ \ \displaystyle {{12}\ {\arctan \left({{{2 \ {\sqrt{3}}\ {\root{3}\of{{a}^{2}}}}-{a \ {\sqrt{3}}}}\over{3 \ a}}\right)}}+{2 \ \pi}$

(13)

Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

fricas

integrate(x^3+x^2/4+x,x)

$\label{eq14}{{1 \over 4}\ {{x}^{4}}}+{{1 \over{12}}\ {{x}^{3}}}+{{1 \over 2}\ {{x}^{2}}}$

(14)

Type: Polynomial(Fraction(Integer))

You cannot integrate Expression Float

fricas

integrate(50*%e^(-0.02*t),t)

   There are 12 exposed and 10 unexposed library operations named 
      integrate having 2 argument(s) but none was determined to be 
      applicable. Use HyperDoc Browse, or issue
                            )display op integrate
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named 
      integrate with argument type(s) 
                              Expression(Float)
                                 Variable(t)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

But symbolic integration works with integer expressions

fricas

integrate(50*%e^(-0.02*t)::Expression Fraction Integer,t)

$\label{eq15}-{{2500}\ {{e}^{-{{1 \over{50}}\ t}}}}$

(15)

Type: Union(Expression(Fraction(Integer)),...)

fricas

integrate(exp(cos(x)),x)

$\label{eq16}\int^{ \displaystyle x}{{{e}^{\cos \left({\%A}\right)}}\ {d \%A}}$

(16)

Type: Union(Expression(Integer),...)

fricas

integrate(sin(x),x)

$\label{eq17}-{\cos \left({x}\right)}$

(17)

Type: Union(Expression(Integer),...)

fricas

integrate(%,x)

$\label{eq18}-{\sin \left({x}\right)}$

(18)

Type: Union(Expression(Integer),...)

fricas

integrate(a/h - c*h/12 + (b/h)*r + (c/h)*r^2,r)

$\label{eq19}{{4 \ c \ {{r}^{3}}}+{6 \ b \ {{r}^{2}}}+{{\left(-{c \ {{h}^{2}}}+{{12}\ a}\right)}\ r}}\over{{12}\ h}$

(19)

Type: Union(Expression(Integer),...)

fricas

integrate(exp(-(a+b*t)^2/2),t)

$\label{eq20}{{\erf \left({{{b \ t}+ a}\over{\sqrt{2}}}\right)}\ {\sqrt{\pi}}}\over{b \ {\sqrt{2}}}$

(20)

Type: Union(Expression(Integer),...)

fricas

integrate(exp(-(a+b*t)^2/t),t)

$\label{eq21}\int^{ \displaystyle t}{{{e}^{{-{{{\%A}^{2}}\ {{b}^{2}}}-{2 \ \%A \ a \ b}-{{a}^{2}}}\over \%A}}\ {d \%A}}$

(21)

Type: Union(Expression(Integer),...)

fricas

integrate(exp(-1/t),t)

$\label{eq22}\int^{ \displaystyle t}{{{e}^{-{1 \over \%A}}}\ {d \%A}}$

(22)

Type: Union(Expression(Integer),...)

fricas

integrate(exp(-1/t),t=1..x)

$\label{eq23}potentialPole$

(23)

Type: Union(pole: potentialPole,...)

Unfortunately, there is currently no easy way to make "assumptions" about variables. Thus, The following won't work:

  \begin{axiom}
   assume(x, real)
   integrate(exp(-1/t),t=1..x)
   \end{axiom}

fricas

integrate(t*exp(-(a+b*t)^2/2),t)

$\label{eq24}{-{a \ {\erf \left({{{b \ t}+ a}\over{\sqrt{2}}}\right)}\ {\sqrt{\pi}}}-{{\sqrt{2}}\ {{e}^{{-{{{b}^{2}}\ {{t}^{2}}}-{2 \ a \ b \ t}-{{a}^{2}}}\over 2}}}}\over{{{b}^{2}}\ {\sqrt{2}}}$

(24)

Type: Union(Expression(Integer),...)

fricas

integrate(1/(a+z^3), z=0..1,"noPole")

$\label{eq25}{\left( \begin{array}{@{}l} \displaystyle -{{\sqrt{3}}\ {\log{\left({{3 \ {{a}^{2}}\ {{\root{3}\of{{a}^{2}}}^{2}}}+{{\left(-{2 \ {{a}^{3}}}+{{a}^{2}}\right)}\ {\root{3}\of{{a}^{2}}}}+{{a}^{4}}-{2 \ {{a}^{3}}}}\right)}}}+{2 \ {\sqrt{3}}\ {\log \left({{{\root{3}\of{{a}^{2}}}^{2}}+{2 \ a \ {\root{3}\of{{a}^{2}}}}+{{a}^{2}}}\right)}}+ \ \ \displaystyle {{12}\ {\arctan \left({{{2 \ {\sqrt{3}}\ {\root{3}\of{{a}^{2}}}}-{a \ {\sqrt{3}}}}\over{3 \ a}}\right)}}+{2 \ \pi}$

(25)

Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

From the ReduceProblem?:

fricas

integrate(1/sqrt(2*%pi)*exp(-1/2*log(x)^2),x=0..%plusInfinity)

$\label{eq26}\mbox{\tt "failed"}$

(26)

Type: Union(fail: failed,...)

fricas

integrate(1/sqrt(2*%pi)*exp(-1/2*log(x)^2),x)

$\label{eq27}\int^{ \displaystyle x}{{{{e}^{-{{{\log \left({\%A}\right)}^{2}}\over 2}}}\over{\sqrt{2 \ \pi}}}\ {d \%A}}$

(27)

Type: Union(Expression(Integer),...)

If you would get a result, you could use limit afterwards, of course.

Mathematical Paradox?: Thu, 10 Feb 2005 17:45:57 -0600 reply

Area under the curve:

fricas

integrate(1/x,x=1..%plusInfinity)

$\label{eq28}+ \infty$

(28)

Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

Paradox Part2:

Thu, 10 Feb 2005 17:47:59 -0600 reply

Volume under that curve:

fricas

integrate(%pi*((1/x)^2), x=1..%plusInfinity)

$\label{eq29}\pi$

(29)

Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

Curve has an infinite area...but a finite volume (I think I did this correctly)!

... --unknown, Tue, 17 May 2005 18:52:42 -0500 reply

fricas

integrate(1/x,x)

$\label{eq30}\log \left({x}\right)$

(30)

Type: Union(Expression(Integer),...)

fricas

integrate(sqrt(x),x)

$\label{eq31}{2 \ x \ {\sqrt{x}}}\over 3$

(31)

Type: Union(Expression(Integer),...)

fricas

integrate(sqrt(x^3+x),x)

$\label{eq32}\int^{ \displaystyle x}{{\sqrt{{{\%A}^{3}}+ \%A}}\ {d \%A}}$

(32)

Type: Union(Expression(Integer),...)

a turning moving body --unknown, Sun, 19 Jun 2005 20:16:03 -0500 reply

fricas

integrate(( a*sin( m + n*t + o*t*t/2 ) )/( n + ot ) + ( b*cos( m + n*t + o*t*t/2 ) )/( n + ot ), t)

$\label{eq33}\int^{ \displaystyle t}{{{{a \ {\sin \left({{{{{\%A}^{2}}\ o}+{2 \ \%A \ n}+{2 \ m}}\over 2}\right)}}+{b \ {\cos \left({{{{{\%A}^{2}}\ o}+{2 \ \%A \ n}+{2 \ m}}\over 2}\right)}}}\over{ot + n}}\ {d \%A}}$

(33)

Type: Union(Expression(Integer),...)

a turning accelerating body --unknown, Sun, 19 Jun 2005 23:16:48 -0500 reply

fricas

integrate(( a*cos( m + n*t + o*t*t/2 ) )- ( b*sin( m + n*t + o*t*t/2 ) ), t)

$\label{eq34}\int^{ \displaystyle t}{{\left(-{b \ {\sin \left({{{{{\%A}^{2}}\ o}+{2 \ \%A \ n}+{2 \ m}}\over 2}\right)}}+{a \ {\cos \left({{{{{\%A}^{2}}\ o}+{2 \ \%A \ n}+{2 \ m}}\over 2}\right)}}\right)}\ {d \%A}}$

(34)

Type: Union(Expression(Integer),...)

... --unknown, Sun, 26 Jun 2005 16:02:47 -0500 reply

fricas

integrate(-2*(3-3*t)^2*(3*t),t)

$\label{eq35}-{{{27}\over 2}\ {{t}^{4}}}+{{36}\ {{t}^{3}}}-{{27}\ {{t}^{2}}}$

(35)

Type: Polynomial(Fraction(Integer))

... --unknown, Sun, 30 Oct 2005 09:29:01 -0600 reply

fricas

integrate(1/(1+x^2),x=-u..u)

$\label{eq36}potentialPole$

(36)

Type: Union(pole: potentialPole,...)

fricas

integrate(1/(1+x^2),x=-u..u, "noPole")

$\label{eq37}2 \ {\arctan \left({u}\right)}$

(37)

Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

... --unknown, Tue, 01 Nov 2005 10:35:26 -0600 reply

fricas

integrate(x^6*exp(-x^2), x=0..%plusInfinity)

$\label{eq38}{\Gamma \left({7 \over 2}\right)}\over 2$

(38)

Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

... --unknown, Wed, 02 Nov 2005 17:00:21 -0600 reply

fricas

integrate(1/sqrt(1/x+1),x)

$\label{eq39}{-{\log \left({{\sqrt{{x + 1}\over x}}+ 1}\right)}+{\log \left({{\sqrt{{x + 1}\over x}}- 1}\right)}+{2 \ x \ {\sqrt{{x + 1}\over x}}}}\over 2$

(39)

Type: Union(Expression(Integer),...)

... --unknown, Wed, 23 Nov 2005 08:30:18 -0600 reply

fricas

integrate(sin(sin x), x)

$\label{eq40}\int^{ \displaystyle x}{{\sin \left({\sin \left({\%A}\right)}\right)}\ {d \%A}}$

(40)

Type: Union(Expression(Integer),...)

... --unknown, Fri, 25 Nov 2005 16:28:37 -0600 reply

fricas

integrate(a/2*(1-cos(b*t)),t)

$\label{eq41}{-{a \ {\sin \left({b \ t}\right)}}+{a \ b \ t}}\over{2 \ b}$

(41)

Type: Union(Expression(Integer),...)

yet another test that shall work but not in maple ? --unknown, Thu, 09 Mar 2006 09:21:47 -0600 reply

fricas

integrate(tan(atan(x)/3),x)

$\label{eq42}{\left( \begin{array}{@{}l} \displaystyle {8 \ {\log \left({{3 \ {{\tan \left({{\arctan \left({x}\right)}\over 3}\right)}^{2}}}- 1}\right)}}-{3 \ {{\tan \left({{\arctan \left({x}\right)}\over 3}\right)}^{2}}}+ \ \ \displaystyle {{18}\ x \ {\tan \left({{\arctan \left({x}\right)}\over 3}\right)}}$

(42)

Type: Union(Expression(Integer),...)

... --unknown, Sat, 11 Mar 2006 12:41:33 -0600 reply

fricas

integrate(x, x)

$\label{eq43}{1 \over 2}\ {{x}^{2}}$

(43)

Type: Polynomial(Fraction(Integer))

fricas

simplify((1/(2*z))*z^2)

$\label{eq44}z \over 2$

(44)

Type: Expression(Integer)

fricas

integrate((1/(2*z))*z^2, z)

$\label{eq45}{{z}^{2}}\over 4$

(45)

Type: Union(Expression(Integer),...)

fricas

integrate(log(x),x)

$\label{eq46}{x \ {\log \left({x}\right)}}- x$

(46)

Type: Union(Expression(Integer),...)

fricas

integrate(1/x,x)

$\label{eq47}\log \left({x}\right)$

(47)

Type: Union(Expression(Integer),...)

fricas

integrate(0^0,x)

$\label{eq48}x$

(48)

Type: Polynomial(Fraction(Integer))

from fr.sci.maths --unknown, Tue, 09 May 2006 09:58:11 -0500 reply

fricas

integrate( log(y)^3/(y*(y-1)),y)

$\label{eq49}\int^{ \displaystyle y}{{{{\log \left({\%A}\right)}^{3}}\over{{{\%A}^{2}}- \%A}}\ {d \%A}}$

(49)

Type: Union(Expression(Integer),...)

... --unknown, Mon, 29 May 2006 14:20:28 -0500 reply

fricas

integrate(exp(-x^2),x=0..%plusInfinity)

$\label{eq50}{\sqrt{\pi}}\over 2$

(50)

Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

... --unknown, Mon, 29 May 2006 14:21:39 -0500 reply

fricas

integrate(x^2*exp(-x^2),x=0..%plusInfinity)

$\label{eq51}{\Gamma \left({3 \over 2}\right)}\over 2$

(51)

Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

test --unknown, Wed, 19 Jul 2006 11:59:36 -0500 reply

No ; after command or else output is supressed.

fricas

integrate(exp(%i*2*%pi*f*t), t=0..T)

$\label{eq52}{-{i \ {{e}^{2 \ i \ T \ f \ \pi}}}+ i}\over{2 \ f \ \pi}$

(52)

Type: Union(f1: OrderedCompletion?(Expression(Complex(Integer))),...)

Axiom and Maxima not capable of this integrand --WinnieThePooh?, Tue, 29 May 2007 17:24:44 -0500 reply

int(exp(sin(x)),x);

reduce

$\displaylines{\qdd \int {e^{\sin \(x$

fricas

integrate(exp(sin(x)),x)

$\label{eq53}\int^{ \displaystyle x}{{{e}^{\sin \left({\%A}\right)}}\ {d \%A}}$

(53)

Type: Union(Expression(Integer),...)

An integral which Maxima can't do, but Axiom can --amca01, Tue, 05 Jan 2010 15:08:53 -0800 reply

fricas

integrate(sqrt(x+sqrt(1+x^2))/x,x)

$\label{eq54}\begin{array}{@{}l} \displaystyle -{\log \left({{\sqrt{{\sqrt{{{x}^{2}}+ 1}}+ x}}+ 1}\right)}+{\log \left({{\sqrt{{\sqrt{{{x}^{2}}+ 1}}+ x}}- 1}\right)}- \ \ \displaystyle {2 \ {\arctan \left({\sqrt{{\sqrt{{{x}^{2}}+ 1}}+ x}}\right)}}+{2 \ {\sqrt{{\sqrt{{{x}^{2}}+ 1}}+ x}}}$

(54)

Type: Union(Expression(Integer),...)

Exp Integral --pabjr, Tue, 14 Apr 2020 22:01:58 +0000 reply

\begin(axiom) integrate((1 - x/a)%e^(-(b (x - a))/(x + da)),x) \end(axiom)

Exp Integral --pabjr, Tue, 14 Apr 2020 22:03:00 +0000 reply

integrate((1 - x/a)%e^(-(b (x - a))/(x + da)),x)