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Integration

Let's do some integration examples:

fricas
(1) -> 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}\frac{{{\left(-{{32}\ {\sqrt{-{{x}^{2}}+ 4}}}-{8 \ {{x}^{2}}}+{64}\right)}\ {\arctan \left({\frac{{\sqrt{-{{x}^{2}}+ 4}}- 2}{x}}\right)}}+{{\left(-{{x}^{3}}+{8 \  x}\right)}\ {\sqrt{-{{x}^{2}}+ 4}}}+{4 \ {{x}^{3}}}-{{16}\  x}}{{8 \ {\sqrt{-{{x}^{2}}+ 4}}}+{2 \ {{x}^{2}}}-{16}}(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}\verb#"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 the value from type Variable(a) to PositiveInteger .

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

\label{eq4}\frac{{{\left(-{5 \ {{x}^{2}}}+ 5 \right)}\ {\log \left({x + 1}\right)}}+{{\left({8 \ {{x}^{2}}}- 8 \right)}\ {\log \left({x}\right)}}+{{\left(-{3 \ {{x}^{2}}}+ 3 \right)}\ {\log \left({x - 1}\right)}}-{2 \  x}- 6}{{4 \ {{x}^{2}}}- 4}(4)
Type: Union(Expression(Integer),...)

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

\label{eq5}\frac{{2 \ {\sin \left({x}\right)}\ {\log \left({\frac{\sin \left({x}\right)}{2}}\right)}}-{2 \  x \ {\cos \left({x}\right)}}}{\sin \left({x}\right)}(5)
Type: Union(Expression(Integer),...)

Comparing FriCAS and Reduce:

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

\label{eq6}{x \ {\sin \left({\frac{1}{x}}\right)}}-{Ci \left({\frac{1}{x}}\right)}(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 FriCAS has to give up:

fricas
integrate(sqrt(sin(1/x)),x)
>> Error detected within library code: integrate: implementation incomplete (has polynomial part)

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}\frac{{\erf \left({x}\right)}\ {\sqrt{\pi}}}{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}\frac{\sin \left({x}\right)}{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}\verb#"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}{{\frac{\arctan \left({\frac{\%A}{a}}\right)}{\%A}}\ {d \%A}}(12)
Type: Union(Expression(Integer),...)

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

\label{eq13}\frac{-{{\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)}}+{{12}\ {\arctan \left({\frac{{2 \ {\sqrt{3}}\ {\root{3}\of{{a}^{2}}}}-{a \ {\sqrt{3}}}}{3 \  a}}\right)}}+{{\sqrt{3}}\ {\log \left({{a}^{4}}\right)}}-{2 \ {\sqrt{3}}\ {\log \left({{a}^{2}}\right)}}+{2 \  \pi}}{{12}\ {\sqrt{3}}\ {\root{3}\of{{a}^{2}}}}(13)
Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

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

\label{eq14}{{\frac{1}{4}}\ {{x}^{4}}}+{{\frac{1}{12}}\ {{x}^{3}}}+{{\frac{1}{2}}\ {{x}^{2}}}(14)
Type: Polynomial(Fraction(Integer))

You cannot integrate Expression Float

fricas
integrate(50*%e^(-0.02*t),t)
There are 9 exposed and 11 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}^{-{{\frac{1}{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}\frac{{4 \  c \ {{r}^{3}}}+{6 \  b \ {{r}^{2}}}+{{\left(-{c \ {{h}^{2}}}+{{12}\  a}\right)}\  r}}{{12}\  h}(19)
Type: Union(Expression(Integer),...)

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

\label{eq20}\frac{{\sqrt{\pi}}\ {\erf \left({\frac{{\left({b \  t}+ a \right)}\ {\sqrt{\frac{{b}^{2}}{2}}}}{b}}\right)}}{2 \ {\sqrt{\frac{{b}^{2}}{2}}}}(20)
Type: Union(Expression(Integer),...)

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

\label{eq21}\int^{
\displaystyle
t}{{{e}^{\frac{-{{{\%A}^{2}}\ {{b}^{2}}}-{2 \  \%A \  a \  b}-{{a}^{2}}}{\%A}}}\ {d \%A}}(21)
Type: Union(Expression(Integer),...)

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

\label{eq22}{t \ {{e}^{-{\frac{1}{t}}}}}+{Ei \left({-{\frac{1}{t}}}\right)}(22)
Type: Union(Expression(Integer),...)

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

\label{eq23}\verb#"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}\frac{-{a \  b \ {\sqrt{\pi}}\ {\erf \left({\frac{{\left({b \  t}+ a \right)}\ {\sqrt{\frac{{b}^{2}}{2}}}}{b}}\right)}}-{2 \ {{e}^{\frac{-{{{b}^{2}}\ {{t}^{2}}}-{2 \  a \  b \  t}-{{a}^{2}}}{2}}}\ {\sqrt{\frac{{b}^{2}}{2}}}}}{2 \ {{b}^{2}}\ {\sqrt{\frac{{b}^{2}}{2}}}}(24)
Type: Union(Expression(Integer),...)

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

\label{eq25}\frac{-{{\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)}}+{{12}\ {\arctan \left({\frac{{2 \ {\sqrt{3}}\ {\root{3}\of{{a}^{2}}}}-{a \ {\sqrt{3}}}}{3 \  a}}\right)}}+{{\sqrt{3}}\ {\log \left({{a}^{4}}\right)}}-{2 \ {\sqrt{3}}\ {\log \left({{a}^{2}}\right)}}+{2 \  \pi}}{{12}\ {\sqrt{3}}\ {\root{3}\of{{a}^{2}}}}(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}\frac{{{e}^{\frac{1}{2}}}\ {\sqrt{2}}\ {\sqrt{\pi}}}{\sqrt{2 \  \pi}}(26)
Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)
fricas
integrate(1/sqrt(2*%pi)*exp(-1/2*log(x)^2),x)

\label{eq27}\frac{{{e}^{\frac{1}{2}}}\ {\sqrt{2}}\ {\sqrt{\pi}}\ {\erf \left({\frac{{\log \left({x}\right)}- 1}{\sqrt{2}}}\right)}}{2 \ {\sqrt{2 \  \pi}}}(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 replyVolume 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)!

fricas
integrate(1/x,x)

\label{eq30}\log \left({x}\right)(30)
Type: Union(Expression(Integer),...)
fricas
integrate(sqrt(x),x)

\label{eq31}\frac{2 \  x \ {\sqrt{x}}}{3}(31)
Type: Union(Expression(Integer),...)
fricas
integrate(sqrt(x^3+x),x)

\label{eq32}\frac{-{4 \ {weierstrassZeta \left({- 4, \: 0, \:{weierstrassPInverse \left({- 4, \: 0, \: x}\right)}}\right)}}+{2 \  x \ {\sqrt{{{x}^{3}}+ x}}}}{5}(32)
Type: Union(Expression(Integer),...)

a turning moving body --unknown, Sun, 19 Jun 2005 20:16:03 -0500 reply
fricas
)set output tex off
 
fricas
)set output algebra on
integrate(( a*sin( m + n*t + o*t*t/2 ) )/( n + ot ) + ( b*cos( m + n*t + o*t*t/2 ) )/( n + ot ), t)
(33) +---+ | o 2 2 (o t + n) |--- 2 m o - n 2 m o - n \|%pi (- b sin(----------) + a cos(----------))fresnelS(---------------) 2 o 2 o o + +---+ | o 2 2 (o t + n) |--- 2 m o - n 2 m o - n \|%pi (a sin(----------) + b cos(----------))fresnelC(---------------) 2 o 2 o o / +---+ | o (ot + n) |--- \|%pi
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)
(34) +---+ | o 2 2 (o t + n) |--- 2 m o - n 2 m o - n \|%pi (- a sin(----------) - b cos(----------))fresnelS(---------------) 2 o 2 o o + +---+ | o 2 2 (o t + n) |--- 2 m o - n 2 m o - n \|%pi (- b sin(----------) + a cos(----------))fresnelC(---------------) 2 o 2 o o / +---+ | o |--- \|%pi
Type: Union(Expression(Integer),...)
fricas
)set output algebra off
 
fricas
)set output tex on

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

\label{eq33}-{{\frac{27}{2}}\ {{t}^{4}}}+{{36}\ {{t}^{3}}}-{{27}\ {{t}^{2}}}(33)
Type: Polynomial(Fraction(Integer))

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

\label{eq34}\verb#"potentialPole"#(34)
Type: Union(pole: potentialPole,...)
fricas
integrate(1/(1+x^2),x=-u..u, "noPole")

\label{eq35}2 \ {\arctan \left({u}\right)}(35)
Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

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

\label{eq36}\frac{{15}\ {\sqrt{\pi}}}{16}(36)
Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

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

\label{eq37}\frac{-{\log \left({{\sqrt{\frac{x + 1}{x}}}+ 1}\right)}+{\log \left({{\sqrt{\frac{x + 1}{x}}}- 1}\right)}+{2 \  x \ {\sqrt{\frac{x + 1}{x}}}}}{2}(37)
Type: Union(Expression(Integer),...)

fricas
integrate(sin(sin x), x)

\label{eq38}\int^{
\displaystyle
x}{{\sin \left({\sin \left({\%A}\right)}\right)}\ {d \%A}}(38)
Type: Union(Expression(Integer),...)

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

\label{eq39}\frac{-{a \ {\sin \left({b \  t}\right)}}+{a \  b \  t}}{2 \  b}(39)
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{eq40}\frac{{8 \ {\log \left({{3 \ {{\tan \left({\frac{\arctan \left({x}\right)}{3}}\right)}^{2}}}- 1}\right)}}-{3 \ {{\tan \left({\frac{\arctan \left({x}\right)}{3}}\right)}^{2}}}+{{18}\  x \ {\tan \left({\frac{\arctan \left({x}\right)}{3}}\right)}}}{1
8}(40)
Type: Union(Expression(Integer),...)

fricas
integrate(x, x)

\label{eq41}{\frac{1}{2}}\ {{x}^{2}}(41)
Type: Polynomial(Fraction(Integer))
fricas
simplify((1/(2*z))*z^2)

\label{eq42}\frac{z}{2}(42)
Type: Expression(Integer)
fricas
integrate((1/(2*z))*z^2, z)

\label{eq43}\frac{{z}^{2}}{4}(43)
Type: Union(Expression(Integer),...)
fricas
integrate(log(x),x)

\label{eq44}{x \ {\log \left({x}\right)}}- x(44)
Type: Union(Expression(Integer),...)
fricas
integrate(1/x,x)

\label{eq45}\log \left({x}\right)(45)
Type: Union(Expression(Integer),...)
fricas
integrate(0^0,x)

\label{eq46}x(46)
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{eq47}\int^{
\displaystyle
y}{{\frac{{\log \left({\%A}\right)}^{3}}{{{\%A}^{2}}- \%A}}\ {d \%A}}(47)
Type: Union(Expression(Integer),...)

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

\label{eq48}\frac{\sqrt{\pi}}{2}(48)
Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

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

\label{eq49}\frac{\sqrt{\pi}}{4}(49)
Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

No ; after command or else output is supressed.
fricas
integrate(exp(%i*2*%pi*f*t), t=0..T)

\label{eq50}\frac{-{i \ {{e}^{2 \  i \  T \  f \  \pi}}}+ i}{2 \  f \  \pi}(50)
Type: Union(f1: OrderedCompletion?(Expression(Complex(Integer))),...)

FriCAS 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{eq51}\int^{
\displaystyle
x}{{{e}^{\sin \left({\%A}\right)}}\ {d \%A}}(51)
Type: Union(Expression(Integer),...)

An integral which Maxima can't do, but FriCAS can --amca01, Tue, 05 Jan 2010 15:08:53 -0800 reply
fricas
integrate(sqrt(x+sqrt(1+x^2))/x,x)

\label{eq52}\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}}}
(52)
Type: Union(Expression(Integer),...)

fricas
)set output algebra on
r2:=integrate((1 - x)*%e^((-b*(x - 1))/(x + d)),x)
(55) - b x + b --------- 2 2 x + d (- x + (- b d - b + 2)x + (- b + 1)d + (- b + 2)d)%e + 2 2 2 2 b d + b - b ((b - 2 b)d + (2 b - 4 b)d + b - 2 b)Ei(-------)%e x + d / 2

\label{eq53}\frac{{{\left(-{{x}^{2}}+{{\left(-{b \  d}- b + 2 \right)}\  x}+{{\left(- b + 1 \right)}\ {{d}^{2}}}+{{\left(- b + 2 \right)}\  d}\right)}\ {{e}^{\frac{-{b \  x}+ b}{x + d}}}}+{{\left({{\left({{b}^{2}}-{2 \  b}\right)}\ {{d}^{2}}}+{{\left({2 \ {{b}^{2}}}-{4 \  b}\right)}\  d}+{{b}^{2}}-{2 \  b}\right)}\ {Ei \left({\frac{{b \  d}+ b}{x + d}}\right)}\ {{e}^{- b}}}}{2}(53)
Type: Union(Expression(Integer),...)
fricas
unparse(r2::InputForm)
(56) "(((-1)*x^2+((-1)*b*d+((-1)*b+2))*x+(((-1)*b+1)*d^2+((-1)*b+2)*d))*exp(((-1)* b*x+b)/(x+d))+((b^2+(-2)*b)*d^2+(2*b^2+(-4)*b)*d+(b^2+(-2)*b))*Ei((b*d+b)/(x+ d))*exp((-1)*b))/2"

\label{eq54}\verb#"(((-1)<em>x^2+((-1)</em>b<em>d+((-1)</em>b+2))<em>x+(((-1)</em>b+1)<em>d^2+((-1)</em>b+2)<em>d))</em>exp(((-1)<em>b</em>x+b)/(x+d))+((b^2+(-2)<em>b)</em>d^2+(2<em>b^2+(-4)</em>b)<em>d+(b^2+(-2)</em>b))<em>Ei((b</em>d+b)/(x+d))<em>exp((-1)</em>b))/2"#(54)
Type: String

Test

fricas
ii := integral(sin(x), x)
x ++ (57) | sin(%A)d%A ++

\label{eq55}\int^{
\displaystyle
x}{{\sin \left({\%A}\right)}\ {d \%A}}(55)
Type: Expression(Integer)

fricas
integrate((7*x^13+10*x^8+4*x^7-7*x^6-4*x^3-4*x^2+3*x+3)/(x^14-2*x^8-2*x^7-2*x^4-4*x^3-x^2+2*x+1),x)
(58) 14 8 7 4 3 2 log(x - 2 x - 2 x - 2 x - 4 x - x + 2 x + 1) + +-+ \|2 * log 9 8 3 2 +-+ 14 8 7 (- 2 x - 2 x + 2 x + 4 x + 2 x)\|2 + x - 2 x - 2 x + 4 3 2 2 x + 4 x + 3 x + 2 x + 1 / 14 8 7 4 3 2 x - 2 x - 2 x - 2 x - 4 x - x + 2 x + 1 / 2

\label{eq56}\frac{{\log \left({{{x}^{14}}-{2 \ {{x}^{8}}}-{2 \ {{x}^{7}}}-{2 \ {{x}^{4}}}-{4 \ {{x}^{3}}}-{{x}^{2}}+{2 \  x}+ 1}\right)}+{{\sqrt{2}}\ {\log \left({\frac{{{\left(-{2 \ {{x}^{9}}}-{2 \ {{x}^{8}}}+{2 \ {{x}^{3}}}+{4 \ {{x}^{2}}}+{2 \  x}\right)}\ {\sqrt{2}}}+{{x}^{14}}-{2 \ {{x}^{8}}}-{2 \ {{x}^{7}}}+{2 \ {{x}^{4}}}+{4 \ {{x}^{3}}}+{3 \ {{x}^{2}}}+{2 \  x}+ 1}{{{x}^{14}}-{2 \ {{x}^{8}}}-{2 \ {{x}^{7}}}-{2 \ {{x}^{4}}}-{4 \ {{x}^{3}}}-{{x}^{2}}+{2 \  x}+ 1}}\right)}}}{2}(56)
Type: Union(Expression(Integer),...)




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