Using CauchyPrincipalValue =true:

`assuming`([int(exp(I*x*t)/((x-a)*(x+a)), x = -infinity .. infinity, CauchyPrincipalValue = true)], [a > 0, t > 0])

#(1/2*I)*Pi*(exp(I*a*t)-exp(-I*a*t))/a

int.mw

Exectuted in Maple 2018.

remove[flatten](x-> x = 0, [seq(sin((1/4)*k*Pi), k = 1 .. 8)]);

The function RootOf is a placeholder for representing all the roots of an equation in one variable.  In particular, it is the standard representation for Maple algebraic numbers, algebraic functions.

For more info execute this code in Maple:

?RootOf

sol := solve({x^2+y^2 = 3, x^2+2*y^2 = 3}, {x, y});
allvalues(sol);
evalf(%);

#{x = sqrt(3), y = 0}, {x = -sqrt(3), y = 0}
#{x = 1.732050808, y = 0.}, {x = -1.732050808, y = 0.}

I don't know way it's happens,but if we convert to rational(exact) numbers then works:

eq := [190.0315457*d^6-7.601261828*d^4+344.0*c^2*d^2-2.677685950*c^4, 0.6830134554e-2*c^4+2084.317025*d^8-166.7453620*d^6+3.334907240*d^4-c^2*d^2];
eliminate(convert(eq, rational, exact), c);

#[{c = -(1/14265)*sqrt(13071106675-17435*sqrt(9667390500*d^2+561670891405))*d}, {121*d^4*(582104796124362625620*d^4-46433010457684955790*d^2+2932165094782*sqrt(9667390500*d^2+561670891405)-1272306087125807465)}], [{c = (1/14265)*sqrt(13071106675-17435*sqrt(9667390500*d^2+561670891405))*d}, {121*d^4*(582104796124362625620*d^4-46433010457684955790*d^2+2932165094782*sqrt(9667390500*d^2+561670891405)-1272306087125807465)}], [{c = -(1/14265)*sqrt(13071106675+17435*sqrt(9667390500*d^2+561670891405))*d}, {-121*d^4*(-582104796124362625620*d^4+46433010457684955790*d^2+2932165094782*sqrt(9667390500*d^2+561670891405)+1272306087125807465)}], [{c = (1/14265)*sqrt(13071106675+17435*sqrt(9667390500*d^2+561670891405))*d}, {-121*d^4*(-582104796124362625620*d^4+46433010457684955790*d^2+2932165094782*sqrt(9667390500*d^2+561670891405)+1272306087125807465)}]

Download Latex_Sum.mw

pdsolve([diff(f2(lam, mu, l, m), m)-(diff(f1(lam, mu, l, m), l))-(diff(f4(lam, mu, l, m), mu))+diff(f3(lam, mu, l, m), lam) = 0, diff(f1(lam, mu, l, m), m)+diff(f2(lam, mu, l, m), l)-(diff(f3(lam, mu, l, m), mu))-(diff(f4(lam, mu, l, m), lam)) = 0], HINT = `+`);

#A simple solutions.

pdsolve([diff(f2(lam, mu, l, m), m)-(diff(f1(lam, mu, l, m), l))-(diff(f4(lam, mu, l, m), mu))+diff(f3(lam, mu, l, m), lam) = 0, diff(f1(lam, mu, l, m), m)+diff(f2(lam, mu, l, m), l)-(diff(f3(lam, mu, l, m), mu))-(diff(f4(lam, mu, l, m), lam)) = 0], HINT = TWS(sin));

#Not simple solution.For more info. execute this code below?

?pdsolve

Probably you want:

with(Fractals:-LSystem); with(LSystemExamples):

PlotExample(DragonCurve, 15);

Lindenmayer System Plot Generator:

states := "FX";

rules := ["Y" = "FX-Y", "X" = "X+YF"];

cons := ["F" = "draw:1", "+" = "turn:-90", "-" = "turn:90"];

newstate1 := Iterate(states, rules, 10);

LSystemPlot(newstate1, cons);

EDITED:

Code from book Geometry of Curves and Surfaces with MAPLE. https://books.google.pl/books/about/Geometry_of_Curves_and_Surfaces_with_MAP.html?id=78w0CseXgvMC&redir_esc=y

dragon := proc (k::algebraic, N::integer) local t, i, q1, q2, q3, q4, d; global p; q2 := [k, 0]; q3 := [1-k, 0]; d := evalm(q3-q2); p[0] := plot([[q2[1], q2[2]], [q3[1], q3[2]]]); for i to N do if `mod`(i, 2) = 0 then t[i] := t[(1/2)*i] else t[i] := (`mod`(i, 4))-2 end if; q4 := evalm(q3+k*d/(1-2*k)); d := evalm(t[i]*[d[2], -d[1]]); q1 := evalm(q3); q2 := evalm(q4+k*d/(1-2*k)); q3 := evalm(q2+d); p[i] := plot([[q1[1], q1[2]], [q2[1], q2[2]], [q3[1], q3[2]]]) end do; return plots:-display([seq(p[i], i = 0 .. N)]) end proc;

dragon(0.1, 500);

f := x^2*y^2+3*x*y^2;

fdiff(f, [x, y], {x = 1, y = 2});

See in Help for more information,exectute code below:

?fdiff 

restart;

de := diff(y(t), t, t) = -1; ic := y(0) = 1, (D(y))(0) = 0;

Events := [y(t), diff(y(t), t) = -.5*(diff(y(t), t))];

Events2 := [y(t) = -2*(1/1000), halt];

dsol := dsolve({de, ic}, numeric, events = [Events, Events2], range = 0 .. 5);

plots[odeplot](dsol, thickness = 3, color = red)

You can do it numerically:

c[1]:=1;
c[2]:=1;
int(12.*x^3*c[2]+6.*x^2*c[1]+x^2*exp(x^3*c[2])*exp(x^2*c[1]), x = 0. .. 1.,numeric);

#6.155281446

If you looking a analyticaly solution it's probably not possible.

See :https://en.wikipedia.org/wiki/Integral#Symbolic

PDE := diff(f(x, y), x, x)+diff(f(x, y), y, y) = 0;
sol := PDEtools:-SimilaritySolutions(PDE);
sol[1];

OR:

pdsolve(PDE, HINT = `+`, build);
pdsolve(PDE, HINT = `*`, build);

I doubt there's a closed form for the integral for general variables.

Only for a=0,1 and k=-1,0,1,2,3. can be found:

By Maple 2018.

sol := x*a*k*((x-g)/b)^(a-1)*(1+((x-g)/b)^a)^(k+1)/b;

`assuming`([[seq([int(eval(sol, a = j), x = g .. t)], j = 0 .. 1)]], [a > 0, b > 0, k in integer, k > 0, g > 0, t > 0]);

#[[0], [k*(-g*k+b^(-k)*(b-g+t)^k*k*t-2*b^(-k-1)*(b-g+t)^k*g*k*t+2*b^(-k-1)*(b-g+t)^k*t^2*k+b^(-k-2)*(b-g+t)^k*g^2*k*t-2*b^(-k-2)*(b-g+t)^k*t^2*k*g+b^(-k-2)*(b-g+t)^k*t^3*k+b-3*g-b^(-k+1)*(b-g+t)^k+3*b^(-k)*(b-g+t)^k*g-3*b^(-k-1)*(b-g+t)^k*g^2+3*b^(-k-1)*(b-g+t)^k*t^2+b^(-k-2)*(b-g+t)^k*g^3-3*b^(-k-2)*(b-g+t)^k*t^2*g+2*b^(-k-2)*(b-g+t)^k*t^3)/(k^2+5*k+6)]]


`assuming`([[seq([int(eval(sol, k = j), x = g .. t)], j = -1 .. 3)]], [a > 0, b > 0, g > 0, t > 0, a > 0]);

#[[-b^(-a)*(t-g)^a*(a*t+g)/(a+1)], [0], [(6*b^(-3*a)*(t-g)^(3*a)*a^3*t+2*b^(-3*a)*(t-g)^(3*a)*a^2*g+9*b^(-3*a)*(t-g)^(3*a)*a^2*t+18*b^(-2*a)*(t-g)^(2*a)*a^3*t+3*b^(-3*a)*(t-g)^(3*a)*a*g+3*b^(-3*a)*(t-g)^(3*a)*a*t+9*b^(-2*a)*(t-g)^(2*a)*a^2*g+24*b^(-2*a)*(t-g)^(2*a)*a^2*t+18*b^(-a)*(t-g)^a*a^3*t+b^(-3*a)*(t-g)^(3*a)*g+12*b^(-2*a)*(t-g)^(2*a)*a*g+6*b^(-2*a)*(t-g)^(2*a)*a*t+18*b^(-a)*(t-g)^a*a^2*g+15*b^(-a)*(t-g)^a*a^2*t+3*g*b^(-2*a)*(t-g)^(2*a)+15*b^(-a)*(t-g)^a*a*g+3*b^(-a)*(t-g)^a*a*t+3*g*b^(-a)*(t-g)^a)/(3*(6*a^3+11*a^2+6*a+1))], [(168*b^(-3*a)*(t-g)^(3*a)*a^3*t+56*b^(-3*a)*(t-g)^(3*a)*a^2*g+84*b^(-3*a)*(t-g)^(3*a)*a^2*t+228*b^(-2*a)*(t-g)^(2*a)*a^3*t+28*b^(-3*a)*(t-g)^(3*a)*a*g+12*b^(-3*a)*(t-g)^(3*a)*a*t+114*b^(-2*a)*(t-g)^(2*a)*a^2*g+96*b^(-2*a)*(t-g)^(2*a)*a^2*t+104*b^(-a)*(t-g)^a*a^3*t+48*b^(-2*a)*(t-g)^(2*a)*a*g+12*b^(-2*a)*(t-g)^(2*a)*a*t+104*b^(-a)*(t-g)^a*a^2*g+36*b^(-a)*(t-g)^a*a^2*t+36*b^(-a)*(t-g)^a*a*g+4*b^(-a)*(t-g)^a*a*t+4*b^(-3*a)*(t-g)^(3*a)*g+6*g*b^(-2*a)*(t-g)^(2*a)+4*g*b^(-a)*(t-g)^a+11*b^(-4*a)*(t-g)^(4*a)*a^2*g+24*b^(-4*a)*(t-g)^(4*a)*a^2*t+6*b^(-4*a)*(t-g)^(4*a)*a*g+4*b^(-4*a)*(t-g)^(4*a)*a*t+96*b^(-a)*(t-g)^a*a^3*g+24*b^(-4*a)*(t-g)^(4*a)*a^4*t+6*b^(-4*a)*(t-g)^(4*a)*a^3*g+44*b^(-4*a)*(t-g)^(4*a)*a^3*t+96*b^(-3*a)*(t-g)^(3*a)*a^4*t+32*b^(-3*a)*(t-g)^(3*a)*a^3*g+144*b^(-2*a)*(t-g)^(2*a)*a^4*t+72*b^(-2*a)*(t-g)^(2*a)*a^3*g+96*b^(-a)*(t-g)^a*a^4*t+b^(-4*a)*(t-g)^(4*a)*g)/(2*(24*a^4+50*a^3+35*a^2+10*a+1))], [(3*(b^(-5*a)*(t-g)^(5*a)*g+600*b^(-a)*(t-g)^a*a^4*g+600*b^(-a)*(t-g)^a*a^5*t+120*b^(-5*a)*(t-g)^(5*a)*a^5*t+1470*b^(-3*a)*(t-g)^(3*a)*a^3*t+490*b^(-3*a)*(t-g)^(3*a)*a^2*g+360*b^(-3*a)*(t-g)^(3*a)*a^2*t+1180*b^(-2*a)*(t-g)^(2*a)*a^3*t+120*b^(-3*a)*(t-g)^(3*a)*a*g+30*b^(-3*a)*(t-g)^(3*a)*a*t+590*b^(-2*a)*(t-g)^(2*a)*a^2*g+260*b^(-2*a)*(t-g)^(2*a)*a^2*t+355*b^(-a)*(t-g)^a*a^3*t+130*b^(-2*a)*(t-g)^(2*a)*a*g+20*b^(-2*a)*(t-g)^(2*a)*a*t+355*b^(-a)*(t-g)^a*a^2*g+70*b^(-a)*(t-g)^a*a^2*t+70*b^(-a)*(t-g)^a*a*g+5*b^(-a)*(t-g)^a*a*t+10*b^(-3*a)*(t-g)^(3*a)*g+10*g*b^(-2*a)*(t-g)^(2*a)+5*g*b^(-a)*(t-g)^a+205*b^(-4*a)*(t-g)^(4*a)*a^2*g+220*b^(-4*a)*(t-g)^(4*a)*a^2*t+55*b^(-4*a)*(t-g)^(4*a)*a*g+20*b^(-4*a)*(t-g)^(4*a)*a*t+770*b^(-a)*(t-g)^a*a^3*g+1220*b^(-4*a)*(t-g)^(4*a)*a^4*t+305*b^(-4*a)*(t-g)^(4*a)*a^3*g+820*b^(-4*a)*(t-g)^(4*a)*a^3*t+2340*b^(-3*a)*(t-g)^(3*a)*a^4*t+780*b^(-3*a)*(t-g)^(3*a)*a^3*g+2140*b^(-2*a)*(t-g)^(2*a)*a^4*t+1070*b^(-2*a)*(t-g)^(2*a)*a^3*g+770*b^(-a)*(t-g)^a*a^4*t+5*b^(-4*a)*(t-g)^(4*a)*g+24*b^(-5*a)*(t-g)^(5*a)*a^4*g+250*b^(-5*a)*(t-g)^(5*a)*a^4*t+50*b^(-5*a)*(t-g)^(5*a)*a^3*g+175*b^(-5*a)*(t-g)^(5*a)*a^3*t+35*b^(-5*a)*(t-g)^(5*a)*a^2*g+50*b^(-5*a)*(t-g)^(5*a)*a^2*t+10*b^(-5*a)*(t-g)^(5*a)*a*g+5*b^(-5*a)*(t-g)^(5*a)*a*t+600*b^(-4*a)*(t-g)^(4*a)*a^5*t+150*b^(-4*a)*(t-g)^(4*a)*a^4*g+1200*b^(-3*a)*(t-g)^(3*a)*a^5*t+400*b^(-3*a)*(t-g)^(3*a)*a^4*g+1200*b^(-2*a)*(t-g)^(2*a)*a^5*t+600*b^(-2*a)*(t-g)^(2*a)*a^4*g))/(5*(120*a^5+274*a^4+225*a^3+85*a^2+15*a+1))]]

Mathematica 11.3 can find solution for: - infinity > a >= 2 and -1<= k < infinity

Download The_Hard_Spring_by_Maple_2018.mw

sol := `assuming`([dsolve(Eq2 = 0)], [phi(xi) > 0]);
with(DEtools);
odeadvisor(op([2, 2], sol)[1, 1]);

dsolve can't solve symbolical Abel's equation. Try numerically.

 I invert that laplace transform numerically,but answer is different than :(section 4.1 page 13629, data used for FIG. 3 )

Download FracPDE_vers_2.mw