kernelopts(version)
#`Maple 2022.1, X86 64 WINDOWS, May 26 2022, Build ID 1619613`

(int(sin(x)*exp(-2*I*Pi*f*x)/x, x = -infinity .. infinity) assuming (0 < f));

#signum(0, -2*Pi*f + 1, 0)*Pi/4 + Pi/2 - signum(0, 2*Pi*f - 1, 0)*Pi/4

convert(%, piecewise);

#piecewise(f < 1/(2*Pi), Pi, f = 1/(2*Pi), Pi/2, 1/(2*Pi) < f, 0)

Order := 50;
ode := diff(x(u), u, u) + 1/4*exp(-u^2)*x(u) = 0;
sol := convert(dsolve([ode, x(0) = 1, D(x)(0) = 0], x(u), 'series'), polynom);
plot([rhs(sol), diff(rhs(sol), u)], u = 0 .. 2, view = [0 .. 2, -1/2 .. 1])

Looks like a bug in integrate.

int(x^n*exp(x), x = 0 .. 2, method = _RETURNVERBOSE); #Can't solve.

I have only a workround.

See attached file:2_ver1.mw

As a infinite sum:

Int(sin(sqrt(-x^2 + 1)), x) = Sum(2^(-1/2 - k)*sqrt(Pi)*x^(1 + 2*k)*BesselY(-1/2 + k, 1)/((1 + 2*k)*GAMMA(1 + k)), k = 0 .. infinity) + C

for: -1<=x<=1

Check:

int(sin(sqrt(-x^2 + 1)), x = 0 .. 1, numeric);
evalf(add(eval(2^(-1/2 - k)*sqrt(Pi)*x^(1 + 2*k)*BesselY(-1/2 + k, 1)/((1 + 2*k)*GAMMA(1 + k)), x = 1), k = 0 .. 2000));

Download SystemRecursive_new2.mw

.

f := (x, y) -> piecewise(x <> 0 and y <> 0, (y^2 + x^2)^x, x = 0 and y = 0, 1);
DX := diff(f(x, y), x);
DY := diff(f(x, y), y);
[eval(DX, [x = 1, y = 1]), eval(DY, [x = 1, y = 1])]

#at x=1 and y=1;
#[2*ln(2) + 2, 2]

Download Arctan_ver2.mw

restart;
with(LinearAlgebra);
w := (2*Pi)/14;
v := Vector([1, sin(w*t), cos(w*t)]);
simplify(eval(sum(v . (Transpose(v)), t = k .. k + n), n = 13));

restart;
ODE := diff(W(x), x $ 2) + (A*ohm^2*ro - kd)*W(x)/T = p;
sol1 := dsolve(ODE, W(x));
assign(sol1);

subs(_C1 = 0, W(x));

#or:

eval(W(x), _C1 = 0);


#sin(sqrt(A*ohm^2*ro - kd)*x/sqrt(T))*_C2 + p*T/(A*ohm^2*ro - kd)

int(exp(-2*r)*cos(theta)^3*r^2*sin(theta), phi = 0 .. 2*Pi, r = 0 .. infinity, theta = 0 .. Pi);
#O

We can find solution with series representation:

restart;
odeSystem := {diff(y1(x), x) = -x*y2(x) - (1 + x)*y3(x), diff(y2(x), x) = -x*y1(x) - (1 + x)*y4(x), diff(y3(x), x) = -x*y1(x) - (1 + x)*y4(x) - 5*x*cos(1/2*x^2), diff(y4(x), x) = -x*y2(x) - (1 + x)*y3(x) + 5*x*sin(1/2*x^2), y1(0) = 5, y2(0) = 1, y3(0) = -1, y4(0) = 0};
Order := 10;
systemSol := dsolve(odeSystem, [y1(x), y2(x), y3(x), y4(x)], series);
F := convert(systemSol, polynom);
plot([rhs(F[1]), rhs(F[2]), rhs(F[3]), rhs(F[4])], x = 0 .. 1, color = [red, blue, green, gold], legend = [y1(x), y2(x), y3(x), y4(x)]);

A workaround with Mathematica:

f[k_] := Sum[(-1)^i (k - i + 1)^(2 k + 4)/((i!)*((2 k - i + 2)!)), {i,0, k}];
L = Table[f[k], {k, 0, 1000}];
FindSequenceFunction[L, k]

(*1/12 (k + 3 k^2 + 2 k^3)*)

With Maple:

restart;
with(gfun):
f := k -> sum((-1)^i*(k - i + 1)^(2*k + 4)/(i!*(2*k - i + 2)!), i = 0 .. k);
l := [seq(f(k), k = 0 .. 1000)];
rec := listtorec(l, u(k), [ogf]);
rsolve(op(1, rec), u(k));

#1/6*k^3 + 3/4*k^2 + 13/12*k + 1/2

See attached file.

integral.mw

Sum((-1)^k*sin(k*x)/k, k = -infinity .. infinity) = evalc(sum((-1)^k*sin(k*x)/k, k = -infinity .. infinity, formal))

f := x -> convert(sin(theta), x);
map(f, [cos, tan, cot, csc, sec]);

#[-cos(theta + Pi/2), 2*tan(theta/2)/(1 + tan(theta/2)^2), 2*cot(theta/2)/(cot(theta/2)^2 + 1), 1/csc(theta), -1/sec(theta + Pi/2)]