int = 
        Integrate[
          (1/Cos)[αα]^3, αα];
f[α_] := 
      int /. αα -> α
FullSimplify[f[α]]
Plot[
        f[α] - (1/2) Log[Tan[(π/4) + (α/2)]] - 
          Sin[α]/(2  Cos[α]^2), {α, 
          0, (π/4)}];
Simplify[f[(π/4)] - f[0]]
Cos[(π/4)]
tm = (1/2) (Sqrt[2 - (Sqrt[2])])
tp = (1/2) (Sqrt[2 + (Sqrt[2])])
FullSimplify[Sin[(π/8)] - tm]
FullSimplify[Cos[(π/8)] - tp]
FullSimplify[Sin[((3  π)/8)] - tp]
FullSimplify[Cos[((3  π)/8)] - tm]
FullSimplify[(∫_0^1)(∫_0^1)(Sqrt[ (x1^2) + 
                    (x2^2)]) \[DifferentialD]x1 \[DifferentialD]x2 - (1/3) (Sqrt[2]) - (1/3) Log[1 + (Sqrt[2])]]
N[(1/3) (Sqrt[2]) + (1/3) Log[1 + (Sqrt[2])], 21]

(∫_(-(π/4))^((π/4)))(1/Cos)[α]^3 \[DifferentialD]α
FullSimplify[% == (Sqrt[2]) + Log[1 + (Sqrt[2])]]

<< Graphics`ImplicitPlot`
p = 
        ImplicitPlot[Abs[x1 x2] == 1, {x1, -5, 5}, 
          PlotRange -> {-5, 5}, PlotPoints -> 100, 
          ImageSize -> 500, Axes -> None, 
          Ticks -> None];
\(f[y1, y2] := 1\/\@\(1 + \(y1\^2\) y2\^2\)\%4\)
{{(∂_y1)( 
            f[y1, y2] y1), (∂_y2) (f[y1, y2] 
              y1)}, {(∂_y1) (f[y1, y2] 
              y2), (∂_y2) (f[y1, y2] 
              y2)}}
Simplify[
      Det[{{(∂_y1)( 
                f[y1, y2] y1), (∂_y2) (f[y1, y2] 
                  y1)}, {(∂_y1) (f[y1, y2] 
                  y2), (∂_y2) (f[y1, y2] y2)}}], {Im[
            y1] == 0, Im[y2] == 0}]
p1[k_] := 
        ParametricPlot[
          Evaluate[f[y1, y2] {y1, y2} /. y2 -> k], {y1, -5, 5}, 
          PlotRange -> All, 
          DisplayFunction -> Identity];
pp1[l_] := 
        Table[p1[k], {k, -5, 5, 0.25}];
p2[k_] := 
        ParametricPlot[
          Evaluate[f[y1, y2] {y1, y2} /. y1 -> k], {y2, -5, 5}, 
          PlotRange -> All, 
          DisplayFunction -> Identity];
pp2[l_] := 
        Table[p2[k], {k, -5, 5, 0.25}];
Show[p, pp1[3], pp2[3], AspectRatio -> Automatic, 
        DisplayFunction -> $DisplayFunction, 
        ImageSize -> 400];