reduce-historical/r38/packages/tri/tritst.tex

\TRIexa{Integration}{TeXindent}{1000}{int(1+x+x**2,x);}
$$\displaylines{\qdd
\frac{x\cdot 
      \(2\cdot x^{2}
        +3\cdot x
        +6
      \)
      }{
      6}
\cr}$$
\TRIexa{Integration}{TeXindent}{1000}{int(x**2*(2*x**2+x)**2,x);}
$$\displaylines{\qdd
\frac{x^{5}\cdot 
      \(60\cdot x^{2}
        +70\cdot x
        +21
      \)
      }{
      105}
\cr}$$
\TRIexa{Integration}{TeXindent}{1000}{int(x*(x**2+2*x+1),x);}
$$\displaylines{\qdd
\frac{x^{2}\cdot 
      \(3\cdot x^{2}
        +8\cdot x
        +6
      \)
      }{
      12}
\cr}$$
\TRIexa{Integration}{TeXindent}{1000}{int(1/x,x);}
$$\displaylines{\qdd
\ln 
\(x
\)
\cr}$$
\TRIexa{Integration}{TeXindent}{1000}{int((x+1)**3/(x-1)**4,x);}
$$\displaylines{\qdd
\(3\cdot \ln 
  \(x
    -1
  \)
  \cdot x^{3}
  -9\cdot \ln 
  \(x
    -1
  \)
  \cdot x^{2}
  +9\cdot \ln 
  \(x
    -1
  \)
  \cdot x
  -3\cdot \ln 
  \(x
    -1
  \)
  -6\cdot x^{3}
  -2
\)
/\nl 
\(3\cdot 
  \(x^{3}
    -3\cdot x^{2}
    +3\cdot x
    -1
  \)
\)
\nl}$$
\TRIexa{Integration}{TeXindent}{1000}{int(1/(x*(x-1)*(x+1)**2),x);}
$$\displaylines{\qdd
\(\ln 
  \(x
    -1
  \)
  \cdot x
  +\ln 
  \(x
    -1
  \)
  +3\cdot \ln 
  \(x
    +1
  \)
  \cdot x\nl 
  \off{327680}
  +3\cdot \ln 
  \(x
    +1
  \)
  -4\cdot \ln 
  \(x
  \)
  \cdot x
  -4\cdot \ln 
  \(x
  \)
  +2\cdot x
\)
/\nl 
\(4\cdot 
  \(x
    +1
  \)
\)
\nl}$$
\TRIexa{Integration}{TeXindent}{1000}{int((a*x+b)/((x-p)*(x-q)),x);}
$$\displaylines{\qdd
\frac{\ln 
      \(p
        -x
      \)
      \cdot a\cdot p
      +\ln 
      \(p
        -x
      \)
      \cdot b
      -\ln 
      \(q
        -x
      \)
      \cdot a\cdot q
      -\ln 
      \(q
        -x
      \)
      \cdot b}{
      p
      -q}
\cr}$$
\TRIexa{Integration}{TeXindent}{1000}{int(1/(a*x**2+b*x+c),x);}
$$\displaylines{\qdd
\frac{2\cdot 
      \sqrt{4\cdot a\cdot c
            -b^{2}}\cdot \arctan 
      \(\frac{2\cdot a\cdot x
              +b}{
              \sqrt{4\cdot a\cdot c
                    -b^{2}}}
      \)
      }{
      4\cdot a\cdot c
      -b^{2}}
\cr}$$
\TRIexa{Integration}{TeXindent}{1000}{int((a*x+b)/(1+x**2),x);}
$$\displaylines{\qdd
\frac{2\cdot \arctan 
      \(x
      \)
      \cdot b
      +\ln 
      \(x^{2}
        +1
      \)
      \cdot a}{
      2}
\cr}$$
\TRIexa{Integration}{TeXindent}{1000}{int(1/(x**2-2*x+3),x);}
$$\displaylines{\qdd
\frac{\sqrt{2}
      \cdot \arctan 
      \(\frac{x
              -1}{
              \sqrt{2}}
      \)
      }{
      2}
\cr}$$
\TRIexa{Integration}{TeXindent}{1000}{int(1/((x-1)*(x**2+1))**2,x);}
$$\displaylines{\qdd
\(\arctan 
  \(x
  \)
  \cdot x^{3}
  -\arctan 
  \(x
  \)
  \cdot x^{2}
  +\arctan 
  \(x
  \)
  \cdot x\nl 
  \off{327680}
  -\arctan 
  \(x
  \)
  +\ln 
  \(x^{2}
    +1
  \)
  \cdot x^{3}
  -\ln 
  \(x^{2}
    +1
  \)
  \cdot x^{2}\nl 
  \off{327680}
  +\ln 
  \(x^{2}
    +1
  \)
  \cdot x
  -\ln 
  \(x^{2}
    +1
  \)
  -2\cdot \ln 
  \(x
    -1
  \)
  \cdot x^{3}\nl 
  \off{327680}
  +2\cdot \ln 
  \(x
    -1
  \)
  \cdot x^{2}
  -2\cdot \ln 
  \(x
    -1
  \)
  \cdot x
  +2\cdot \ln 
  \(x
    -1
  \)
  -x^{3}
  -2\cdot x
  +1
\)
/\nl 
\(4\cdot 
  \(x^{3}
    -x^{2}
    +x
    -1
  \)
\)
\nl}$$
\TRIexa{Integration}{TeXindent}{1000}{int(x/((x-a)*(x-b)*(x-c)),x);}
$$\displaylines{\qdd
\(\ln 
  \(a
    -x
  \)
  \cdot a\cdot b
  -\ln 
  \(a
    -x
  \)
  \cdot a\cdot c
  -\ln 
  \(b
    -x
  \)
  \cdot a\cdot b\nl 
  \off{327680}
  +\ln 
  \(b
    -x
  \)
  \cdot b\cdot c
  +\ln 
  \(c
    -x
  \)
  \cdot a\cdot c
  -\ln 
  \(c
    -x
  \)
  \cdot b\cdot c
\)
/\nl 
\(a^{2}\cdot b
  -a^{2}\cdot c
  -a\cdot b^{2}
  +a\cdot c^{2}
  +b^{2}\cdot c
  -b\cdot c^{2}
\)
\nl}$$
\TRIexa{Integration}{TeXindent}{1000}{int(x/((x**2+a**2)*(x**2+b**2)),x);}
$$\displaylines{\qdd
\frac{-\ln 
      \(a^{2}
        +x^{2}
      \)
      +\ln 
      \(b^{2}
        +x^{2}
      \)
      }{
      2\cdot 
      \(a^{2}
        -b^{2}
      \)
      }
\cr}$$
\TRIexa{Integration}{TeXindent}{1000}{int(x**2/((x**2+a**2)*(x**2+b**2)),x);}
$$\displaylines{\qdd
\frac{\arctan 
      \(\frac{x}{
              a}
      \)
      \cdot a
      -\arctan 
      \(\frac{x}{
              b}
      \)
      \cdot b}{
      a^{2}
      -b^{2}}
\cr}$$
\TRIexa{Integration}{TeXindent}{1000}{int(x/((x-1)*(x**2+1)),x);}
$$\displaylines{\qdd
\frac{2\cdot \arctan 
      \(x
      \)
      -\ln 
      \(x^{2}
        +1
      \)
      +2\cdot \ln 
      \(x
        -1
      \)
      }{
      4}
\cr}$$
\TRIexa{Integration}{TeXindent}{1000}{int(x/(1+x**3),x);}
$$\displaylines{\qdd
\frac{2\cdot 
      \sqrt{3}\cdot \arctan 
      \(\frac{2\cdot x
              -1}{
              \sqrt{3}}
      \)
      +\ln 
      \(x^{2}
        -x
        +1
      \)
      -2\cdot \ln 
      \(x
        +1
      \)
      }{
      6}
\cr}$$
\TRIexa{Integration}{TeXindent}{1000}{int(x**3/((x-1)**2*(x**3+1)),x);}
$$\displaylines{\qdd
\(-
  \(4\cdot \ln 
    \(x^{2}
      -x
      +1
    \)
    \cdot x
  \)
  +4\cdot \ln 
  \(x^{2}
    -x
    +1
  \)
  \nl 
  \off{327680}
  +9\cdot \ln 
  \(x
    -1
  \)
  \cdot x
  -9\cdot \ln 
  \(x
    -1
  \)
  -\ln 
  \(x
    +1
  \)
  \cdot x
  +\ln 
  \(x
    +1
  \)
  -6\cdot x
\)
/\nl 
\(12\cdot 
  \(x
    -1
  \)
\)
\nl}$$
\TRIexa{Integration}{TeXindent}{1000}{int(1/(1+x**4),x);}
$$\displaylines{\qdd
\(\sqrt{2}\cdot 
  \(-
    \(2\cdot \arctan 
      \(\frac{\sqrt{2}
              -2\cdot x}{
              \sqrt{2}}
      \)
    \)
    +2\cdot \arctan 
    \(\frac{\sqrt{2}
            +2\cdot x}{
            \sqrt{2}}
    \)
    \nl 
    \off{1277951}
    -\ln 
    \(-
      \(\sqrt{2}\cdot x
      \)
      +x^{2}
      +1
    \)
    +\ln 
    \(\sqrt{2}\cdot x
      +x^{2}
      +1
    \)
  \)
\)
/8
\nl}$$
\TRIexa{Integration}{TeXindent}{1000}{int(x**2/(1+x**4),x);}
$$\displaylines{\qdd
\(\sqrt{2}\cdot 
  \(-
    \(2\cdot \arctan 
      \(\frac{\sqrt{2}
              -2\cdot x}{
              \sqrt{2}}
      \)
    \)
    +2\cdot \arctan 
    \(\frac{\sqrt{2}
            +2\cdot x}{
            \sqrt{2}}
    \)
    \nl 
    \off{1277951}
    +\ln 
    \(-
      \(\sqrt{2}\cdot x
      \)
      +x^{2}
      +1
    \)
    -\ln 
    \(\sqrt{2}\cdot x
      +x^{2}
      +1
    \)
  \)
\)
/8
\nl}$$
\TRIexa{Integration}{TeXindent}{1000}{int(1/(1+x**2+x**4),x);}
$$\displaylines{\qdd
\(2\cdot 
  \sqrt{3}\cdot \arctan 
  \(\frac{2\cdot x
          -1}{
          \sqrt{3}}
  \)
  +2\cdot 
  \sqrt{3}\cdot \arctan 
  \(\frac{2\cdot x
          +1}{
          \sqrt{3}}
  \)
  \nl 
  \off{327680}
  -3\cdot \ln 
  \(x^{2}
    -x
    +1
  \)
  +3\cdot \ln 
  \(x^{2}
    +x
    +1
  \)
\)
/12
\nl}$$
\TRIexa{Integration}{TeXindent}{1000}{int(sin x**2/x,x);}
$$\displaylines{\qdd
\frac{-ci
      \(2\cdot x
      \)
      +\ln 
      \(x
      \)
      }{
      2}
\cr}$$
\TRIexa{Integration}{TeXindent}{1000}{int(x*cos(xi/sin(x))*cos(x)/sin(x)**2,x);}
$$\displaylines{\qdd
\int 
\frac{\cos 
      \(\frac{\xi }{
              \sin 
              \(x
              \)
              }
      \)
      \cdot \cos 
      \(x
      \)
      \cdot x}{
      \sin 
      \(x
      \)
      ^{2}}\,dx
\cr}$$