Integrand size = 21, antiderivative size = 253 \[ \int \frac {1}{1+2 x+\sqrt {-2+8 x-5 x^2}} \, dx=\frac {2}{9} \sqrt {5} \arctan \left (\frac {\sqrt {5} \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}\right )+\frac {26 \arctan \left (\frac {6+\frac {\left (12-13 \sqrt {6}\right ) \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}}{\sqrt {138}}\right )}{9 \sqrt {23}}-\frac {2}{9} \log \left (\frac {2 \left (3-2 \sqrt {6}\right )+5 \sqrt {6} x}{\left (4-\sqrt {6}-5 x\right )^2}\right )+\frac {2}{9} \log \left (\frac {2 \left (3-2 \sqrt {6}\right )+12 x-3 \sqrt {6} x+10 \sqrt {6} x^2+6 \sqrt {-2+8 x-5 x^2}-4 \sqrt {6} \sqrt {-2+8 x-5 x^2}+5 \sqrt {6} x \sqrt {-2+8 x-5 x^2}}{\left (4-\sqrt {6}-5 x\right )^2}\right ) \] Output:
2/9*arctan(5^(1/2)*(-5*x^2+8*x-2)^(1/2)/(4-6^(1/2)-5*x))*5^(1/2)+26/207*ar ctan(1/138*(6+(12-13*6^(1/2))*(-5*x^2+8*x-2)^(1/2)/(4-6^(1/2)-5*x))*138^(1 /2))*23^(1/2)-2/9*ln((6-4*6^(1/2)+5*x*6^(1/2))/(4-6^(1/2)-5*x)^2)+2/9*ln(( 6-4*6^(1/2)+12*x-3*x*6^(1/2)+10*6^(1/2)*x^2+6*(-5*x^2+8*x-2)^(1/2)-4*6^(1/ 2)*(-5*x^2+8*x-2)^(1/2)+5*6^(1/2)*x*(-5*x^2+8*x-2)^(1/2))/(4-6^(1/2)-5*x)^ 2)
Time = 1.20 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.74 \[ \int \frac {1}{1+2 x+\sqrt {-2+8 x-5 x^2}} \, dx=-\frac {2}{9} \sqrt {5} \arctan \left (\frac {4+\sqrt {6}-5 x}{\sqrt {5} \sqrt {-2+8 x-5 x^2}}\right )+\frac {26 \arctan \left (\frac {\sqrt {23} \left (4+\sqrt {6}-5 x\right )}{6+4 \sqrt {6}-5 \sqrt {6} x+13 \sqrt {-2+8 x-5 x^2}+2 \sqrt {6} \sqrt {-2+8 x-5 x^2}}\right )}{9 \sqrt {23}}+\frac {2}{9} \left (-\log \left (-6-4 \sqrt {6}+5 \sqrt {6} x\right )+\log \left (\left (-6-4 \sqrt {6}+5 \sqrt {6} x\right ) \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )\right )\right ) \] Input:
Integrate[(1 + 2*x + Sqrt[-2 + 8*x - 5*x^2])^(-1),x]
Output:
(-2*Sqrt[5]*ArcTan[(4 + Sqrt[6] - 5*x)/(Sqrt[5]*Sqrt[-2 + 8*x - 5*x^2])])/ 9 + (26*ArcTan[(Sqrt[23]*(4 + Sqrt[6] - 5*x))/(6 + 4*Sqrt[6] - 5*Sqrt[6]*x + 13*Sqrt[-2 + 8*x - 5*x^2] + 2*Sqrt[6]*Sqrt[-2 + 8*x - 5*x^2])])/(9*Sqrt [23]) + (2*(-Log[-6 - 4*Sqrt[6] + 5*Sqrt[6]*x] + Log[(-6 - 4*Sqrt[6] + 5*S qrt[6]*x)*(1 + 2*x + Sqrt[-2 + 8*x - 5*x^2])]))/9
Time = 0.38 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.47, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{\sqrt {-5 x^2+8 x-2}+2 x+1} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 x+1}{9 x^2-4 x+3}-\frac {\sqrt {-5 x^2+8 x-2}}{9 x^2-4 x+3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{9} \sqrt {5} \arcsin \left (\frac {4-5 x}{\sqrt {6}}\right )+\frac {13 \arctan \left (\frac {8-13 x}{\sqrt {23} \sqrt {-5 x^2+8 x-2}}\right )}{9 \sqrt {23}}-\frac {13 \arctan \left (\frac {2-9 x}{\sqrt {23}}\right )}{9 \sqrt {23}}+\frac {2}{9} \text {arctanh}\left (\frac {2 x+1}{\sqrt {-5 x^2+8 x-2}}\right )+\frac {1}{9} \log \left (9 x^2-4 x+3\right )\) |
Input:
Int[(1 + 2*x + Sqrt[-2 + 8*x - 5*x^2])^(-1),x]
Output:
-1/9*(Sqrt[5]*ArcSin[(4 - 5*x)/Sqrt[6]]) - (13*ArcTan[(2 - 9*x)/Sqrt[23]]) /(9*Sqrt[23]) + (13*ArcTan[(8 - 13*x)/(Sqrt[23]*Sqrt[-2 + 8*x - 5*x^2])])/ (9*Sqrt[23]) + (2*ArcTanh[(1 + 2*x)/Sqrt[-2 + 8*x - 5*x^2]])/9 + Log[3 - 4 *x + 9*x^2]/9
Leaf count of result is larger than twice the leaf count of optimal. \(458\) vs. \(2(198)=396\).
Time = 0.97 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.81
| method | result | size |
| default | \(\frac {\sqrt {5}\, \arcsin \left (\frac {5 \sqrt {6}\, \left (x -\frac {4}{5}\right )}{6}\right )}{9}-\frac {5 \sqrt {29}\, \sqrt {676}\, \sqrt {\frac {696 \left (x +\frac {1}{2}\right )^{2}}{\left (\frac {8}{13}-x \right )^{2}}-4901}\, \left (7 \sqrt {23}\, \arctan \left (\frac {\sqrt {\frac {696 \left (x +\frac {1}{2}\right )^{2}}{\left (\frac {8}{13}-x \right )^{2}}-4901}\, \sqrt {23}}{377}\right )+230 \,\operatorname {arctanh}\left (\frac {58 x +29}{\left (\frac {8}{13}-x \right ) \sqrt {\frac {696 \left (x +\frac {1}{2}\right )^{2}}{\left (\frac {8}{13}-x \right )^{2}}-4901}}\right )\right )}{4526262 \sqrt {\frac {\frac {24 \left (x +\frac {1}{2}\right )^{2}}{\left (\frac {8}{13}-x \right )^{2}}-169}{\left (\frac {x +\frac {1}{2}}{\frac {8}{13}-x}+1\right )^{2}}}\, \left (\frac {x +\frac {1}{2}}{\frac {8}{13}-x}+1\right )}-\frac {4 \sqrt {29}\, \sqrt {676}\, \sqrt {\frac {696 \left (x +\frac {1}{2}\right )^{2}}{\left (\frac {8}{13}-x \right )^{2}}-4901}\, \left (8 \sqrt {23}\, \arctan \left (\frac {\sqrt {\frac {696 \left (x +\frac {1}{2}\right )^{2}}{\left (\frac {8}{13}-x \right )^{2}}-4901}\, \sqrt {23}}{377}\right )-23 \,\operatorname {arctanh}\left (\frac {58 x +29}{\left (\frac {8}{13}-x \right ) \sqrt {\frac {696 \left (x +\frac {1}{2}\right )^{2}}{\left (\frac {8}{13}-x \right )^{2}}-4901}}\right )\right )}{251459 \sqrt {\frac {\frac {24 \left (x +\frac {1}{2}\right )^{2}}{\left (\frac {8}{13}-x \right )^{2}}-169}{\left (\frac {x +\frac {1}{2}}{\frac {8}{13}-x}+1\right )^{2}}}\, \left (\frac {x +\frac {1}{2}}{\frac {8}{13}-x}+1\right )}+\frac {\sqrt {29}\, \sqrt {676}\, \sqrt {\frac {696 \left (x +\frac {1}{2}\right )^{2}}{\left (\frac {8}{13}-x \right )^{2}}-4901}\, \left (13 \sqrt {23}\, \arctan \left (\frac {\sqrt {\frac {696 \left (x +\frac {1}{2}\right )^{2}}{\left (\frac {8}{13}-x \right )^{2}}-4901}\, \sqrt {23}}{377}\right )+46 \,\operatorname {arctanh}\left (\frac {58 x +29}{\left (\frac {8}{13}-x \right ) \sqrt {\frac {696 \left (x +\frac {1}{2}\right )^{2}}{\left (\frac {8}{13}-x \right )^{2}}-4901}}\right )\right )}{251459 \sqrt {\frac {\frac {24 \left (x +\frac {1}{2}\right )^{2}}{\left (\frac {8}{13}-x \right )^{2}}-169}{\left (\frac {x +\frac {1}{2}}{\frac {8}{13}-x}+1\right )^{2}}}\, \left (\frac {x +\frac {1}{2}}{\frac {8}{13}-x}+1\right )}+\frac {13 \sqrt {23}\, \arctan \left (\frac {\left (18 x -4\right ) \sqrt {23}}{46}\right )}{207}+\frac {\ln \left (9 x^{2}-4 x +3\right )}{9}\) | \(459\) |
| trager | \(\text {Expression too large to display}\) | \(1151\) |
Input:
int(1/(1+2*x+(-5*x^2+8*x-2)^(1/2)),x,method=_RETURNVERBOSE)
Output:
1/9*5^(1/2)*arcsin(5/6*6^(1/2)*(x-4/5))-5/4526262*29^(1/2)*676^(1/2)*(696* (x+1/2)^2/(8/13-x)^2-4901)^(1/2)*(7*23^(1/2)*arctan(1/377*(696*(x+1/2)^2/( 8/13-x)^2-4901)^(1/2)*23^(1/2))+230*arctanh(58*(x+1/2)/(8/13-x)/(696*(x+1/ 2)^2/(8/13-x)^2-4901)^(1/2)))/((24*(x+1/2)^2/(8/13-x)^2-169)/((x+1/2)/(8/1 3-x)+1)^2)^(1/2)/((x+1/2)/(8/13-x)+1)-4/251459*29^(1/2)*676^(1/2)*(696*(x+ 1/2)^2/(8/13-x)^2-4901)^(1/2)*(8*23^(1/2)*arctan(1/377*(696*(x+1/2)^2/(8/1 3-x)^2-4901)^(1/2)*23^(1/2))-23*arctanh(58*(x+1/2)/(8/13-x)/(696*(x+1/2)^2 /(8/13-x)^2-4901)^(1/2)))/((24*(x+1/2)^2/(8/13-x)^2-169)/((x+1/2)/(8/13-x) +1)^2)^(1/2)/((x+1/2)/(8/13-x)+1)+1/251459*29^(1/2)*676^(1/2)*(696*(x+1/2) ^2/(8/13-x)^2-4901)^(1/2)*(13*23^(1/2)*arctan(1/377*(696*(x+1/2)^2/(8/13-x )^2-4901)^(1/2)*23^(1/2))+46*arctanh(58*(x+1/2)/(8/13-x)/(696*(x+1/2)^2/(8 /13-x)^2-4901)^(1/2)))/((24*(x+1/2)^2/(8/13-x)^2-169)/((x+1/2)/(8/13-x)+1) ^2)^(1/2)/((x+1/2)/(8/13-x)+1)+13/207*23^(1/2)*arctan(1/46*(18*x-4)*23^(1/ 2))+1/9*ln(9*x^2-4*x+3)
Time = 0.08 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00 \[ \int \frac {1}{1+2 x+\sqrt {-2+8 x-5 x^2}} \, dx=\frac {13}{207} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (9 \, x - 2\right )}\right ) - \frac {1}{9} \, \sqrt {5} \arctan \left (\frac {\sqrt {5} \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (5 \, x - 4\right )}}{5 \, {\left (5 \, x^{2} - 8 \, x + 2\right )}}\right ) + \frac {13}{414} \, \sqrt {23} \arctan \left (\frac {\sqrt {23} \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (13 \, x - 8\right )} + 2 \, \sqrt {23} {\left (2 \, x^{2} - 3 \, x\right )}}{23 \, {\left (7 \, x^{2} - 8 \, x + 2\right )}}\right ) + \frac {13}{414} \, \sqrt {23} \arctan \left (\frac {\sqrt {23} \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (13 \, x - 8\right )} - 2 \, \sqrt {23} {\left (2 \, x^{2} - 3 \, x\right )}}{23 \, {\left (7 \, x^{2} - 8 \, x + 2\right )}}\right ) + \frac {1}{9} \, \log \left (9 \, x^{2} - 4 \, x + 3\right ) - \frac {1}{18} \, \log \left (-\frac {x^{2} + 2 \, \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (2 \, x + 1\right )} - 12 \, x + 1}{x^{2}}\right ) + \frac {1}{18} \, \log \left (-\frac {x^{2} - 2 \, \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (2 \, x + 1\right )} - 12 \, x + 1}{x^{2}}\right ) \] Input:
integrate(1/(1+2*x+(-5*x^2+8*x-2)^(1/2)),x, algorithm="fricas")
Output:
13/207*sqrt(23)*arctan(1/23*sqrt(23)*(9*x - 2)) - 1/9*sqrt(5)*arctan(1/5*s qrt(5)*sqrt(-5*x^2 + 8*x - 2)*(5*x - 4)/(5*x^2 - 8*x + 2)) + 13/414*sqrt(2 3)*arctan(1/23*(sqrt(23)*sqrt(-5*x^2 + 8*x - 2)*(13*x - 8) + 2*sqrt(23)*(2 *x^2 - 3*x))/(7*x^2 - 8*x + 2)) + 13/414*sqrt(23)*arctan(1/23*(sqrt(23)*sq rt(-5*x^2 + 8*x - 2)*(13*x - 8) - 2*sqrt(23)*(2*x^2 - 3*x))/(7*x^2 - 8*x + 2)) + 1/9*log(9*x^2 - 4*x + 3) - 1/18*log(-(x^2 + 2*sqrt(-5*x^2 + 8*x - 2 )*(2*x + 1) - 12*x + 1)/x^2) + 1/18*log(-(x^2 - 2*sqrt(-5*x^2 + 8*x - 2)*( 2*x + 1) - 12*x + 1)/x^2)
\[ \int \frac {1}{1+2 x+\sqrt {-2+8 x-5 x^2}} \, dx=\int \frac {1}{2 x + \sqrt {- 5 x^{2} + 8 x - 2} + 1}\, dx \] Input:
integrate(1/(1+2*x+(-5*x**2+8*x-2)**(1/2)),x)
Output:
Integral(1/(2*x + sqrt(-5*x**2 + 8*x - 2) + 1), x)
\[ \int \frac {1}{1+2 x+\sqrt {-2+8 x-5 x^2}} \, dx=\int { \frac {1}{2 \, x + \sqrt {-5 \, x^{2} + 8 \, x - 2} + 1} \,d x } \] Input:
integrate(1/(1+2*x+(-5*x^2+8*x-2)^(1/2)),x, algorithm="maxima")
Output:
integrate(1/(2*x + sqrt(-5*x^2 + 8*x - 2) + 1), x)
Leaf count of result is larger than twice the leaf count of optimal. 383 vs. \(2 (191) = 382\).
Time = 0.16 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.51 \[ \int \frac {1}{1+2 x+\sqrt {-2+8 x-5 x^2}} \, dx=\frac {1}{9} \, \sqrt {5} \arcsin \left (\frac {1}{6} \, \sqrt {6} {\left (5 \, x - 4\right )}\right ) + \frac {13}{207} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (9 \, x - 2\right )}\right ) + \frac {13 \, {\left (5 \, \sqrt {6} + 13 \, \sqrt {5}\right )} \arctan \left (-\frac {26 \, \sqrt {6} + 12 \, \sqrt {5} - \frac {139 \, {\left (\sqrt {5} \sqrt {-5 \, x^{2} + 8 \, x - 2} - \sqrt {6}\right )}}{5 \, x - 4}}{5 \, \sqrt {138} + 13 \, \sqrt {115}}\right )}{9 \, {\left (5 \, \sqrt {138} + 13 \, \sqrt {115}\right )}} + \frac {13 \, {\left (5 \, \sqrt {6} - 13 \, \sqrt {5}\right )} \arctan \left (\frac {26 \, \sqrt {6} - 12 \, \sqrt {5} - \frac {139 \, {\left (\sqrt {5} \sqrt {-5 \, x^{2} + 8 \, x - 2} - \sqrt {6}\right )}}{5 \, x - 4}}{5 \, \sqrt {138} - 13 \, \sqrt {115}}\right )}{9 \, {\left (5 \, \sqrt {138} - 13 \, \sqrt {115}\right )}} + \frac {1}{9} \, \log \left (9 \, x^{2} - 4 \, x + 3\right ) + \frac {1}{9} \, \log \left (-\frac {4 \, {\left (\sqrt {5} \sqrt {-5 \, x^{2} + 8 \, x - 2} - \sqrt {6}\right )} {\left (13 \, \sqrt {6} + 6 \, \sqrt {5}\right )}}{5 \, x - 4} + 26 \, \sqrt {30} + \frac {139 \, {\left (\sqrt {5} \sqrt {-5 \, x^{2} + 8 \, x - 2} - \sqrt {6}\right )}^{2}}{{\left (5 \, x - 4\right )}^{2}} + 199\right ) - \frac {1}{9} \, \log \left (-\frac {4 \, {\left (\sqrt {5} \sqrt {-5 \, x^{2} + 8 \, x - 2} - \sqrt {6}\right )} {\left (13 \, \sqrt {6} - 6 \, \sqrt {5}\right )}}{5 \, x - 4} - 26 \, \sqrt {30} + \frac {139 \, {\left (\sqrt {5} \sqrt {-5 \, x^{2} + 8 \, x - 2} - \sqrt {6}\right )}^{2}}{{\left (5 \, x - 4\right )}^{2}} + 199\right ) \] Input:
integrate(1/(1+2*x+(-5*x^2+8*x-2)^(1/2)),x, algorithm="giac")
Output:
1/9*sqrt(5)*arcsin(1/6*sqrt(6)*(5*x - 4)) + 13/207*sqrt(23)*arctan(1/23*sq rt(23)*(9*x - 2)) + 13/9*(5*sqrt(6) + 13*sqrt(5))*arctan(-(26*sqrt(6) + 12 *sqrt(5) - 139*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))/(5*x - 4))/(5*sq rt(138) + 13*sqrt(115)))/(5*sqrt(138) + 13*sqrt(115)) + 13/9*(5*sqrt(6) - 13*sqrt(5))*arctan((26*sqrt(6) - 12*sqrt(5) - 139*(sqrt(5)*sqrt(-5*x^2 + 8 *x - 2) - sqrt(6))/(5*x - 4))/(5*sqrt(138) - 13*sqrt(115)))/(5*sqrt(138) - 13*sqrt(115)) + 1/9*log(9*x^2 - 4*x + 3) + 1/9*log(-4*(sqrt(5)*sqrt(-5*x^ 2 + 8*x - 2) - sqrt(6))*(13*sqrt(6) + 6*sqrt(5))/(5*x - 4) + 26*sqrt(30) + 139*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^2/(5*x - 4)^2 + 199) - 1/9 *log(-4*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))*(13*sqrt(6) - 6*sqrt(5) )/(5*x - 4) - 26*sqrt(30) + 139*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6)) ^2/(5*x - 4)^2 + 199)
Timed out. \[ \int \frac {1}{1+2 x+\sqrt {-2+8 x-5 x^2}} \, dx=\int \frac {1}{2\,x+\sqrt {-5\,x^2+8\,x-2}+1} \,d x \] Input:
int(1/(2*x + (8*x - 5*x^2 - 2)^(1/2) + 1),x)
Output:
int(1/(2*x + (8*x - 5*x^2 - 2)^(1/2) + 1), x)
\[ \int \frac {1}{1+2 x+\sqrt {-2+8 x-5 x^2}} \, dx=\frac {13 \sqrt {23}\, \mathit {atan} \left (\frac {9 x -2}{\sqrt {23}}\right )}{207}-\frac {\sqrt {5}\, \mathit {atan} \left (\frac {5 \sqrt {-5 x^{2}+8 x -2}\, \sqrt {5}\, x -4 \sqrt {-5 x^{2}+8 x -2}\, \sqrt {5}}{25 x^{2}-40 x +10}\right )}{9}-\frac {\left (\int \frac {\sqrt {-5 x^{2}+8 x -2}}{45 x^{4}-92 x^{3}+65 x^{2}-32 x +6}d x \right )}{3}+\frac {52 \left (\int \frac {\sqrt {-5 x^{2}+8 x -2}\, x}{45 x^{4}-92 x^{3}+65 x^{2}-32 x +6}d x \right )}{9}+\frac {\mathrm {log}\left (9 x^{2}-4 x +3\right )}{9} \] Input:
int(1/(1+2*x+(-5*x^2+8*x-2)^(1/2)),x)
Output:
(13*sqrt(23)*atan((9*x - 2)/sqrt(23)) - 23*sqrt(5)*atan((5*sqrt( - 5*x**2 + 8*x - 2)*sqrt(5)*x - 4*sqrt( - 5*x**2 + 8*x - 2)*sqrt(5))/(25*x**2 - 40* x + 10)) - 69*int(sqrt( - 5*x**2 + 8*x - 2)/(45*x**4 - 92*x**3 + 65*x**2 - 32*x + 6),x) + 1196*int((sqrt( - 5*x**2 + 8*x - 2)*x)/(45*x**4 - 92*x**3 + 65*x**2 - 32*x + 6),x) + 23*log(9*x**2 - 4*x + 3))/207