Integrand size = 25, antiderivative size = 262 \[ \int \frac {1}{x \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )} \, dx=\frac {2}{3} \sqrt {2} \arctan \left (\frac {\sqrt {11-4 \sqrt {6}} \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}\right )+\frac {16 \arctan \left (\frac {6+\frac {\left (12-13 \sqrt {6}\right ) \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}}{\sqrt {138}}\right )}{3 \sqrt {23}}+\frac {1}{3} \log \left (\frac {x \left (2 \left (3-2 \sqrt {6}\right )+5 \sqrt {6} x\right )}{\left (4-\sqrt {6}-5 x\right )^2}\right )-\frac {1}{3} \log \left (\frac {2 \left (3-2 \sqrt {6}\right )+12 x-3 \sqrt {6} x+10 \sqrt {6} x^2+6 \sqrt {-2+8 x-5 x^2}-4 \sqrt {6} \sqrt {-2+8 x-5 x^2}+5 \sqrt {6} x \sqrt {-2+8 x-5 x^2}}{\left (4-\sqrt {6}-5 x\right )^2}\right ) \] Output:
2/3*2^(1/2)*arctan((2*2^(1/2)-3^(1/2))*(-5*x^2+8*x-2)^(1/2)/(4-6^(1/2)-5*x ))+16/69*arctan(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)+1/3*ln(x*(6-4*6^(1/2)+5*x*6^(1/2))/(4-6^(1/2)-5*x )^2)-1/3*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 = 2.41 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )} \, dx=\frac {1}{69} \left (-46 \sqrt {2} \arctan \left (\frac {4+\sqrt {6}-5 x}{5 \sqrt {\frac {2-8 x+5 x^2}{-11+4 \sqrt {6}}}}\right )-16 \sqrt {23} \arctan \left (\frac {\sqrt {23} \left (-22-8 \sqrt {6}+5 \sqrt {22+8 \sqrt {6}} x\right )}{48+22 \sqrt {6}-10 \left (3+2 \sqrt {6}\right ) x+64 \sqrt {-2+8 x-5 x^2}+21 \sqrt {6} \sqrt {-2+8 x-5 x^2}}\right )+23 \left (\log \left (x \left (-6-4 \sqrt {6}+5 \sqrt {6} x\right )\right )-\log \left (\left (-24-11 \sqrt {6}+5 \left (3+2 \sqrt {6}\right ) x\right ) \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )\right )\right )\right ) \] Input:
Integrate[1/(x*(1 + 2*x + Sqrt[-2 + 8*x - 5*x^2])),x]
Output:
(-46*Sqrt[2]*ArcTan[(4 + Sqrt[6] - 5*x)/(5*Sqrt[(2 - 8*x + 5*x^2)/(-11 + 4 *Sqrt[6])])] - 16*Sqrt[23]*ArcTan[(Sqrt[23]*(-22 - 8*Sqrt[6] + 5*Sqrt[22 + 8*Sqrt[6]]*x))/(48 + 22*Sqrt[6] - 10*(3 + 2*Sqrt[6])*x + 64*Sqrt[-2 + 8*x - 5*x^2] + 21*Sqrt[6]*Sqrt[-2 + 8*x - 5*x^2])] + 23*(Log[x*(-6 - 4*Sqrt[6 ] + 5*Sqrt[6]*x)] - Log[(-24 - 11*Sqrt[6] + 5*(3 + 2*Sqrt[6])*x)*(1 + 2*x + Sqrt[-2 + 8*x - 5*x^2])]))/69
Time = 0.64 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.53, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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.
| \(\displaystyle \int \frac {1}{x \left (\sqrt {-5 x^2+8 x-2}+2 x+1\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {10-9 x}{3 \left (9 x^2-4 x+3\right )}-\frac {\sqrt {-5 x^2+8 x-2}}{3 x}+\frac {3 x \sqrt {-5 x^2+8 x-2}}{9 x^2-4 x+3}-\frac {4 \sqrt {-5 x^2+8 x-2}}{3 \left (9 x^2-4 x+3\right )}+\frac {1}{3 x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {8 \arctan \left (\frac {8-13 x}{\sqrt {23} \sqrt {-5 x^2+8 x-2}}\right )}{3 \sqrt {23}}-\frac {1}{3} \sqrt {2} \arctan \left (\frac {\sqrt {2} (1-2 x)}{\sqrt {-5 x^2+8 x-2}}\right )-\frac {8 \arctan \left (\frac {2-9 x}{\sqrt {23}}\right )}{3 \sqrt {23}}-\frac {1}{3} \text {arctanh}\left (\frac {2 x+1}{\sqrt {-5 x^2+8 x-2}}\right )-\frac {1}{6} \log \left (9 x^2-4 x+3\right )+\frac {\log (x)}{3}\) |
Input:
Int[1/(x*(1 + 2*x + Sqrt[-2 + 8*x - 5*x^2])),x]
Output:
(-8*ArcTan[(2 - 9*x)/Sqrt[23]])/(3*Sqrt[23]) + (8*ArcTan[(8 - 13*x)/(Sqrt[ 23]*Sqrt[-2 + 8*x - 5*x^2])])/(3*Sqrt[23]) - (Sqrt[2]*ArcTan[(Sqrt[2]*(1 - 2*x))/Sqrt[-2 + 8*x - 5*x^2]])/3 - ArcTanh[(1 + 2*x)/Sqrt[-2 + 8*x - 5*x^ 2]]/3 + Log[x]/3 - Log[3 - 4*x + 9*x^2]/6
Leaf count of result is larger than twice the leaf count of optimal. \(476\) vs. \(2(207)=414\).
Time = 0.77 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.82
| method | result | size |
| default | \(\frac {\ln \left (x \right )}{3}-\frac {\ln \left (9 x^{2}-4 x +3\right )}{6}+\frac {8 \sqrt {23}\, \arctan \left (\frac {\left (18 x -4\right ) \sqrt {23}}{46}\right )}{69}+\frac {\sqrt {2}\, \arctan \left (\frac {\left (8 x -4\right ) \sqrt {2}}{4 \sqrt {-5 x^{2}+8 x -2}}\right )}{3}+\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 (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 )}{502918 \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 (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 {\sqrt {29}\, \sqrt {676}\, \sqrt {\frac {696 \left (x +\frac {1}{2}\right )^{2}}{\left (\frac {8}{13}-x \right )^{2}}-4901}\, \left (20 \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 )-391 \,\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 )}{754377 \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 )}\) | \(477\) |
| trager | \(\text {Expression too large to display}\) | \(1459\) |
Input:
int(1/x/(1+2*x+(-5*x^2+8*x-2)^(1/2)),x,method=_RETURNVERBOSE)
Output:
1/3*ln(x)-1/6*ln(9*x^2-4*x+3)+8/69*23^(1/2)*arctan(1/46*(18*x-4)*23^(1/2)) +1/3*2^(1/2)*arctan(1/4*(8*x-4)*2^(1/2)/(-5*x^2+8*x-2)^(1/2))+5/502918*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/(8/13-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)-4/251459*29^(1/2 )*676^(1/2)*(696*(x+1/2)^2/(8/13-x)^2-4901)^(1/2)*(13*23^(1/2)*arctan(1/37 7*(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)^2-1 69)/((x+1/2)/(8/13-x)+1)^2)^(1/2)/((x+1/2)/(8/13-x)+1)-1/754377*29^(1/2)*6 76^(1/2)*(696*(x+1/2)^2/(8/13-x)^2-4901)^(1/2)*(20*23^(1/2)*arctan(1/377*( 696*(x+1/2)^2/(8/13-x)^2-4901)^(1/2)*23^(1/2))-391*arctanh(58*(x+1/2)/(8/1 3-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)
Time = 0.09 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )} \, dx=\frac {8}{69} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (9 \, x - 2\right )}\right ) - \frac {1}{3} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (2 \, x - 1\right )}}{5 \, x^{2} - 8 \, x + 2}\right ) + \frac {4}{69} \, \sqrt {23} \arctan \left (\frac {\sqrt {23} \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (13 \, x - 8\right )} + 2 \, \sqrt {23} {\left (2 \, x^{2} - 3 \, x\right )}}{23 \, {\left (7 \, x^{2} - 8 \, x + 2\right )}}\right ) + \frac {4}{69} \, \sqrt {23} \arctan \left (\frac {\sqrt {23} \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (13 \, x - 8\right )} - 2 \, \sqrt {23} {\left (2 \, x^{2} - 3 \, x\right )}}{23 \, {\left (7 \, x^{2} - 8 \, x + 2\right )}}\right ) - \frac {1}{6} \, \log \left (9 \, x^{2} - 4 \, x + 3\right ) + \frac {1}{3} \, \log \left (x\right ) + \frac {1}{12} \, \log \left (-\frac {x^{2} + 2 \, \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (2 \, x + 1\right )} - 12 \, x + 1}{x^{2}}\right ) - \frac {1}{12} \, \log \left (-\frac {x^{2} - 2 \, \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (2 \, x + 1\right )} - 12 \, x + 1}{x^{2}}\right ) \] Input:
integrate(1/x/(1+2*x+(-5*x^2+8*x-2)^(1/2)),x, algorithm="fricas")
Output:
8/69*sqrt(23)*arctan(1/23*sqrt(23)*(9*x - 2)) - 1/3*sqrt(2)*arctan(sqrt(2) *sqrt(-5*x^2 + 8*x - 2)*(2*x - 1)/(5*x^2 - 8*x + 2)) + 4/69*sqrt(23)*arcta n(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)) + 4/69*sqrt(23)*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)) - 1/6 *log(9*x^2 - 4*x + 3) + 1/3*log(x) + 1/12*log(-(x^2 + 2*sqrt(-5*x^2 + 8*x - 2)*(2*x + 1) - 12*x + 1)/x^2) - 1/12*log(-(x^2 - 2*sqrt(-5*x^2 + 8*x - 2 )*(2*x + 1) - 12*x + 1)/x^2)
\[ \int \frac {1}{x \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )} \, dx=\int \frac {1}{x \left (2 x + \sqrt {- 5 x^{2} + 8 x - 2} + 1\right )}\, dx \] Input:
integrate(1/x/(1+2*x+(-5*x**2+8*x-2)**(1/2)),x)
Output:
Integral(1/(x*(2*x + sqrt(-5*x**2 + 8*x - 2) + 1)), x)
\[ \int \frac {1}{x \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )} \, dx=\int { \frac {1}{{\left (2 \, x + \sqrt {-5 \, x^{2} + 8 \, x - 2} + 1\right )} x} \,d x } \] Input:
integrate(1/x/(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), x)
Leaf count of result is larger than twice the leaf count of optimal. 421 vs. \(2 (197) = 394\).
Time = 0.18 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.61 \[ \int \frac {1}{x \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )} \, dx=-\frac {2}{15} \, \sqrt {10} \sqrt {5} \arctan \left (-\frac {1}{10} \, \sqrt {10} {\left (\sqrt {6} - \frac {4 \, {\left (\sqrt {5} \sqrt {-5 \, x^{2} + 8 \, x - 2} - \sqrt {6}\right )}}{5 \, x - 4}\right )}\right ) + \frac {8}{69} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (9 \, x - 2\right )}\right ) + \frac {8 \, {\left (5 \, \sqrt {6} + 13 \, \sqrt {5}\right )} \arctan \left (-\frac {26 \, \sqrt {6} + 12 \, \sqrt {5} - \frac {139 \, {\left (\sqrt {5} \sqrt {-5 \, x^{2} + 8 \, x - 2} - \sqrt {6}\right )}}{5 \, x - 4}}{5 \, \sqrt {138} + 13 \, \sqrt {115}}\right )}{3 \, {\left (5 \, \sqrt {138} + 13 \, \sqrt {115}\right )}} + \frac {8 \, {\left (5 \, \sqrt {6} - 13 \, \sqrt {5}\right )} \arctan \left (\frac {26 \, \sqrt {6} - 12 \, \sqrt {5} - \frac {139 \, {\left (\sqrt {5} \sqrt {-5 \, x^{2} + 8 \, x - 2} - \sqrt {6}\right )}}{5 \, x - 4}}{5 \, \sqrt {138} - 13 \, \sqrt {115}}\right )}{3 \, {\left (5 \, \sqrt {138} - 13 \, \sqrt {115}\right )}} - \frac {1}{6} \, \log \left (9 \, x^{2} - 4 \, x + 3\right ) - \frac {1}{6} \, \log \left (-\frac {4 \, {\left (\sqrt {5} \sqrt {-5 \, x^{2} + 8 \, x - 2} - \sqrt {6}\right )} {\left (13 \, \sqrt {6} + 6 \, \sqrt {5}\right )}}{5 \, x - 4} + 26 \, \sqrt {30} + \frac {139 \, {\left (\sqrt {5} \sqrt {-5 \, x^{2} + 8 \, x - 2} - \sqrt {6}\right )}^{2}}{{\left (5 \, x - 4\right )}^{2}} + 199\right ) + \frac {1}{6} \, \log \left (-\frac {4 \, {\left (\sqrt {5} \sqrt {-5 \, x^{2} + 8 \, x - 2} - \sqrt {6}\right )} {\left (13 \, \sqrt {6} - 6 \, \sqrt {5}\right )}}{5 \, x - 4} - 26 \, \sqrt {30} + \frac {139 \, {\left (\sqrt {5} \sqrt {-5 \, x^{2} + 8 \, x - 2} - \sqrt {6}\right )}^{2}}{{\left (5 \, x - 4\right )}^{2}} + 199\right ) + \frac {1}{3} \, \log \left ({\left | x \right |}\right ) \] Input:
integrate(1/x/(1+2*x+(-5*x^2+8*x-2)^(1/2)),x, algorithm="giac")
Output:
-2/15*sqrt(10)*sqrt(5)*arctan(-1/10*sqrt(10)*(sqrt(6) - 4*(sqrt(5)*sqrt(-5 *x^2 + 8*x - 2) - sqrt(6))/(5*x - 4))) + 8/69*sqrt(23)*arctan(1/23*sqrt(23 )*(9*x - 2)) + 8/3*(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)) + 8/3*(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*sqr t(115)) - 1/6*log(9*x^2 - 4*x + 3) - 1/6*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*(s qrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^2/(5*x - 4)^2 + 199) + 1/6*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) + 1/3*log(abs(x))
Timed out. \[ \int \frac {1}{x \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )} \, dx=\int \frac {1}{x\,\left (2\,x+\sqrt {-5\,x^2+8\,x-2}+1\right )} \,d x \] Input:
int(1/(x*(2*x + (8*x - 5*x^2 - 2)^(1/2) + 1)),x)
Output:
int(1/(x*(2*x + (8*x - 5*x^2 - 2)^(1/2) + 1)), x)
\[ \int \frac {1}{x \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )} \, dx=\text {too large to display} \] Input:
int(1/x/(1+2*x+(-5*x^2+8*x-2)^(1/2)),x)
Output:
(329045635848810631332960*sqrt(5)*asin((5*x - 4)/sqrt(6)) + 34933931535011 9874328080*sqrt(6)*asin((5*x - 4)/sqrt(6)) + 141840387171668727314208*sqrt (15)*atan((sqrt(3) - 2*sqrt(2)*tan(asin((5*x - 4)/sqrt(6))/2))/sqrt(5)) + 344805731627951929750560*sqrt(2)*atan((sqrt(3) - 2*sqrt(2)*tan(asin((5*x - 4)/sqrt(6))/2))/sqrt(5)) - 82501749357835074762528*sqrt(15)*atan((sqrt(3) + 2*sqrt(2)*tan(asin((5*x - 4)/sqrt(6))/2))/sqrt(5)) - 988895003621265596 45520*sqrt(2)*atan((sqrt(3) + 2*sqrt(2)*tan(asin((5*x - 4)/sqrt(6))/2))/sq rt(5)) - 28255013935671420745002277920*sqrt(30)*int(tan(asin((5*x - 4)/sqr t(6))/2)**6/(38642*sqrt( - 5*x**2 + 8*x - 2)*tan(asin((5*x - 4)/sqrt(6))/2 )**12 + 164193*sqrt( - 5*x**2 + 8*x - 2)*tan(asin((5*x - 4)/sqrt(6))/2)**1 0 + 416850*sqrt( - 5*x**2 + 8*x - 2)*tan(asin((5*x - 4)/sqrt(6))/2)**8 + 4 42630*sqrt( - 5*x**2 + 8*x - 2)*tan(asin((5*x - 4)/sqrt(6))/2)**6 + 416850 *sqrt( - 5*x**2 + 8*x - 2)*tan(asin((5*x - 4)/sqrt(6))/2)**4 + 164193*sqrt ( - 5*x**2 + 8*x - 2)*tan(asin((5*x - 4)/sqrt(6))/2)**2 + 38642*sqrt( - 5* x**2 + 8*x - 2)),x) - 113408736435798599080596860640*sqrt(30)*int(tan(asin ((5*x - 4)/sqrt(6))/2)**4/(38642*sqrt( - 5*x**2 + 8*x - 2)*tan(asin((5*x - 4)/sqrt(6))/2)**12 + 164193*sqrt( - 5*x**2 + 8*x - 2)*tan(asin((5*x - 4)/ sqrt(6))/2)**10 + 416850*sqrt( - 5*x**2 + 8*x - 2)*tan(asin((5*x - 4)/sqrt (6))/2)**8 + 442630*sqrt( - 5*x**2 + 8*x - 2)*tan(asin((5*x - 4)/sqrt(6))/ 2)**6 + 416850*sqrt( - 5*x**2 + 8*x - 2)*tan(asin((5*x - 4)/sqrt(6))/2)...