Integrand size = 25, antiderivative size = 193 \[ \int \frac {1}{x^2 \left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )} \, dx=-\frac {5+3 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )}{2 \left (3+\left (2 x+\sqrt {-3-2 x+4 x^2}\right )^2\right )}-\frac {5 \arctan \left (\frac {2 x+\sqrt {-3-2 x+4 x^2}}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {9}{4} \arctan \left (\frac {1}{2} \left (1+5 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right )\right )-\frac {1}{4} \log \left (7+3 x-20 x^2-\sqrt {-3-2 x+4 x^2}-10 x \sqrt {-3-2 x+4 x^2}\right )+\frac {1}{4} \log \left (x-4 x^2-2 x \sqrt {-3-2 x+4 x^2}\right ) \] Output:
-1/2*(5+6*x+3*(4*x^2-2*x-3)^(1/2))/(3+(2*x+(4*x^2-2*x-3)^(1/2))^2)-5/6*3^( 1/2)*arctan(1/3*(2*x+(4*x^2-2*x-3)^(1/2))*3^(1/2))+9/4*arctan(1/2+5*x+5/2* (4*x^2-2*x-3)^(1/2))-1/4*ln(7+3*x-20*x^2-(4*x^2-2*x-3)^(1/2)-10*x*(4*x^2-2 *x-3)^(1/2))+1/4*ln(x-4*x^2-2*x*(4*x^2-2*x-3)^(1/2))
Time = 0.42 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x^2 \left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )} \, dx=-\frac {3-3 \sqrt {-3-2 x+4 x^2}+27 x \arctan \left (\frac {3}{2}+x-\frac {1}{2} \sqrt {-3-2 x+4 x^2}\right )+10 \sqrt {3} x \arctan \left (\frac {-2 x+\sqrt {-3-2 x+4 x^2}}{\sqrt {3}}\right )+6 x \text {arctanh}\left (\frac {-5-6 x+3 \sqrt {-3-2 x+4 x^2}}{-5-4 x-8 x^2+(3+4 x) \sqrt {-3-2 x+4 x^2}}\right )}{12 x} \] Input:
Integrate[1/(x^2*(1 + 3*x + Sqrt[-3 - 2*x + 4*x^2])),x]
Output:
-1/12*(3 - 3*Sqrt[-3 - 2*x + 4*x^2] + 27*x*ArcTan[3/2 + x - Sqrt[-3 - 2*x + 4*x^2]/2] + 10*Sqrt[3]*x*ArcTan[(-2*x + Sqrt[-3 - 2*x + 4*x^2])/Sqrt[3]] + 6*x*ArcTanh[(-5 - 6*x + 3*Sqrt[-3 - 2*x + 4*x^2])/(-5 - 4*x - 8*x^2 + ( 3 + 4*x)*Sqrt[-3 - 2*x + 4*x^2])])/x
Time = 0.67 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.94, 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.
| \(\displaystyle \int \frac {1}{x^2 \left (\sqrt {4 x^2-2 x-3}+3 x+1\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {-5 x-13}{4 \left (5 x^2+8 x+4\right )}+\frac {\sqrt {4 x^2-2 x-3}}{2 x}-\frac {\sqrt {4 x^2-2 x-3}}{4 x^2}-\frac {5 x \sqrt {4 x^2-2 x-3}}{2 \left (5 x^2+8 x+4\right )}-\frac {11 \sqrt {4 x^2-2 x-3}}{4 \left (5 x^2+8 x+4\right )}+\frac {1}{4 x^2}+\frac {1}{4 x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \sqrt {3} \arctan \left (\frac {x+3}{\sqrt {3} \sqrt {4 x^2-2 x-3}}\right )-\frac {\arctan \left (\frac {x+3}{\sqrt {3} \sqrt {4 x^2-2 x-3}}\right )}{4 \sqrt {3}}-\frac {9}{8} \arctan \left (\frac {7 x+8}{2 \sqrt {4 x^2-2 x-3}}\right )-\frac {9}{8} \arctan \left (\frac {5 x}{2}+2\right )-\frac {1}{4} \text {arctanh}\left (\frac {3 x+1}{\sqrt {4 x^2-2 x-3}}\right )+\frac {\sqrt {4 x^2-2 x-3}}{4 x}-\frac {1}{8} \log \left (5 x^2+8 x+4\right )-\frac {1}{4 x}+\frac {\log (x)}{4}\) |
Input:
Int[1/(x^2*(1 + 3*x + Sqrt[-3 - 2*x + 4*x^2])),x]
Output:
-1/4*1/x + Sqrt[-3 - 2*x + 4*x^2]/(4*x) - (9*ArcTan[2 + (5*x)/2])/8 - ArcT an[(3 + x)/(Sqrt[3]*Sqrt[-3 - 2*x + 4*x^2])]/(4*Sqrt[3]) + (Sqrt[3]*ArcTan [(3 + x)/(Sqrt[3]*Sqrt[-3 - 2*x + 4*x^2])])/2 - (9*ArcTan[(8 + 7*x)/(2*Sqr t[-3 - 2*x + 4*x^2])])/8 - ArcTanh[(1 + 3*x)/Sqrt[-3 - 2*x + 4*x^2]]/4 + L og[x]/4 - Log[4 + 8*x + 5*x^2]/8
Leaf count of result is larger than twice the leaf count of optimal. \(544\) vs. \(2(160)=320\).
Time = 0.65 (sec) , antiderivative size = 545, normalized size of antiderivative = 2.82
| method | result | size |
| default | \(-\frac {\ln \left (5 x^{2}+8 x +4\right )}{8}-\frac {9 \arctan \left (\frac {5 x}{2}+2\right )}{8}-\frac {1}{4 x}+\frac {\ln \left (x \right )}{4}-\frac {\left (4 x^{2}-2 x -3\right )^{\frac {3}{2}}}{12 x}-\frac {\sqrt {4 x^{2}-2 x -3}}{12}-\frac {5 \sqrt {3}\, \arctan \left (\frac {\left (-6-2 x \right ) \sqrt {3}}{6 \sqrt {4 x^{2}-2 x -3}}\right )}{12}+\frac {\left (8 x -2\right ) \sqrt {4 x^{2}-2 x -3}}{24}+\frac {2 \sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}\, \left (6 \,\operatorname {arctanh}\left (\frac {\sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}}{51}\right )+7 \arctan \left (\frac {7 \sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}\, \left (\frac {8}{7}+x \right )}{2 \left (\frac {49 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}-117\right ) \left (-\frac {1}{3}-x \right )}\right )\right )}{17 \sqrt {-\frac {17 \left (\frac {49 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}-117\right )}{\left (\frac {\frac {8}{7}+x}{-\frac {1}{3}-x}+1\right )^{2}}}\, \left (\frac {\frac {8}{7}+x}{-\frac {1}{3}-x}+1\right )}+\frac {\sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}\, \left (19 \,\operatorname {arctanh}\left (\frac {\sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}}{51}\right )+8 \arctan \left (\frac {7 \sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}\, \left (\frac {8}{7}+x \right )}{2 \left (\frac {49 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}-117\right ) \left (-\frac {1}{3}-x \right )}\right )\right )}{34 \sqrt {-\frac {17 \left (\frac {49 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}-117\right )}{\left (\frac {\frac {8}{7}+x}{-\frac {1}{3}-x}+1\right )^{2}}}\, \left (\frac {\frac {8}{7}+x}{-\frac {1}{3}-x}+1\right )}-\frac {3 \sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}\, \left (46 \,\operatorname {arctanh}\left (\frac {\sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}}{51}\right )-3 \arctan \left (\frac {7 \sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}\, \left (\frac {8}{7}+x \right )}{2 \left (\frac {49 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}-117\right ) \left (-\frac {1}{3}-x \right )}\right )\right )}{136 \sqrt {-\frac {17 \left (\frac {49 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}-117\right )}{\left (\frac {\frac {8}{7}+x}{-\frac {1}{3}-x}+1\right )^{2}}}\, \left (\frac {\frac {8}{7}+x}{-\frac {1}{3}-x}+1\right )}\) | \(545\) |
| trager | \(\text {Expression too large to display}\) | \(1646\) |
Input:
int(1/x^2/(1+3*x+(4*x^2-2*x-3)^(1/2)),x,method=_RETURNVERBOSE)
Output:
-1/8*ln(5*x^2+8*x+4)-9/8*arctan(5/2*x+2)-1/4/x+1/4*ln(x)-1/12/x*(4*x^2-2*x -3)^(3/2)-1/12*(4*x^2-2*x-3)^(1/2)-5/12*3^(1/2)*arctan(1/6*(-6-2*x)*3^(1/2 )/(4*x^2-2*x-3)^(1/2))+1/24*(8*x-2)*(4*x^2-2*x-3)^(1/2)+2/17*(-833*(8/7+x) ^2/(-1/3-x)^2+1989)^(1/2)*(6*arctanh(1/51*(-833*(8/7+x)^2/(-1/3-x)^2+1989) ^(1/2))+7*arctan(7/2/(49*(8/7+x)^2/(-1/3-x)^2-117)*(-833*(8/7+x)^2/(-1/3-x )^2+1989)^(1/2)*(8/7+x)/(-1/3-x)))/(-17*(49*(8/7+x)^2/(-1/3-x)^2-117)/((8/ 7+x)/(-1/3-x)+1)^2)^(1/2)/((8/7+x)/(-1/3-x)+1)+1/34*(-833*(8/7+x)^2/(-1/3- x)^2+1989)^(1/2)*(19*arctanh(1/51*(-833*(8/7+x)^2/(-1/3-x)^2+1989)^(1/2))+ 8*arctan(7/2/(49*(8/7+x)^2/(-1/3-x)^2-117)*(-833*(8/7+x)^2/(-1/3-x)^2+1989 )^(1/2)*(8/7+x)/(-1/3-x)))/(-17*(49*(8/7+x)^2/(-1/3-x)^2-117)/((8/7+x)/(-1 /3-x)+1)^2)^(1/2)/((8/7+x)/(-1/3-x)+1)-3/136*(-833*(8/7+x)^2/(-1/3-x)^2+19 89)^(1/2)*(46*arctanh(1/51*(-833*(8/7+x)^2/(-1/3-x)^2+1989)^(1/2))-3*arcta n(7/2/(49*(8/7+x)^2/(-1/3-x)^2-117)*(-833*(8/7+x)^2/(-1/3-x)^2+1989)^(1/2) *(8/7+x)/(-1/3-x)))/(-17*(49*(8/7+x)^2/(-1/3-x)^2-117)/((8/7+x)/(-1/3-x)+1 )^2)^(1/2)/((8/7+x)/(-1/3-x)+1)
Time = 0.09 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.01 \[ \int \frac {1}{x^2 \left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )} \, dx=-\frac {20 \, \sqrt {3} x \arctan \left (-\frac {2}{3} \, \sqrt {3} x + \frac {1}{3} \, \sqrt {3} \sqrt {4 \, x^{2} - 2 \, x - 3}\right ) + 27 \, x \arctan \left (\frac {5}{2} \, x + 2\right ) - 27 \, x \arctan \left (-x + \frac {1}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {3}{2}\right ) - 27 \, x \arctan \left (-5 \, x + \frac {5}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {1}{2}\right ) - 3 \, x \log \left (20 \, x^{2} - \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (10 \, x + 1\right )} - 3 \, x - 7\right ) + 3 \, x \log \left (5 \, x^{2} + 8 \, x + 4\right ) + 3 \, x \log \left (4 \, x^{2} - \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (2 \, x + 3\right )} + 5 \, x + 5\right ) - 6 \, x \log \left (x\right ) - 12 \, x - 6 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 6}{24 \, x} \] Input:
integrate(1/x^2/(1+3*x+(4*x^2-2*x-3)^(1/2)),x, algorithm="fricas")
Output:
-1/24*(20*sqrt(3)*x*arctan(-2/3*sqrt(3)*x + 1/3*sqrt(3)*sqrt(4*x^2 - 2*x - 3)) + 27*x*arctan(5/2*x + 2) - 27*x*arctan(-x + 1/2*sqrt(4*x^2 - 2*x - 3) - 3/2) - 27*x*arctan(-5*x + 5/2*sqrt(4*x^2 - 2*x - 3) - 1/2) - 3*x*log(20 *x^2 - sqrt(4*x^2 - 2*x - 3)*(10*x + 1) - 3*x - 7) + 3*x*log(5*x^2 + 8*x + 4) + 3*x*log(4*x^2 - sqrt(4*x^2 - 2*x - 3)*(2*x + 3) + 5*x + 5) - 6*x*log (x) - 12*x - 6*sqrt(4*x^2 - 2*x - 3) + 6)/x
\[ \int \frac {1}{x^2 \left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )} \, dx=\int \frac {1}{x^{2} \cdot \left (3 x + \sqrt {4 x^{2} - 2 x - 3} + 1\right )}\, dx \] Input:
integrate(1/x**2/(1+3*x+(4*x**2-2*x-3)**(1/2)),x)
Output:
Integral(1/(x**2*(3*x + sqrt(4*x**2 - 2*x - 3) + 1)), x)
\[ \int \frac {1}{x^2 \left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )} \, dx=\int { \frac {1}{{\left (3 \, x + \sqrt {4 \, x^{2} - 2 \, x - 3} + 1\right )} x^{2}} \,d x } \] Input:
integrate(1/x^2/(1+3*x+(4*x^2-2*x-3)^(1/2)),x, algorithm="maxima")
Output:
integrate(1/((3*x + sqrt(4*x^2 - 2*x - 3) + 1)*x^2), x)
Time = 0.14 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.22 \[ \int \frac {1}{x^2 \left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )} \, dx=-\frac {5}{6} \, \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}\right ) + \frac {2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3} + 6}{2 \, {\left ({\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 3\right )}} - \frac {1}{4 \, x} - \frac {9}{8} \, \arctan \left (\frac {5}{2} \, x + 2\right ) + \frac {9}{8} \, \arctan \left (-x + \frac {1}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {3}{2}\right ) + \frac {9}{8} \, \arctan \left (-5 \, x + \frac {5}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {1}{2}\right ) + \frac {1}{8} \, \log \left (5 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 4 \, x - 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1\right ) - \frac {1}{8} \, \log \left ({\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 12 \, x - 6 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 13\right ) - \frac {1}{8} \, \log \left (5 \, x^{2} + 8 \, x + 4\right ) + \frac {1}{4} \, \log \left ({\left | x \right |}\right ) \] Input:
integrate(1/x^2/(1+3*x+(4*x^2-2*x-3)^(1/2)),x, algorithm="giac")
Output:
-5/6*sqrt(3)*arctan(-1/3*sqrt(3)*(2*x - sqrt(4*x^2 - 2*x - 3))) + 1/2*(2*x - sqrt(4*x^2 - 2*x - 3) + 6)/((2*x - sqrt(4*x^2 - 2*x - 3))^2 + 3) - 1/4/ x - 9/8*arctan(5/2*x + 2) + 9/8*arctan(-x + 1/2*sqrt(4*x^2 - 2*x - 3) - 3/ 2) + 9/8*arctan(-5*x + 5/2*sqrt(4*x^2 - 2*x - 3) - 1/2) + 1/8*log(5*(2*x - sqrt(4*x^2 - 2*x - 3))^2 + 4*x - 2*sqrt(4*x^2 - 2*x - 3) + 1) - 1/8*log(( 2*x - sqrt(4*x^2 - 2*x - 3))^2 + 12*x - 6*sqrt(4*x^2 - 2*x - 3) + 13) - 1/ 8*log(5*x^2 + 8*x + 4) + 1/4*log(abs(x))
Timed out. \[ \int \frac {1}{x^2 \left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )} \, dx=\int \frac {1}{x^2\,\left (3\,x+\sqrt {4\,x^2-2\,x-3}+1\right )} \,d x \] Input:
int(1/(x^2*(3*x + (4*x^2 - 2*x - 3)^(1/2) + 1)),x)
Output:
int(1/(x^2*(3*x + (4*x^2 - 2*x - 3)^(1/2) + 1)), x)
Time = 0.22 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^2 \left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )} \, dx=\frac {27 \mathit {atan} \left (\frac {\sqrt {4 x^{2}-2 x -3}}{2}+x +\frac {3}{2}\right ) x -20 \sqrt {3}\, \mathit {atan} \left (\frac {\sqrt {4 x^{2}-2 x -3}+2 x}{\sqrt {3}}\right ) x +27 \mathit {atan} \left (\frac {5 \sqrt {4 x^{2}-2 x -3}}{2}+5 x +\frac {1}{2}\right ) x -27 \mathit {atan} \left (\frac {5 x}{2}+2\right ) x +6 \sqrt {4 x^{2}-2 x -3}-3 \,\mathrm {log}\left (5 x^{2}+8 x +4\right ) x -3 \,\mathrm {log}\left (\frac {80 \sqrt {4 x^{2}-2 x -3}\, x +8 \sqrt {4 x^{2}-2 x -3}+160 x^{2}-24 x -56}{\sqrt {13}}\right ) x +3 \,\mathrm {log}\left (\frac {16 \sqrt {4 x^{2}-2 x -3}\, x +24 \sqrt {4 x^{2}-2 x -3}+32 x^{2}+40 x +40}{\sqrt {13}}\right ) x +6 \,\mathrm {log}\left (x \right ) x -6}{24 x} \] Input:
int(1/x^2/(1+3*x+(4*x^2-2*x-3)^(1/2)),x)
Output:
(27*atan((sqrt(4*x**2 - 2*x - 3) + 2*x + 3)/2)*x - 20*sqrt(3)*atan((sqrt(4 *x**2 - 2*x - 3) + 2*x)/sqrt(3))*x + 27*atan((5*sqrt(4*x**2 - 2*x - 3) + 1 0*x + 1)/2)*x - 27*atan((5*x + 4)/2)*x + 6*sqrt(4*x**2 - 2*x - 3) - 3*log( 5*x**2 + 8*x + 4)*x - 3*log((80*sqrt(4*x**2 - 2*x - 3)*x + 8*sqrt(4*x**2 - 2*x - 3) + 160*x**2 - 24*x - 56)/sqrt(13))*x + 3*log((16*sqrt(4*x**2 - 2* x - 3)*x + 24*sqrt(4*x**2 - 2*x - 3) + 32*x**2 + 40*x + 40)/sqrt(13))*x + 6*log(x)*x - 6)/(24*x)