Integrand size = 25, antiderivative size = 141 \[ \int \frac {1}{x \left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )} \, dx=\frac {1}{2} \sqrt {3} \arctan \left (\frac {2 x+\sqrt {-3-2 x+4 x^2}}{\sqrt {3}}\right )-2 \arctan \left (\frac {1}{2} \left (1+5 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right )\right )-\frac {1}{4} \log \left (7+3 x-20 x^2-\sqrt {-3-2 x+4 x^2}-10 x \sqrt {-3-2 x+4 x^2}\right )+\frac {1}{4} \log \left (x-4 x^2-2 x \sqrt {-3-2 x+4 x^2}\right ) \] Output:
1/2*3^(1/2)*arctan(1/3*(2*x+(4*x^2-2*x-3)^(1/2))*3^(1/2))-2*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 = 0.40 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x \left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )} \, dx=\frac {1}{2} \left (4 \arctan \left (\frac {3}{2}+x-\frac {1}{2} \sqrt {-3-2 x+4 x^2}\right )+\sqrt {3} \arctan \left (\frac {-2 x+\sqrt {-3-2 x+4 x^2}}{\sqrt {3}}\right )-\text {arctanh}\left (\frac {-5-6 x+3 \sqrt {-3-2 x+4 x^2}}{-5-4 x-8 x^2+(3+4 x) \sqrt {-3-2 x+4 x^2}}\right )\right ) \] Input:
Integrate[1/(x*(1 + 3*x + Sqrt[-3 - 2*x + 4*x^2])),x]
Output:
(4*ArcTan[3/2 + x - Sqrt[-3 - 2*x + 4*x^2]/2] + Sqrt[3]*ArcTan[(-2*x + Sqr t[-3 - 2*x + 4*x^2])/Sqrt[3]] - 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])])/2
Time = 0.60 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.79, 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.
| \(\displaystyle \int \frac {1}{x \left (\sqrt {4 x^2-2 x-3}+3 x+1\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4-5 x}{4 \left (5 x^2+8 x+4\right )}-\frac {\sqrt {4 x^2-2 x-3}}{4 x}+\frac {5 x \sqrt {4 x^2-2 x-3}}{4 \left (5 x^2+8 x+4\right )}+\frac {2 \sqrt {4 x^2-2 x-3}}{5 x^2+8 x+4}+\frac {1}{4 x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{4} \sqrt {3} \arctan \left (\frac {x+3}{\sqrt {3} \sqrt {4 x^2-2 x-3}}\right )+\arctan \left (\frac {7 x+8}{2 \sqrt {4 x^2-2 x-3}}\right )+\arctan \left (\frac {5 x}{2}+2\right )-\frac {1}{4} \text {arctanh}\left (\frac {3 x+1}{\sqrt {4 x^2-2 x-3}}\right )-\frac {1}{8} \log \left (5 x^2+8 x+4\right )+\frac {\log (x)}{4}\) |
Input:
Int[1/(x*(1 + 3*x + Sqrt[-3 - 2*x + 4*x^2])),x]
Output:
ArcTan[2 + (5*x)/2] - (Sqrt[3]*ArcTan[(3 + x)/(Sqrt[3]*Sqrt[-3 - 2*x + 4*x ^2])])/4 + ArcTan[(8 + 7*x)/(2*Sqrt[-3 - 2*x + 4*x^2])] - ArcTanh[(1 + 3*x )/Sqrt[-3 - 2*x + 4*x^2]]/4 + Log[x]/4 - Log[4 + 8*x + 5*x^2]/8
Leaf count of result is larger than twice the leaf count of optimal. \(485\) vs. \(2(117)=234\).
Time = 0.54 (sec) , antiderivative size = 486, normalized size of antiderivative = 3.45
| method | result | size |
| default | \(-\frac {\ln \left (5 x^{2}+8 x +4\right )}{8}+\arctan \left (\frac {5 x}{2}+2\right )+\frac {\ln \left (x \right )}{4}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (-6-2 x \right ) \sqrt {3}}{6 \sqrt {4 x^{2}-2 x -3}}\right )}{4}-\frac {4 \sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}\, \left (\operatorname {arctanh}\left (\frac {\sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}}{51}\right )+4 \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 (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 {3 \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 )}{68 \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 )}\) | \(486\) |
| trager | \(\text {Expression too large to display}\) | \(1620\) |
Input:
int(1/x/(1+3*x+(4*x^2-2*x-3)^(1/2)),x,method=_RETURNVERBOSE)
Output:
-1/8*ln(5*x^2+8*x+4)+arctan(5/2*x+2)+1/4*ln(x)+1/4*3^(1/2)*arctan(1/6*(-6- 2*x)*3^(1/2)/(4*x^2-2*x-3)^(1/2))-4/17*(-833*(8/7+x)^2/(-1/3-x)^2+1989)^(1 /2)*(arctanh(1/51*(-833*(8/7+x)^2/(-1/3-x)^2+1989)^(1/2))+4*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)-1/17*(-833*(8/7+x)^2/(-1/3-x)^2+1989)^(1/2)*(6*arct anh(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)+3/68*(-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)
Time = 0.08 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.15 \[ \int \frac {1}{x \left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )} \, dx=\frac {1}{2} \, \sqrt {3} \arctan \left (-\frac {2}{3} \, \sqrt {3} x + \frac {1}{3} \, \sqrt {3} \sqrt {4 \, x^{2} - 2 \, x - 3}\right ) + \arctan \left (\frac {5}{2} \, x + 2\right ) - \arctan \left (-x + \frac {1}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {3}{2}\right ) - \arctan \left (-5 \, x + \frac {5}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {1}{2}\right ) + \frac {1}{8} \, \log \left (20 \, x^{2} - \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (10 \, x + 1\right )} - 3 \, x - 7\right ) - \frac {1}{8} \, \log \left (5 \, x^{2} + 8 \, x + 4\right ) - \frac {1}{8} \, \log \left (4 \, x^{2} - \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (2 \, x + 3\right )} + 5 \, x + 5\right ) + \frac {1}{4} \, \log \left (x\right ) \] Input:
integrate(1/x/(1+3*x+(4*x^2-2*x-3)^(1/2)),x, algorithm="fricas")
Output:
1/2*sqrt(3)*arctan(-2/3*sqrt(3)*x + 1/3*sqrt(3)*sqrt(4*x^2 - 2*x - 3)) + a rctan(5/2*x + 2) - arctan(-x + 1/2*sqrt(4*x^2 - 2*x - 3) - 3/2) - arctan(- 5*x + 5/2*sqrt(4*x^2 - 2*x - 3) - 1/2) + 1/8*log(20*x^2 - sqrt(4*x^2 - 2*x - 3)*(10*x + 1) - 3*x - 7) - 1/8*log(5*x^2 + 8*x + 4) - 1/8*log(4*x^2 - s qrt(4*x^2 - 2*x - 3)*(2*x + 3) + 5*x + 5) + 1/4*log(x)
\[ \int \frac {1}{x \left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )} \, dx=\int \frac {1}{x \left (3 x + \sqrt {4 x^{2} - 2 x - 3} + 1\right )}\, dx \] Input:
integrate(1/x/(1+3*x+(4*x**2-2*x-3)**(1/2)),x)
Output:
Integral(1/(x*(3*x + sqrt(4*x**2 - 2*x - 3) + 1)), x)
\[ \int \frac {1}{x \left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )} \, dx=\int { \frac {1}{{\left (3 \, x + \sqrt {4 \, x^{2} - 2 \, x - 3} + 1\right )} x} \,d x } \] Input:
integrate(1/x/(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), x)
Time = 0.15 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.30 \[ \int \frac {1}{x \left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )} \, dx=\frac {1}{2} \, \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}\right ) + \arctan \left (\frac {5}{2} \, x + 2\right ) - \arctan \left (-x + \frac {1}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {3}{2}\right ) - \arctan \left (-5 \, x + \frac {5}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {1}{2}\right ) + \frac {1}{8} \, \log \left (5 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 4 \, x - 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1\right ) - \frac {1}{8} \, \log \left ({\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 12 \, x - 6 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 13\right ) - \frac {1}{8} \, \log \left (5 \, x^{2} + 8 \, x + 4\right ) + \frac {1}{4} \, \log \left ({\left | x \right |}\right ) \] Input:
integrate(1/x/(1+3*x+(4*x^2-2*x-3)^(1/2)),x, algorithm="giac")
Output:
1/2*sqrt(3)*arctan(-1/3*sqrt(3)*(2*x - sqrt(4*x^2 - 2*x - 3))) + arctan(5/ 2*x + 2) - arctan(-x + 1/2*sqrt(4*x^2 - 2*x - 3) - 3/2) - 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. \[ \int \frac {1}{x \left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )} \, dx=\int \frac {1}{x\,\left (3\,x+\sqrt {4\,x^2-2\,x-3}+1\right )} \,d x \] Input:
int(1/(x*(3*x + (4*x^2 - 2*x - 3)^(1/2) + 1)),x)
Output:
int(1/(x*(3*x + (4*x^2 - 2*x - 3)^(1/2) + 1)), x)
Time = 0.37 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.27 \[ \int \frac {1}{x \left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )} \, dx=-\mathit {atan} \left (\frac {\sqrt {4 x^{2}-2 x -3}}{2}+x +\frac {3}{2}\right )+\frac {\sqrt {3}\, \mathit {atan} \left (\frac {\sqrt {4 x^{2}-2 x -3}+2 x}{\sqrt {3}}\right )}{2}-\mathit {atan} \left (\frac {5 \sqrt {4 x^{2}-2 x -3}}{2}+5 x +\frac {1}{2}\right )+\mathit {atan} \left (\frac {5 x}{2}+2\right )-\frac {\mathrm {log}\left (5 x^{2}+8 x +4\right )}{8}-\frac {\mathrm {log}\left (\frac {80 \sqrt {4 x^{2}-2 x -3}\, x +8 \sqrt {4 x^{2}-2 x -3}+160 x^{2}-24 x -56}{\sqrt {13}}\right )}{8}+\frac {\mathrm {log}\left (\frac {16 \sqrt {4 x^{2}-2 x -3}\, x +24 \sqrt {4 x^{2}-2 x -3}+32 x^{2}+40 x +40}{\sqrt {13}}\right )}{8}+\frac {\mathrm {log}\left (x \right )}{4} \] Input:
int(1/x/(1+3*x+(4*x^2-2*x-3)^(1/2)),x)
Output:
( - 8*atan((sqrt(4*x**2 - 2*x - 3) + 2*x + 3)/2) + 4*sqrt(3)*atan((sqrt(4* x**2 - 2*x - 3) + 2*x)/sqrt(3)) - 8*atan((5*sqrt(4*x**2 - 2*x - 3) + 10*x + 1)/2) + 8*atan((5*x + 4)/2) - log(5*x**2 + 8*x + 4) - 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) ) + 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)) + 2*log(x))/8