Integrand size = 25, antiderivative size = 308 \[ \int \frac {1}{x^2 \left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^2} \, dx=-\frac {97+45 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )}{8 \left (1+2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )+5 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )^2\right )}+\frac {23-7 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )}{\left (3+\left (2 x+\sqrt {-3-2 x+4 x^2}\right )^2\right ) \left (1+2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )+5 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )^2\right )}-\frac {\arctan \left (\frac {2 x+\sqrt {-3-2 x+4 x^2}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {37}{16} \arctan \left (\frac {1}{2} \left (1+5 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right )\right )-\frac {3}{4} \log \left (7+3 x-20 x^2-\sqrt {-3-2 x+4 x^2}-10 x \sqrt {-3-2 x+4 x^2}\right )+\frac {3}{4} \log \left (x-4 x^2-2 x \sqrt {-3-2 x+4 x^2}\right ) \] Output:
-1/8*(97+90*x+45*(4*x^2-2*x-3)^(1/2))/(1+4*x+2*(4*x^2-2*x-3)^(1/2)+5*(2*x+ (4*x^2-2*x-3)^(1/2))^2)+(23-14*x-7*(4*x^2-2*x-3)^(1/2))/(3+(2*x+(4*x^2-2*x -3)^(1/2))^2)/(1+4*x+2*(4*x^2-2*x-3)^(1/2)+5*(2*x+(4*x^2-2*x-3)^(1/2))^2)- 1/6*3^(1/2)*arctan(1/3*(2*x+(4*x^2-2*x-3)^(1/2))*3^(1/2))-37/16*arctan(1/2 +5*x+5/2*(4*x^2-2*x-3)^(1/2))-3/4*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))+3/4*ln(x-4*x^2-2*x*(4*x^2-2*x-3)^(1/2))
Time = 0.70 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.63 \[ \int \frac {1}{x^2 \left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^2} \, dx=\frac {1}{48} \left (\frac {3 \sqrt {-3-2 x+4 x^2} \left (8+48 x+55 x^2\right )}{x \left (4+8 x+5 x^2\right )}+\frac {24+540 x+783 x^2+330 x^3}{4 x+8 x^2+5 x^3}+111 \arctan \left (\frac {3}{2}+x-\frac {1}{2} \sqrt {-3-2 x+4 x^2}\right )-8 \sqrt {3} \arctan \left (\frac {-2 x+\sqrt {-3-2 x+4 x^2}}{\sqrt {3}}\right )-72 \text {arctanh}\left (\frac {-5-6 x+3 \sqrt {-3-2 x+4 x^2}}{-5-4 x-8 x^2+(3+4 x) \sqrt {-3-2 x+4 x^2}}\right )\right ) \] Input:
Integrate[1/(x^2*(1 + 3*x + Sqrt[-3 - 2*x + 4*x^2])^2),x]
Output:
((3*Sqrt[-3 - 2*x + 4*x^2]*(8 + 48*x + 55*x^2))/(x*(4 + 8*x + 5*x^2)) + (2 4 + 540*x + 783*x^2 + 330*x^3)/(4*x + 8*x^2 + 5*x^3) + 111*ArcTan[3/2 + x - Sqrt[-3 - 2*x + 4*x^2]/2] - 8*Sqrt[3]*ArcTan[(-2*x + Sqrt[-3 - 2*x + 4*x ^2])/Sqrt[3]] - 72*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])])/48
Time = 0.92 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.88, 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.
| \(\displaystyle \int \frac {1}{x^2 \left (\sqrt {4 x^2-2 x-3}+3 x+1\right )^2} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {-30 x-43}{8 \left (5 x^2+8 x+4\right )}+\frac {\sqrt {4 x^2-2 x-3}}{8 x}-\frac {\sqrt {4 x^2-2 x-3}}{8 x^2}-\frac {5 x \sqrt {4 x^2-2 x-3}}{8 \left (5 x^2+8 x+4\right )}-\frac {3 \sqrt {4 x^2-2 x-3}}{8 \left (5 x^2+8 x+4\right )}-\frac {1}{8 x^2}+\frac {-20 x-1}{2 \left (5 x^2+8 x+4\right )^2}+\frac {5 x \sqrt {4 x^2-2 x-3}}{2 \left (5 x^2+8 x+4\right )^2}+\frac {13 \sqrt {4 x^2-2 x-3}}{2 \left (5 x^2+8 x+4\right )^2}+\frac {3}{4 x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{8} \sqrt {3} \arctan \left (\frac {x+3}{\sqrt {3} \sqrt {4 x^2-2 x-3}}\right )-\frac {\arctan \left (\frac {x+3}{\sqrt {3} \sqrt {4 x^2-2 x-3}}\right )}{8 \sqrt {3}}+\frac {37}{32} \arctan \left (\frac {7 x+8}{2 \sqrt {4 x^2-2 x-3}}\right )+\frac {37}{32} \arctan \left (\frac {5 x}{2}+2\right )-\frac {3}{4} \text {arctanh}\left (\frac {3 x+1}{\sqrt {4 x^2-2 x-3}}\right )-\frac {5 \sqrt {4 x^2-2 x-3} (x+1)}{4 \left (5 x^2+8 x+4\right )}+\frac {\sqrt {4 x^2-2 x-3}}{8 x}+\frac {75 x+76}{16 \left (5 x^2+8 x+4\right )}+\frac {13 (5 x+4) \sqrt {4 x^2-2 x-3}}{16 \left (5 x^2+8 x+4\right )}-\frac {3}{8} \log \left (5 x^2+8 x+4\right )+\frac {1}{8 x}+\frac {3 \log (x)}{4}\) |
Input:
Int[1/(x^2*(1 + 3*x + Sqrt[-3 - 2*x + 4*x^2])^2),x]
Output:
1/(8*x) + Sqrt[-3 - 2*x + 4*x^2]/(8*x) + (76 + 75*x)/(16*(4 + 8*x + 5*x^2) ) - (5*(1 + x)*Sqrt[-3 - 2*x + 4*x^2])/(4*(4 + 8*x + 5*x^2)) + (13*(4 + 5* x)*Sqrt[-3 - 2*x + 4*x^2])/(16*(4 + 8*x + 5*x^2)) + (37*ArcTan[2 + (5*x)/2 ])/32 - ArcTan[(3 + x)/(Sqrt[3]*Sqrt[-3 - 2*x + 4*x^2])]/(8*Sqrt[3]) + (Sq rt[3]*ArcTan[(3 + x)/(Sqrt[3]*Sqrt[-3 - 2*x + 4*x^2])])/8 + (37*ArcTan[(8 + 7*x)/(2*Sqrt[-3 - 2*x + 4*x^2])])/32 - (3*ArcTanh[(1 + 3*x)/Sqrt[-3 - 2* x + 4*x^2]])/4 + (3*Log[x])/4 - (3*Log[4 + 8*x + 5*x^2])/8
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.72 (sec) , antiderivative size = 1687, normalized size of antiderivative = 5.48
| method | result | size |
| trager | \(\text {Expression too large to display}\) | \(1687\) |
| default | \(\text {Expression too large to display}\) | \(3547\) |
Input:
int(1/x^2/(1+3*x+(4*x^2-2*x-3)^(1/2))^2,x,method=_RETURNVERBOSE)
Output:
-1/272*(x-1)*(925*x^2+960*x+136)/x/(5*x^2+8*x+4)+1/16*(55*x^2+48*x+8)/x/(5 *x^2+8*x+4)*(4*x^2-2*x-3)^(1/2)+17/16*RootOf(867*_Z^2-1224*_Z+448)*ln((-70 91684589*RootOf(867*_Z^2-1224*_Z+448)^2*RootOf(1156*_Z^2+1632*_Z+1945)^2*x +7091684589*RootOf(867*_Z^2-1224*_Z+448)^2*RootOf(1156*_Z^2+1632*_Z+1945)^ 2-28957005828*RootOf(867*_Z^2-1224*_Z+448)^2*RootOf(1156*_Z^2+1632*_Z+1945 )*x+8994799008*RootOf(867*_Z^2-1224*_Z+448)*RootOf(1156*_Z^2+1632*_Z+1945) ^2*x+739100160*RootOf(867*_Z^2-1224*_Z+448)*RootOf(1156*_Z^2+1632*_Z+1945) *(4*x^2-2*x-3)^(1/2)+28957005828*RootOf(867*_Z^2-1224*_Z+448)^2*RootOf(115 6*_Z^2+1632*_Z+1945)-26743884288*RootOf(867*_Z^2-1224*_Z+448)^2*x-89947990 08*RootOf(867*_Z^2-1224*_Z+448)*RootOf(1156*_Z^2+1632*_Z+1945)^2+412822547 08*RootOf(867*_Z^2-1224*_Z+448)*RootOf(1156*_Z^2+1632*_Z+1945)*x-282300748 8*RootOf(1156*_Z^2+1632*_Z+1945)^2*x+4626004402*RootOf(867*_Z^2-1224*_Z+44 8)*(4*x^2-2*x-3)^(1/2)-1115735296*(4*x^2-2*x-3)^(1/2)*RootOf(1156*_Z^2+163 2*_Z+1945)+26743884288*RootOf(867*_Z^2-1224*_Z+448)^2-31871003232*RootOf(8 67*_Z^2-1224*_Z+448)*RootOf(1156*_Z^2+1632*_Z+1945)+38819732560*RootOf(867 *_Z^2-1224*_Z+448)*x+2823007488*RootOf(1156*_Z^2+1632*_Z+1945)^2-139676624 00*RootOf(1156*_Z^2+1632*_Z+1945)*x-4089671568*(4*x^2-2*x-3)^(1/2)-2553568 5312*RootOf(867*_Z^2-1224*_Z+448)+8602624512*RootOf(1156*_Z^2+1632*_Z+1945 )-12035612608*x+6073635072)/(68*RootOf(1156*_Z^2+1632*_Z+1945)*x-68*RootOf (1156*_Z^2+1632*_Z+1945)-285*x-344))+3/2*ln(-(7091684589*RootOf(867*_Z^...
Time = 0.09 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^2 \left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^2} \, dx=\frac {660 \, x^{3} - 16 \, \sqrt {3} {\left (5 \, x^{3} + 8 \, x^{2} + 4 \, x\right )} \arctan \left (-\frac {2}{3} \, \sqrt {3} x + \frac {1}{3} \, \sqrt {3} \sqrt {4 \, x^{2} - 2 \, x - 3}\right ) + 1566 \, x^{2} + 111 \, {\left (5 \, x^{3} + 8 \, x^{2} + 4 \, x\right )} \arctan \left (\frac {5}{2} \, x + 2\right ) - 111 \, {\left (5 \, x^{3} + 8 \, x^{2} + 4 \, x\right )} \arctan \left (-x + \frac {1}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {3}{2}\right ) - 111 \, {\left (5 \, x^{3} + 8 \, x^{2} + 4 \, x\right )} \arctan \left (-5 \, x + \frac {5}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {1}{2}\right ) + 36 \, {\left (5 \, x^{3} + 8 \, x^{2} + 4 \, x\right )} \log \left (20 \, x^{2} - \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (10 \, x + 1\right )} - 3 \, x - 7\right ) - 36 \, {\left (5 \, x^{3} + 8 \, x^{2} + 4 \, x\right )} \log \left (5 \, x^{2} + 8 \, x + 4\right ) - 36 \, {\left (5 \, x^{3} + 8 \, x^{2} + 4 \, x\right )} \log \left (4 \, x^{2} - \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (2 \, x + 3\right )} + 5 \, x + 5\right ) + 72 \, {\left (5 \, x^{3} + 8 \, x^{2} + 4 \, x\right )} \log \left (x\right ) + 6 \, {\left (55 \, x^{2} + 48 \, x + 8\right )} \sqrt {4 \, x^{2} - 2 \, x - 3} + 1080 \, x + 48}{96 \, {\left (5 \, x^{3} + 8 \, x^{2} + 4 \, x\right )}} \] Input:
integrate(1/x^2/(1+3*x+(4*x^2-2*x-3)^(1/2))^2,x, algorithm="fricas")
Output:
1/96*(660*x^3 - 16*sqrt(3)*(5*x^3 + 8*x^2 + 4*x)*arctan(-2/3*sqrt(3)*x + 1 /3*sqrt(3)*sqrt(4*x^2 - 2*x - 3)) + 1566*x^2 + 111*(5*x^3 + 8*x^2 + 4*x)*a rctan(5/2*x + 2) - 111*(5*x^3 + 8*x^2 + 4*x)*arctan(-x + 1/2*sqrt(4*x^2 - 2*x - 3) - 3/2) - 111*(5*x^3 + 8*x^2 + 4*x)*arctan(-5*x + 5/2*sqrt(4*x^2 - 2*x - 3) - 1/2) + 36*(5*x^3 + 8*x^2 + 4*x)*log(20*x^2 - sqrt(4*x^2 - 2*x - 3)*(10*x + 1) - 3*x - 7) - 36*(5*x^3 + 8*x^2 + 4*x)*log(5*x^2 + 8*x + 4) - 36*(5*x^3 + 8*x^2 + 4*x)*log(4*x^2 - sqrt(4*x^2 - 2*x - 3)*(2*x + 3) + 5*x + 5) + 72*(5*x^3 + 8*x^2 + 4*x)*log(x) + 6*(55*x^2 + 48*x + 8)*sqrt(4* x^2 - 2*x - 3) + 1080*x + 48)/(5*x^3 + 8*x^2 + 4*x)
\[ \int \frac {1}{x^2 \left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^2} \, dx=\int \frac {1}{x^{2} \left (3 x + \sqrt {4 x^{2} - 2 x - 3} + 1\right )^{2}}\, dx \] Input:
integrate(1/x**2/(1+3*x+(4*x**2-2*x-3)**(1/2))**2,x)
Output:
Integral(1/(x**2*(3*x + sqrt(4*x**2 - 2*x - 3) + 1)**2), x)
\[ \int \frac {1}{x^2 \left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^2} \, dx=\int { \frac {1}{{\left (3 \, x + \sqrt {4 \, x^{2} - 2 \, x - 3} + 1\right )}^{2} x^{2}} \,d x } \] Input:
integrate(1/x^2/(1+3*x+(4*x^2-2*x-3)^(1/2))^2,x, algorithm="maxima")
Output:
integrate(1/((3*x + sqrt(4*x^2 - 2*x - 3) + 1)^2*x^2), x)
Time = 0.14 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.47 \[ \int \frac {1}{x^2 \left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^2} \, dx=-\frac {1}{6} \, \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}\right ) + \frac {215 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{5} + 1018 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{4} + 1898 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{3} + 4012 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 5278 \, x - 2639 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1146}{8 \, {\left (5 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{6} + 32 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{5} + 93 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{4} + 128 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{3} + 247 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 192 \, x - 96 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 39\right )}} + \frac {85 \, x^{2} + 92 \, x + 8}{16 \, {\left (5 \, x^{3} + 8 \, x^{2} + 4 \, x\right )}} + \frac {37}{32} \, \arctan \left (\frac {5}{2} \, x + 2\right ) - \frac {37}{32} \, \arctan \left (-x + \frac {1}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {3}{2}\right ) - \frac {37}{32} \, \arctan \left (-5 \, x + \frac {5}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {1}{2}\right ) + \frac {3}{8} \, \log \left (5 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 4 \, x - 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1\right ) - \frac {3}{8} \, \log \left ({\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 12 \, x - 6 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 13\right ) - \frac {3}{8} \, \log \left (5 \, x^{2} + 8 \, x + 4\right ) + \frac {3}{4} \, \log \left ({\left | x \right |}\right ) \] Input:
integrate(1/x^2/(1+3*x+(4*x^2-2*x-3)^(1/2))^2,x, algorithm="giac")
Output:
-1/6*sqrt(3)*arctan(-1/3*sqrt(3)*(2*x - sqrt(4*x^2 - 2*x - 3))) + 1/8*(215 *(2*x - sqrt(4*x^2 - 2*x - 3))^5 + 1018*(2*x - sqrt(4*x^2 - 2*x - 3))^4 + 1898*(2*x - sqrt(4*x^2 - 2*x - 3))^3 + 4012*(2*x - sqrt(4*x^2 - 2*x - 3))^ 2 + 5278*x - 2639*sqrt(4*x^2 - 2*x - 3) + 1146)/(5*(2*x - sqrt(4*x^2 - 2*x - 3))^6 + 32*(2*x - sqrt(4*x^2 - 2*x - 3))^5 + 93*(2*x - sqrt(4*x^2 - 2*x - 3))^4 + 128*(2*x - sqrt(4*x^2 - 2*x - 3))^3 + 247*(2*x - sqrt(4*x^2 - 2 *x - 3))^2 + 192*x - 96*sqrt(4*x^2 - 2*x - 3) + 39) + 1/16*(85*x^2 + 92*x + 8)/(5*x^3 + 8*x^2 + 4*x) + 37/32*arctan(5/2*x + 2) - 37/32*arctan(-x + 1 /2*sqrt(4*x^2 - 2*x - 3) - 3/2) - 37/32*arctan(-5*x + 5/2*sqrt(4*x^2 - 2*x - 3) - 1/2) + 3/8*log(5*(2*x - sqrt(4*x^2 - 2*x - 3))^2 + 4*x - 2*sqrt(4* x^2 - 2*x - 3) + 1) - 3/8*log((2*x - sqrt(4*x^2 - 2*x - 3))^2 + 12*x - 6*s qrt(4*x^2 - 2*x - 3) + 13) - 3/8*log(5*x^2 + 8*x + 4) + 3/4*log(abs(x))
Timed out. \[ \int \frac {1}{x^2 \left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^2} \, dx=\int \frac {1}{x^2\,{\left (3\,x+\sqrt {4\,x^2-2\,x-3}+1\right )}^2} \,d x \] Input:
int(1/(x^2*(3*x + (4*x^2 - 2*x - 3)^(1/2) + 1)^2),x)
Output:
int(1/(x^2*(3*x + (4*x^2 - 2*x - 3)^(1/2) + 1)^2), x)
Time = 0.59 (sec) , antiderivative size = 671, normalized size of antiderivative = 2.18 \[ \int \frac {1}{x^2 \left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^2} \, dx =\text {Too large to display} \] Input:
int(1/x^2/(1+3*x+(4*x^2-2*x-3)^(1/2))^2,x)
Output:
( - 1753722300*atan((sqrt(4*x**2 - 2*x - 3) + 2*x + 3)/2)*x**3 - 280595568 0*atan((sqrt(4*x**2 - 2*x - 3) + 2*x + 3)/2)*x**2 - 1402977840*atan((sqrt( 4*x**2 - 2*x - 3) + 2*x + 3)/2)*x - 252788800*sqrt(3)*atan((sqrt(4*x**2 - 2*x - 3) + 2*x)/sqrt(3))*x**3 - 404462080*sqrt(3)*atan((sqrt(4*x**2 - 2*x - 3) + 2*x)/sqrt(3))*x**2 - 202231040*sqrt(3)*atan((sqrt(4*x**2 - 2*x - 3) + 2*x)/sqrt(3))*x - 1753722300*atan((5*sqrt(4*x**2 - 2*x - 3) + 10*x + 1) /2)*x**3 - 2805955680*atan((5*sqrt(4*x**2 - 2*x - 3) + 10*x + 1)/2)*x**2 - 1402977840*atan((5*sqrt(4*x**2 - 2*x - 3) + 10*x + 1)/2)*x + 1753722300*a tan((5*x + 4)/2)*x**3 + 2805955680*atan((5*x + 4)/2)*x**2 + 1402977840*ata n((5*x + 4)/2)*x + 1042753800*sqrt(4*x**2 - 2*x - 3)*x**2 + 910039680*sqrt (4*x**2 - 2*x - 3)*x + 151673280*sqrt(4*x**2 - 2*x - 3) - 568774800*log(5* x**2 + 8*x + 4)*x**3 - 910039680*log(5*x**2 + 8*x + 4)*x**2 - 455019840*lo g(5*x**2 + 8*x + 4)*x - 568774800*log((80*sqrt(4*x**2 - 2*x - 3)*x + 8*sqr t(4*x**2 - 2*x - 3) + 160*x**2 - 24*x - 56)/sqrt(13))*x**3 - 910039680*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**2 - 455019840*log((80*sqrt(4*x**2 - 2*x - 3)*x + 8*sqr t(4*x**2 - 2*x - 3) + 160*x**2 - 24*x - 56)/sqrt(13))*x + 568774800*log((1 6*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**3 + 910039680*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**2 + 455019840*log...