Integrand size = 25, antiderivative size = 103 \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )} \, dx=-\frac {1}{4 x}+\frac {\sqrt {-3-2 x+4 x^2}}{4 x}+\frac {7 \arctan \left (\frac {3+x}{\sqrt {3} \sqrt {-3-2 x+4 x^2}}\right )}{8 \sqrt {3}}-\frac {1}{8} \text {arctanh}\left (\frac {7+11 x}{\sqrt {-3-2 x+4 x^2}}\right )+\frac {\log (x)}{8}-\frac {1}{8} \log (2+3 x) \] Output:
-1/4/x+1/4*(4*x^2-2*x-3)^(1/2)/x+7/24*3^(1/2)*arctan(1/3*(3+x)*3^(1/2)/(4* x^2-2*x-3)^(1/2))-1/8*arctanh((7+11*x)/(4*x^2-2*x-3)^(1/2))+1/8*ln(x)-1/8* ln(2+3*x)
Time = 0.29 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )} \, dx=-\frac {6-6 \sqrt {-3-2 x+4 x^2}+14 \sqrt {3} x \arctan \left (\frac {-2 x+\sqrt {-3-2 x+4 x^2}}{\sqrt {3}}\right )-3 x \log \left (x \left (-1+4 x-2 \sqrt {-3-2 x+4 x^2}\right )\right )+6 x \log \left (-5-6 x+3 \sqrt {-3-2 x+4 x^2}\right )}{24 x} \] Input:
Integrate[1/(x^2*(1 + 2*x + Sqrt[-3 - 2*x + 4*x^2])),x]
Output:
-1/24*(6 - 6*Sqrt[-3 - 2*x + 4*x^2] + 14*Sqrt[3]*x*ArcTan[(-2*x + Sqrt[-3 - 2*x + 4*x^2])/Sqrt[3]] - 3*x*Log[x*(-1 + 4*x - 2*Sqrt[-3 - 2*x + 4*x^2]) ] + 6*x*Log[-5 - 6*x + 3*Sqrt[-3 - 2*x + 4*x^2]])/x
Time = 0.46 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.32, 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.
\(\displaystyle \int \frac {1}{x^2 \left (\sqrt {4 x^2-2 x-3}+2 x+1\right )} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {3 \sqrt {4 x^2-2 x-3}}{8 x}-\frac {9 \sqrt {4 x^2-2 x-3}}{8 (3 x+2)}-\frac {\sqrt {4 x^2-2 x-3}}{4 x^2}+\frac {1}{4 x^2}+\frac {1}{8 x}-\frac {3}{8 (3 x+2)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3}{8} \sqrt {3} \arctan \left (\frac {x+3}{\sqrt {3} \sqrt {4 x^2-2 x-3}}\right )-\frac {\arctan \left (\frac {x+3}{\sqrt {3} \sqrt {4 x^2-2 x-3}}\right )}{4 \sqrt {3}}-\frac {1}{8} \text {arctanh}\left (\frac {11 x+7}{\sqrt {4 x^2-2 x-3}}\right )+\frac {\sqrt {4 x^2-2 x-3}}{4 x}-\frac {1}{4 x}+\frac {\log (x)}{8}-\frac {1}{8} \log (3 x+2)\) |
Input:
Int[1/(x^2*(1 + 2*x + Sqrt[-3 - 2*x + 4*x^2])),x]
Output:
-1/4*1/x + Sqrt[-3 - 2*x + 4*x^2]/(4*x) - ArcTan[(3 + x)/(Sqrt[3]*Sqrt[-3 - 2*x + 4*x^2])]/(4*Sqrt[3]) + (3*Sqrt[3]*ArcTan[(3 + x)/(Sqrt[3]*Sqrt[-3 - 2*x + 4*x^2])])/8 - ArcTanh[(7 + 11*x)/Sqrt[-3 - 2*x + 4*x^2]]/8 + Log[x ]/8 - Log[2 + 3*x]/8
Leaf count of result is larger than twice the leaf count of optimal. \(196\) vs. \(2(82)=164\).
Time = 0.17 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.91
method | result | size |
default | \(\frac {\ln \left (x \right )}{8}-\frac {\ln \left (2+3 x \right )}{8}-\frac {1}{4 x}-\frac {\left (4 x^{2}-2 x -3\right )^{\frac {3}{2}}}{12 x}+\frac {7 \sqrt {4 x^{2}-2 x -3}}{24}-\frac {11 \ln \left (\frac {\left (4 x -1\right ) \sqrt {4}}{4}+\sqrt {4 x^{2}-2 x -3}\right ) \sqrt {4}}{32}-\frac {7 \sqrt {3}\, \arctan \left (\frac {\left (-6-2 x \right ) \sqrt {3}}{6 \sqrt {4 x^{2}-2 x -3}}\right )}{24}+\frac {\left (8 x -2\right ) \sqrt {4 x^{2}-2 x -3}}{24}-\frac {\sqrt {36 \left (x +\frac {2}{3}\right )^{2}-66 x -43}}{8}+\frac {11 \ln \left (\frac {\left (4 x -1\right ) \sqrt {4}}{4}+\sqrt {4 \left (x +\frac {2}{3}\right )^{2}-\frac {22 x}{3}-\frac {43}{9}}\right ) \sqrt {4}}{32}+\frac {\operatorname {arctanh}\left (\frac {-21-33 x}{\sqrt {36 \left (x +\frac {2}{3}\right )^{2}-66 x -43}}\right )}{8}\) | \(197\) |
trager | \(\frac {x -1}{4 x}+\frac {\sqrt {4 x^{2}-2 x -3}}{4 x}-\frac {\ln \left (\frac {3 \operatorname {RootOf}\left (3 \textit {\_Z}^{2}-3 \textit {\_Z} +13\right )^{2} x -3 \operatorname {RootOf}\left (3 \textit {\_Z}^{2}-3 \textit {\_Z} +13\right )^{2}-259 \operatorname {RootOf}\left (3 \textit {\_Z}^{2}-3 \textit {\_Z} +13\right ) \sqrt {4 x^{2}-2 x -3}+25 \operatorname {RootOf}\left (3 \textit {\_Z}^{2}-3 \textit {\_Z} +13\right ) x +108 \operatorname {RootOf}\left (3 \textit {\_Z}^{2}-3 \textit {\_Z} +13\right )+7 \sqrt {4 x^{2}-2 x -3}-1422 x -972}{x}\right )}{4}+\frac {\operatorname {RootOf}\left (3 \textit {\_Z}^{2}-3 \textit {\_Z} +13\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (3 \textit {\_Z}^{2}-3 \textit {\_Z} +13\right ) x +6 \operatorname {RootOf}\left (3 \textit {\_Z}^{2}-3 \textit {\_Z} +13\right )-7 \sqrt {4 x^{2}-2 x -3}-x -3}{x}\right )}{4}\) | \(201\) |
Input:
int(1/x^2/(1+2*x+(4*x^2-2*x-3)^(1/2)),x,method=_RETURNVERBOSE)
Output:
1/8*ln(x)-1/8*ln(2+3*x)-1/4/x-1/12/x*(4*x^2-2*x-3)^(3/2)+7/24*(4*x^2-2*x-3 )^(1/2)-11/32*ln(1/4*(4*x-1)*4^(1/2)+(4*x^2-2*x-3)^(1/2))*4^(1/2)-7/24*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)-1/8*(36*(x+2/3)^2-66*x-43)^(1/2)+11/32*ln(1/4*(4*x-1)*4^(1/2) +(4*(x+2/3)^2-22/3*x-43/9)^(1/2))*4^(1/2)+1/8*arctanh(9/2*(-14/3-22/3*x)/( 36*(x+2/3)^2-66*x-43)^(1/2))
Time = 0.09 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )} \, dx=-\frac {14 \, \sqrt {3} x \arctan \left (-\frac {2}{3} \, \sqrt {3} x + \frac {1}{3} \, \sqrt {3} \sqrt {4 \, x^{2} - 2 \, x - 3}\right ) + 3 \, x \log \left (3 \, x + 2\right ) - 3 \, x \log \left (x\right ) - 3 \, x \log \left (-2 \, x + \sqrt {4 \, x^{2} - 2 \, x - 3} - 1\right ) + 3 \, x \log \left (-6 \, x + 3 \, \sqrt {4 \, x^{2} - 2 \, x - 3} - 5\right ) - 12 \, x - 6 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 6}{24 \, x} \] Input:
integrate(1/x^2/(1+2*x+(4*x^2-2*x-3)^(1/2)),x, algorithm="fricas")
Output:
-1/24*(14*sqrt(3)*x*arctan(-2/3*sqrt(3)*x + 1/3*sqrt(3)*sqrt(4*x^2 - 2*x - 3)) + 3*x*log(3*x + 2) - 3*x*log(x) - 3*x*log(-2*x + sqrt(4*x^2 - 2*x - 3 ) - 1) + 3*x*log(-6*x + 3*sqrt(4*x^2 - 2*x - 3) - 5) - 12*x - 6*sqrt(4*x^2 - 2*x - 3) + 6)/x
\[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )} \, dx=\int \frac {1}{x^{2} \cdot \left (2 x + \sqrt {4 x^{2} - 2 x - 3} + 1\right )}\, dx \] Input:
integrate(1/x**2/(1+2*x+(4*x**2-2*x-3)**(1/2)),x)
Output:
Integral(1/(x**2*(2*x + sqrt(4*x**2 - 2*x - 3) + 1)), x)
\[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )} \, dx=\int { \frac {1}{{\left (2 \, x + \sqrt {4 \, x^{2} - 2 \, x - 3} + 1\right )} x^{2}} \,d x } \] Input:
integrate(1/x^2/(1+2*x+(4*x^2-2*x-3)^(1/2)),x, algorithm="maxima")
Output:
integrate(1/((2*x + sqrt(4*x^2 - 2*x - 3) + 1)*x^2), x)
Time = 0.14 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.34 \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )} \, dx=-\frac {7}{12} \, \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}\right ) + \frac {2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3} + 6}{2 \, {\left ({\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 3\right )}} - \frac {1}{4 \, x} - \frac {1}{8} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac {1}{8} \, \log \left ({\left | x \right |}\right ) + \frac {1}{8} \, \log \left ({\left | -2 \, x + \sqrt {4 \, x^{2} - 2 \, x - 3} - 1 \right |}\right ) - \frac {1}{8} \, \log \left ({\left | -6 \, x + 3 \, \sqrt {4 \, x^{2} - 2 \, x - 3} - 5 \right |}\right ) \] Input:
integrate(1/x^2/(1+2*x+(4*x^2-2*x-3)^(1/2)),x, algorithm="giac")
Output:
-7/12*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 - 1/8*log(abs(3*x + 2)) + 1/8*log(abs(x)) + 1/8*log(abs(-2*x + sqrt(4*x ^2 - 2*x - 3) - 1)) - 1/8*log(abs(-6*x + 3*sqrt(4*x^2 - 2*x - 3) - 5))
Timed out. \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )} \, dx=-\int \frac {\sqrt {4\,x^2-2\,x-3}}{2\,\left (3\,x^3+2\,x^2\right )} \,d x-\frac {1}{4\,x}+\frac {\mathrm {atan}\left (x\,3{}\mathrm {i}+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4} \] Input:
int(1/(x^2*(2*x + (4*x^2 - 2*x - 3)^(1/2) + 1)),x)
Output:
(atan(x*3i + 1i)*1i)/4 - int((4*x^2 - 2*x - 3)^(1/2)/(2*(2*x^2 + 3*x^3)), x) - 1/(4*x)
Time = 0.18 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )} \, dx=\frac {-14 \sqrt {3}\, \mathit {atan} \left (\frac {\sqrt {4 x^{2}-2 x -3}+2 x}{\sqrt {3}}\right ) x +6 \sqrt {4 x^{2}-2 x -3}-3 \,\mathrm {log}\left (3 x +2\right ) x -3 \,\mathrm {log}\left (\frac {26 \sqrt {4 x^{2}-2 x -3}+52 x +26}{\sqrt {13}}\right ) x +3 \,\mathrm {log}\left (\frac {6 \sqrt {4 x^{2}-2 x -3}+12 x +10}{\sqrt {13}}\right ) x +3 \,\mathrm {log}\left (x \right ) x -6}{24 x} \] Input:
int(1/x^2/(1+2*x+(4*x^2-2*x-3)^(1/2)),x)
Output:
( - 14*sqrt(3)*atan((sqrt(4*x**2 - 2*x - 3) + 2*x)/sqrt(3))*x + 6*sqrt(4*x **2 - 2*x - 3) - 3*log(3*x + 2)*x - 3*log((26*sqrt(4*x**2 - 2*x - 3) + 52* x + 26)/sqrt(13))*x + 3*log((6*sqrt(4*x**2 - 2*x - 3) + 12*x + 10)/sqrt(13 ))*x + 3*log(x)*x - 6)/(24*x)