Integrand size = 15, antiderivative size = 71 \[ \int \frac {1}{\left (2 x+\sqrt {1+x^2}\right )^2} \, dx=-\frac {4 x}{3 \left (-1+3 x^2\right )}+\frac {2 \sqrt {1+x^2}}{3 \left (-1+3 x^2\right )}-\frac {2 \text {arctanh}\left (\frac {x}{\sqrt {3}}-\frac {\sqrt {1+x^2}}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
-4*x/(9*x^2-3)+2*(x^2+1)^(1/2)/(9*x^2-3)-2/9*arctanh(1/3*x*3^(1/2)-1/3*(x^ 2+1)^(1/2)*3^(1/2))*3^(1/2)
Time = 0.20 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\left (2 x+\sqrt {1+x^2}\right )^2} \, dx=\frac {6 \left (-2 x+\sqrt {1+x^2}\right )-2 \sqrt {3} \left (-1+3 x^2\right ) \text {arctanh}\left (\frac {x-\sqrt {1+x^2}}{\sqrt {3}}\right )}{-9+27 x^2} \]
(6*(-2*x + Sqrt[1 + x^2]) - 2*Sqrt[3]*(-1 + 3*x^2)*ArcTanh[(x - Sqrt[1 + x ^2])/Sqrt[3]])/(-9 + 27*x^2)
Time = 0.29 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.15, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, 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.
\(\displaystyle \int \frac {1}{\left (\sqrt {x^2+1}+2 x\right )^2} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {4 \sqrt {x^2+1} x}{\left (3 x^2-1\right )^2}+\frac {5}{3 \left (3 x^2-1\right )}+\frac {8}{3 \left (3 x^2-1\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\text {arctanh}\left (\frac {1}{2} \sqrt {3} \sqrt {x^2+1}\right )}{3 \sqrt {3}}-\frac {\text {arctanh}\left (\sqrt {3} x\right )}{3 \sqrt {3}}+\frac {4 x}{3 \left (1-3 x^2\right )}-\frac {2 \sqrt {x^2+1}}{3 \left (1-3 x^2\right )}\) |
(4*x)/(3*(1 - 3*x^2)) - (2*Sqrt[1 + x^2])/(3*(1 - 3*x^2)) - ArcTanh[Sqrt[3 ]*x]/(3*Sqrt[3]) + ArcTanh[(Sqrt[3]*Sqrt[1 + x^2])/2]/(3*Sqrt[3])
3.10.42.3.1 Defintions of rubi rules used
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.42 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.03
method | result | size |
trager | \(-\frac {4 x}{3 \left (3 x^{2}-1\right )}+\frac {2 \sqrt {x^{2}+1}}{3 \left (3 x^{2}-1\right )}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {3 \sqrt {x^{2}+1}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )}{\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +1}\right )}{9}\) | \(73\) |
default | \(-\frac {x}{2 \left (3 x^{2}-1\right )}-\frac {\sqrt {3}\, \operatorname {arctanh}\left (x \sqrt {3}\right )}{9}-\frac {5 x}{18 \left (x^{2}-\frac {1}{3}\right )}-\sqrt {3}\, \left (-\frac {\left (\left (x -\frac {\sqrt {3}}{3}\right )^{2}+\frac {2 \sqrt {3}\, \left (x -\frac {\sqrt {3}}{3}\right )}{3}+\frac {4}{3}\right )^{\frac {3}{2}}}{12 \left (x -\frac {\sqrt {3}}{3}\right )}+\frac {\sqrt {3}\, \left (\frac {\sqrt {9 \left (x -\frac {\sqrt {3}}{3}\right )^{2}+6 \sqrt {3}\, \left (x -\frac {\sqrt {3}}{3}\right )+12}}{3}+\frac {\sqrt {3}\, \operatorname {arcsinh}\left (x \right )}{3}-\frac {2 \sqrt {3}\, \operatorname {arctanh}\left (\frac {3 \left (\frac {8}{3}+\frac {2 \sqrt {3}\, \left (x -\frac {\sqrt {3}}{3}\right )}{3}\right ) \sqrt {3}}{4 \sqrt {9 \left (x -\frac {\sqrt {3}}{3}\right )^{2}+6 \sqrt {3}\, \left (x -\frac {\sqrt {3}}{3}\right )+12}}\right )}{3}\right )}{36}+\frac {x \sqrt {\left (x -\frac {\sqrt {3}}{3}\right )^{2}+\frac {2 \sqrt {3}\, \left (x -\frac {\sqrt {3}}{3}\right )}{3}+\frac {4}{3}}}{12}+\frac {\operatorname {arcsinh}\left (x \right )}{12}\right )+\sqrt {3}\, \left (-\frac {\left (\left (x +\frac {\sqrt {3}}{3}\right )^{2}-\frac {2 \sqrt {3}\, \left (x +\frac {\sqrt {3}}{3}\right )}{3}+\frac {4}{3}\right )^{\frac {3}{2}}}{12 \left (x +\frac {\sqrt {3}}{3}\right )}-\frac {\sqrt {3}\, \left (\frac {\sqrt {9 \left (x +\frac {\sqrt {3}}{3}\right )^{2}-6 \sqrt {3}\, \left (x +\frac {\sqrt {3}}{3}\right )+12}}{3}-\frac {\sqrt {3}\, \operatorname {arcsinh}\left (x \right )}{3}-\frac {2 \sqrt {3}\, \operatorname {arctanh}\left (\frac {3 \left (\frac {8}{3}-\frac {2 \sqrt {3}\, \left (x +\frac {\sqrt {3}}{3}\right )}{3}\right ) \sqrt {3}}{4 \sqrt {9 \left (x +\frac {\sqrt {3}}{3}\right )^{2}-6 \sqrt {3}\, \left (x +\frac {\sqrt {3}}{3}\right )+12}}\right )}{3}\right )}{36}+\frac {x \sqrt {\left (x +\frac {\sqrt {3}}{3}\right )^{2}-\frac {2 \sqrt {3}\, \left (x +\frac {\sqrt {3}}{3}\right )}{3}+\frac {4}{3}}}{12}+\frac {\operatorname {arcsinh}\left (x \right )}{12}\right )\) | \(370\) |
-4/3*x/(3*x^2-1)+2/3/(3*x^2-1)*(x^2+1)^(1/2)+1/9*RootOf(_Z^2-3)*ln(-(3*(x^ 2+1)^(1/2)+2*RootOf(_Z^2-3))/(RootOf(_Z^2-3)*x+1))
Time = 0.27 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.41 \[ \int \frac {1}{\left (2 x+\sqrt {1+x^2}\right )^2} \, dx=\frac {\sqrt {3} {\left (3 \, x^{2} - 1\right )} \log \left (\frac {3 \, x^{2} - 2 \, \sqrt {3} x + 1}{3 \, x^{2} - 1}\right ) + \sqrt {3} {\left (3 \, x^{2} - 1\right )} \log \left (\frac {3 \, x^{2} + 4 \, \sqrt {3} \sqrt {x^{2} + 1} + 7}{3 \, x^{2} - 1}\right ) - 24 \, x + 12 \, \sqrt {x^{2} + 1}}{18 \, {\left (3 \, x^{2} - 1\right )}} \]
1/18*(sqrt(3)*(3*x^2 - 1)*log((3*x^2 - 2*sqrt(3)*x + 1)/(3*x^2 - 1)) + sqr t(3)*(3*x^2 - 1)*log((3*x^2 + 4*sqrt(3)*sqrt(x^2 + 1) + 7)/(3*x^2 - 1)) - 24*x + 12*sqrt(x^2 + 1))/(3*x^2 - 1)
\[ \int \frac {1}{\left (2 x+\sqrt {1+x^2}\right )^2} \, dx=\int \frac {1}{\left (2 x + \sqrt {x^{2} + 1}\right )^{2}}\, dx \]
\[ \int \frac {1}{\left (2 x+\sqrt {1+x^2}\right )^2} \, dx=\int { \frac {1}{{\left (2 \, x + \sqrt {x^{2} + 1}\right )}^{2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 177 vs. \(2 (56) = 112\).
Time = 0.27 (sec) , antiderivative size = 177, normalized size of antiderivative = 2.49 \[ \int \frac {1}{\left (2 x+\sqrt {1+x^2}\right )^2} \, dx=\frac {1}{18} \, \sqrt {3} \log \left (\frac {{\left | 6 \, x - 2 \, \sqrt {3} \right |}}{{\left | 6 \, x + 2 \, \sqrt {3} \right |}}\right ) - \frac {1}{18} \, \sqrt {3} \log \left (-\frac {{\left | -6 \, x - 8 \, \sqrt {3} + 6 \, \sqrt {x^{2} + 1} - \frac {6}{x - \sqrt {x^{2} + 1}} \right |}}{2 \, {\left (3 \, x - 4 \, \sqrt {3} - 3 \, \sqrt {x^{2} + 1} + \frac {3}{x - \sqrt {x^{2} + 1}}\right )}}\right ) - \frac {4 \, {\left (x - \sqrt {x^{2} + 1} + \frac {1}{x - \sqrt {x^{2} + 1}}\right )}}{3 \, {\left (3 \, {\left (x - \sqrt {x^{2} + 1} + \frac {1}{x - \sqrt {x^{2} + 1}}\right )}^{2} - 16\right )}} - \frac {4 \, x}{3 \, {\left (3 \, x^{2} - 1\right )}} \]
1/18*sqrt(3)*log(abs(6*x - 2*sqrt(3))/abs(6*x + 2*sqrt(3))) - 1/18*sqrt(3) *log(-1/2*abs(-6*x - 8*sqrt(3) + 6*sqrt(x^2 + 1) - 6/(x - sqrt(x^2 + 1)))/ (3*x - 4*sqrt(3) - 3*sqrt(x^2 + 1) + 3/(x - sqrt(x^2 + 1)))) - 4/3*(x - sq rt(x^2 + 1) + 1/(x - sqrt(x^2 + 1)))/(3*(x - sqrt(x^2 + 1) + 1/(x - sqrt(x ^2 + 1)))^2 - 16) - 4/3*x/(3*x^2 - 1)
Time = 0.24 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.87 \[ \int \frac {1}{\left (2 x+\sqrt {1+x^2}\right )^2} \, dx=\frac {\sqrt {3}\,\left (\ln \left (x-\frac {\sqrt {3}}{3}\right )-\ln \left (x+\sqrt {3}+2\,\sqrt {x^2+1}\right )\right )}{18}-\frac {4\,x}{9\,\left (x^2-\frac {1}{3}\right )}+\frac {\sqrt {3}\,\left (\ln \left (x+\frac {\sqrt {3}}{3}\right )-\ln \left (x-\sqrt {3}-2\,\sqrt {x^2+1}\right )\right )}{18}-\frac {\sqrt {3}\,\left (6\,\ln \left (x-\frac {\sqrt {3}}{3}\right )-6\,\ln \left (x+\sqrt {3}+2\,\sqrt {x^2+1}\right )\right )}{54}-\frac {\sqrt {3}\,\left (6\,\ln \left (x+\frac {\sqrt {3}}{3}\right )-6\,\ln \left (x-\sqrt {3}-2\,\sqrt {x^2+1}\right )\right )}{54}+\frac {\sqrt {3}\,\sqrt {x^2+1}}{9\,\left (x-\frac {\sqrt {3}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {x^2+1}}{9\,\left (x+\frac {\sqrt {3}}{3}\right )}+\frac {\sqrt {3}\,\mathrm {atan}\left (\sqrt {3}\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{9} \]
(3^(1/2)*(log(x - 3^(1/2)/3) - log(x + 3^(1/2) + 2*(x^2 + 1)^(1/2))))/18 + (3^(1/2)*atan(3^(1/2)*x*1i)*1i)/9 - (4*x)/(9*(x^2 - 1/3)) + (3^(1/2)*(log (x + 3^(1/2)/3) - log(x - 3^(1/2) - 2*(x^2 + 1)^(1/2))))/18 - (3^(1/2)*(6* log(x - 3^(1/2)/3) - 6*log(x + 3^(1/2) + 2*(x^2 + 1)^(1/2))))/54 - (3^(1/2 )*(6*log(x + 3^(1/2)/3) - 6*log(x - 3^(1/2) - 2*(x^2 + 1)^(1/2))))/54 + (3 ^(1/2)*(x^2 + 1)^(1/2))/(9*(x - 3^(1/2)/3)) - (3^(1/2)*(x^2 + 1)^(1/2))/(9 *(x + 3^(1/2)/3))