Optimal. Leaf size=82 \[ \frac {4 x}{3 \left (1-3 x^2\right )}-\frac {2 \sqrt {x^2+1}}{3 \left (1-3 x^2\right )}+\frac {\tanh ^{-1}\left (\frac {1}{2} \sqrt {3} \sqrt {x^2+1}\right )}{3 \sqrt {3}}-\frac {\tanh ^{-1}\left (\sqrt {3} x\right )}{3 \sqrt {3}} \]
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Rubi [A] time = 0.08, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6742, 199, 207, 444, 47, 63} \[ \frac {4 x}{3 \left (1-3 x^2\right )}-\frac {2 \sqrt {x^2+1}}{3 \left (1-3 x^2\right )}+\frac {\tanh ^{-1}\left (\frac {1}{2} \sqrt {3} \sqrt {x^2+1}\right )}{3 \sqrt {3}}-\frac {\tanh ^{-1}\left (\sqrt {3} x\right )}{3 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 199
Rule 207
Rule 444
Rule 6742
Rubi steps
\begin {align*} \int \frac {1}{\left (2 x+\sqrt {1+x^2}\right )^2} \, dx &=\int \left (\frac {8}{3 \left (-1+3 x^2\right )^2}-\frac {4 x \sqrt {1+x^2}}{\left (-1+3 x^2\right )^2}+\frac {5}{3 \left (-1+3 x^2\right )}\right ) \, dx\\ &=\frac {5}{3} \int \frac {1}{-1+3 x^2} \, dx+\frac {8}{3} \int \frac {1}{\left (-1+3 x^2\right )^2} \, dx-4 \int \frac {x \sqrt {1+x^2}}{\left (-1+3 x^2\right )^2} \, dx\\ &=\frac {4 x}{3 \left (1-3 x^2\right )}-\frac {5 \tanh ^{-1}\left (\sqrt {3} x\right )}{3 \sqrt {3}}-\frac {4}{3} \int \frac {1}{-1+3 x^2} \, dx-2 \operatorname {Subst}\left (\int \frac {\sqrt {1+x}}{(-1+3 x)^2} \, dx,x,x^2\right )\\ &=\frac {4 x}{3 \left (1-3 x^2\right )}-\frac {2 \sqrt {1+x^2}}{3 \left (1-3 x^2\right )}-\frac {\tanh ^{-1}\left (\sqrt {3} x\right )}{3 \sqrt {3}}-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x} (-1+3 x)} \, dx,x,x^2\right )\\ &=\frac {4 x}{3 \left (1-3 x^2\right )}-\frac {2 \sqrt {1+x^2}}{3 \left (1-3 x^2\right )}-\frac {\tanh ^{-1}\left (\sqrt {3} x\right )}{3 \sqrt {3}}-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{-4+3 x^2} \, dx,x,\sqrt {1+x^2}\right )\\ &=\frac {4 x}{3 \left (1-3 x^2\right )}-\frac {2 \sqrt {1+x^2}}{3 \left (1-3 x^2\right )}-\frac {\tanh ^{-1}\left (\sqrt {3} x\right )}{3 \sqrt {3}}+\frac {\tanh ^{-1}\left (\frac {1}{2} \sqrt {3} \sqrt {1+x^2}\right )}{3 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 99, normalized size = 1.21 \[ \frac {1}{9} \left (\frac {12 x}{1-3 x^2}-\frac {\frac {6 x^2+6}{1-3 x^2}+\sqrt {3} \sqrt {-x^2-1} \tan ^{-1}\left (\frac {1}{2} \sqrt {3} \sqrt {-x^2-1}\right )}{\sqrt {x^2+1}}-\sqrt {3} \tanh ^{-1}\left (\sqrt {3} x\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.17, size = 100, normalized size = 1.22 \[ \frac {\sqrt {3} {\left (3 \, x^{2} - 1\right )} \log \left (\frac {3 \, x^{2} - 2 \, \sqrt {3} x + 1}{3 \, x^{2} - 1}\right ) + \sqrt {3} {\left (3 \, x^{2} - 1\right )} \log \left (\frac {3 \, x^{2} + 4 \, \sqrt {3} \sqrt {x^{2} + 1} + 7}{3 \, x^{2} - 1}\right ) - 24 \, x + 12 \, \sqrt {x^{2} + 1}}{18 \, {\left (3 \, x^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.91, size = 177, normalized size = 2.16 \[ \frac {1}{18} \, \sqrt {3} \log \left (\frac {{\left | 6 \, x - 2 \, \sqrt {3} \right |}}{{\left | 6 \, x + 2 \, \sqrt {3} \right |}}\right ) - \frac {1}{18} \, \sqrt {3} \log \left (-\frac {{\left | -6 \, x - 8 \, \sqrt {3} + 6 \, \sqrt {x^{2} + 1} - \frac {6}{x - \sqrt {x^{2} + 1}} \right |}}{2 \, {\left (3 \, x - 4 \, \sqrt {3} - 3 \, \sqrt {x^{2} + 1} + \frac {3}{x - \sqrt {x^{2} + 1}}\right )}}\right ) - \frac {4 \, {\left (x - \sqrt {x^{2} + 1} + \frac {1}{x - \sqrt {x^{2} + 1}}\right )}}{3 \, {\left (3 \, {\left (x - \sqrt {x^{2} + 1} + \frac {1}{x - \sqrt {x^{2} + 1}}\right )}^{2} - 16\right )}} - \frac {4 \, x}{3 \, {\left (3 \, x^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 370, normalized size = 4.51 \[ -\frac {x}{2 \left (3 x^{2}-1\right )}-\frac {5 x}{18 \left (x^{2}-\frac {1}{3}\right )}-\frac {\sqrt {3}\, \arctanh \left (\sqrt {3}\, x \right )}{9}-\sqrt {3}\, \left (\frac {\sqrt {\left (x -\frac {\sqrt {3}}{3}\right )^{2}+\frac {2 \sqrt {3}\, \left (x -\frac {\sqrt {3}}{3}\right )}{3}+\frac {4}{3}}\, x}{12}+\frac {\arcsinh \relax (x )}{12}-\frac {\left (\left (x -\frac {\sqrt {3}}{3}\right )^{2}+\frac {2 \sqrt {3}\, \left (x -\frac {\sqrt {3}}{3}\right )}{3}+\frac {4}{3}\right )^{\frac {3}{2}}}{12 \left (x -\frac {\sqrt {3}}{3}\right )}+\frac {\sqrt {3}\, \left (\frac {\sqrt {3}\, \arcsinh \relax (x )}{3}-\frac {2 \sqrt {3}\, \arctanh \left (\frac {3 \left (\frac {8}{3}+\frac {2 \sqrt {3}\, \left (x -\frac {\sqrt {3}}{3}\right )}{3}\right ) \sqrt {3}}{4 \sqrt {9 \left (x -\frac {\sqrt {3}}{3}\right )^{2}+6 \sqrt {3}\, \left (x -\frac {\sqrt {3}}{3}\right )+12}}\right )}{3}+\frac {\sqrt {9 \left (x -\frac {\sqrt {3}}{3}\right )^{2}+6 \sqrt {3}\, \left (x -\frac {\sqrt {3}}{3}\right )+12}}{3}\right )}{36}\right )+\sqrt {3}\, \left (\frac {\sqrt {\left (x +\frac {\sqrt {3}}{3}\right )^{2}-\frac {2 \sqrt {3}\, \left (x +\frac {\sqrt {3}}{3}\right )}{3}+\frac {4}{3}}\, x}{12}+\frac {\arcsinh \relax (x )}{12}-\frac {\left (\left (x +\frac {\sqrt {3}}{3}\right )^{2}-\frac {2 \sqrt {3}\, \left (x +\frac {\sqrt {3}}{3}\right )}{3}+\frac {4}{3}\right )^{\frac {3}{2}}}{12 \left (x +\frac {\sqrt {3}}{3}\right )}-\frac {\sqrt {3}\, \left (-\frac {\sqrt {3}\, \arcsinh \relax (x )}{3}-\frac {2 \sqrt {3}\, \arctanh \left (\frac {3 \left (\frac {8}{3}-\frac {2 \sqrt {3}\, \left (x +\frac {\sqrt {3}}{3}\right )}{3}\right ) \sqrt {3}}{4 \sqrt {9 \left (x +\frac {\sqrt {3}}{3}\right )^{2}-6 \sqrt {3}\, \left (x +\frac {\sqrt {3}}{3}\right )+12}}\right )}{3}+\frac {\sqrt {9 \left (x +\frac {\sqrt {3}}{3}\right )^{2}-6 \sqrt {3}\, \left (x +\frac {\sqrt {3}}{3}\right )+12}}{3}\right )}{36}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (2 \, x + \sqrt {x^{2} + 1}\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.57, size = 204, normalized size = 2.49 \[ \frac {\sqrt {3}\,\left (\ln \left (x-\frac {\sqrt {3}}{3}\right )-\ln \left (x+\sqrt {3}+2\,\sqrt {x^2+1}\right )\right )}{18}-\frac {4\,x}{9\,\left (x^2-\frac {1}{3}\right )}+\frac {\sqrt {3}\,\left (\ln \left (x+\frac {\sqrt {3}}{3}\right )-\ln \left (x-\sqrt {3}-2\,\sqrt {x^2+1}\right )\right )}{18}-\frac {\sqrt {3}\,\left (6\,\ln \left (x-\frac {\sqrt {3}}{3}\right )-6\,\ln \left (x+\sqrt {3}+2\,\sqrt {x^2+1}\right )\right )}{54}-\frac {\sqrt {3}\,\left (6\,\ln \left (x+\frac {\sqrt {3}}{3}\right )-6\,\ln \left (x-\sqrt {3}-2\,\sqrt {x^2+1}\right )\right )}{54}+\frac {\sqrt {3}\,\sqrt {x^2+1}}{9\,\left (x-\frac {\sqrt {3}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {x^2+1}}{9\,\left (x+\frac {\sqrt {3}}{3}\right )}+\frac {\sqrt {3}\,\mathrm {atan}\left (\sqrt {3}\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (2 x + \sqrt {x^{2} + 1}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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