Optimal. Leaf size=49 \[ -\frac {1}{2} \tan ^{-1}\left (\sqrt {x^2+2 x}\right )-\frac {\tanh ^{-1}\left (\frac {2 x+1}{\sqrt {3} \sqrt {x^2+2 x}}\right )}{2 \sqrt {3}} \]
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Rubi [A] time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {984, 688, 204, 724, 206} \[ -\frac {1}{2} \tan ^{-1}\left (\sqrt {x^2+2 x}\right )-\frac {\tanh ^{-1}\left (\frac {2 x+1}{\sqrt {3} \sqrt {x^2+2 x}}\right )}{2 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 206
Rule 688
Rule 724
Rule 984
Rubi steps
\begin {align*} \int \frac {1}{\left (-1+x^2\right ) \sqrt {2 x+x^2}} \, dx &=\frac {1}{2} \int \frac {1}{(-1-x) \sqrt {2 x+x^2}} \, dx+\frac {1}{2} \int \frac {1}{(-1+x) \sqrt {2 x+x^2}} \, dx\\ &=2 \operatorname {Subst}\left (\int \frac {1}{-4-4 x^2} \, dx,x,\sqrt {2 x+x^2}\right )-\operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {2+4 x}{\sqrt {2 x+x^2}}\right )\\ &=-\frac {1}{2} \tan ^{-1}\left (\sqrt {2 x+x^2}\right )-\frac {\tanh ^{-1}\left (\frac {2+4 x}{2 \sqrt {3} \sqrt {2 x+x^2}}\right )}{2 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 64, normalized size = 1.31 \[ -\frac {\sqrt {x} \sqrt {x+2} \left (3 \tan ^{-1}\left (\sqrt {\frac {x}{x+2}}\right )+\sqrt {3} \tanh ^{-1}\left (\sqrt {3} \sqrt {\frac {x}{x+2}}\right )\right )}{3 \sqrt {x (x+2)}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.19, size = 57, normalized size = 1.16 \[ \tan ^{-1}\left (-\sqrt {x^2+2 x}+x+1\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {x^2+2 x}}{\sqrt {3}}-\frac {x}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 62, normalized size = 1.27 \[ \frac {1}{6} \, \sqrt {3} \log \left (-\frac {\sqrt {3} {\left (2 \, x + 1\right )} + \sqrt {x^{2} + 2 \, x} {\left (2 \, \sqrt {3} - 3\right )} - 4 \, x - 2}{x - 1}\right ) - \arctan \left (-x + \sqrt {x^{2} + 2 \, x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.71, size = 71, normalized size = 1.45 \[ \frac {1}{6} \, \sqrt {3} \log \left (\frac {{\left | -2 \, x - 2 \, \sqrt {3} + 2 \, \sqrt {x^{2} + 2 \, x} + 2 \right |}}{{\left | -2 \, x + 2 \, \sqrt {3} + 2 \, \sqrt {x^{2} + 2 \, x} + 2 \right |}}\right ) - \arctan \left (-x + \sqrt {x^{2} + 2 \, x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 42, normalized size = 0.86
method | result | size |
default | \(-\frac {\sqrt {3}\, \arctanh \left (\frac {\left (2+4 x \right ) \sqrt {3}}{6 \sqrt {\left (-1+x \right )^{2}-1+4 x}}\right )}{6}+\frac {\arctan \left (\frac {1}{\sqrt {\left (1+x \right )^{2}-1}}\right )}{2}\) | \(42\) |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {-2 \RootOf \left (\textit {\_Z}^{2}-3\right ) x +3 \sqrt {x^{2}+2 x}-\RootOf \left (\textit {\_Z}^{2}-3\right )}{-1+x}\right )}{6}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{2}+2 x}}{1+x}\right )}{2}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 54, normalized size = 1.10 \[ -\frac {1}{6} \, \sqrt {3} \log \left (\frac {2 \, \sqrt {3} \sqrt {x^{2} + 2 \, x}}{{\left | 2 \, x - 2 \right |}} + \frac {6}{{\left | 2 \, x - 2 \right |}} + 2\right ) + \frac {1}{2} \, \arcsin \left (\frac {2}{{\left | 2 \, x + 2 \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {x^2+2\,x}\,\left (x^2-1\right )} \,d x \]
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 \[ \int \frac {1}{\sqrt {x \left (x + 2\right )} \left (x - 1\right ) \left (x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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