Optimal. Leaf size=28 \[ \frac {\tan ^{-1}\left (\frac {x+1}{\sqrt {3} \sqrt {x^2+2 x+5}}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {982, 204} \[ \frac {\tan ^{-1}\left (\frac {x+1}{\sqrt {3} \sqrt {x^2+2 x+5}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 982
Rubi steps
\begin {align*} \int \frac {1}{\left (4+2 x+x^2\right ) \sqrt {5+2 x+x^2}} \, dx &=-\left (4 \operatorname {Subst}\left (\int \frac {1}{-24-2 x^2} \, dx,x,\frac {2+2 x}{\sqrt {5+2 x+x^2}}\right )\right )\\ &=\frac {\tan ^{-1}\left (\frac {1+x}{\sqrt {3} \sqrt {5+2 x+x^2}}\right )}{\sqrt {3}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 84, normalized size = 3.00 \[ -\frac {i \left (\tanh ^{-1}\left (\frac {-i \sqrt {3} x-i \sqrt {3}+4}{\sqrt {x^2+2 x+5}}\right )-\tanh ^{-1}\left (\frac {i \sqrt {3} x+i \sqrt {3}+4}{\sqrt {x^2+2 x+5}}\right )\right )}{2 \sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 38, normalized size = 1.36 \[ \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} \sqrt {x^{2} + 2 \, x + 5} {\left (x + 1\right )} - \frac {1}{3} \, \sqrt {3} {\left (x^{2} + 2 \, x + 4\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 52, normalized size = 1.86 \[ -\frac {1}{3} \, \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} {\left (x - \sqrt {x^{2} + 2 \, x + 5} + 2\right )}\right ) + \frac {1}{3} \, \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} {\left (x - \sqrt {x^{2} + 2 \, x + 5}\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 27, normalized size = 0.96 \[ \frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2 x +2\right )}{6 \sqrt {x^{2}+2 x +5}}\right )}{3} \]
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 \[ \int \frac {1}{\sqrt {x^{2} + 2 \, x + 5} {\left (x^{2} + 2 \, x + 4\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{\left (x^2+2\,x+4\right )\,\sqrt {x^2+2\,x+5}} \,d x \]
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 \[ \int \frac {1}{\left (x^{2} + 2 x + 4\right ) \sqrt {x^{2} + 2 x + 5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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