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 = 0.0172824, antiderivative size = 28, normalized size of antiderivative = 1., 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 982
Rule 204
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*}
Mathematica [C] time = 0.0677789, size = 84, normalized size = 3. \[ -\frac{i \left (\tanh ^{-1}\left (\frac{-i \sqrt{3} x-i \sqrt{3}+4}{\sqrt{x^2+2 x+5}}\right )-\tanh ^{-1}\left (\frac{i \sqrt{3} x+i \sqrt{3}+4}{\sqrt{x^2+2 x+5}}\right )\right )}{2 \sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 27, normalized size = 1. \begin{align*}{\frac{\sqrt{3}}{3}\arctan \left ({\frac{\sqrt{3} \left ( 2\,x+2 \right ) }{6}{\frac{1}{\sqrt{{x}^{2}+2\,x+5}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x^{2} + 2 \, x + 5}{\left (x^{2} + 2 \, x + 4\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.03401, size = 123, normalized size = 4.39 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (x^{2} + 2 x + 4\right ) \sqrt{x^{2} + 2 x + 5}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14215, size = 70, normalized size = 2.5 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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