Optimal. Leaf size=15 \[ -\tanh ^{-1}\left (\sqrt {x^2+2 x+5}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1024, 206} \begin {gather*} -\tanh ^{-1}\left (\sqrt {x^2+2 x+5}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 1024
Rubi steps
\begin {align*} \int \frac {1+x}{\left (4+2 x+x^2\right ) \sqrt {5+2 x+x^2}} \, dx &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{2-2 x^2} \, dx,x,\sqrt {5+2 x+x^2}\right )\right )\\ &=-\tanh ^{-1}\left (\sqrt {5+2 x+x^2}\right )\\ \end {align*}
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Mathematica [C] time = 0.04, size = 79, normalized size = 5.27 \begin {gather*} \frac {1}{2} \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 ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.30, size = 15, normalized size = 1.00 \begin {gather*} -\tanh ^{-1}\left (\sqrt {x^2+2 x+5}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 49, normalized size = 3.27 \begin {gather*} \frac {1}{2} \, \log \left (x^{2} - \sqrt {x^{2} + 2 \, x + 5} {\left (x + 2\right )} + 3 \, x + 6\right ) - \frac {1}{2} \, \log \left (x^{2} - \sqrt {x^{2} + 2 \, x + 5} x + x + 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.35, size = 31, normalized size = 2.07 \begin {gather*} -\frac {1}{2} \, \log \left (\sqrt {x^{2} + 2 \, x + 5} + 1\right ) + \frac {1}{2} \, \log \left (\sqrt {x^{2} + 2 \, x + 5} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 14, normalized size = 0.93 \begin {gather*} -\arctanh \left (\sqrt {x^{2}+2 x +5}\right ) \end {gather*}
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 \begin {gather*} \int \frac {x + 1}{\sqrt {x^{2} + 2 \, x + 5} {\left (x^{2} + 2 \, x + 4\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.76, size = 13, normalized size = 0.87 \begin {gather*} -\mathrm {atanh}\left (\sqrt {x^2+2\,x+5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 7.00, size = 36, normalized size = 2.40 \begin {gather*} \frac {\log {\left (-1 + \frac {1}{\sqrt {x^{2} + 2 x + 5}} \right )}}{2} - \frac {\log {\left (1 + \frac {1}{\sqrt {x^{2} + 2 x + 5}} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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