Optimal. Leaf size=44 \[ \sqrt {3} \tan ^{-1}\left (\frac {x+1}{\sqrt {3} \sqrt {x^2+2 x+5}}\right )-\tanh ^{-1}\left (\sqrt {x^2+2 x+5}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1025, 982, 204, 1024, 206} \begin {gather*} \sqrt {3} \tan ^{-1}\left (\frac {x+1}{\sqrt {3} \sqrt {x^2+2 x+5}}\right )-\tanh ^{-1}\left (\sqrt {x^2+2 x+5}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 206
Rule 982
Rule 1024
Rule 1025
Rubi steps
\begin {align*} \int \frac {4+x}{\left (4+2 x+x^2\right ) \sqrt {5+2 x+x^2}} \, dx &=\frac {1}{2} \int \frac {2+2 x}{\left (4+2 x+x^2\right ) \sqrt {5+2 x+x^2}} \, dx+3 \int \frac {1}{\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 )-12 \operatorname {Subst}\left (\int \frac {1}{-24-2 x^2} \, dx,x,\frac {2+2 x}{\sqrt {5+2 x+x^2}}\right )\\ &=\sqrt {3} \tan ^{-1}\left (\frac {1+x}{\sqrt {3} \sqrt {5+2 x+x^2}}\right )-\tanh ^{-1}\left (\sqrt {5+2 x+x^2}\right )\\ \end {align*}
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Mathematica [C] time = 0.05, size = 101, normalized size = 2.30 \begin {gather*} -\frac {1}{2} \left (1+i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {-i \sqrt {3} x-i \sqrt {3}+4}{\sqrt {x^2+2 x+5}}\right )-\frac {1}{2} \left (1-i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {i \sqrt {3} x+i \sqrt {3}+4}{\sqrt {x^2+2 x+5}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.35, size = 71, normalized size = 1.61 \begin {gather*} -\sqrt {3} \tan ^{-1}\left (\frac {x^2}{\sqrt {3}}-\frac {(x+1) \sqrt {x^2+2 x+5}}{\sqrt {3}}+\frac {2 x}{\sqrt {3}}+\frac {4}{\sqrt {3}}\right )-\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 = 106, normalized size = 2.41 \begin {gather*} -\sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} {\left (x + 2\right )} + \frac {1}{3} \, \sqrt {3} \sqrt {x^{2} + 2 \, x + 5}\right ) + \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} x + \frac {1}{3} \, \sqrt {3} \sqrt {x^{2} + 2 \, x + 5}\right ) + \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 = 0.37, size = 108, normalized size = 2.45 \begin {gather*} -\sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} {\left (x - \sqrt {x^{2} + 2 \, x + 5} + 2\right )}\right ) + \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} {\left (x - \sqrt {x^{2} + 2 \, x + 5}\right )}\right ) + \frac {1}{2} \, \log \left ({\left (x - \sqrt {x^{2} + 2 \, x + 5}\right )}^{2} + 4 \, x - 4 \, \sqrt {x^{2} + 2 \, x + 5} + 7\right ) - \frac {1}{2} \, \log \left ({\left (x - \sqrt {x^{2} + 2 \, x + 5}\right )}^{2} + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 40, normalized size = 0.91 \begin {gather*} -\arctanh \left (\sqrt {x^{2}+2 x +5}\right )+\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2 x +2\right )}{6 \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 + 4}{\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 [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x+4}{\left (x^2+2\,x+4\right )\,\sqrt {x^2+2\,x+5}} \,d x \end {gather*}
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 \begin {gather*} \int \frac {x + 4}{\left (x^{2} + 2 x + 4\right ) \sqrt {x^{2} + 2 x + 5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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