Optimal. Leaf size=28 \[ -\frac {1}{x^2+2 x+2}+\log \left (x^2+2 x+2\right )-\tan ^{-1}(x+1) \]
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Rubi [A] time = 0.02, antiderivative size = 28, 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 = {1660, 634, 617, 204, 628} \begin {gather*} -\frac {1}{x^2+2 x+2}+\log \left (x^2+2 x+2\right )-\tan ^{-1}(x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 617
Rule 628
Rule 634
Rule 1660
Rubi steps
\begin {align*} \int \frac {4+8 x+5 x^2+2 x^3}{\left (2+2 x+x^2\right )^2} \, dx &=-\frac {1}{2+2 x+x^2}+\frac {1}{4} \int \frac {4+8 x}{2+2 x+x^2} \, dx\\ &=-\frac {1}{2+2 x+x^2}-\int \frac {1}{2+2 x+x^2} \, dx+\int \frac {2+2 x}{2+2 x+x^2} \, dx\\ &=-\frac {1}{2+2 x+x^2}+\log \left (2+2 x+x^2\right )+\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+x\right )\\ &=-\frac {1}{2+2 x+x^2}-\tan ^{-1}(1+x)+\log \left (2+2 x+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 1.00 \begin {gather*} -\frac {1}{x^2+2 x+2}+\log \left (x^2+2 x+2\right )-\tan ^{-1}(x+1) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4+8 x+5 x^2+2 x^3}{\left (2+2 x+x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.99, size = 46, normalized size = 1.64 \begin {gather*} -\frac {{\left (x^{2} + 2 \, x + 2\right )} \arctan \left (x + 1\right ) - {\left (x^{2} + 2 \, x + 2\right )} \log \left (x^{2} + 2 \, x + 2\right ) + 1}{x^{2} + 2 \, x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 28, normalized size = 1.00 \begin {gather*} -\frac {1}{x^{2} + 2 \, x + 2} - \arctan \left (x + 1\right ) + \log \left (x^{2} + 2 \, x + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 29, normalized size = 1.04 \begin {gather*} -\arctan \left (x +1\right )+\ln \left (x^{2}+2 x +2\right )-\frac {1}{x^{2}+2 x +2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.66, size = 28, normalized size = 1.00 \begin {gather*} -\frac {1}{x^{2} + 2 \, x + 2} - \arctan \left (x + 1\right ) + \log \left (x^{2} + 2 \, x + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 28, normalized size = 1.00 \begin {gather*} \ln \left (x^2+2\,x+2\right )-\mathrm {atan}\left (x+1\right )-\frac {1}{x^2+2\,x+2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 24, normalized size = 0.86 \begin {gather*} \log {\left (x^{2} + 2 x + 2 \right )} - \operatorname {atan}{\left (x + 1 \right )} - \frac {1}{x^{2} + 2 x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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