Optimal. Leaf size=45 \[ \frac {47 x+67}{18 \left (x^2+4 x+13\right )}+\frac {1}{2} \log \left (x^2+4 x+13\right )-\frac {61}{54} \tan ^{-1}\left (\frac {x+2}{3}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1660, 634, 618, 204, 628} \begin {gather*} \frac {47 x+67}{18 \left (x^2+4 x+13\right )}+\frac {1}{2} \log \left (x^2+4 x+13\right )-\frac {61}{54} \tan ^{-1}\left (\frac {x+2}{3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1660
Rubi steps
\begin {align*} \int \frac {1+x^3}{\left (13+4 x+x^2\right )^2} \, dx &=\frac {67+47 x}{18 \left (13+4 x+x^2\right )}+\frac {1}{36} \int \frac {-50+36 x}{13+4 x+x^2} \, dx\\ &=\frac {67+47 x}{18 \left (13+4 x+x^2\right )}+\frac {1}{2} \int \frac {4+2 x}{13+4 x+x^2} \, dx-\frac {61}{18} \int \frac {1}{13+4 x+x^2} \, dx\\ &=\frac {67+47 x}{18 \left (13+4 x+x^2\right )}+\frac {1}{2} \log \left (13+4 x+x^2\right )+\frac {61}{9} \operatorname {Subst}\left (\int \frac {1}{-36-x^2} \, dx,x,4+2 x\right )\\ &=\frac {67+47 x}{18 \left (13+4 x+x^2\right )}-\frac {61}{54} \tan ^{-1}\left (\frac {2+x}{3}\right )+\frac {1}{2} \log \left (13+4 x+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 45, normalized size = 1.00 \begin {gather*} \frac {47 x+67}{18 \left (x^2+4 x+13\right )}+\frac {1}{2} \log \left (x^2+4 x+13\right )-\frac {61}{54} \tan ^{-1}\left (\frac {x+2}{3}\right ) \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 {1+x^3}{\left (13+4 x+x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.74, size = 52, normalized size = 1.16 \begin {gather*} -\frac {61 \, {\left (x^{2} + 4 \, x + 13\right )} \arctan \left (\frac {1}{3} \, x + \frac {2}{3}\right ) - 27 \, {\left (x^{2} + 4 \, x + 13\right )} \log \left (x^{2} + 4 \, x + 13\right ) - 141 \, x - 201}{54 \, {\left (x^{2} + 4 \, x + 13\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 37, normalized size = 0.82 \begin {gather*} \frac {47 \, x + 67}{18 \, {\left (x^{2} + 4 \, x + 13\right )}} - \frac {61}{54} \, \arctan \left (\frac {1}{3} \, x + \frac {2}{3}\right ) + \frac {1}{2} \, \log \left (x^{2} + 4 \, x + 13\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 37, normalized size = 0.82 \begin {gather*} -\frac {61 \arctan \left (\frac {x}{3}+\frac {2}{3}\right )}{54}+\frac {\ln \left (x^{2}+4 x +13\right )}{2}+\frac {\frac {47 x}{18}+\frac {67}{18}}{x^{2}+4 x +13} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.84, size = 37, normalized size = 0.82 \begin {gather*} \frac {47 \, x + 67}{18 \, {\left (x^{2} + 4 \, x + 13\right )}} - \frac {61}{54} \, \arctan \left (\frac {1}{3} \, x + \frac {2}{3}\right ) + \frac {1}{2} \, \log \left (x^{2} + 4 \, x + 13\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 49, normalized size = 1.09 \begin {gather*} \frac {\ln \left (x^2+4\,x+13\right )}{2}-\frac {61\,\mathrm {atan}\left (\frac {x}{3}+\frac {2}{3}\right )}{54}+\frac {47\,x}{18\,\left (x^2+4\,x+13\right )}+\frac {67}{18\,\left (x^2+4\,x+13\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 37, normalized size = 0.82 \begin {gather*} \frac {47 x + 67}{18 x^{2} + 72 x + 234} + \frac {\log {\left (x^{2} + 4 x + 13 \right )}}{2} - \frac {61 \operatorname {atan}{\left (\frac {x}{3} + \frac {2}{3} \right )}}{54} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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