Optimal. Leaf size=49 \[ \frac{56}{221} \log \left (x^2-6 x+10\right )+\frac{109}{442} \log \left (2 x^2-2 x+1\right )-\frac{261}{221} \tan ^{-1}(1-2 x)-\frac{1026}{221} \tan ^{-1}(3-x) \]
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Rubi [A] time = 0.142056, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {6728, 634, 618, 204, 628, 617} \[ \frac{56}{221} \log \left (x^2-6 x+10\right )+\frac{109}{442} \log \left (2 x^2-2 x+1\right )-\frac{261}{221} \tan ^{-1}(1-2 x)-\frac{1026}{221} \tan ^{-1}(3-x) \]
Antiderivative was successfully verified.
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Rule 6728
Rule 634
Rule 618
Rule 204
Rule 628
Rule 617
Rubi steps
\begin{align*} \int \frac{5+x^3}{\left (10-6 x+x^2\right ) \left (\frac{1}{2}-x+x^2\right )} \, dx &=\int \left (\frac{2 (345+56 x)}{221 \left (10-6 x+x^2\right )}+\frac{2 (76+109 x)}{221 \left (1-2 x+2 x^2\right )}\right ) \, dx\\ &=\frac{2}{221} \int \frac{345+56 x}{10-6 x+x^2} \, dx+\frac{2}{221} \int \frac{76+109 x}{1-2 x+2 x^2} \, dx\\ &=\frac{109}{442} \int \frac{-2+4 x}{1-2 x+2 x^2} \, dx+\frac{56}{221} \int \frac{-6+2 x}{10-6 x+x^2} \, dx+\frac{261}{221} \int \frac{1}{1-2 x+2 x^2} \, dx+\frac{1026}{221} \int \frac{1}{10-6 x+x^2} \, dx\\ &=\frac{56}{221} \log \left (10-6 x+x^2\right )+\frac{109}{442} \log \left (1-2 x+2 x^2\right )+\frac{261}{221} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-2 x\right )-\frac{2052}{221} \operatorname{Subst}\left (\int \frac{1}{-4-x^2} \, dx,x,-6+2 x\right )\\ &=-\frac{261}{221} \tan ^{-1}(1-2 x)-\frac{1026}{221} \tan ^{-1}(3-x)+\frac{56}{221} \log \left (10-6 x+x^2\right )+\frac{109}{442} \log \left (1-2 x+2 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0147152, size = 49, normalized size = 1. \[ \frac{56}{221} \log \left (x^2-6 x+10\right )+\frac{109}{442} \log \left (2 x^2-2 x+1\right )-\frac{261}{221} \tan ^{-1}(1-2 x)-\frac{1026}{221} \tan ^{-1}(3-x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 40, normalized size = 0.8 \begin{align*}{\frac{261\,\arctan \left ( 2\,x-1 \right ) }{221}}+{\frac{1026\,\arctan \left ( -3+x \right ) }{221}}+{\frac{56\,\ln \left ({x}^{2}-6\,x+10 \right ) }{221}}+{\frac{109\,\ln \left ( 2\,{x}^{2}-2\,x+1 \right ) }{442}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.70234, size = 53, normalized size = 1.08 \begin{align*} \frac{261}{221} \, \arctan \left (2 \, x - 1\right ) + \frac{1026}{221} \, \arctan \left (x - 3\right ) + \frac{109}{442} \, \log \left (2 \, x^{2} - 2 \, x + 1\right ) + \frac{56}{221} \, \log \left (x^{2} - 6 \, x + 10\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50995, size = 146, normalized size = 2.98 \begin{align*} \frac{261}{221} \, \arctan \left (2 \, x - 1\right ) + \frac{1026}{221} \, \arctan \left (x - 3\right ) + \frac{109}{442} \, \log \left (x^{2} - x + \frac{1}{2}\right ) + \frac{56}{221} \, \log \left (x^{2} - 6 \, x + 10\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.197824, size = 44, normalized size = 0.9 \begin{align*} \frac{56 \log{\left (x^{2} - 6 x + 10 \right )}}{221} + \frac{109 \log{\left (x^{2} - x + \frac{1}{2} \right )}}{442} + \frac{1026 \operatorname{atan}{\left (x - 3 \right )}}{221} + \frac{261 \operatorname{atan}{\left (2 x - 1 \right )}}{221} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11396, size = 53, normalized size = 1.08 \begin{align*} \frac{261}{221} \, \arctan \left (2 \, x - 1\right ) + \frac{1026}{221} \, \arctan \left (x - 3\right ) + \frac{109}{442} \, \log \left (2 \, x^{2} - 2 \, x + 1\right ) + \frac{56}{221} \, \log \left (x^{2} - 6 \, x + 10\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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