Optimal. Leaf size=23 \[ 3 \log \left (x^2-6 x+25\right )+x-2 \tan ^{-1}\left (\frac{x-3}{4}\right ) \]
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Rubi [A] time = 0.0349412, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {1657, 634, 618, 204, 628} \[ 3 \log \left (x^2-6 x+25\right )+x-2 \tan ^{-1}\left (\frac{x-3}{4}\right ) \]
Antiderivative was successfully verified.
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Rule 1657
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{-1+x^2}{25-6 x+x^2} \, dx &=\int \left (1-\frac{2 (13-3 x)}{25-6 x+x^2}\right ) \, dx\\ &=x-2 \int \frac{13-3 x}{25-6 x+x^2} \, dx\\ &=x+3 \int \frac{-6+2 x}{25-6 x+x^2} \, dx-8 \int \frac{1}{25-6 x+x^2} \, dx\\ &=x+3 \log \left (25-6 x+x^2\right )+16 \operatorname{Subst}\left (\int \frac{1}{-64-x^2} \, dx,x,-6+2 x\right )\\ &=x-2 \tan ^{-1}\left (\frac{1}{4} (-3+x)\right )+3 \log \left (25-6 x+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0050814, size = 23, normalized size = 1. \[ 3 \log \left (x^2-6 x+25\right )+x-2 \tan ^{-1}\left (\frac{x-3}{4}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 22, normalized size = 1. \begin{align*} x-2\,\arctan \left ( -3/4+x/4 \right ) +3\,\ln \left ({x}^{2}-6\,x+25 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4844, size = 28, normalized size = 1.22 \begin{align*} x - 2 \, \arctan \left (\frac{1}{4} \, x - \frac{3}{4}\right ) + 3 \, \log \left (x^{2} - 6 \, x + 25\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70121, size = 69, normalized size = 3. \begin{align*} x - 2 \, \arctan \left (\frac{1}{4} \, x - \frac{3}{4}\right ) + 3 \, \log \left (x^{2} - 6 \, x + 25\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.100963, size = 22, normalized size = 0.96 \begin{align*} x + 3 \log{\left (x^{2} - 6 x + 25 \right )} - 2 \operatorname{atan}{\left (\frac{x}{4} - \frac{3}{4} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33842, size = 28, normalized size = 1.22 \begin{align*} x - 2 \, \arctan \left (\frac{1}{4} \, x - \frac{3}{4}\right ) + 3 \, \log \left (x^{2} - 6 \, x + 25\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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