Optimal. Leaf size=25 \[ -\frac{2 \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{6-x}}{2}\right ),\frac{4}{5}\right )}{\sqrt{5}} \]
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Rubi [A] time = 0.0163658, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {718, 419} \[ -\frac{2 F\left (\sin ^{-1}\left (\frac{\sqrt{6-x}}{2}\right )|\frac{4}{5}\right )}{\sqrt{5}} \]
Antiderivative was successfully verified.
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Rule 718
Rule 419
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-1+x} \sqrt{-12+8 x-x^2}} \, dx &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1-\frac{4 x^2}{5}}} \, dx,x,\frac{\sqrt{12-2 x}}{2 \sqrt{2}}\right )}{\sqrt{5}}\\ &=-\frac{2 F\left (\sin ^{-1}\left (\frac{\sqrt{6-x}}{2}\right )|\frac{4}{5}\right )}{\sqrt{5}}\\ \end{align*}
Mathematica [B] time = 0.0700341, size = 68, normalized size = 2.72 \[ -\frac{2 \sqrt{\frac{x-6}{x-1}} \sqrt{\frac{x-2}{x-1}} (x-1) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{5}}{\sqrt{x-1}}\right ),\frac{1}{5}\right )}{\sqrt{5} \sqrt{-x^2+8 x-12}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 55, normalized size = 2.2 \begin{align*}{\frac{2\,\sqrt{5}}{5\,{x}^{2}-40\,x+60}{\it EllipticF} \left ({\frac{1}{2}\sqrt{6-x}},{\frac{2\,\sqrt{5}}{5}} \right ) \sqrt{-2+x}\sqrt{6-x}\sqrt{-{x}^{2}+8\,x-12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-x^{2} + 8 \, x - 12} \sqrt{x - 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-x^{2} + 8 \, x - 12} \sqrt{x - 1}}{x^{3} - 9 \, x^{2} + 20 \, x - 12}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- \left (x - 6\right ) \left (x - 2\right )} \sqrt{x - 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-x^{2} + 8 \, x - 12} \sqrt{x - 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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