Optimal. Leaf size=144 \[ \frac{\sqrt{\sqrt{a+4}+1} \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right ) F\left (\tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{\sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}} \]
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Rubi [A] time = 0.103659, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1106, 1104, 418} \[ \frac{\sqrt{\sqrt{a+4}+1} \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right ) F\left (\tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{\sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}} \]
Antiderivative was successfully verified.
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Rule 1106
Rule 1104
Rule 418
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+8 x-8 x^2+4 x^3-x^4}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{3+a-2 x^2-x^4}} \, dx,x,-1+x\right )\\ &=\frac{\left (\sqrt{1-\frac{2 (-1+x)^2}{-2-2 \sqrt{4+a}}} \sqrt{1-\frac{2 (-1+x)^2}{-2+2 \sqrt{4+a}}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{2 x^2}{-2-2 \sqrt{4+a}}} \sqrt{1-\frac{2 x^2}{-2+2 \sqrt{4+a}}}} \, dx,x,-1+x\right )}{\sqrt{3+a-2 (-1+x)^2-(-1+x)^4}}\\ &=-\frac{\sqrt{1+\sqrt{4+a}} \left (1+\frac{(1-x)^2}{1-\sqrt{4+a}}\right ) F\left (\tan ^{-1}\left (\frac{1-x}{\sqrt{1+\sqrt{4+a}}}\right )|-\frac{2 \sqrt{4+a}}{1-\sqrt{4+a}}\right )}{\sqrt{\frac{1+\frac{(1-x)^2}{1-\sqrt{4+a}}}{1+\frac{(1-x)^2}{1+\sqrt{4+a}}}} \sqrt{3+a-2 (1-x)^2-(1-x)^4}}\\ \end{align*}
Mathematica [B] time = 1.4009, size = 540, normalized size = 3.75 \[ \frac{2 \left (\sqrt{-\sqrt{a+4}-1}-x+1\right ) \sqrt{\frac{\sqrt{-\sqrt{a+4}-1} \left (\sqrt{\sqrt{a+4}-1}-x+1\right )}{\left (\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right ) \left (\sqrt{-\sqrt{a+4}-1}-x+1\right )}} \left (\sqrt{-\sqrt{a+4}-1}+x-1\right ) \sqrt{\frac{\sqrt{-\sqrt{a+4}-1} \left (\sqrt{\sqrt{a+4}-1}+x-1\right )}{\left (\sqrt{\sqrt{a+4}-1}-\sqrt{-\sqrt{a+4}-1}\right ) \left (\sqrt{-\sqrt{a+4}-1}-x+1\right )}} F\left (\sin ^{-1}\left (\sqrt{\frac{\left (\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right ) \left (x+\sqrt{-\sqrt{a+4}-1}-1\right )}{\left (\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right ) \left (-x+\sqrt{-\sqrt{a+4}-1}+1\right )}}\right )|\frac{\left (\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right )^2}{\left (\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right )^2}\right )}{\sqrt{-\sqrt{a+4}-1} \sqrt{\frac{\left (\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right ) \left (\sqrt{-\sqrt{a+4}-1}+x-1\right )}{\left (\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right ) \left (\sqrt{-\sqrt{a+4}-1}-x+1\right )}} \sqrt{a-x \left (x^3-4 x^2+8 x-8\right )}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.015, size = 530, normalized size = 3.7 \begin{align*} -{ \left ( \sqrt{-1-\sqrt{4+a}}+\sqrt{-1+\sqrt{4+a}} \right ) \sqrt{{ \left ( -\sqrt{-1-\sqrt{4+a}}+\sqrt{-1+\sqrt{4+a}} \right ) \left ( x-1-\sqrt{-1+\sqrt{4+a}} \right ) \left ( -\sqrt{-1-\sqrt{4+a}}-\sqrt{-1+\sqrt{4+a}} \right ) ^{-1} \left ( x-1+\sqrt{-1+\sqrt{4+a}} \right ) ^{-1}}} \left ( x-1+\sqrt{-1+\sqrt{4+a}} \right ) ^{2}\sqrt{-2\,{\frac{\sqrt{-1+\sqrt{4+a}} \left ( x-1-\sqrt{-1-\sqrt{4+a}} \right ) }{ \left ( \sqrt{-1-\sqrt{4+a}}-\sqrt{-1+\sqrt{4+a}} \right ) \left ( x-1+\sqrt{-1+\sqrt{4+a}} \right ) }}}\sqrt{-2\,{\frac{\sqrt{-1+\sqrt{4+a}} \left ( x-1+\sqrt{-1-\sqrt{4+a}} \right ) }{ \left ( -\sqrt{-1-\sqrt{4+a}}-\sqrt{-1+\sqrt{4+a}} \right ) \left ( x-1+\sqrt{-1+\sqrt{4+a}} \right ) }}}{\it EllipticF} \left ( \sqrt{{ \left ( -\sqrt{-1-\sqrt{4+a}}+\sqrt{-1+\sqrt{4+a}} \right ) \left ( x-1-\sqrt{-1+\sqrt{4+a}} \right ) \left ( -\sqrt{-1-\sqrt{4+a}}-\sqrt{-1+\sqrt{4+a}} \right ) ^{-1} \left ( x-1+\sqrt{-1+\sqrt{4+a}} \right ) ^{-1}}},\sqrt{{ \left ( -\sqrt{-1-\sqrt{4+a}}-\sqrt{-1+\sqrt{4+a}} \right ) \left ( \sqrt{-1-\sqrt{4+a}}+\sqrt{-1+\sqrt{4+a}} \right ) \left ( -\sqrt{-1-\sqrt{4+a}}+\sqrt{-1+\sqrt{4+a}} \right ) ^{-1} \left ( \sqrt{-1-\sqrt{4+a}}-\sqrt{-1+\sqrt{4+a}} \right ) ^{-1}}} \right ) \left ( -\sqrt{-1-\sqrt{4+a}}+\sqrt{-1+\sqrt{4+a}} \right ) ^{-1}{\frac{1}{\sqrt{-1+\sqrt{4+a}}}}{\frac{1}{\sqrt{- \left ( x-1-\sqrt{-1+\sqrt{4+a}} \right ) \left ( x-1+\sqrt{-1+\sqrt{4+a}} \right ) \left ( x-1-\sqrt{-1-\sqrt{4+a}} \right ) \left ( x-1+\sqrt{-1-\sqrt{4+a}} \right ) }}}} \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^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x}}\,{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^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x}}{x^{4} - 4 \, x^{3} + 8 \, x^{2} - a - 8 \, x}, 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{a - x^{4} + 4 x^{3} - 8 x^{2} + 8 x}}\, 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^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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