Optimal. Leaf size=122 \[ -\frac{\sqrt{3} \log \left (-\sqrt{3} \sqrt [4]{a} x+\sqrt{a}+x^2\right )}{4 \sqrt [4]{a}}+\frac{\sqrt{3} \log \left (\sqrt{3} \sqrt [4]{a} x+\sqrt{a}+x^2\right )}{4 \sqrt [4]{a}}-\frac{\tan ^{-1}\left (\sqrt{3}-\frac{2 x}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}}+\frac{\tan ^{-1}\left (\frac{2 x}{\sqrt [4]{a}}+\sqrt{3}\right )}{2 \sqrt [4]{a}} \]
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Rubi [A] time = 0.165027, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161 \[ -\frac{\sqrt{3} \log \left (-\sqrt{3} \sqrt [4]{a} x+\sqrt{a}+x^2\right )}{4 \sqrt [4]{a}}+\frac{\sqrt{3} \log \left (\sqrt{3} \sqrt [4]{a} x+\sqrt{a}+x^2\right )}{4 \sqrt [4]{a}}-\frac{\tan ^{-1}\left (\sqrt{3}-\frac{2 x}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}}+\frac{\tan ^{-1}\left (\frac{2 x}{\sqrt [4]{a}}+\sqrt{3}\right )}{2 \sqrt [4]{a}} \]
Antiderivative was successfully verified.
[In] Int[(2*Sqrt[a] - x^2)/(a - Sqrt[a]*x^2 + x^4),x]
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Rubi in Sympy [A] time = 39.6504, size = 134, normalized size = 1.1 \[ - \frac{\sqrt{3} \log{\left (- \sqrt{3} \sqrt [4]{a} x + \sqrt{a} + x^{2} \right )}}{4 \sqrt [4]{a}} + \frac{\sqrt{3} \log{\left (\sqrt{3} \sqrt [4]{a} x + \sqrt{a} + x^{2} \right )}}{4 \sqrt [4]{a}} - \frac{\operatorname{atan}{\left (\frac{\sqrt{3} \left (\sqrt [4]{a} - \frac{2 \sqrt{3} x}{3}\right )}{\sqrt [4]{a}} \right )}}{2 \sqrt [4]{a}} + \frac{\operatorname{atan}{\left (\frac{\sqrt{3} \left (\sqrt [4]{a} + \frac{2 \sqrt{3} x}{3}\right )}{\sqrt [4]{a}} \right )}}{2 \sqrt [4]{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-x**2+2*a**(1/2))/(a+x**4-x**2*a**(1/2)),x)
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Mathematica [C] time = 0.303101, size = 115, normalized size = 0.94 \[ \frac{\sqrt [4]{-1} \left (\sqrt{\sqrt{3}-i} \left (\sqrt{3}-3 i\right ) \tanh ^{-1}\left (\frac{(1+i) x}{\sqrt{\sqrt{3}+i} \sqrt [4]{a}}\right )-\sqrt{\sqrt{3}+i} \left (\sqrt{3}+3 i\right ) \tan ^{-1}\left (\frac{(1+i) x}{\sqrt{\sqrt{3}-i} \sqrt [4]{a}}\right )\right )}{2 \sqrt{6} \sqrt [4]{a}} \]
Antiderivative was successfully verified.
[In] Integrate[(2*Sqrt[a] - x^2)/(a - Sqrt[a]*x^2 + x^4),x]
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Maple [A] time = 0.05, size = 96, normalized size = 0.8 \[ -{\frac{\sqrt{3}}{4}\ln \left ( \sqrt{3}\sqrt [4]{a}x-{x}^{2}-\sqrt{a} \right ){\frac{1}{\sqrt [4]{a}}}}-{\frac{1}{2}\arctan \left ({1 \left ( \sqrt{3}\sqrt [4]{a}-2\,x \right ){\frac{1}{\sqrt [4]{a}}}} \right ){\frac{1}{\sqrt [4]{a}}}}+{\frac{\sqrt{3}}{4}\ln \left ({x}^{2}+\sqrt{3}\sqrt [4]{a}x+\sqrt{a} \right ){\frac{1}{\sqrt [4]{a}}}}+{\frac{1}{2}\arctan \left ({1 \left ( 2\,x+\sqrt{3}\sqrt [4]{a} \right ){\frac{1}{\sqrt [4]{a}}}} \right ){\frac{1}{\sqrt [4]{a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-x^2+2*a^(1/2))/(a+x^4-a^(1/2)*x^2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{x^{2} - 2 \, \sqrt{a}}{x^{4} - \sqrt{a} x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^2 - 2*sqrt(a))/(x^4 - sqrt(a)*x^2 + a),x, algorithm="maxima")
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Fricas [A] time = 0.313, size = 350, normalized size = 2.87 \[ \frac{1}{2} \, \sqrt{\frac{1}{2}} \sqrt{\frac{\sqrt{3} a \sqrt{-\frac{1}{a}} + \sqrt{a}}{a}} \log \left (\sqrt{\frac{1}{2}} a \sqrt{\frac{\sqrt{3} a \sqrt{-\frac{1}{a}} + \sqrt{a}}{a}} + \sqrt{a} x\right ) - \frac{1}{2} \, \sqrt{\frac{1}{2}} \sqrt{\frac{\sqrt{3} a \sqrt{-\frac{1}{a}} + \sqrt{a}}{a}} \log \left (-\sqrt{\frac{1}{2}} a \sqrt{\frac{\sqrt{3} a \sqrt{-\frac{1}{a}} + \sqrt{a}}{a}} + \sqrt{a} x\right ) + \frac{1}{2} \, \sqrt{\frac{1}{2}} \sqrt{-\frac{\sqrt{3} a \sqrt{-\frac{1}{a}} - \sqrt{a}}{a}} \log \left (\sqrt{\frac{1}{2}} a \sqrt{-\frac{\sqrt{3} a \sqrt{-\frac{1}{a}} - \sqrt{a}}{a}} + \sqrt{a} x\right ) - \frac{1}{2} \, \sqrt{\frac{1}{2}} \sqrt{-\frac{\sqrt{3} a \sqrt{-\frac{1}{a}} - \sqrt{a}}{a}} \log \left (-\sqrt{\frac{1}{2}} a \sqrt{-\frac{\sqrt{3} a \sqrt{-\frac{1}{a}} - \sqrt{a}}{a}} + \sqrt{a} x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^2 - 2*sqrt(a))/(x^4 - sqrt(a)*x^2 + a),x, algorithm="fricas")
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: PolynomialError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x**2+2*a**(1/2))/(a+x**4-x**2*a**(1/2)),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^2 - 2*sqrt(a))/(x^4 - sqrt(a)*x^2 + a),x, algorithm="giac")
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