Optimal. Leaf size=79 \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{-a-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-a-b}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{-a-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-a-b}} \]
[Out]
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Rubi [A] time = 0.062781, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{-a-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-a-b}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{-a-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-a-b}} \]
Antiderivative was successfully verified.
[In] Int[x^2/(2*(a + b) + x^4),x]
[Out]
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Rubi in Sympy [A] time = 11.1388, size = 66, normalized size = 0.84 \[ \frac{2^{\frac{3}{4}} \operatorname{atan}{\left (\frac{2^{\frac{3}{4}} x}{2 \sqrt [4]{- a - b}} \right )}}{4 \sqrt [4]{- a - b}} - \frac{2^{\frac{3}{4}} \operatorname{atanh}{\left (\frac{2^{\frac{3}{4}} x}{2 \sqrt [4]{- a - b}} \right )}}{4 \sqrt [4]{- a - b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(x**4+2*a+2*b),x)
[Out]
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Mathematica [A] time = 0.0205576, size = 128, normalized size = 1.62 \[ \frac{\log \left (-2 \sqrt [4]{2} x \sqrt [4]{a+b}+2 \sqrt{a+b}+\sqrt{2} x^2\right )-\log \left (2 \sqrt [4]{2} x \sqrt [4]{a+b}+2 \sqrt{a+b}+\sqrt{2} x^2\right )-2 \tan ^{-1}\left (1-\frac{\sqrt [4]{2} x}{\sqrt [4]{a+b}}\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{a+b}}+1\right )}{4\ 2^{3/4} \sqrt [4]{a+b}} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/(2*(a + b) + x^4),x]
[Out]
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Maple [B] time = 0., size = 137, normalized size = 1.7 \[{\frac{\sqrt{2}}{8}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{2\,a+2\,b}x\sqrt{2}+\sqrt{2\,a+2\,b} \right ) \left ({x}^{2}+\sqrt [4]{2\,a+2\,b}x\sqrt{2}+\sqrt{2\,a+2\,b} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{2\,a+2\,b}}}}+{\frac{\sqrt{2}}{4}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{2\,a+2\,b}}}}+1 \right ){\frac{1}{\sqrt [4]{2\,a+2\,b}}}}+{\frac{\sqrt{2}}{4}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{2\,a+2\,b}}}}-1 \right ){\frac{1}{\sqrt [4]{2\,a+2\,b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(x^4+2*a+2*b),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(x^4 + 2*a + 2*b),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.234493, size = 171, normalized size = 2.16 \[ \left (\frac{1}{2}\right )^{\frac{1}{4}} \left (-\frac{1}{a + b}\right )^{\frac{1}{4}} \arctan \left (\frac{2 \, \left (\frac{1}{2}\right )^{\frac{3}{4}}{\left (a + b\right )} \left (-\frac{1}{a + b}\right )^{\frac{3}{4}}}{x + \sqrt{x^{2} - 2 \, \sqrt{\frac{1}{2}}{\left (a + b\right )} \sqrt{-\frac{1}{a + b}}}}\right ) + \frac{1}{4} \, \left (\frac{1}{2}\right )^{\frac{1}{4}} \left (-\frac{1}{a + b}\right )^{\frac{1}{4}} \log \left (2 \, \left (\frac{1}{2}\right )^{\frac{3}{4}}{\left (a + b\right )} \left (-\frac{1}{a + b}\right )^{\frac{3}{4}} + x\right ) - \frac{1}{4} \, \left (\frac{1}{2}\right )^{\frac{1}{4}} \left (-\frac{1}{a + b}\right )^{\frac{1}{4}} \log \left (-2 \, \left (\frac{1}{2}\right )^{\frac{3}{4}}{\left (a + b\right )} \left (-\frac{1}{a + b}\right )^{\frac{3}{4}} + x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(x^4 + 2*a + 2*b),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.500326, size = 29, normalized size = 0.37 \[ \operatorname{RootSum}{\left (t^{4} \left (512 a + 512 b\right ) + 1, \left ( t \mapsto t \log{\left (128 t^{3} a + 128 t^{3} b + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(x**4+2*a+2*b),x)
[Out]
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GIAC/XCAS [A] time = 0.22104, size = 296, normalized size = 3.75 \[ \frac{{\left (2 \, a + 2 \, b\right )}^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2}{\left (2 \, a + 2 \, b\right )}^{\frac{1}{4}}\right )}}{2 \,{\left (2 \, a + 2 \, b\right )}^{\frac{1}{4}}}\right )}{4 \,{\left (\sqrt{2} a + \sqrt{2} b\right )}} + \frac{{\left (2 \, a + 2 \, b\right )}^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2}{\left (2 \, a + 2 \, b\right )}^{\frac{1}{4}}\right )}}{2 \,{\left (2 \, a + 2 \, b\right )}^{\frac{1}{4}}}\right )}{4 \,{\left (\sqrt{2} a + \sqrt{2} b\right )}} - \frac{{\left (2 \, a + 2 \, b\right )}^{\frac{3}{4}}{\rm ln}\left (x^{2} + \sqrt{2}{\left (2 \, a + 2 \, b\right )}^{\frac{1}{4}} x + \sqrt{2 \, a + 2 \, b}\right )}{8 \,{\left (\sqrt{2} a + \sqrt{2} b\right )}} + \frac{{\left (2 \, a + 2 \, b\right )}^{\frac{3}{4}}{\rm ln}\left (x^{2} - \sqrt{2}{\left (2 \, a + 2 \, b\right )}^{\frac{1}{4}} x + \sqrt{2 \, a + 2 \, b}\right )}{8 \,{\left (\sqrt{2} a + \sqrt{2} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(x^4 + 2*a + 2*b),x, algorithm="giac")
[Out]