Optimal. Leaf size=79 \[ -\frac{\tan ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{-a-b}}\right )}{2\ 2^{3/4} (-a-b)^{3/4}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{-a-b}}\right )}{2\ 2^{3/4} (-a-b)^{3/4}} \]
[Out]
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Rubi [A] time = 0.0726998, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{\tan ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{-a-b}}\right )}{2\ 2^{3/4} (-a-b)^{3/4}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{-a-b}}\right )}{2\ 2^{3/4} (-a-b)^{3/4}} \]
Antiderivative was successfully verified.
[In] Int[(2*(a + b) + x^4)^(-1),x]
[Out]
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Rubi in Sympy [A] time = 10.1798, size = 68, normalized size = 0.86 \[ - \frac{\sqrt [4]{2} \operatorname{atan}{\left (\frac{2^{\frac{3}{4}} x}{2 \sqrt [4]{- a - b}} \right )}}{4 \left (- a - b\right )^{\frac{3}{4}}} - \frac{\sqrt [4]{2} \operatorname{atanh}{\left (\frac{2^{\frac{3}{4}} x}{2 \sqrt [4]{- a - b}} \right )}}{4 \left (- a - b\right )^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(x**4+2*a+2*b),x)
[Out]
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Mathematica [A] time = 0.0213, size = 128, normalized size = 1.62 \[ \frac{-\log \left (-2 \sqrt [4]{2} x \sqrt [4]{a+b}+2 \sqrt{a+b}+\sqrt{2} x^2\right )+\log \left (2 \sqrt [4]{2} x \sqrt [4]{a+b}+2 \sqrt{a+b}+\sqrt{2} x^2\right )-2 \tan ^{-1}\left (1-\frac{\sqrt [4]{2} x}{\sqrt [4]{a+b}}\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{a+b}}+1\right )}{8 \sqrt [4]{2} (a+b)^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(2*(a + b) + x^4)^(-1),x]
[Out]
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Maple [B] time = 0.001, size = 137, normalized size = 1.7 \[{\frac{\sqrt{2}}{8}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{2\,a+2\,b}x\sqrt{2}+\sqrt{2\,a+2\,b} \right ) \left ({x}^{2}-\sqrt [4]{2\,a+2\,b}x\sqrt{2}+\sqrt{2\,a+2\,b} \right ) ^{-1}} \right ) \left ( 2\,a+2\,b \right ) ^{-{\frac{3}{4}}}}+{\frac{\sqrt{2}}{4}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{2\,a+2\,b}}}}+1 \right ) \left ( 2\,a+2\,b \right ) ^{-{\frac{3}{4}}}}+{\frac{\sqrt{2}}{4}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{2\,a+2\,b}}}}-1 \right ) \left ( 2\,a+2\,b \right ) ^{-{\frac{3}{4}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(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(1/(x^4 + 2*a + 2*b),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.24146, size = 335, normalized size = 4.24 \[ -\left (\frac{1}{8}\right )^{\frac{1}{4}} \left (-\frac{1}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}}\right )^{\frac{1}{4}} \arctan \left (\frac{2 \, \left (\frac{1}{8}\right )^{\frac{1}{4}}{\left (a + b\right )} \left (-\frac{1}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}}\right )^{\frac{1}{4}}}{x + \sqrt{x^{2} + 2 \, \sqrt{\frac{1}{2}}{\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt{-\frac{1}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}}}}}\right ) + \frac{1}{4} \, \left (\frac{1}{8}\right )^{\frac{1}{4}} \left (-\frac{1}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}}\right )^{\frac{1}{4}} \log \left (2 \, \left (\frac{1}{8}\right )^{\frac{1}{4}}{\left (a + b\right )} \left (-\frac{1}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}}\right )^{\frac{1}{4}} + x\right ) - \frac{1}{4} \, \left (\frac{1}{8}\right )^{\frac{1}{4}} \left (-\frac{1}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}}\right )^{\frac{1}{4}} \log \left (-2 \, \left (\frac{1}{8}\right )^{\frac{1}{4}}{\left (a + b\right )} \left (-\frac{1}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}}\right )^{\frac{1}{4}} + x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x^4 + 2*a + 2*b),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.620501, size = 42, normalized size = 0.53 \[ \operatorname{RootSum}{\left (t^{4} \left (2048 a^{3} + 6144 a^{2} b + 6144 a b^{2} + 2048 b^{3}\right ) + 1, \left ( t \mapsto t \log{\left (8 t a + 8 t b + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x**4+2*a+2*b),x)
[Out]
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GIAC/XCAS [A] time = 0.218594, size = 296, normalized size = 3.75 \[ \frac{{\left (2 \, a + 2 \, b\right )}^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2}{\left (2 \, a + 2 \, b\right )}^{\frac{1}{4}}\right )}}{2 \,{\left (2 \, a + 2 \, b\right )}^{\frac{1}{4}}}\right )}{4 \,{\left (\sqrt{2} a + \sqrt{2} b\right )}} + \frac{{\left (2 \, a + 2 \, b\right )}^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2}{\left (2 \, a + 2 \, b\right )}^{\frac{1}{4}}\right )}}{2 \,{\left (2 \, a + 2 \, b\right )}^{\frac{1}{4}}}\right )}{4 \,{\left (\sqrt{2} a + \sqrt{2} b\right )}} + \frac{{\left (2 \, a + 2 \, b\right )}^{\frac{1}{4}}{\rm ln}\left (x^{2} + \sqrt{2}{\left (2 \, a + 2 \, b\right )}^{\frac{1}{4}} x + \sqrt{2 \, a + 2 \, b}\right )}{8 \,{\left (\sqrt{2} a + \sqrt{2} b\right )}} - \frac{{\left (2 \, a + 2 \, b\right )}^{\frac{1}{4}}{\rm ln}\left (x^{2} - \sqrt{2}{\left (2 \, a + 2 \, b\right )}^{\frac{1}{4}} x + \sqrt{2 \, a + 2 \, b}\right )}{8 \,{\left (\sqrt{2} a + \sqrt{2} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x^4 + 2*a + 2*b),x, algorithm="giac")
[Out]