Optimal. Leaf size=53 \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )}{2\ 2^{3/4}}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )}{2\ 2^{3/4}} \]
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Rubi [A] time = 0.0568344, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )}{2\ 2^{3/4}}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )}{2\ 2^{3/4}} \]
Antiderivative was successfully verified.
[In] Int[1/((1 + x^4)^(1/4)*(2 + x^4)),x]
[Out]
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Rubi in Sympy [A] time = 4.79372, size = 49, normalized size = 0.92 \[ \frac{\sqrt [4]{2} \operatorname{atan}{\left (\frac{2^{\frac{3}{4}} x}{2 \sqrt [4]{x^{4} + 1}} \right )}}{4} + \frac{\sqrt [4]{2} \operatorname{atanh}{\left (\frac{2^{\frac{3}{4}} x}{2 \sqrt [4]{x^{4} + 1}} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(x**4+1)**(1/4)/(x**4+2),x)
[Out]
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Mathematica [A] time = 0.08512, size = 70, normalized size = 1.32 \[ \frac{-\log \left (2-\frac{2^{3/4} x}{\sqrt [4]{x^4+1}}\right )+\log \left (\frac{2^{3/4} x}{\sqrt [4]{x^4+1}}+2\right )+2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )}{4\ 2^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((1 + x^4)^(1/4)*(2 + x^4)),x]
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Maple [F] time = 0.038, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4}+2}{\frac{1}{\sqrt [4]{{x}^{4}+1}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(x^4+1)^(1/4)/(x^4+2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{4} + 2\right )}{\left (x^{4} + 1\right )}^{\frac{1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^4 + 2)*(x^4 + 1)^(1/4)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 4.31309, size = 273, normalized size = 5.15 \[ \frac{1}{64} \cdot 8^{\frac{3}{4}}{\left (4 \, \arctan \left (-\frac{4 \, \sqrt{x^{4} + 1} x^{2} - \sqrt{2}{\left (3 \, x^{4} + 2\right )}}{2 \cdot 8^{\frac{1}{4}}{\left (x^{4} + 1\right )}^{\frac{1}{4}} x^{3} - 8^{\frac{3}{4}}{\left (x^{4} + 1\right )}^{\frac{3}{4}} x - \sqrt{2}{\left (x^{4} + 2\right )}}\right ) + \log \left (\frac{2 \,{\left (2 \cdot 8^{\frac{1}{4}}{\left (x^{4} + 1\right )}^{\frac{1}{4}} x^{3} + 8^{\frac{3}{4}}{\left (x^{4} + 1\right )}^{\frac{3}{4}} x + 4 \, \sqrt{x^{4} + 1} x^{2} + \sqrt{2}{\left (3 \, x^{4} + 2\right )}\right )}}{x^{4} + 2}\right ) - \log \left (\frac{2 \,{\left (2 \cdot 8^{\frac{1}{4}}{\left (x^{4} + 1\right )}^{\frac{1}{4}} x^{3} + 8^{\frac{3}{4}}{\left (x^{4} + 1\right )}^{\frac{3}{4}} x - 4 \, \sqrt{x^{4} + 1} x^{2} - \sqrt{2}{\left (3 \, x^{4} + 2\right )}\right )}}{x^{4} + 2}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^4 + 2)*(x^4 + 1)^(1/4)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt [4]{x^{4} + 1} \left (x^{4} + 2\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x**4+1)**(1/4)/(x**4+2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{4} + 2\right )}{\left (x^{4} + 1\right )}^{\frac{1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^4 + 2)*(x^4 + 1)^(1/4)),x, algorithm="giac")
[Out]