Optimal. Leaf size=74 \[ \frac{\left (x^4+1\right )^{3/4} x}{8 \left (x^4+2\right )}+\frac{3 \tan ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )}{16\ 2^{3/4}}+\frac{3 \tanh ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )}{16\ 2^{3/4}} \]
[Out]
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Rubi [A] time = 0.0641195, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{\left (x^4+1\right )^{3/4} x}{8 \left (x^4+2\right )}+\frac{3 \tan ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )}{16\ 2^{3/4}}+\frac{3 \tanh ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )}{16\ 2^{3/4}} \]
Antiderivative was successfully verified.
[In] Int[(1 + x^4)^(3/4)/(2 + x^4)^2,x]
[Out]
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Rubi in Sympy [A] time = 3.68551, size = 70, normalized size = 0.95 \[ \frac{x \left (x^{4} + 1\right )^{\frac{3}{4}}}{8 \left (x^{4} + 2\right )} + \frac{3 \sqrt [4]{2} \operatorname{atan}{\left (\frac{2^{\frac{3}{4}} x}{2 \sqrt [4]{x^{4} + 1}} \right )}}{32} + \frac{3 \sqrt [4]{2} \operatorname{atanh}{\left (\frac{2^{\frac{3}{4}} x}{2 \sqrt [4]{x^{4} + 1}} \right )}}{32} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x**4+1)**(3/4)/(x**4+2)**2,x)
[Out]
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Mathematica [A] time = 0.160765, size = 92, normalized size = 1.24 \[ \frac{\left (x^4+1\right )^{3/4} x}{8 \left (x^4+2\right )}+\frac{3 \left (-\log \left (2-\frac{2^{3/4} x}{\sqrt [4]{x^4+1}}\right )+\log \left (\frac{2^{3/4} x}{\sqrt [4]{x^4+1}}+2\right )+2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )\right )}{32\ 2^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(1 + x^4)^(3/4)/(2 + x^4)^2,x]
[Out]
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Maple [F] time = 0.055, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ({x}^{4}+2 \right ) ^{2}} \left ({x}^{4}+1 \right ) ^{{\frac{3}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x^4+1)^(3/4)/(x^4+2)^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x^{4} + 1\right )}^{\frac{3}{4}}}{{\left (x^{4} + 2\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^4 + 1)^(3/4)/(x^4 + 2)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 4.28261, size = 323, normalized size = 4.36 \[ \frac{8^{\frac{3}{4}}{\left (8 \cdot 8^{\frac{1}{4}}{\left (x^{4} + 1\right )}^{\frac{3}{4}} x + 12 \,{\left (x^{4} + 2\right )} \arctan \left (-\frac{4 \, \sqrt{x^{4} + 1} x^{2} - \sqrt{2}{\left (3 \, x^{4} + 2\right )}}{2 \cdot 8^{\frac{1}{4}}{\left (x^{4} + 1\right )}^{\frac{1}{4}} x^{3} - 8^{\frac{3}{4}}{\left (x^{4} + 1\right )}^{\frac{3}{4}} x - \sqrt{2}{\left (x^{4} + 2\right )}}\right ) + 3 \,{\left (x^{4} + 2\right )} \log \left (\frac{2 \,{\left (2 \cdot 8^{\frac{1}{4}}{\left (x^{4} + 1\right )}^{\frac{1}{4}} x^{3} + 8^{\frac{3}{4}}{\left (x^{4} + 1\right )}^{\frac{3}{4}} x + 4 \, \sqrt{x^{4} + 1} x^{2} + \sqrt{2}{\left (3 \, x^{4} + 2\right )}\right )}}{x^{4} + 2}\right ) - 3 \,{\left (x^{4} + 2\right )} \log \left (\frac{2 \,{\left (2 \cdot 8^{\frac{1}{4}}{\left (x^{4} + 1\right )}^{\frac{1}{4}} x^{3} + 8^{\frac{3}{4}}{\left (x^{4} + 1\right )}^{\frac{3}{4}} x - 4 \, \sqrt{x^{4} + 1} x^{2} - \sqrt{2}{\left (3 \, x^{4} + 2\right )}\right )}}{x^{4} + 2}\right )\right )}}{512 \,{\left (x^{4} + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^4 + 1)^(3/4)/(x^4 + 2)^2,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x^{4} + 1\right )^{\frac{3}{4}}}{\left (x^{4} + 2\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x**4+1)**(3/4)/(x**4+2)**2,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x^{4} + 1\right )}^{\frac{3}{4}}}{{\left (x^{4} + 2\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^4 + 1)^(3/4)/(x^4 + 2)^2,x, algorithm="giac")
[Out]