Optimal. Leaf size=103 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{x^4+1}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{x^4+1}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}} \]
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Rubi [A] time = 1.41132, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 46, number of rules used = 19, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.792 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{x^4+1}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{x^4+1}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}} \]
Antiderivative was successfully verified.
[In] Int[(1 + x^3)/((1 - x^4)*(1 + x^4)^(1/4)),x]
[Out]
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Rubi in Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: GeneratorsError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x**3+1)/(-x**4+1)/(x**4+1)**(1/4),x)
[Out]
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Mathematica [C] time = 0.316525, size = 166, normalized size = 1.61 \[ \frac{-\log \left (1-\frac{\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )+\log \left (\frac{\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}+1\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{4 \sqrt [4]{2}}-\frac{2 x^4 F_1\left (1;\frac{1}{4},1;2;-x^4,x^4\right )}{\left (x^4-1\right ) \sqrt [4]{x^4+1} \left (x^4 \left (4 F_1\left (2;\frac{1}{4},2;3;-x^4,x^4\right )-F_1\left (2;\frac{5}{4},1;3;-x^4,x^4\right )\right )+8 F_1\left (1;\frac{1}{4},1;2;-x^4,x^4\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(1 + x^3)/((1 - x^4)*(1 + x^4)^(1/4)),x]
[Out]
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Maple [F] time = 0.068, size = 0, normalized size = 0. \[ \int{\frac{{x}^{3}+1}{-{x}^{4}+1}{\frac{1}{\sqrt [4]{{x}^{4}+1}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x^3+1)/(-x^4+1)/(x^4+1)^(1/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{x^{3} + 1}{{\left (x^{4} + 1\right )}^{\frac{1}{4}}{\left (x^{4} - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^3 + 1)/((x^4 + 1)^(1/4)*(x^4 - 1)),x, algorithm="maxima")
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^3 + 1)/((x^4 + 1)^(1/4)*(x^4 - 1)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{x}{x^{3} \sqrt [4]{x^{4} + 1} - x^{2} \sqrt [4]{x^{4} + 1} + x \sqrt [4]{x^{4} + 1} - \sqrt [4]{x^{4} + 1}}\right )\, dx - \int \frac{x^{2}}{x^{3} \sqrt [4]{x^{4} + 1} - x^{2} \sqrt [4]{x^{4} + 1} + x \sqrt [4]{x^{4} + 1} - \sqrt [4]{x^{4} + 1}}\, dx - \int \frac{1}{x^{3} \sqrt [4]{x^{4} + 1} - x^{2} \sqrt [4]{x^{4} + 1} + x \sqrt [4]{x^{4} + 1} - \sqrt [4]{x^{4} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x**3+1)/(-x**4+1)/(x**4+1)**(1/4),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{x^{3} + 1}{{\left (x^{4} + 1\right )}^{\frac{1}{4}}{\left (x^{4} - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^3 + 1)/((x^4 + 1)^(1/4)*(x^4 - 1)),x, algorithm="giac")
[Out]