Optimal. Leaf size=58 \[ \frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{4 \sqrt{x^4+1}}-\frac{x}{2 \sqrt{x^4+1}} \]
[Out]
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Rubi [A] time = 0.0300925, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{4 \sqrt{x^4+1}}-\frac{x}{2 \sqrt{x^4+1}} \]
Antiderivative was successfully verified.
[In] Int[x^4/(1 + x^4)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 2.82101, size = 49, normalized size = 0.84 \[ - \frac{x}{2 \sqrt{x^{4} + 1}} + \frac{\sqrt{\frac{x^{4} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{4 \sqrt{x^{4} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4/(x**4+1)**(3/2),x)
[Out]
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Mathematica [C] time = 0.037637, size = 38, normalized size = 0.66 \[ -\frac{x}{2 \sqrt{x^4+1}}-\frac{1}{2} \sqrt [4]{-1} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^4/(1 + x^4)^(3/2),x]
[Out]
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Maple [C] time = 0.011, size = 72, normalized size = 1.2 \[ -{\frac{x}{2}{\frac{1}{\sqrt{{x}^{4}+1}}}}+{\frac{{\it EllipticF} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) }{\sqrt{2}+i\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4/(x^4+1)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (x^{4} + 1\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(x^4 + 1)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{4}}{{\left (x^{4} + 1\right )}^{\frac{3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(x^4 + 1)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.92442, size = 29, normalized size = 0.5 \[ \frac{x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{3}{2} \\ \frac{9}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4/(x**4+1)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (x^{4} + 1\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(x^4 + 1)^(3/2),x, algorithm="giac")
[Out]