Optimal. Leaf size=76 \[ -\frac{5 \sqrt{x^4+1}}{6 x^3}+\frac{1}{2 x^3 \sqrt{x^4+1}}-\frac{5 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{12 \sqrt{x^4+1}} \]
[Out]
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Rubi [A] time = 0.0448991, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{5 \sqrt{x^4+1}}{6 x^3}+\frac{1}{2 x^3 \sqrt{x^4+1}}-\frac{5 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{12 \sqrt{x^4+1}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^4*(1 + x^4)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 4.18499, size = 70, normalized size = 0.92 \[ - \frac{5 \sqrt{\frac{x^{4} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{12 \sqrt{x^{4} + 1}} - \frac{5 \sqrt{x^{4} + 1}}{6 x^{3}} + \frac{1}{2 x^{3} \sqrt{x^{4} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**4/(x**4+1)**(3/2),x)
[Out]
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Mathematica [C] time = 0.0664525, size = 46, normalized size = 0.61 \[ \frac{1}{6} \left (\frac{-5 x^4-2}{x^3 \sqrt{x^4+1}}+5 \sqrt [4]{-1} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^4*(1 + x^4)^(3/2)),x]
[Out]
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Maple [C] time = 0.017, size = 84, normalized size = 1.1 \[ -{\frac{x}{2}{\frac{1}{\sqrt{{x}^{4}+1}}}}-{\frac{1}{3\,{x}^{3}}\sqrt{{x}^{4}+1}}-{\frac{5\,{\it EllipticF} \left ( x \left ( 1/2\,\sqrt{2}+i/2\sqrt{2} \right ) ,i \right ) }{3\,\sqrt{2}+3\,i\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^4/(x^4+1)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{4} + 1\right )}^{\frac{3}{2}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^4 + 1)^(3/2)*x^4),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (x^{8} + x^{4}\right )} \sqrt{x^{4} + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^4 + 1)^(3/2)*x^4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.83842, size = 32, normalized size = 0.42 \[ \frac{\Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{3}{2} \\ \frac{1}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 x^{3} \Gamma \left (\frac{1}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**4/(x**4+1)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{4} + 1\right )}^{\frac{3}{2}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^4 + 1)^(3/2)*x^4),x, algorithm="giac")
[Out]