Optimal. Leaf size=49 \[ -\frac{1}{6 \sqrt{x^4+1} x^6}+\frac{4 x^2}{3 \sqrt{x^4+1}}+\frac{2}{3 \sqrt{x^4+1} x^2} \]
[Out]
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Rubi [A] time = 0.03545, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{1}{6 \sqrt{x^4+1} x^6}+\frac{4 x^2}{3 \sqrt{x^4+1}}+\frac{2}{3 \sqrt{x^4+1} x^2} \]
Antiderivative was successfully verified.
[In] Int[1/(x^7*(1 + x^4)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 3.7896, size = 44, normalized size = 0.9 \[ \frac{4 x^{2}}{3 \sqrt{x^{4} + 1}} + \frac{2}{3 x^{2} \sqrt{x^{4} + 1}} - \frac{1}{6 x^{6} \sqrt{x^{4} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**7/(x**4+1)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0155781, size = 28, normalized size = 0.57 \[ \frac{8 x^8+4 x^4-1}{6 x^6 \sqrt{x^4+1}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^7*(1 + x^4)^(3/2)),x]
[Out]
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Maple [A] time = 0.006, size = 25, normalized size = 0.5 \[{\frac{8\,{x}^{8}+4\,{x}^{4}-1}{6\,{x}^{6}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^7/(x^4+1)^(3/2),x)
[Out]
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Maxima [A] time = 1.48871, size = 49, normalized size = 1. \[ \frac{x^{2}}{2 \, \sqrt{x^{4} + 1}} + \frac{\sqrt{x^{4} + 1}}{x^{2}} - \frac{{\left (x^{4} + 1\right )}^{\frac{3}{2}}}{6 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^4 + 1)^(3/2)*x^7),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.264475, size = 84, normalized size = 1.71 \[ \frac{4 \, x^{4} - 4 \, \sqrt{x^{4} + 1} x^{2} + 1}{6 \,{\left (8 \, x^{16} + 12 \, x^{12} + 4 \, x^{8} -{\left (8 \, x^{14} + 8 \, x^{10} + x^{6}\right )} \sqrt{x^{4} + 1}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^4 + 1)^(3/2)*x^7),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.06567, size = 70, normalized size = 1.43 \[ \frac{8 x^{8} \sqrt{1 + \frac{1}{x^{4}}}}{6 x^{8} + 6 x^{4}} + \frac{4 x^{4} \sqrt{1 + \frac{1}{x^{4}}}}{6 x^{8} + 6 x^{4}} - \frac{\sqrt{1 + \frac{1}{x^{4}}}}{6 x^{8} + 6 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**7/(x**4+1)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.233008, size = 39, normalized size = 0.8 \[ \frac{x^{2}}{2 \, \sqrt{x^{4} + 1}} - \frac{1}{6} \,{\left (\frac{1}{x^{4}} + 1\right )}^{\frac{3}{2}} + \sqrt{\frac{1}{x^{4}} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^4 + 1)^(3/2)*x^7),x, algorithm="giac")
[Out]