Optimal. Leaf size=21 \[ x \sqrt{x^2+1} \left (-\sqrt{\frac{a}{x^4}}\right ) \]
[Out]
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Rubi [A] time = 0.01534, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ x \sqrt{x^2+1} \left (-\sqrt{\frac{a}{x^4}}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a/x^4]/Sqrt[1 + x^2],x]
[Out]
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Rubi in Sympy [A] time = 7.87076, size = 19, normalized size = 0.9 \[ - x \sqrt{\frac{a}{x^{4}}} \sqrt{x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a/x**4)**(1/2)/(x**2+1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.00980076, size = 21, normalized size = 1. \[ x \sqrt{x^2+1} \left (-\sqrt{\frac{a}{x^4}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a/x^4]/Sqrt[1 + x^2],x]
[Out]
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Maple [A] time = 0.006, size = 18, normalized size = 0.9 \[ -x\sqrt{{\frac{a}{{x}^{4}}}}\sqrt{{x}^{2}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a/x^4)^(1/2)/(x^2+1)^(1/2),x)
[Out]
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Maxima [A] time = 0.775118, size = 31, normalized size = 1.48 \[ -\frac{\sqrt{a} x^{2} + \sqrt{a}}{\sqrt{x^{2} + 1} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a/x^4)/sqrt(x^2 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.289448, size = 66, normalized size = 3.14 \[ \frac{x^{2} \sqrt{\frac{a}{x^{4}}} - \sqrt{x^{2} + 1} x \sqrt{\frac{a}{x^{4}}}}{2 \, x^{2} - 2 \, \sqrt{x^{2} + 1} x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a/x^4)/sqrt(x^2 + 1),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\frac{a}{x^{4}}}}{\sqrt{x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a/x**4)**(1/2)/(x**2+1)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.262143, size = 30, normalized size = 1.43 \[ \frac{2 \, \sqrt{a}}{{\left (x - \sqrt{x^{2} + 1}\right )}^{2} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a/x^4)/sqrt(x^2 + 1),x, algorithm="giac")
[Out]