Optimal. Leaf size=31 \[ -\frac{x^2}{\sqrt{x^4+1}}-\frac{1}{2 \sqrt{x^4+1} x^2} \]
[Out]
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Rubi [A] time = 0.0223854, antiderivative size = 31, 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 \[ -\frac{x^2}{\sqrt{x^4+1}}-\frac{1}{2 \sqrt{x^4+1} x^2} \]
Antiderivative was successfully verified.
[In] Int[1/(x^3*(1 + x^4)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 2.85681, size = 27, normalized size = 0.87 \[ - \frac{x^{2}}{\sqrt{x^{4} + 1}} - \frac{1}{2 x^{2} \sqrt{x^{4} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(x**4+1)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0130547, size = 23, normalized size = 0.74 \[ -\frac{2 x^4+1}{2 x^2 \sqrt{x^4+1}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^3*(1 + x^4)^(3/2)),x]
[Out]
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Maple [A] time = 0.005, size = 20, normalized size = 0.7 \[ -{\frac{2\,{x}^{4}+1}{2\,{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(x^4+1)^(3/2),x)
[Out]
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Maxima [A] time = 1.46181, size = 34, normalized size = 1.1 \[ -\frac{x^{2}}{2 \, \sqrt{x^{4} + 1}} - \frac{\sqrt{x^{4} + 1}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^4 + 1)^(3/2)*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.265342, size = 45, normalized size = 1.45 \[ \frac{1}{2 \,{\left (2 \, x^{8} + 2 \, x^{4} -{\left (2 \, x^{6} + x^{2}\right )} \sqrt{x^{4} + 1}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^4 + 1)^(3/2)*x^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.3531, size = 42, normalized size = 1.35 \[ - \frac{2 x^{4} \sqrt{x^{4} + 1}}{2 x^{6} + 2 x^{2}} - \frac{\sqrt{x^{4} + 1}}{2 x^{6} + 2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**3/(x**4+1)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.233895, size = 30, normalized size = 0.97 \[ -\frac{x^{2}}{2 \, \sqrt{x^{4} + 1}} - \frac{1}{2} \, \sqrt{\frac{1}{x^{4}} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^4 + 1)^(3/2)*x^3),x, algorithm="giac")
[Out]