Optimal. Leaf size=41 \[ -\frac{3}{4} \sinh ^{-1}\left (x^2\right )-\frac{x^6}{2 \sqrt{x^4+1}}+\frac{3}{4} \sqrt{x^4+1} x^2 \]
[Out]
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Rubi [A] time = 0.0475946, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{3}{4} \sinh ^{-1}\left (x^2\right )-\frac{x^6}{2 \sqrt{x^4+1}}+\frac{3}{4} \sqrt{x^4+1} x^2 \]
Antiderivative was successfully verified.
[In] Int[x^9/(1 + x^4)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 6.23299, size = 36, normalized size = 0.88 \[ - \frac{x^{6}}{2 \sqrt{x^{4} + 1}} + \frac{3 x^{2} \sqrt{x^{4} + 1}}{4} - \frac{3 \operatorname{asinh}{\left (x^{2} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**9/(x**4+1)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0310787, size = 37, normalized size = 0.9 \[ \frac{x^6+3 x^2-3 \sqrt{x^4+1} \sinh ^{-1}\left (x^2\right )}{4 \sqrt{x^4+1}} \]
Antiderivative was successfully verified.
[In] Integrate[x^9/(1 + x^4)^(3/2),x]
[Out]
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Maple [A] time = 0.015, size = 32, normalized size = 0.8 \[{\frac{{x}^{6}}{4}{\frac{1}{\sqrt{{x}^{4}+1}}}}+{\frac{3\,{x}^{2}}{4}{\frac{1}{\sqrt{{x}^{4}+1}}}}-{\frac{3\,{\it Arcsinh} \left ({x}^{2} \right ) }{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^9/(x^4+1)^(3/2),x)
[Out]
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Maxima [A] time = 1.42306, size = 99, normalized size = 2.41 \[ -\frac{\frac{3 \,{\left (x^{4} + 1\right )}}{x^{4}} - 2}{4 \,{\left (\frac{\sqrt{x^{4} + 1}}{x^{2}} - \frac{{\left (x^{4} + 1\right )}^{\frac{3}{2}}}{x^{6}}\right )}} - \frac{3}{8} \, \log \left (\frac{\sqrt{x^{4} + 1}}{x^{2}} + 1\right ) + \frac{3}{8} \, \log \left (\frac{\sqrt{x^{4} + 1}}{x^{2}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^9/(x^4 + 1)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.252759, size = 170, normalized size = 4.15 \[ -\frac{4 \, x^{12} + 7 \, x^{8} - x^{4} - 3 \,{\left (4 \, x^{8} + 5 \, x^{4} -{\left (4 \, x^{6} + 3 \, x^{2}\right )} \sqrt{x^{4} + 1} + 1\right )} \log \left (-x^{2} + \sqrt{x^{4} + 1}\right ) -{\left (4 \, x^{10} + 5 \, x^{6} - 3 \, x^{2}\right )} \sqrt{x^{4} + 1} - 2}{4 \,{\left (4 \, x^{8} + 5 \, x^{4} -{\left (4 \, x^{6} + 3 \, x^{2}\right )} \sqrt{x^{4} + 1} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^9/(x^4 + 1)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 9.2043, size = 36, normalized size = 0.88 \[ \frac{x^{6}}{4 \sqrt{x^{4} + 1}} + \frac{3 x^{2}}{4 \sqrt{x^{4} + 1}} - \frac{3 \operatorname{asinh}{\left (x^{2} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**9/(x**4+1)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.226761, size = 46, normalized size = 1.12 \[ \frac{{\left (x^{4} + 3\right )} x^{2}}{4 \, \sqrt{x^{4} + 1}} + \frac{3}{4} \,{\rm ln}\left (-x^{2} + \sqrt{x^{4} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^9/(x^4 + 1)^(3/2),x, algorithm="giac")
[Out]