Optimal. Leaf size=25 \[ \frac{1}{8} x^4 \sqrt{x^8+1}-\frac{1}{8} \sinh ^{-1}\left (x^4\right ) \]
[Out]
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Rubi [A] time = 0.0364963, antiderivative size = 25, 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 \[ \frac{1}{8} x^4 \sqrt{x^8+1}-\frac{1}{8} \sinh ^{-1}\left (x^4\right ) \]
Antiderivative was successfully verified.
[In] Int[x^11/Sqrt[1 + x^8],x]
[Out]
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Rubi in Sympy [A] time = 4.66603, size = 19, normalized size = 0.76 \[ \frac{x^{4} \sqrt{x^{8} + 1}}{8} - \frac{\operatorname{asinh}{\left (x^{4} \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**11/(x**8+1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0127126, size = 25, normalized size = 1. \[ \frac{1}{8} x^4 \sqrt{x^8+1}-\frac{1}{8} \sinh ^{-1}\left (x^4\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^11/Sqrt[1 + x^8],x]
[Out]
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Maple [A] time = 0.035, size = 20, normalized size = 0.8 \[ -{\frac{{\it Arcsinh} \left ({x}^{4} \right ) }{8}}+{\frac{{x}^{4}}{8}\sqrt{{x}^{8}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^11/(x^8+1)^(1/2),x)
[Out]
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Maxima [A] time = 1.44347, size = 78, normalized size = 3.12 \[ \frac{\sqrt{x^{8} + 1}}{8 \, x^{4}{\left (\frac{x^{8} + 1}{x^{8}} - 1\right )}} - \frac{1}{16} \, \log \left (\frac{\sqrt{x^{8} + 1}}{x^{4}} + 1\right ) + \frac{1}{16} \, \log \left (\frac{\sqrt{x^{8} + 1}}{x^{4}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^11/sqrt(x^8 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.221649, size = 117, normalized size = 4.68 \[ -\frac{2 \, x^{16} + 2 \, x^{8} -{\left (2 \, x^{8} - 2 \, \sqrt{x^{8} + 1} x^{4} + 1\right )} \log \left (-x^{4} + \sqrt{x^{8} + 1}\right ) -{\left (2 \, x^{12} + x^{4}\right )} \sqrt{x^{8} + 1}}{8 \,{\left (2 \, x^{8} - 2 \, \sqrt{x^{8} + 1} x^{4} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^11/sqrt(x^8 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 8.38976, size = 19, normalized size = 0.76 \[ \frac{x^{4} \sqrt{x^{8} + 1}}{8} - \frac{\operatorname{asinh}{\left (x^{4} \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**11/(x**8+1)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.243473, size = 50, normalized size = 2. \[ \frac{1}{8} \, \sqrt{x^{8} + 1} x^{4} - \frac{1}{16} \,{\rm ln}\left (\sqrt{\frac{1}{x^{8}} + 1} + 1\right ) + \frac{1}{16} \,{\rm ln}\left (\sqrt{\frac{1}{x^{8}} + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^11/sqrt(x^8 + 1),x, algorithm="giac")
[Out]