Optimal. Leaf size=25 \[ \frac{1}{8} x^4 \sqrt{x^8+1}-\frac{1}{8} \sinh ^{-1}\left (x^4\right ) \]
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Rubi [A] time = 0.0095592, 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, Rules used = {275, 321, 215} \[ \frac{1}{8} x^4 \sqrt{x^8+1}-\frac{1}{8} \sinh ^{-1}\left (x^4\right ) \]
Antiderivative was successfully verified.
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Rule 275
Rule 321
Rule 215
Rubi steps
\begin{align*} \int \frac{x^{11}}{\sqrt{1+x^8}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+x^2}} \, dx,x,x^4\right )\\ &=\frac{1}{8} x^4 \sqrt{1+x^8}-\frac{1}{8} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,x^4\right )\\ &=\frac{1}{8} x^4 \sqrt{1+x^8}-\frac{1}{8} \sinh ^{-1}\left (x^4\right )\\ \end{align*}
Mathematica [A] time = 0.0047353, 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.
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Maple [A] time = 0.023, size = 20, normalized size = 0.8 \begin{align*} -{\frac{{\it Arcsinh} \left ({x}^{4} \right ) }{8}}+{\frac{{x}^{4}}{8}\sqrt{{x}^{8}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.95863, size = 78, normalized size = 3.12 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.2078, size = 74, normalized size = 2.96 \begin{align*} \frac{1}{8} \, \sqrt{x^{8} + 1} x^{4} + \frac{1}{8} \, \log \left (-x^{4} + \sqrt{x^{8} + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.17746, size = 19, normalized size = 0.76 \begin{align*} \frac{x^{4} \sqrt{x^{8} + 1}}{8} - \frac{\operatorname{asinh}{\left (x^{4} \right )}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2332, size = 50, normalized size = 2. \begin{align*} \frac{1}{8} \, \sqrt{x^{8} + 1} x^{4} - \frac{1}{16} \, \log \left (\sqrt{\frac{1}{x^{8}} + 1} + 1\right ) + \frac{1}{16} \, \log \left (\sqrt{\frac{1}{x^{8}} + 1} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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