Optimal. Leaf size=38 \[ \frac{1}{6} \left (x^4+1\right )^{3/2}-\sqrt{x^4+1}-\frac{1}{2 \sqrt{x^4+1}} \]
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Rubi [A] time = 0.0137624, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{1}{6} \left (x^4+1\right )^{3/2}-\sqrt{x^4+1}-\frac{1}{2 \sqrt{x^4+1}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{11}}{\left (1+x^4\right )^{3/2}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^2}{(1+x)^{3/2}} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{1}{(1+x)^{3/2}}-\frac{2}{\sqrt{1+x}}+\sqrt{1+x}\right ) \, dx,x,x^4\right )\\ &=-\frac{1}{2 \sqrt{1+x^4}}-\sqrt{1+x^4}+\frac{1}{6} \left (1+x^4\right )^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0078431, size = 23, normalized size = 0.61 \[ \frac{x^8-4 x^4-8}{6 \sqrt{x^4+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.074, size = 20, normalized size = 0.5 \begin{align*}{\frac{{x}^{8}-4\,{x}^{4}-8}{6}{\frac{1}{\sqrt{{x}^{4}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.993598, size = 38, normalized size = 1. \begin{align*} \frac{1}{6} \,{\left (x^{4} + 1\right )}^{\frac{3}{2}} - \sqrt{x^{4} + 1} - \frac{1}{2 \, \sqrt{x^{4} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43311, size = 50, normalized size = 1.32 \begin{align*} \frac{x^{8} - 4 \, x^{4} - 8}{6 \, \sqrt{x^{4} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.22573, size = 39, normalized size = 1.03 \begin{align*} \frac{x^{8}}{6 \sqrt{x^{4} + 1}} - \frac{2 x^{4}}{3 \sqrt{x^{4} + 1}} - \frac{4}{3 \sqrt{x^{4} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15364, size = 38, normalized size = 1. \begin{align*} \frac{1}{6} \,{\left (x^{4} + 1\right )}^{\frac{3}{2}} - \sqrt{x^{4} + 1} - \frac{1}{2 \, \sqrt{x^{4} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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