Optimal. Leaf size=40 \[ \frac{1}{10} \left (x^4+1\right )^{5/2}-\frac{1}{3} \left (x^4+1\right )^{3/2}+\frac{\sqrt{x^4+1}}{2} \]
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Rubi [A] time = 0.0143273, antiderivative size = 40, 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} \[ \frac{1}{10} \left (x^4+1\right )^{5/2}-\frac{1}{3} \left (x^4+1\right )^{3/2}+\frac{\sqrt{x^4+1}}{2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{11}}{\sqrt{1+x^4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+x}} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{1}{\sqrt{1+x}}-2 \sqrt{1+x}+(1+x)^{3/2}\right ) \, dx,x,x^4\right )\\ &=\frac{\sqrt{1+x^4}}{2}-\frac{1}{3} \left (1+x^4\right )^{3/2}+\frac{1}{10} \left (1+x^4\right )^{5/2}\\ \end{align*}
Mathematica [A] time = 0.0069453, size = 25, normalized size = 0.62 \[ \frac{1}{30} \sqrt{x^4+1} \left (3 x^8-4 x^4+8\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 22, normalized size = 0.6 \begin{align*}{\frac{3\,{x}^{8}-4\,{x}^{4}+8}{30}\sqrt{{x}^{4}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.987845, size = 38, normalized size = 0.95 \begin{align*} \frac{1}{10} \,{\left (x^{4} + 1\right )}^{\frac{5}{2}} - \frac{1}{3} \,{\left (x^{4} + 1\right )}^{\frac{3}{2}} + \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 = 1.5019, size = 54, normalized size = 1.35 \begin{align*} \frac{1}{30} \,{\left (3 \, x^{8} - 4 \, x^{4} + 8\right )} \sqrt{x^{4} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.27381, size = 39, normalized size = 0.98 \begin{align*} \frac{x^{8} \sqrt{x^{4} + 1}}{10} - \frac{2 x^{4} \sqrt{x^{4} + 1}}{15} + \frac{4 \sqrt{x^{4} + 1}}{15} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18601, size = 38, normalized size = 0.95 \begin{align*} \frac{1}{10} \,{\left (x^{4} + 1\right )}^{\frac{5}{2}} - \frac{1}{3} \,{\left (x^{4} + 1\right )}^{\frac{3}{2}} + \frac{1}{2} \, \sqrt{x^{4} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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