Optimal. Leaf size=31 \[ \frac{1}{6} \left (1-x^4\right )^{3/2}-\frac{\sqrt{1-x^4}}{2} \]
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Rubi [A] time = 0.0146928, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{1}{6} \left (1-x^4\right )^{3/2}-\frac{\sqrt{1-x^4}}{2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^7}{\sqrt{1-x^4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-x}} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{1}{\sqrt{1-x}}-\sqrt{1-x}\right ) \, dx,x,x^4\right )\\ &=-\frac{1}{2} \sqrt{1-x^4}+\frac{1}{6} \left (1-x^4\right )^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0058819, size = 20, normalized size = 0.65 \[ -\frac{1}{6} \sqrt{1-x^4} \left (x^4+2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 28, normalized size = 0.9 \begin{align*}{\frac{ \left ( -1+x \right ) \left ( 1+x \right ) \left ({x}^{2}+1 \right ) \left ({x}^{4}+2 \right ) }{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 = 0.975793, size = 31, normalized size = 1. \begin{align*} \frac{1}{6} \,{\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 = 1.48706, size = 42, normalized size = 1.35 \begin{align*} -\frac{1}{6} \,{\left (x^{4} + 2\right )} \sqrt{-x^{4} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.609201, size = 24, normalized size = 0.77 \begin{align*} - \frac{x^{4} \sqrt{1 - x^{4}}}{6} - \frac{\sqrt{1 - x^{4}}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12525, size = 31, normalized size = 1. \begin{align*} \frac{1}{6} \,{\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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