Optimal. Leaf size=31 \[ \frac{\sqrt{1-x^4}}{2}+\frac{1}{2 \sqrt{1-x^4}} \]
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Rubi [A] time = 0.0138459, 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} \[ \frac{\sqrt{1-x^4}}{2}+\frac{1}{2 \sqrt{1-x^4}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^7}{\left (1-x^4\right )^{3/2}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x}{(1-x)^{3/2}} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{1}{(1-x)^{3/2}}-\frac{1}{\sqrt{1-x}}\right ) \, dx,x,x^4\right )\\ &=\frac{1}{2 \sqrt{1-x^4}}+\frac{\sqrt{1-x^4}}{2}\\ \end{align*}
Mathematica [A] time = 0.0061457, size = 22, normalized size = 0.71 \[ \frac{2-x^4}{2 \sqrt{1-x^4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, 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 ) }{2} \left ( -{x}^{4}+1 \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01243, size = 31, normalized size = 1. \begin{align*} \frac{1}{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 = 1.46528, size = 54, normalized size = 1.74 \begin{align*} \frac{{\left (x^{4} - 2\right )} \sqrt{-x^{4} + 1}}{2 \,{\left (x^{4} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.915426, size = 22, normalized size = 0.71 \begin{align*} - \frac{x^{4}}{2 \sqrt{1 - x^{4}}} + \frac{1}{\sqrt{1 - x^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20726, size = 31, normalized size = 1. \begin{align*} \frac{1}{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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