Optimal. Leaf size=50 \[ -\frac{1}{4 x^4 \sqrt{1-x^4}}+\frac{3}{4 \sqrt{1-x^4}}-\frac{3}{4} \tanh ^{-1}\left (\sqrt{1-x^4}\right ) \]
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Rubi [A] time = 0.0222809, antiderivative size = 53, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 51, 63, 206} \[ -\frac{3 \sqrt{1-x^4}}{4 x^4}+\frac{1}{2 x^4 \sqrt{1-x^4}}-\frac{3}{4} \tanh ^{-1}\left (\sqrt{1-x^4}\right ) \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x^5 \left (1-x^4\right )^{3/2}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{(1-x)^{3/2} x^2} \, dx,x,x^4\right )\\ &=\frac{1}{2 x^4 \sqrt{1-x^4}}+\frac{3}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x^2} \, dx,x,x^4\right )\\ &=\frac{1}{2 x^4 \sqrt{1-x^4}}-\frac{3 \sqrt{1-x^4}}{4 x^4}+\frac{3}{8} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x} \, dx,x,x^4\right )\\ &=\frac{1}{2 x^4 \sqrt{1-x^4}}-\frac{3 \sqrt{1-x^4}}{4 x^4}-\frac{3}{4} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1-x^4}\right )\\ &=\frac{1}{2 x^4 \sqrt{1-x^4}}-\frac{3 \sqrt{1-x^4}}{4 x^4}-\frac{3}{4} \tanh ^{-1}\left (\sqrt{1-x^4}\right )\\ \end{align*}
Mathematica [C] time = 0.0052701, size = 30, normalized size = 0.6 \[ \frac{\, _2F_1\left (-\frac{1}{2},2;\frac{1}{2};1-x^4\right )}{2 \sqrt{1-x^4}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.014, size = 82, normalized size = 1.6 \begin{align*} -{\frac{3}{4}{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{4}+1}}} \right ) }+{\frac{1}{4\,{x}^{2}+4}\sqrt{- \left ({x}^{2}+1 \right ) ^{2}+2+2\,{x}^{2}}}-{\frac{1}{4\,{x}^{2}-4}\sqrt{- \left ({x}^{2}-1 \right ) ^{2}+2-2\,{x}^{2}}}-{\frac{1}{4\,{x}^{4}}\sqrt{-{x}^{4}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.98178, size = 82, normalized size = 1.64 \begin{align*} -\frac{3 \, x^{4} - 1}{4 \,{\left ({\left (-x^{4} + 1\right )}^{\frac{3}{2}} - \sqrt{-x^{4} + 1}\right )}} - \frac{3}{8} \, \log \left (\sqrt{-x^{4} + 1} + 1\right ) + \frac{3}{8} \, \log \left (\sqrt{-x^{4} + 1} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.48874, size = 174, normalized size = 3.48 \begin{align*} -\frac{3 \,{\left (x^{8} - x^{4}\right )} \log \left (\sqrt{-x^{4} + 1} + 1\right ) - 3 \,{\left (x^{8} - x^{4}\right )} \log \left (\sqrt{-x^{4} + 1} - 1\right ) + 2 \,{\left (3 \, x^{4} - 1\right )} \sqrt{-x^{4} + 1}}{8 \,{\left (x^{8} - x^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.67922, size = 95, normalized size = 1.9 \begin{align*} \begin{cases} - \frac{3 \operatorname{acosh}{\left (\frac{1}{x^{2}} \right )}}{4} + \frac{3}{4 x^{2} \sqrt{-1 + \frac{1}{x^{4}}}} - \frac{1}{4 x^{6} \sqrt{-1 + \frac{1}{x^{4}}}} & \text{for}\: \frac{1}{\left |{x^{4}}\right |} > 1 \\\frac{3 i \operatorname{asin}{\left (\frac{1}{x^{2}} \right )}}{4} - \frac{3 i}{4 x^{2} \sqrt{1 - \frac{1}{x^{4}}}} + \frac{i}{4 x^{6} \sqrt{1 - \frac{1}{x^{4}}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15365, size = 85, normalized size = 1.7 \begin{align*} -\frac{3 \, x^{4} - 1}{4 \,{\left ({\left (-x^{4} + 1\right )}^{\frac{3}{2}} - \sqrt{-x^{4} + 1}\right )}} - \frac{3}{8} \, \log \left (\sqrt{-x^{4} + 1} + 1\right ) + \frac{3}{8} \, \log \left (-\sqrt{-x^{4} + 1} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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