Optimal. Leaf size=53 \[ -\frac{\sqrt{1-x^3}}{4 x^3}-\frac{\sqrt{1-x^3}}{6 x^6}-\frac{1}{4} \tanh ^{-1}\left (\sqrt{1-x^3}\right ) \]
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Rubi [A] time = 0.0204685, antiderivative size = 53, normalized size of antiderivative = 1., 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} \[ -\frac{\sqrt{1-x^3}}{4 x^3}-\frac{\sqrt{1-x^3}}{6 x^6}-\frac{1}{4} \tanh ^{-1}\left (\sqrt{1-x^3}\right ) \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x^7 \sqrt{1-x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x^3} \, dx,x,x^3\right )\\ &=-\frac{\sqrt{1-x^3}}{6 x^6}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x^2} \, dx,x,x^3\right )\\ &=-\frac{\sqrt{1-x^3}}{6 x^6}-\frac{\sqrt{1-x^3}}{4 x^3}+\frac{1}{8} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x} \, dx,x,x^3\right )\\ &=-\frac{\sqrt{1-x^3}}{6 x^6}-\frac{\sqrt{1-x^3}}{4 x^3}-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1-x^3}\right )\\ &=-\frac{\sqrt{1-x^3}}{6 x^6}-\frac{\sqrt{1-x^3}}{4 x^3}-\frac{1}{4} \tanh ^{-1}\left (\sqrt{1-x^3}\right )\\ \end{align*}
Mathematica [C] time = 0.0048277, size = 30, normalized size = 0.57 \[ -\frac{2}{3} \sqrt{1-x^3} \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};1-x^3\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 42, normalized size = 0.8 \begin{align*} -{\frac{1}{4}{\it Artanh} \left ( \sqrt{-{x}^{3}+1} \right ) }-{\frac{1}{6\,{x}^{6}}\sqrt{-{x}^{3}+1}}-{\frac{1}{4\,{x}^{3}}\sqrt{-{x}^{3}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05768, size = 95, normalized size = 1.79 \begin{align*} \frac{3 \,{\left (-x^{3} + 1\right )}^{\frac{3}{2}} - 5 \, \sqrt{-x^{3} + 1}}{12 \,{\left (2 \, x^{3} +{\left (x^{3} - 1\right )}^{2} - 1\right )}} - \frac{1}{8} \, \log \left (\sqrt{-x^{3} + 1} + 1\right ) + \frac{1}{8} \, \log \left (\sqrt{-x^{3} + 1} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48859, size = 143, normalized size = 2.7 \begin{align*} -\frac{3 \, x^{6} \log \left (\sqrt{-x^{3} + 1} + 1\right ) - 3 \, x^{6} \log \left (\sqrt{-x^{3} + 1} - 1\right ) + 2 \,{\left (3 \, x^{3} + 2\right )} \sqrt{-x^{3} + 1}}{24 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.83352, size = 138, normalized size = 2.6 \begin{align*} \begin{cases} - \frac{\operatorname{acosh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{4} + \frac{1}{4 x^{\frac{3}{2}} \sqrt{-1 + \frac{1}{x^{3}}}} - \frac{1}{12 x^{\frac{9}{2}} \sqrt{-1 + \frac{1}{x^{3}}}} - \frac{1}{6 x^{\frac{15}{2}} \sqrt{-1 + \frac{1}{x^{3}}}} & \text{for}\: \frac{1}{\left |{x^{3}}\right |} > 1 \\\frac{i \operatorname{asin}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{4} - \frac{i}{4 x^{\frac{3}{2}} \sqrt{1 - \frac{1}{x^{3}}}} + \frac{i}{12 x^{\frac{9}{2}} \sqrt{1 - \frac{1}{x^{3}}}} + \frac{i}{6 x^{\frac{15}{2}} \sqrt{1 - \frac{1}{x^{3}}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1768, size = 78, normalized size = 1.47 \begin{align*} \frac{3 \,{\left (-x^{3} + 1\right )}^{\frac{3}{2}} - 5 \, \sqrt{-x^{3} + 1}}{12 \, x^{6}} - \frac{1}{8} \, \log \left (\sqrt{-x^{3} + 1} + 1\right ) + \frac{1}{8} \, \log \left ({\left | \sqrt{-x^{3} + 1} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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