Optimal. Leaf size=35 \[ \frac {\sqrt {-x^3-1}}{3 x^3}-\frac {1}{3} \tan ^{-1}\left (\sqrt {-x^3-1}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {266, 51, 63, 204} \[ \frac {\sqrt {-x^3-1}}{3 x^3}-\frac {1}{3} \tan ^{-1}\left (\sqrt {-x^3-1}\right ) \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 204
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x^4 \sqrt {-1-x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-x} x^2} \, dx,x,x^3\right )\\ &=\frac {\sqrt {-1-x^3}}{3 x^3}-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-x} x} \, dx,x,x^3\right )\\ &=\frac {\sqrt {-1-x^3}}{3 x^3}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {-1-x^3}\right )\\ &=\frac {\sqrt {-1-x^3}}{3 x^3}-\frac {1}{3} \tan ^{-1}\left (\sqrt {-1-x^3}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 40, normalized size = 1.14 \[ \frac {1}{3} \sqrt {-x^3-1} \left (\frac {1}{x^3}-\frac {\tanh ^{-1}\left (\sqrt {x^3+1}\right )}{\sqrt {x^3+1}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 31, normalized size = 0.89 \[ -\frac {x^{3} \arctan \left (\sqrt {-x^{3} - 1}\right ) - \sqrt {-x^{3} - 1}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 27, normalized size = 0.77 \[ \frac {\sqrt {-x^{3} - 1}}{3 \, x^{3}} - \frac {1}{3} \, \arctan \left (\sqrt {-x^{3} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 28, normalized size = 0.80 \[ -\frac {\arctan \left (\sqrt {-x^{3}-1}\right )}{3}+\frac {\sqrt {-x^{3}-1}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.98, size = 27, normalized size = 0.77 \[ \frac {\sqrt {-x^{3} - 1}}{3 \, x^{3}} - \frac {1}{3} \, \arctan \left (\sqrt {-x^{3} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.98, size = 194, normalized size = 5.54 \[ \frac {\sqrt {-x^3-1}}{3\,x^3}+\frac {\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {x^3+1}\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{\sqrt {-x^3-1}\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.15, size = 29, normalized size = 0.83 \[ - \frac {i \operatorname {asinh}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{3} + \frac {i \sqrt {1 + \frac {1}{x^{3}}}}{3 x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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