Optimal. Leaf size=16 \[ \frac {2}{3} \tan ^{-1}\left (\sqrt {-1-x^3}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {272, 65, 210}
\begin {gather*} \frac {2}{3} \text {ArcTan}\left (\sqrt {-x^3-1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 210
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {-1-x^3}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt {-1-x} x} \, dx,x,x^3\right )\\ &=-\left (\frac {2}{3} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {-1-x^3}\right )\right )\\ &=\frac {2}{3} \tan ^{-1}\left (\sqrt {-1-x^3}\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} \frac {2}{3} \tan ^{-1}\left (\sqrt {-1-x^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.31, size = 13, normalized size = 0.81
method | result | size |
default | \(\frac {2 \arctan \left (\sqrt {-x^{3}-1}\right )}{3}\) | \(13\) |
elliptic | \(\frac {2 \arctan \left (\sqrt {-x^{3}-1}\right )}{3}\) | \(13\) |
meijerg | \(-\frac {i \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{3}+1}}{2}\right )+\left (-2 \ln \left (2\right )+3 \ln \left (x \right )\right ) \sqrt {\pi }\right )}{3 \sqrt {\pi }}\) | \(38\) |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {x^{3} \RootOf \left (\textit {\_Z}^{2}+1\right )+2 \RootOf \left (\textit {\_Z}^{2}+1\right )+2 \sqrt {-x^{3}-1}}{x^{3}}\right )}{3}\) | \(45\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 12, normalized size = 0.75 \begin {gather*} \frac {2}{3} \, \arctan \left (\sqrt {-x^{3} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 12, normalized size = 0.75 \begin {gather*} \frac {2}{3} \, \arctan \left (\sqrt {-x^{3} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.41, size = 12, normalized size = 0.75 \begin {gather*} \frac {2 i \operatorname {asinh}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.75, size = 12, normalized size = 0.75 \begin {gather*} \frac {2}{3} \, \arctan \left (\sqrt {-x^{3} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 180, normalized size = 11.25 \begin {gather*} -\frac {2\,\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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