Optimal. Leaf size=16 \[ -\frac {2}{3} \tanh ^{-1}\left (\sqrt {1-x^3}\right ) \]
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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, 212}
\begin {gather*} -\frac {2}{3} \tanh ^{-1}\left (\sqrt {1-x^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 212
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} \tanh ^{-1}\left (\sqrt {1-x^3}\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 16, normalized size = 1.00 \begin {gather*} -\frac {2}{3} \tanh ^{-1}\left (\sqrt {1-x^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 13, normalized size = 0.81
method | result | size |
default | \(-\frac {2 \arctanh \left (\sqrt {-x^{3}+1}\right )}{3}\) | \(13\) |
elliptic | \(-\frac {2 \arctanh \left (\sqrt {-x^{3}+1}\right )}{3}\) | \(13\) |
trager | \(-\frac {\ln \left (-\frac {-x^{3}+2 \sqrt {-x^{3}+1}+2}{x^{3}}\right )}{3}\) | \(27\) |
meijerg | \(\frac {-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{3}+1}}{2}\right )+\left (-2 \ln \left (2\right )+3 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{3 \sqrt {\pi }}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 29 vs.
\(2 (12) = 24\).
time = 0.32, size = 29, normalized size = 1.81 \begin {gather*} -\frac {1}{3} \, \log \left (\sqrt {-x^{3} + 1} + 1\right ) + \frac {1}{3} \, \log \left (\sqrt {-x^{3} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 29 vs.
\(2 (12) = 24\).
time = 0.36, size = 29, normalized size = 1.81 \begin {gather*} -\frac {1}{3} \, \log \left (\sqrt {-x^{3} + 1} + 1\right ) + \frac {1}{3} \, \log \left (\sqrt {-x^{3} + 1} - 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 = 0.43, size = 31, normalized size = 1.94 \begin {gather*} \begin {cases} - \frac {2 \operatorname {acosh}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{3} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\\frac {2 i \operatorname {asin}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{3} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 30 vs.
\(2 (12) = 24\).
time = 2.42, size = 30, normalized size = 1.88 \begin {gather*} -\frac {1}{3} \, \log \left (\sqrt {-x^{3} + 1} + 1\right ) + \frac {1}{3} \, \log \left ({\left | \sqrt {-x^{3} + 1} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, 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+\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}}}\,\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 {1-x^3}\,\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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