Optimal. Leaf size=35 \[ -\frac {\sqrt {1-x^3}}{3 x^3}-\frac {1}{3} \tanh ^{-1}\left (\sqrt {1-x^3}\right ) \]
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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 = {272, 44, 65,
212} \begin {gather*} -\frac {\sqrt {1-x^3}}{3 x^3}-\frac {1}{3} \tanh ^{-1}\left (\sqrt {1-x^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 65
Rule 212
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x^4 \sqrt {1-x^3}} \, dx &=\frac {1}{3} \text {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} \text {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} \text {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} \tanh ^{-1}\left (\sqrt {1-x^3}\right )\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 35, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {1-x^3}}{3 x^3}-\frac {1}{3} \tanh ^{-1}\left (\sqrt {1-x^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 28, normalized size = 0.80
method | result | size |
default | \(-\frac {\arctanh \left (\sqrt {-x^{3}+1}\right )}{3}-\frac {\sqrt {-x^{3}+1}}{3 x^{3}}\) | \(28\) |
elliptic | \(-\frac {\arctanh \left (\sqrt {-x^{3}+1}\right )}{3}-\frac {\sqrt {-x^{3}+1}}{3 x^{3}}\) | \(28\) |
risch | \(\frac {x^{3}-1}{3 x^{3} \sqrt {-x^{3}+1}}-\frac {\arctanh \left (\sqrt {-x^{3}+1}\right )}{3}\) | \(33\) |
trager | \(-\frac {\sqrt {-x^{3}+1}}{3 x^{3}}-\frac {\ln \left (-\frac {-x^{3}+2 \sqrt {-x^{3}+1}+2}{x^{3}}\right )}{6}\) | \(42\) |
meijerg | \(-\frac {-\frac {\sqrt {\pi }\, \left (-4 x^{3}+8\right )}{8 x^{3}}+\frac {\sqrt {\pi }\, \sqrt {-x^{3}+1}}{x^{3}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{3}+1}}{2}\right )-\frac {\left (1-2 \ln \left (2\right )+3 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{2}+\frac {\sqrt {\pi }}{x^{3}}}{3 \sqrt {\pi }}\) | \(82\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 43, normalized size = 1.23 \begin {gather*} -\frac {\sqrt {-x^{3} + 1}}{3 \, x^{3}} - \frac {1}{6} \, \log \left (\sqrt {-x^{3} + 1} + 1\right ) + \frac {1}{6} \, \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 [A]
time = 0.36, size = 50, normalized size = 1.43 \begin {gather*} -\frac {x^{3} \log \left (\sqrt {-x^{3} + 1} + 1\right ) - x^{3} \log \left (\sqrt {-x^{3} + 1} - 1\right ) + 2 \, \sqrt {-x^{3} + 1}}{6 \, x^{3}} \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 = 0.94, size = 82, normalized size = 2.34 \begin {gather*} \begin {cases} - \frac {\operatorname {acosh}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{3} + \frac {1}{3 x^{\frac {3}{2}} \sqrt {-1 + \frac {1}{x^{3}}}} - \frac {1}{3 x^{\frac {9}{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 )}}{3} - \frac {i \sqrt {1 - \frac {1}{x^{3}}}}{3 x^{\frac {3}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.68, size = 44, normalized size = 1.26 \begin {gather*} -\frac {\sqrt {-x^{3} + 1}}{3 \, x^{3}} - \frac {1}{6} \, \log \left (\sqrt {-x^{3} + 1} + 1\right ) + \frac {1}{6} \, \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 = 0.07, size = 195, normalized size = 5.57 \begin {gather*} -\frac {\sqrt {1-x^3}}{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+\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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