Optimal. Leaf size=31 \[ \frac {\sqrt {x^3-1}}{3 x^3}+\frac {1}{3} \tan ^{-1}\left (\sqrt {x^3-1}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {266, 51, 63, 203} \begin {gather*} \frac {\sqrt {x^3-1}}{3 x^3}+\frac {1}{3} \tan ^{-1}\left (\sqrt {x^3-1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 203
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 = 41, normalized size = 1.32 \begin {gather*} \frac {1}{3} \sqrt {x^3-1} \left (\frac {1}{x^3}+\frac {\tanh ^{-1}\left (\sqrt {1-x^3}\right )}{\sqrt {1-x^3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 31, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1+x^3}}{3 x^3}+\frac {1}{3} \tan ^{-1}\left (\sqrt {-1+x^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 25, normalized size = 0.81 \begin {gather*} \frac {x^{3} \arctan \left (\sqrt {x^{3} - 1}\right ) + \sqrt {x^{3} - 1}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 23, normalized size = 0.74 \begin {gather*} \frac {\sqrt {x^{3} - 1}}{3 \, x^{3}} + \frac {1}{3} \, \arctan \left (\sqrt {x^{3} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.37, size = 24, normalized size = 0.77
method | result | size |
default | \(\frac {\sqrt {x^{3}-1}}{3 x^{3}}+\frac {\arctan \left (\sqrt {x^{3}-1}\right )}{3}\) | \(24\) |
risch | \(\frac {\sqrt {x^{3}-1}}{3 x^{3}}+\frac {\arctan \left (\sqrt {x^{3}-1}\right )}{3}\) | \(24\) |
elliptic | \(\frac {\sqrt {x^{3}-1}}{3 x^{3}}+\frac {\arctan \left (\sqrt {x^{3}-1}\right )}{3}\) | \(24\) |
trager | \(\frac {\sqrt {x^{3}-1}}{3 x^{3}}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \sqrt {x^{3}-1}+2 \RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{3}}\right )}{6}\) | \(57\) |
meijerg | \(-\frac {\sqrt {-\mathrm {signum}\left (x^{3}-1\right )}\, \left (\frac {\sqrt {\pi }}{x^{3}}-\frac {\left (1-2 \ln \relax (2)+3 \ln \relax (x )+i \pi \right ) \sqrt {\pi }}{2}-\frac {\sqrt {\pi }\, \left (-4 x^{3}+8\right )}{8 x^{3}}+\frac {\sqrt {\pi }\, \sqrt {-x^{3}+1}}{x^{3}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{3}+1}}{2}\right )\right )}{3 \sqrt {\mathrm {signum}\left (x^{3}-1\right )}\, \sqrt {\pi }}\) | \(100\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 23, normalized size = 0.74 \begin {gather*} \frac {\sqrt {x^{3} - 1}}{3 \, x^{3}} + \frac {1}{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 = 0.17, size = 177, normalized size = 5.71 \begin {gather*} \frac {\sqrt {x^3-1}}{3\,x^3}-\frac {\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\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 {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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sympy [A] time = 1.45, size = 82, normalized size = 2.65 \begin {gather*} \begin {cases} \frac {i \operatorname {acosh}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{3} + \frac {i \sqrt {-1 + \frac {1}{x^{3}}}}{3 x^{\frac {3}{2}}} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\- \frac {\operatorname {asin}{\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 {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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