Optimal. Leaf size=38 \[ \frac {1}{4} \tan ^{-1}\left (\sqrt {x^3-1}\right )+\frac {\sqrt {x^3-1} \left (3 x^3+2\right )}{12 x^6} \]
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Rubi [A] time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.24, number of steps used = 5, 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}}{4 x^3}+\frac {1}{4} \tan ^{-1}\left (\sqrt {x^3-1}\right )+\frac {\sqrt {x^3-1}}{6 x^6} \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^7 \sqrt {-1+x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x^3} \, dx,x,x^3\right )\\ &=\frac {\sqrt {-1+x^3}}{6 x^6}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x^2} \, dx,x,x^3\right )\\ &=\frac {\sqrt {-1+x^3}}{6 x^6}+\frac {\sqrt {-1+x^3}}{4 x^3}+\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^3\right )\\ &=\frac {\sqrt {-1+x^3}}{6 x^6}+\frac {\sqrt {-1+x^3}}{4 x^3}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^3}\right )\\ &=\frac {\sqrt {-1+x^3}}{6 x^6}+\frac {\sqrt {-1+x^3}}{4 x^3}+\frac {1}{4} \tan ^{-1}\left (\sqrt {-1+x^3}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 28, normalized size = 0.74 \begin {gather*} \frac {2}{3} \sqrt {x^3-1} \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};1-x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.00, size = 38, normalized size = 1.00 \begin {gather*} \frac {1}{4} \tan ^{-1}\left (\sqrt {x^3-1}\right )+\frac {\sqrt {x^3-1} \left (3 x^3+2\right )}{12 x^6} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 34, normalized size = 0.89 \begin {gather*} \frac {3 \, x^{6} \arctan \left (\sqrt {x^{3} - 1}\right ) + {\left (3 \, x^{3} + 2\right )} \sqrt {x^{3} - 1}}{12 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 35, normalized size = 0.92 \begin {gather*} \frac {3 \, {\left (x^{3} - 1\right )}^{\frac {3}{2}} + 5 \, \sqrt {x^{3} - 1}}{12 \, x^{6}} + \frac {1}{4} \, \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 = 0.02, size = 36, normalized size = 0.95 \begin {gather*} \frac {\sqrt {x^{3}-1}}{6 x^{6}}+\frac {\sqrt {x^{3}-1}}{4 x^{3}}+\frac {\arctan \left (\sqrt {x^{3}-1}\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 48, normalized size = 1.26 \begin {gather*} \frac {3 \, {\left (x^{3} - 1\right )}^{\frac {3}{2}} + 5 \, \sqrt {x^{3} - 1}}{12 \, {\left (2 \, x^{3} + {\left (x^{3} - 1\right )}^{2} - 1\right )}} + \frac {1}{4} \, \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 = 0.20, size = 189, normalized size = 4.97 \begin {gather*} \frac {\sqrt {x^3-1}}{4\,x^3}+\frac {\sqrt {x^3-1}}{6\,x^6}-\frac {3\,\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 )}{4\,\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 [B] time = 2.43, size = 138, normalized size = 3.63 \begin {gather*} \begin {cases} \frac {i \operatorname {acosh}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{4} - \frac {i}{4 x^{\frac {3}{2}} \sqrt {-1 + \frac {1}{x^{3}}}} + \frac {i}{12 x^{\frac {9}{2}} \sqrt {-1 + \frac {1}{x^{3}}}} + \frac {i}{6 x^{\frac {15}{2}} \sqrt {-1 + \frac {1}{x^{3}}}} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\- \frac {\operatorname {asin}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{4} + \frac {1}{4 x^{\frac {3}{2}} \sqrt {1 - \frac {1}{x^{3}}}} - \frac {1}{12 x^{\frac {9}{2}} \sqrt {1 - \frac {1}{x^{3}}}} - \frac {1}{6 x^{\frac {15}{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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