Optimal. Leaf size=61 \[ \frac {1}{36} \sqrt {x^6-1} \left (4 x^6-13\right )+\frac {1}{3} \tan ^{-1}\left (\sqrt {x^6-1}\right )-\frac {\tan ^{-1}\left (\frac {2 \sqrt {x^6-1}}{\sqrt {3}}\right )}{8 \sqrt {3}} \]
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Rubi [A] time = 0.10, antiderivative size = 67, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {573, 178, 88, 63, 203} \begin {gather*} \frac {1}{9} \left (x^6-1\right )^{3/2}-\frac {\sqrt {x^6-1}}{4}+\frac {1}{3} \tan ^{-1}\left (\sqrt {x^6-1}\right )-\frac {\tan ^{-1}\left (\frac {2 \sqrt {x^6-1}}{\sqrt {3}}\right )}{8 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 88
Rule 178
Rule 203
Rule 573
Rubi steps
\begin {align*} \int \frac {\sqrt {-1+x^6} \left (-1+2 x^6\right )^2}{x \left (-1+4 x^6\right )} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x} (-1+2 x)^2}{x (-1+4 x)} \, dx,x,x^6\right )\\ &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {(-1+2 x)^2}{\sqrt {-1+x} x} \, dx,x,x^6\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {(-1+2 x)^2}{\sqrt {-1+x} (-1+4 x)} \, dx,x,x^6\right )\\ &=\frac {1}{6} \operatorname {Subst}\left (\int \left (4 \sqrt {-1+x}+\frac {1}{\sqrt {-1+x} x}\right ) \, dx,x,x^6\right )-\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{4 \sqrt {-1+x}}+\sqrt {-1+x}+\frac {1}{4 \sqrt {-1+x} (-1+4 x)}\right ) \, dx,x,x^6\right )\\ &=-\frac {1}{4} \sqrt {-1+x^6}+\frac {1}{9} \left (-1+x^6\right )^{3/2}-\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} (-1+4 x)} \, dx,x,x^6\right )+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right )\\ &=-\frac {1}{4} \sqrt {-1+x^6}+\frac {1}{9} \left (-1+x^6\right )^{3/2}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{3+4 x^2} \, dx,x,\sqrt {-1+x^6}\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right )\\ &=-\frac {1}{4} \sqrt {-1+x^6}+\frac {1}{9} \left (-1+x^6\right )^{3/2}+\frac {1}{3} \tan ^{-1}\left (\sqrt {-1+x^6}\right )-\frac {\tan ^{-1}\left (\frac {2 \sqrt {-1+x^6}}{\sqrt {3}}\right )}{8 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 100, normalized size = 1.64 \begin {gather*} \frac {1}{144} \left (16 \sqrt {x^6-1} x^6-52 \sqrt {x^6-1}+48 \tan ^{-1}\left (\sqrt {x^6-1}\right )+3 \sqrt {3} \tan ^{-1}\left (\frac {2-x^3}{\sqrt {3} \sqrt {x^6-1}}\right )+3 \sqrt {3} \tan ^{-1}\left (\frac {x^3+2}{\sqrt {3} \sqrt {x^6-1}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 61, normalized size = 1.00 \begin {gather*} \frac {1}{36} \sqrt {x^6-1} \left (4 x^6-13\right )+\frac {1}{3} \tan ^{-1}\left (\sqrt {x^6-1}\right )-\frac {\tan ^{-1}\left (\frac {2 \sqrt {x^6-1}}{\sqrt {3}}\right )}{8 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 45, normalized size = 0.74 \begin {gather*} \frac {1}{36} \, {\left (4 \, x^{6} - 13\right )} \sqrt {x^{6} - 1} - \frac {1}{24} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \sqrt {x^{6} - 1}\right ) + \frac {1}{3} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 47, normalized size = 0.77 \begin {gather*} \frac {1}{9} \, {\left (x^{6} - 1\right )}^{\frac {3}{2}} - \frac {1}{24} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \sqrt {x^{6} - 1}\right ) - \frac {1}{4} \, \sqrt {x^{6} - 1} + \frac {1}{3} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.46, size = 104, normalized size = 1.70 \begin {gather*} \left (\frac {x^{6}}{9}-\frac {13}{36}\right ) \sqrt {x^{6}-1}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right )-\sqrt {x^{6}-1}}{x^{3}}\right )}{3}-\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {-4 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{6}+12 \sqrt {x^{6}-1}+7 \RootOf \left (\textit {\_Z}^{2}+3\right )}{\left (2 x^{3}-1\right ) \left (2 x^{3}+1\right )}\right )}{48} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{6} - 1\right )}^{2} \sqrt {x^{6} - 1}}{{\left (4 \, x^{6} - 1\right )} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.78, size = 47, normalized size = 0.77 \begin {gather*} \frac {\mathrm {atan}\left (\sqrt {x^6-1}\right )}{3}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {2\,\sqrt {3}\,\sqrt {x^6-1}}{3}\right )}{24}-\frac {\sqrt {x^6-1}}{4}+\frac {{\left (x^6-1\right )}^{3/2}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 43.42, size = 56, normalized size = 0.92 \begin {gather*} \frac {\left (x^{6} - 1\right )^{\frac {3}{2}}}{9} - \frac {\sqrt {x^{6} - 1}}{4} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x^{6} - 1}}{3} \right )}}{24} + \frac {\operatorname {atan}{\left (\sqrt {x^{6} - 1} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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