Optimal. Leaf size=58 \[ \frac {1}{12} \sqrt {4 x^6-1} x^9-\frac {1}{96} \sqrt {4 x^6-1} x^3-\frac {1}{192} \tanh ^{-1}\left (\frac {2 x^3}{\sqrt {4 x^6-1}}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {275, 279, 321, 217, 206} \[ \frac {1}{12} \sqrt {4 x^6-1} x^9-\frac {1}{96} \sqrt {4 x^6-1} x^3-\frac {1}{192} \tanh ^{-1}\left (\frac {2 x^3}{\sqrt {4 x^6-1}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 275
Rule 279
Rule 321
Rubi steps
\begin {align*} \int x^8 \sqrt {-1+4 x^6} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int x^2 \sqrt {-1+4 x^2} \, dx,x,x^3\right )\\ &=\frac {1}{12} x^9 \sqrt {-1+4 x^6}-\frac {1}{12} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-1+4 x^2}} \, dx,x,x^3\right )\\ &=-\frac {1}{96} x^3 \sqrt {-1+4 x^6}+\frac {1}{12} x^9 \sqrt {-1+4 x^6}-\frac {1}{96} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+4 x^2}} \, dx,x,x^3\right )\\ &=-\frac {1}{96} x^3 \sqrt {-1+4 x^6}+\frac {1}{12} x^9 \sqrt {-1+4 x^6}-\frac {1}{96} \operatorname {Subst}\left (\int \frac {1}{1-4 x^2} \, dx,x,\frac {x^3}{\sqrt {-1+4 x^6}}\right )\\ &=-\frac {1}{96} x^3 \sqrt {-1+4 x^6}+\frac {1}{12} x^9 \sqrt {-1+4 x^6}-\frac {1}{192} \tanh ^{-1}\left (\frac {2 x^3}{\sqrt {-1+4 x^6}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 56, normalized size = 0.97 \[ \frac {\left (4 x^6-1\right ) \left (\sin ^{-1}\left (2 x^3\right )+2 \sqrt {1-4 x^6} \left (8 x^6-1\right ) x^3\right )}{192 \sqrt {-\left (1-4 x^6\right )^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 41, normalized size = 0.71 \[ \frac {1}{96} \, {\left (8 \, x^{9} - x^{3}\right )} \sqrt {4 \, x^{6} - 1} + \frac {1}{192} \, \log \left (-2 \, x^{3} + \sqrt {4 \, x^{6} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {4 \, x^{6} - 1} x^{8}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.16, size = 53, normalized size = 0.91 \[ \frac {\left (8 x^{6}-1\right ) \sqrt {4 x^{6}-1}\, x^{3}}{96}-\frac {\sqrt {-\mathrm {signum}\left (4 x^{6}-1\right )}\, \arcsin \left (2 x^{3}\right )}{192 \sqrt {\mathrm {signum}\left (4 x^{6}-1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.06, size = 97, normalized size = 1.67 \[ -\frac {\frac {4 \, \sqrt {4 \, x^{6} - 1}}{x^{3}} + \frac {{\left (4 \, x^{6} - 1\right )}^{\frac {3}{2}}}{x^{9}}}{96 \, {\left (\frac {8 \, {\left (4 \, x^{6} - 1\right )}}{x^{6}} - \frac {{\left (4 \, x^{6} - 1\right )}^{2}}{x^{12}} - 16\right )}} - \frac {1}{384} \, \log \left (\frac {\sqrt {4 \, x^{6} - 1}}{x^{3}} + 2\right ) + \frac {1}{384} \, \log \left (\frac {\sqrt {4 \, x^{6} - 1}}{x^{3}} - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^8\,\sqrt {4\,x^6-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.61, size = 119, normalized size = 2.05 \[ \begin {cases} \frac {x^{15}}{3 \sqrt {4 x^{6} - 1}} - \frac {x^{9}}{8 \sqrt {4 x^{6} - 1}} + \frac {x^{3}}{96 \sqrt {4 x^{6} - 1}} - \frac {\operatorname {acosh}{\left (2 x^{3} \right )}}{192} & \text {for}\: 4 \left |{x^{6}}\right | > 1 \\- \frac {i x^{15}}{3 \sqrt {1 - 4 x^{6}}} + \frac {i x^{9}}{8 \sqrt {1 - 4 x^{6}}} - \frac {i x^{3}}{96 \sqrt {1 - 4 x^{6}}} + \frac {i \operatorname {asin}{\left (2 x^{3} \right )}}{192} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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