Optimal. Leaf size=43 \[ \frac {1}{24} \sqrt {-1+x^6} \left (-x^3+2 x^9\right )-\frac {1}{24} \log \left (x^3+\sqrt {-1+x^6}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 51, normalized size of antiderivative = 1.19, number of steps
used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {281, 285, 327,
223, 212} \begin {gather*} \frac {1}{12} \sqrt {x^6-1} x^9-\frac {1}{24} \sqrt {x^6-1} x^3-\frac {1}{24} \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 281
Rule 285
Rule 327
Rubi steps
\begin {align*} \int x^8 \sqrt {-1+x^6} \, dx &=\frac {1}{3} \text {Subst}\left (\int x^2 \sqrt {-1+x^2} \, dx,x,x^3\right )\\ &=\frac {1}{12} x^9 \sqrt {-1+x^6}-\frac {1}{12} \text {Subst}\left (\int \frac {x^2}{\sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=-\frac {1}{24} x^3 \sqrt {-1+x^6}+\frac {1}{12} x^9 \sqrt {-1+x^6}-\frac {1}{24} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=-\frac {1}{24} x^3 \sqrt {-1+x^6}+\frac {1}{12} x^9 \sqrt {-1+x^6}-\frac {1}{24} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right )\\ &=-\frac {1}{24} x^3 \sqrt {-1+x^6}+\frac {1}{12} x^9 \sqrt {-1+x^6}-\frac {1}{24} \tanh ^{-1}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 42, normalized size = 0.98 \begin {gather*} \frac {1}{24} x^3 \sqrt {-1+x^6} \left (-1+2 x^6\right )-\frac {1}{24} \tanh ^{-1}\left (\frac {\sqrt {-1+x^6}}{x^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.43, size = 37, normalized size = 0.86
method | result | size |
trager | \(\frac {x^{3} \left (2 x^{6}-1\right ) \sqrt {x^{6}-1}}{24}+\frac {\ln \left (x^{3}-\sqrt {x^{6}-1}\right )}{24}\) | \(37\) |
risch | \(\frac {x^{3} \left (2 x^{6}-1\right ) \sqrt {x^{6}-1}}{24}-\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \arcsin \left (x^{3}\right )}{24 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}}\) | \(45\) |
meijerg | \(-\frac {i \sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, \left (-\frac {i \sqrt {\pi }\, x^{3} \left (-6 x^{6}+3\right ) \sqrt {-x^{6}+1}}{6}+\frac {i \sqrt {\pi }\, \arcsin \left (x^{3}\right )}{2}\right )}{12 \sqrt {\pi }\, \sqrt {-\mathrm {signum}\left (x^{6}-1\right )}}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 84 vs.
\(2 (35) = 70\).
time = 0.25, size = 84, normalized size = 1.95 \begin {gather*} -\frac {\frac {\sqrt {x^{6} - 1}}{x^{3}} + \frac {{\left (x^{6} - 1\right )}^{\frac {3}{2}}}{x^{9}}}{24 \, {\left (\frac {2 \, {\left (x^{6} - 1\right )}}{x^{6}} - \frac {{\left (x^{6} - 1\right )}^{2}}{x^{12}} - 1\right )}} - \frac {1}{48} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} + 1\right ) + \frac {1}{48} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 37, normalized size = 0.86 \begin {gather*} \frac {1}{24} \, {\left (2 \, x^{9} - x^{3}\right )} \sqrt {x^{6} - 1} + \frac {1}{24} \, \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) \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 = 2.11, size = 104, normalized size = 2.42 \begin {gather*} \begin {cases} \frac {x^{15}}{12 \sqrt {x^{6} - 1}} - \frac {x^{9}}{8 \sqrt {x^{6} - 1}} + \frac {x^{3}}{24 \sqrt {x^{6} - 1}} - \frac {\operatorname {acosh}{\left (x^{3} \right )}}{24} & \text {for}\: \left |{x^{6}}\right | > 1 \\- \frac {i x^{15}}{12 \sqrt {1 - x^{6}}} + \frac {i x^{9}}{8 \sqrt {1 - x^{6}}} - \frac {i x^{3}}{24 \sqrt {1 - x^{6}}} + \frac {i \operatorname {asin}{\left (x^{3} \right )}}{24} & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^8\,\sqrt {x^6-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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