Optimal. Leaf size=35 \[ \frac {1}{6} x^3 \sqrt {-1+x^6}+\frac {1}{6} \log \left (x^3+\sqrt {-1+x^6}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 35, 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 = {281, 327, 223,
212} \begin {gather*} \frac {1}{6} \sqrt {x^6-1} x^3+\frac {1}{6} \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 327
Rubi steps
\begin {align*} \int \frac {x^8}{\sqrt {-1+x^6}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^2}{\sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=\frac {1}{6} x^3 \sqrt {-1+x^6}+\frac {1}{6} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=\frac {1}{6} x^3 \sqrt {-1+x^6}+\frac {1}{6} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right )\\ &=\frac {1}{6} x^3 \sqrt {-1+x^6}+\frac {1}{6} \tanh ^{-1}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 32, normalized size = 0.91 \begin {gather*} \frac {1}{6} \left (x^3 \sqrt {-1+x^6}+\tanh ^{-1}\left (\frac {\sqrt {-1+x^6}}{x^3}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.44, size = 28, normalized size = 0.80
method | result | size |
trager | \(\frac {x^{3} \sqrt {x^{6}-1}}{6}+\frac {\ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{6}\) | \(28\) |
risch | \(\frac {x^{3} \sqrt {x^{6}-1}}{6}+\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \arcsin \left (x^{3}\right )}{6 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}}\) | \(38\) |
meijerg | \(\frac {i \sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \left (i \sqrt {\pi }\, x^{3} \sqrt {-x^{6}+1}-i \sqrt {\pi }\, \arcsin \left (x^{3}\right )\right )}{6 \sqrt {\pi }\, \sqrt {\mathrm {signum}\left (x^{6}-1\right )}}\) | \(54\) |
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. 58 vs.
\(2 (27) = 54\).
time = 0.25, size = 58, normalized size = 1.66 \begin {gather*} -\frac {\sqrt {x^{6} - 1}}{6 \, x^{3} {\left (\frac {x^{6} - 1}{x^{6}} - 1\right )}} + \frac {1}{12} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} + 1\right ) - \frac {1}{12} \, \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 = 0.35, size = 29, normalized size = 0.83 \begin {gather*} \frac {1}{6} \, \sqrt {x^{6} - 1} x^{3} - \frac {1}{6} \, \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 = 1.42, size = 61, normalized size = 1.74 \begin {gather*} \begin {cases} \frac {x^{3} \sqrt {x^{6} - 1}}{6} + \frac {\operatorname {acosh}{\left (x^{3} \right )}}{6} & \text {for}\: \left |{x^{6}}\right | > 1 \\- \frac {i x^{9}}{6 \sqrt {1 - x^{6}}} + \frac {i x^{3}}{6 \sqrt {1 - x^{6}}} - \frac {i \operatorname {asin}{\left (x^{3} \right )}}{6} & \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 = -0.03 \begin {gather*} \int \frac {x^8}{\sqrt {x^6-1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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