Optimal. Leaf size=45 \[ -\frac {x^9}{3 \sqrt {2+x^6}}+\frac {1}{2} x^3 \sqrt {2+x^6}-\sinh ^{-1}\left (\frac {x^3}{\sqrt {2}}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 45, 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, 294, 327,
221} \begin {gather*} -\sinh ^{-1}\left (\frac {x^3}{\sqrt {2}}\right )-\frac {x^9}{3 \sqrt {x^6+2}}+\frac {1}{2} \sqrt {x^6+2} x^3 \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 281
Rule 294
Rule 327
Rubi steps
\begin {align*} \int \frac {x^{14}}{\left (2+x^6\right )^{3/2}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^4}{\left (2+x^2\right )^{3/2}} \, dx,x,x^3\right )\\ &=-\frac {x^9}{3 \sqrt {2+x^6}}+\text {Subst}\left (\int \frac {x^2}{\sqrt {2+x^2}} \, dx,x,x^3\right )\\ &=-\frac {x^9}{3 \sqrt {2+x^6}}+\frac {1}{2} x^3 \sqrt {2+x^6}-\text {Subst}\left (\int \frac {1}{\sqrt {2+x^2}} \, dx,x,x^3\right )\\ &=-\frac {x^9}{3 \sqrt {2+x^6}}+\frac {1}{2} x^3 \sqrt {2+x^6}-\sinh ^{-1}\left (\frac {x^3}{\sqrt {2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 38, normalized size = 0.84 \begin {gather*} \frac {x^3 \left (6+x^6\right )}{6 \sqrt {2+x^6}}-\tanh ^{-1}\left (\frac {x^3}{\sqrt {2+x^6}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 30, normalized size = 0.67
method | result | size |
risch | \(\frac {x^{3} \left (x^{6}+6\right )}{6 \sqrt {x^{6}+2}}-\arcsinh \left (\frac {x^{3} \sqrt {2}}{2}\right )\) | \(30\) |
trager | \(\frac {x^{3} \left (x^{6}+6\right )}{6 \sqrt {x^{6}+2}}+\ln \left (x^{3}-\sqrt {x^{6}+2}\right )\) | \(33\) |
meijerg | \(\frac {\frac {\sqrt {\pi }\, x^{3} \sqrt {2}\, \left (\frac {5 x^{6}}{2}+15\right )}{30 \sqrt {1+\frac {x^{6}}{2}}}-\sqrt {\pi }\, \arcsinh \left (\frac {x^{3} \sqrt {2}}{2}\right )}{\sqrt {\pi }}\) | \(48\) |
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. 73 vs.
\(2 (36) = 72\).
time = 0.29, size = 73, normalized size = 1.62 \begin {gather*} -\frac {\frac {3 \, {\left (x^{6} + 2\right )}}{x^{6}} - 2}{3 \, {\left (\frac {\sqrt {x^{6} + 2}}{x^{3}} - \frac {{\left (x^{6} + 2\right )}^{\frac {3}{2}}}{x^{9}}\right )}} - \frac {1}{2} \, \log \left (\frac {\sqrt {x^{6} + 2}}{x^{3}} + 1\right ) + \frac {1}{2} \, \log \left (\frac {\sqrt {x^{6} + 2}}{x^{3}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 54, normalized size = 1.20 \begin {gather*} \frac {4 \, x^{6} + 6 \, {\left (x^{6} + 2\right )} \log \left (-x^{3} + \sqrt {x^{6} + 2}\right ) + {\left (x^{9} + 6 \, x^{3}\right )} \sqrt {x^{6} + 2} + 8}{6 \, {\left (x^{6} + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.67, size = 36, normalized size = 0.80 \begin {gather*} \frac {x^{9}}{6 \sqrt {x^{6} + 2}} + \frac {x^{3}}{\sqrt {x^{6} + 2}} - \operatorname {asinh}{\left (\frac {\sqrt {2} x^{3}}{2} \right )} \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 \frac {x^{14}}{{\left (x^6+2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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