Optimal. Leaf size=43 \[ \frac {1}{8} \log \left (\sqrt {x^6+1}+x^3\right )+\frac {1}{24} \sqrt {x^6+1} \left (2 x^9-3 x^3\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 41, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {275, 321, 215} \begin {gather*} \frac {1}{8} \sinh ^{-1}\left (x^3\right )+\frac {1}{12} \sqrt {x^6+1} x^9-\frac {1}{8} \sqrt {x^6+1} x^3 \end {gather*}
Antiderivative was successfully verified.
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Rule 215
Rule 275
Rule 321
Rubi steps
\begin {align*} \int \frac {x^{14}}{\sqrt {1+x^6}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {1+x^2}} \, dx,x,x^3\right )\\ &=\frac {1}{12} x^9 \sqrt {1+x^6}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+x^2}} \, dx,x,x^3\right )\\ &=-\frac {1}{8} x^3 \sqrt {1+x^6}+\frac {1}{12} x^9 \sqrt {1+x^6}+\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^3\right )\\ &=-\frac {1}{8} x^3 \sqrt {1+x^6}+\frac {1}{12} x^9 \sqrt {1+x^6}+\frac {1}{8} \sinh ^{-1}\left (x^3\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 0.72 \begin {gather*} \frac {1}{24} \left (3 \sinh ^{-1}\left (x^3\right )+\sqrt {x^6+1} \left (2 x^6-3\right ) x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.16, size = 43, normalized size = 1.00 \begin {gather*} \frac {1}{24} \sqrt {1+x^6} \left (-3 x^3+2 x^9\right )+\frac {1}{8} \log \left (x^3+\sqrt {1+x^6}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 37, normalized size = 0.86 \begin {gather*} \frac {1}{24} \, {\left (2 \, x^{9} - 3 \, x^{3}\right )} \sqrt {x^{6} + 1} - \frac {1}{8} \, \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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{14}}{\sqrt {x^{6} + 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 27, normalized size = 0.63
method | result | size |
risch | \(\frac {x^{3} \left (2 x^{6}-3\right ) \sqrt {x^{6}+1}}{24}+\frac {\arcsinh \left (x^{3}\right )}{8}\) | \(27\) |
trager | \(\frac {x^{3} \left (2 x^{6}-3\right ) \sqrt {x^{6}+1}}{24}-\frac {\ln \left (x^{3}-\sqrt {x^{6}+1}\right )}{8}\) | \(37\) |
meijerg | \(\frac {-\frac {\sqrt {\pi }\, x^{3} \left (-10 x^{6}+15\right ) \sqrt {x^{6}+1}}{20}+\frac {3 \sqrt {\pi }\, \arcsinh \left (x^{3}\right )}{4}}{6 \sqrt {\pi }}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 86, normalized size = 2.00 \begin {gather*} -\frac {\frac {5 \, \sqrt {x^{6} + 1}}{x^{3}} - \frac {3 \, {\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}{16} \, \log \left (\frac {\sqrt {x^{6} + 1}}{x^{3}} + 1\right ) - \frac {1}{16} \, \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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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^{14}}{\sqrt {x^6+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.74, size = 46, normalized size = 1.07 \begin {gather*} \frac {x^{15}}{12 \sqrt {x^{6} + 1}} - \frac {x^{9}}{24 \sqrt {x^{6} + 1}} - \frac {x^{3}}{8 \sqrt {x^{6} + 1}} + \frac {\operatorname {asinh}{\left (x^{3} \right )}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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