Optimal. Leaf size=34 \[ -\frac {\tanh ^{-1}\left (\frac {2 x^4+1}{\sqrt {3} \sqrt {2 x^8+1}}\right )}{4 \sqrt {3}} \]
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Rubi [A] time = 0.03, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1469, 725, 206} \[ -\frac {\tanh ^{-1}\left (\frac {2 x^4+1}{\sqrt {3} \sqrt {2 x^8+1}}\right )}{4 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 725
Rule 1469
Rubi steps
\begin {align*} \int \frac {x^3}{\left (-1+x^4\right ) \sqrt {1+2 x^8}} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {1+2 x^2}} \, dx,x,x^4\right )\\ &=-\left (\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{3-x^2} \, dx,x,\frac {1+2 x^4}{\sqrt {1+2 x^8}}\right )\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {1+2 x^4}{\sqrt {3} \sqrt {1+2 x^8}}\right )}{4 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 0.85 \[ -\frac {\tanh ^{-1}\left (\frac {2 x^4+1}{\sqrt {6 x^8+3}}\right )}{4 \sqrt {3}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.42, size = 48, normalized size = 1.41 \[ \frac {\tanh ^{-1}\left (-\frac {\sqrt {2 x^8+1}}{\sqrt {3}}-\sqrt {\frac {2}{3}} x^4+\sqrt {\frac {2}{3}}\right )}{2 \sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 49, normalized size = 1.44 \[ \frac {1}{12} \, \sqrt {3} \log \left (\frac {2 \, x^{4} - \sqrt {3} {\left (2 \, x^{4} + 1\right )} - \sqrt {2 \, x^{8} + 1} {\left (\sqrt {3} - 3\right )} + 1}{x^{4} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.66, size = 70, normalized size = 2.06 \[ \frac {1}{12} \, \sqrt {3} \log \left (-\frac {{\left | -2 \, \sqrt {2} x^{4} - 2 \, \sqrt {3} + 2 \, \sqrt {2} + 2 \, \sqrt {2 \, x^{8} + 1} \right |}}{2 \, {\left (\sqrt {2} x^{4} - \sqrt {3} - \sqrt {2} - \sqrt {2 \, x^{8} + 1}\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.30, size = 58, normalized size = 1.71
method | result | size |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{4}+\RootOf \left (\textit {\_Z}^{2}-3\right )+3 \sqrt {2 x^{8}+1}}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{12}\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\sqrt {2 \, x^{8} + 1} {\left (x^{4} - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.52, size = 35, normalized size = 1.03 \[ -\frac {\sqrt {3}\,\left (\ln \left (x^4+\frac {\sqrt {2}\,\sqrt {3}\,\sqrt {x^8+\frac {1}{2}}}{2}+\frac {1}{2}\right )-\ln \left (x^4-1\right )\right )}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \sqrt {2 x^{8} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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