Optimal. Leaf size=23 \[ -\frac{\tanh ^{-1}\left (\frac{2 x^4+3}{\sqrt{5}}\right )}{2 \sqrt{5}} \]
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Rubi [A] time = 0.0256436, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {1352, 618, 206} \[ -\frac{\tanh ^{-1}\left (\frac{2 x^4+3}{\sqrt{5}}\right )}{2 \sqrt{5}} \]
Antiderivative was successfully verified.
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Rule 1352
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{x^3}{1+3 x^4+x^8} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{1+3 x+x^2} \, dx,x,x^4\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{5-x^2} \, dx,x,3+2 x^4\right )\right )\\ &=-\frac{\tanh ^{-1}\left (\frac{3+2 x^4}{\sqrt{5}}\right )}{2 \sqrt{5}}\\ \end{align*}
Mathematica [A] time = 0.0100449, size = 38, normalized size = 1.65 \[ \frac{\log \left (-2 x^4+\sqrt{5}-3\right )-\log \left (2 x^4+\sqrt{5}+3\right )}{4 \sqrt{5}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 19, normalized size = 0.8 \begin{align*} -{\frac{\sqrt{5}}{10}{\it Artanh} \left ({\frac{ \left ( 2\,{x}^{4}+3 \right ) \sqrt{5}}{5}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49359, size = 42, normalized size = 1.83 \begin{align*} \frac{1}{20} \, \sqrt{5} \log \left (\frac{2 \, x^{4} - \sqrt{5} + 3}{2 \, x^{4} + \sqrt{5} + 3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.45554, size = 107, normalized size = 4.65 \begin{align*} \frac{1}{20} \, \sqrt{5} \log \left (\frac{2 \, x^{8} + 6 \, x^{4} - \sqrt{5}{\left (2 \, x^{4} + 3\right )} + 7}{x^{8} + 3 \, x^{4} + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.121867, size = 42, normalized size = 1.83 \begin{align*} \frac{\sqrt{5} \log{\left (x^{4} - \frac{\sqrt{5}}{2} + \frac{3}{2} \right )}}{20} - \frac{\sqrt{5} \log{\left (x^{4} + \frac{\sqrt{5}}{2} + \frac{3}{2} \right )}}{20} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29294, size = 42, normalized size = 1.83 \begin{align*} \frac{1}{20} \, \sqrt{5} \log \left (\frac{2 \, x^{4} - \sqrt{5} + 3}{2 \, x^{4} + \sqrt{5} + 3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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