Optimal. Leaf size=49 \[ \frac{4 x^2}{3 \sqrt{x^4+1}}+\frac{2}{3 \sqrt{x^4+1} x^2}-\frac{1}{6 \sqrt{x^4+1} x^6} \]
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Rubi [A] time = 0.010478, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {271, 264} \[ \frac{4 x^2}{3 \sqrt{x^4+1}}+\frac{2}{3 \sqrt{x^4+1} x^2}-\frac{1}{6 \sqrt{x^4+1} x^6} \]
Antiderivative was successfully verified.
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Rule 271
Rule 264
Rubi steps
\begin{align*} \int \frac{1}{x^7 \left (1+x^4\right )^{3/2}} \, dx &=-\frac{1}{6 x^6 \sqrt{1+x^4}}-\frac{4}{3} \int \frac{1}{x^3 \left (1+x^4\right )^{3/2}} \, dx\\ &=-\frac{1}{6 x^6 \sqrt{1+x^4}}+\frac{2}{3 x^2 \sqrt{1+x^4}}+\frac{8}{3} \int \frac{x}{\left (1+x^4\right )^{3/2}} \, dx\\ &=-\frac{1}{6 x^6 \sqrt{1+x^4}}+\frac{2}{3 x^2 \sqrt{1+x^4}}+\frac{4 x^2}{3 \sqrt{1+x^4}}\\ \end{align*}
Mathematica [A] time = 0.0048583, size = 28, normalized size = 0.57 \[ -\frac{-8 x^8-4 x^4+1}{6 x^6 \sqrt{x^4+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 25, normalized size = 0.5 \begin{align*}{\frac{8\,{x}^{8}+4\,{x}^{4}-1}{6\,{x}^{6}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.970691, size = 49, normalized size = 1. \begin{align*} \frac{x^{2}}{2 \, \sqrt{x^{4} + 1}} + \frac{\sqrt{x^{4} + 1}}{x^{2}} - \frac{{\left (x^{4} + 1\right )}^{\frac{3}{2}}}{6 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72803, size = 96, normalized size = 1.96 \begin{align*} \frac{8 \, x^{10} + 8 \, x^{6} +{\left (8 \, x^{8} + 4 \, x^{4} - 1\right )} \sqrt{x^{4} + 1}}{6 \,{\left (x^{10} + x^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.66169, size = 70, normalized size = 1.43 \begin{align*} \frac{8 x^{8} \sqrt{1 + \frac{1}{x^{4}}}}{6 x^{8} + 6 x^{4}} + \frac{4 x^{4} \sqrt{1 + \frac{1}{x^{4}}}}{6 x^{8} + 6 x^{4}} - \frac{\sqrt{1 + \frac{1}{x^{4}}}}{6 x^{8} + 6 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22252, size = 39, normalized size = 0.8 \begin{align*} \frac{x^{2}}{2 \, \sqrt{x^{4} + 1}} - \frac{1}{6} \,{\left (\frac{1}{x^{4}} + 1\right )}^{\frac{3}{2}} + \sqrt{\frac{1}{x^{4}} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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