Optimal. Leaf size=8 \[ \frac{1}{4} \sinh ^{-1}\left (x^4\right ) \]
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Rubi [A] time = 0.0033625, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {275, 215} \[ \frac{1}{4} \sinh ^{-1}\left (x^4\right ) \]
Antiderivative was successfully verified.
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Rule 275
Rule 215
Rubi steps
\begin{align*} \int \frac{x^3}{\sqrt{1+x^8}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,x^4\right )\\ &=\frac{1}{4} \sinh ^{-1}\left (x^4\right )\\ \end{align*}
Mathematica [A] time = 0.0020947, size = 8, normalized size = 1. \[ \frac{1}{4} \sinh ^{-1}\left (x^4\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 7, normalized size = 0.9 \begin{align*}{\frac{{\it Arcsinh} \left ({x}^{4} \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.959081, size = 45, normalized size = 5.62 \begin{align*} \frac{1}{8} \, \log \left (\frac{\sqrt{x^{8} + 1}}{x^{4}} + 1\right ) - \frac{1}{8} \, \log \left (\frac{\sqrt{x^{8} + 1}}{x^{4}} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.26535, size = 43, normalized size = 5.38 \begin{align*} -\frac{1}{4} \, \log \left (-x^{4} + \sqrt{x^{8} + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.93421, size = 5, normalized size = 0.62 \begin{align*} \frac{\operatorname{asinh}{\left (x^{4} \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16165, size = 22, normalized size = 2.75 \begin{align*} -\frac{1}{4} \, \log \left (-x^{4} + \sqrt{x^{8} + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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