Optimal. Leaf size=25 \[ \frac{1}{8} \sqrt{x^8+1} x^4+\frac{1}{8} \sinh ^{-1}\left (x^4\right ) \]
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Rubi [A] time = 0.0076866, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {275, 195, 215} \[ \frac{1}{8} \sqrt{x^8+1} x^4+\frac{1}{8} \sinh ^{-1}\left (x^4\right ) \]
Antiderivative was successfully verified.
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Rule 275
Rule 195
Rule 215
Rubi steps
\begin{align*} \int x^3 \sqrt{1+x^8} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \sqrt{1+x^2} \, dx,x,x^4\right )\\ &=\frac{1}{8} x^4 \sqrt{1+x^8}+\frac{1}{8} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,x^4\right )\\ &=\frac{1}{8} x^4 \sqrt{1+x^8}+\frac{1}{8} \sinh ^{-1}\left (x^4\right )\\ \end{align*}
Mathematica [A] time = 0.006428, size = 22, normalized size = 0.88 \[ \frac{1}{8} \left (\sqrt{x^8+1} x^4+\sinh ^{-1}\left (x^4\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 20, normalized size = 0.8 \begin{align*}{\frac{{\it Arcsinh} \left ({x}^{4} \right ) }{8}}+{\frac{{x}^{4}}{8}\sqrt{{x}^{8}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.969173, size = 78, normalized size = 3.12 \begin{align*} \frac{\sqrt{x^{8} + 1}}{8 \, x^{4}{\left (\frac{x^{8} + 1}{x^{8}} - 1\right )}} + \frac{1}{16} \, \log \left (\frac{\sqrt{x^{8} + 1}}{x^{4}} + 1\right ) - \frac{1}{16} \, \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 [A] time = 1.23788, size = 74, normalized size = 2.96 \begin{align*} \frac{1}{8} \, \sqrt{x^{8} + 1} x^{4} - \frac{1}{8} \, \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 = 1.45465, size = 19, normalized size = 0.76 \begin{align*} \frac{x^{4} \sqrt{x^{8} + 1}}{8} + \frac{\operatorname{asinh}{\left (x^{4} \right )}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19585, size = 39, normalized size = 1.56 \begin{align*} \frac{1}{8} \, \sqrt{x^{8} + 1} x^{4} - \frac{1}{8} \, \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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