Optimal. Leaf size=25 \[ \frac{1}{4} x^2 \sqrt{x^4+1}-\frac{1}{4} \sinh ^{-1}\left (x^2\right ) \]
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Rubi [A] time = 0.0087406, 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, 321, 215} \[ \frac{1}{4} x^2 \sqrt{x^4+1}-\frac{1}{4} \sinh ^{-1}\left (x^2\right ) \]
Antiderivative was successfully verified.
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Rule 275
Rule 321
Rule 215
Rubi steps
\begin{align*} \int \frac{x^5}{\sqrt{1+x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+x^2}} \, dx,x,x^2\right )\\ &=\frac{1}{4} x^2 \sqrt{1+x^4}-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,x^2\right )\\ &=\frac{1}{4} x^2 \sqrt{1+x^4}-\frac{1}{4} \sinh ^{-1}\left (x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0048921, size = 25, normalized size = 1. \[ \frac{1}{4} x^2 \sqrt{x^4+1}-\frac{1}{4} \sinh ^{-1}\left (x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.123, size = 20, normalized size = 0.8 \begin{align*} -{\frac{{\it Arcsinh} \left ({x}^{2} \right ) }{4}}+{\frac{{x}^{2}}{4}\sqrt{{x}^{4}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.985439, size = 78, normalized size = 3.12 \begin{align*} \frac{\sqrt{x^{4} + 1}}{4 \, x^{2}{\left (\frac{x^{4} + 1}{x^{4}} - 1\right )}} - \frac{1}{8} \, \log \left (\frac{\sqrt{x^{4} + 1}}{x^{2}} + 1\right ) + \frac{1}{8} \, \log \left (\frac{\sqrt{x^{4} + 1}}{x^{2}} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48901, size = 74, normalized size = 2.96 \begin{align*} \frac{1}{4} \, \sqrt{x^{4} + 1} x^{2} + \frac{1}{4} \, \log \left (-x^{2} + \sqrt{x^{4} + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.32869, size = 19, normalized size = 0.76 \begin{align*} \frac{x^{2} \sqrt{x^{4} + 1}}{4} - \frac{\operatorname{asinh}{\left (x^{2} \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15236, size = 39, normalized size = 1.56 \begin{align*} \frac{1}{4} \, \sqrt{x^{4} + 1} x^{2} + \frac{1}{4} \, \log \left (-x^{2} + \sqrt{x^{4} + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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