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 = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, 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 215
Rule 275
Rule 321
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*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.00 \[ \frac {1}{4} x^2 \sqrt {x^4+1}-\frac {1}{4} \sinh ^{-1}\left (x^2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 29, normalized size = 1.16 \[ \frac {1}{4} \, \sqrt {x^{4} + 1} x^{2} + \frac {1}{4} \, \log \left (-x^{2} + \sqrt {x^{4} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 29, normalized size = 1.16 \[ \frac {1}{4} \, \sqrt {x^{4} + 1} x^{2} + \frac {1}{4} \, \log \left (-x^{2} + \sqrt {x^{4} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 20, normalized size = 0.80 \[ \frac {\sqrt {x^{4}+1}\, x^{2}}{4}-\frac {\arcsinh \left (x^{2}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.42, size = 58, normalized size = 2.32 \[ \frac {\sqrt {x^{4} + 1}}{4 \, x^{2} {\left (\frac {x^{4} + 1}{x^{4}} - 1\right )}} - \frac {1}{8} \, \log \left (\frac {\sqrt {x^{4} + 1}}{x^{2}} + 1\right ) + \frac {1}{8} \, \log \left (\frac {\sqrt {x^{4} + 1}}{x^{2}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {x^5}{\sqrt {x^4+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.12, size = 19, normalized size = 0.76 \[ \frac {x^{2} \sqrt {x^{4} + 1}}{4} - \frac {\operatorname {asinh}{\left (x^{2} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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