Optimal. Leaf size=25 \[ \frac {1}{4} x^2 \sqrt {1+x^4}-\frac {1}{4} \sinh ^{-1}\left (x^2\right ) \]
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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 = {281, 327, 221}
\begin {gather*} \frac {1}{4} x^2 \sqrt {x^4+1}-\frac {1}{4} \sinh ^{-1}\left (x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 281
Rule 327
Rubi steps
\begin {align*} \int \frac {x^5}{\sqrt {1+x^4}} \, dx &=\frac {1}{2} \text {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} \text {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 = 0.10, size = 35, normalized size = 1.40 \begin {gather*} \frac {1}{4} x^2 \sqrt {1+x^4}-\frac {1}{4} \tanh ^{-1}\left (\frac {x^2}{\sqrt {1+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 20, normalized size = 0.80
method | result | size |
default | \(-\frac {\arcsinh \left (x^{2}\right )}{4}+\frac {x^{2} \sqrt {x^{4}+1}}{4}\) | \(20\) |
risch | \(-\frac {\arcsinh \left (x^{2}\right )}{4}+\frac {x^{2} \sqrt {x^{4}+1}}{4}\) | \(20\) |
elliptic | \(-\frac {\arcsinh \left (x^{2}\right )}{4}+\frac {x^{2} \sqrt {x^{4}+1}}{4}\) | \(20\) |
trager | \(\frac {x^{2} \sqrt {x^{4}+1}}{4}-\frac {\ln \left (x^{2}+\sqrt {x^{4}+1}\right )}{4}\) | \(28\) |
meijerg | \(\frac {\sqrt {\pi }\, x^{2} \sqrt {x^{4}+1}-\sqrt {\pi }\, \arcsinh \left (x^{2}\right )}{4 \sqrt {\pi }}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 58 vs.
\(2 (19) = 38\).
time = 0.29, size = 58, normalized size = 2.32 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 29, normalized size = 1.16 \begin {gather*} \frac {1}{4} \, \sqrt {x^{4} + 1} x^{2} + \frac {1}{4} \, \log \left (-x^{2} + \sqrt {x^{4} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.88, size = 19, normalized size = 0.76 \begin {gather*} \frac {x^{2} \sqrt {x^{4} + 1}}{4} - \frac {\operatorname {asinh}{\left (x^{2} \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.15, size = 29, normalized size = 1.16 \begin {gather*} \frac {1}{4} \, \sqrt {x^{4} + 1} x^{2} + \frac {1}{4} \, \log \left (-x^{2} + \sqrt {x^{4} + 1}\right ) \end {gather*}
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 \begin {gather*} \int \frac {x^5}{\sqrt {x^4+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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