Optimal. Leaf size=25 \[ -\frac {x^2}{2 \sqrt {1+x^4}}+\frac {1}{2} \sinh ^{-1}\left (x^2\right ) \]
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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, 294, 221}
\begin {gather*} \frac {1}{2} \sinh ^{-1}\left (x^2\right )-\frac {x^2}{2 \sqrt {x^4+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 281
Rule 294
Rubi steps
\begin {align*} \int \frac {x^5}{\left (1+x^4\right )^{3/2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{\left (1+x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=-\frac {x^2}{2 \sqrt {1+x^4}}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^2\right )\\ &=-\frac {x^2}{2 \sqrt {1+x^4}}+\frac {1}{2} \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 {x^2}{2 \sqrt {1+x^4}}+\frac {1}{2} \tanh ^{-1}\left (\frac {x^2}{\sqrt {1+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 20, normalized size = 0.80
method | result | size |
default | \(\frac {\arcsinh \left (x^{2}\right )}{2}-\frac {x^{2}}{2 \sqrt {x^{4}+1}}\) | \(20\) |
risch | \(\frac {\arcsinh \left (x^{2}\right )}{2}-\frac {x^{2}}{2 \sqrt {x^{4}+1}}\) | \(20\) |
elliptic | \(\frac {\arcsinh \left (x^{2}\right )}{2}-\frac {x^{2}}{2 \sqrt {x^{4}+1}}\) | \(20\) |
trager | \(-\frac {x^{2}}{2 \sqrt {x^{4}+1}}-\frac {\ln \left (x^{2}-\sqrt {x^{4}+1}\right )}{2}\) | \(30\) |
meijerg | \(\frac {-\frac {\sqrt {\pi }\, x^{2}}{\sqrt {x^{4}+1}}+\sqrt {\pi }\, \arcsinh \left (x^{2}\right )}{2 \sqrt {\pi }}\) | \(30\) |
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. 45 vs.
\(2 (19) = 38\).
time = 0.29, size = 45, normalized size = 1.80 \begin {gather*} -\frac {x^{2}}{2 \, \sqrt {x^{4} + 1}} + \frac {1}{4} \, \log \left (\frac {\sqrt {x^{4} + 1}}{x^{2}} + 1\right ) - \frac {1}{4} \, \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 45 vs.
\(2 (19) = 38\).
time = 0.34, size = 45, normalized size = 1.80 \begin {gather*} -\frac {x^{4} + \sqrt {x^{4} + 1} x^{2} + {\left (x^{4} + 1\right )} \log \left (-x^{2} + \sqrt {x^{4} + 1}\right ) + 1}{2 \, {\left (x^{4} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.64, size = 19, normalized size = 0.76 \begin {gather*} - \frac {x^{2}}{2 \sqrt {x^{4} + 1}} + \frac {\operatorname {asinh}{\left (x^{2} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.12, size = 29, normalized size = 1.16 \begin {gather*} -\frac {x^{2}}{2 \, \sqrt {x^{4} + 1}} - \frac {1}{2} \, \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}{{\left (x^4+1\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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