Optimal. Leaf size=18 \[ \frac {1}{5} \tanh ^{-1}\left (\frac {x^5}{\sqrt {-2+x^{10}}}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 18, 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, 223, 212}
\begin {gather*} \frac {1}{5} \tanh ^{-1}\left (\frac {x^5}{\sqrt {x^{10}-2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 281
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt {-2+x^{10}}} \, dx &=\frac {1}{5} \text {Subst}\left (\int \frac {1}{\sqrt {-2+x^2}} \, dx,x,x^5\right )\\ &=\frac {1}{5} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^5}{\sqrt {-2+x^{10}}}\right )\\ &=\frac {1}{5} \tanh ^{-1}\left (\frac {x^5}{\sqrt {-2+x^{10}}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 18, normalized size = 1.00 \begin {gather*} \frac {1}{5} \tanh ^{-1}\left (\frac {\sqrt {-2+x^{10}}}{x^5}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 15, normalized size = 0.83
method | result | size |
trager | \(\frac {\ln \left (x^{5}+\sqrt {x^{10}-2}\right )}{5}\) | \(15\) |
meijerg | \(\frac {\sqrt {-\mathrm {signum}\left (-1+\frac {x^{10}}{2}\right )}\, \arcsin \left (\frac {x^{5} \sqrt {2}}{2}\right )}{5 \sqrt {\mathrm {signum}\left (-1+\frac {x^{10}}{2}\right )}}\) | \(34\) |
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. 33 vs.
\(2 (14) = 28\).
time = 0.29, size = 33, normalized size = 1.83 \begin {gather*} \frac {1}{10} \, \log \left (\frac {\sqrt {x^{10} - 2}}{x^{5}} + 1\right ) - \frac {1}{10} \, \log \left (\frac {\sqrt {x^{10} - 2}}{x^{5}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 16, normalized size = 0.89 \begin {gather*} -\frac {1}{5} \, \log \left (-x^{5} + \sqrt {x^{10} - 2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.45, size = 32, normalized size = 1.78 \begin {gather*} \begin {cases} \frac {\operatorname {acosh}{\left (\frac {\sqrt {2} x^{5}}{2} \right )}}{5} & \text {for}\: \left |{x^{10}}\right | > 2 \\- \frac {i \operatorname {asin}{\left (\frac {\sqrt {2} x^{5}}{2} \right )}}{5} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 30 vs.
\(2 (14) = 28\).
time = 1.45, size = 30, normalized size = 1.67 \begin {gather*} \frac {1}{10} \, \sqrt {x^{10} - 2} x^{5} + \frac {1}{5} \, \log \left ({\left | -x^{5} + \sqrt {x^{10} - 2} \right |}\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 = -0.06 \begin {gather*} \int \frac {x^4}{\sqrt {x^{10}-2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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