Optimal. Leaf size=20 \[ -\frac {1}{3} \text {ArcTan}\left (\frac {x}{\sqrt {-x^2+x^8}}\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2033, 209}
\begin {gather*} -\frac {1}{3} \text {ArcTan}\left (\frac {x}{\sqrt {x^8-x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 2033
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-x^2+x^8}} \, dx &=-\left (\frac {1}{3} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt {-x^2+x^8}}\right )\right )\\ &=-\frac {1}{3} \tan ^{-1}\left (\frac {x}{\sqrt {-x^2+x^8}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 37, normalized size = 1.85 \begin {gather*} \frac {x \sqrt {-1+x^6} \text {ArcTan}\left (\sqrt {-1+x^6}\right )}{3 \sqrt {x^2 \left (-1+x^6\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.74, size = 26, normalized size = 1.30
method | result | size |
default | \(-\frac {x \sqrt {x^{6}-1}\, \arcsin \left (\frac {1}{x^{3}}\right )}{3 \sqrt {x^{8}-x^{2}}}\) | \(26\) |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {x \RootOf \left (\textit {\_Z}^{2}+1\right )-\sqrt {x^{8}-x^{2}}}{x^{4}}\right )}{3}\) | \(37\) |
meijerg | \(\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )+\left (-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }\right )}{6 \sqrt {\pi }\, \sqrt {\mathrm {signum}\left (x^{6}-1\right )}}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 18, normalized size = 0.90 \begin {gather*} \frac {1}{3} \, \arctan \left (\frac {\sqrt {x^{8} - x^{2}}}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {x^{8} - x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \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 = -0.05 \begin {gather*} \int \frac {1}{\sqrt {x^8-x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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