Optimal. Leaf size=20 \[ -\frac {1}{8} \tanh ^{-1}\left (\frac {\sqrt {16-x^4}}{4}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {272, 65, 212}
\begin {gather*} -\frac {1}{8} \tanh ^{-1}\left (\frac {\sqrt {16-x^4}}{4}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 212
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {16-x^4}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {1}{\sqrt {16-x} x} \, dx,x,x^4\right )\\ &=-\left (\frac {1}{2} \text {Subst}\left (\int \frac {1}{16-x^2} \, dx,x,\sqrt {16-x^4}\right )\right )\\ &=-\frac {1}{8} \tanh ^{-1}\left (\frac {\sqrt {16-x^4}}{4}\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 20, normalized size = 1.00 \begin {gather*} -\frac {1}{8} \tanh ^{-1}\left (\frac {\sqrt {16-x^4}}{4}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 15, normalized size = 0.75
method | result | size |
default | \(-\frac {\arctanh \left (\frac {4}{\sqrt {-x^{4}+16}}\right )}{8}\) | \(15\) |
elliptic | \(-\frac {\arctanh \left (\frac {4}{\sqrt {-x^{4}+16}}\right )}{8}\) | \(15\) |
trager | \(-\frac {\ln \left (\frac {\sqrt {-x^{4}+16}+4}{x^{2}}\right )}{8}\) | \(19\) |
meijerg | \(\frac {-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1-\frac {x^{4}}{16}}}{2}\right )+\left (-6 \ln \left (2\right )+4 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{16 \sqrt {\pi }}\) | \(43\) |
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. 29 vs.
\(2 (14) = 28\).
time = 0.30, size = 29, normalized size = 1.45 \begin {gather*} -\frac {1}{16} \, \log \left (\sqrt {-x^{4} + 16} + 4\right ) + \frac {1}{16} \, \log \left (\sqrt {-x^{4} + 16} - 4\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. 29 vs.
\(2 (14) = 28\).
time = 0.36, size = 29, normalized size = 1.45 \begin {gather*} -\frac {1}{16} \, \log \left (\sqrt {-x^{4} + 16} + 4\right ) + \frac {1}{16} \, \log \left (\sqrt {-x^{4} + 16} - 4\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 = 0.44, size = 26, normalized size = 1.30 \begin {gather*} \begin {cases} - \frac {\operatorname {acosh}{\left (\frac {4}{x^{2}} \right )}}{8} & \text {for}\: \frac {1}{\left |{x^{4}}\right |} > \frac {1}{16} \\\frac {i \operatorname {asin}{\left (\frac {4}{x^{2}} \right )}}{8} & \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. 31 vs.
\(2 (14) = 28\).
time = 1.90, size = 31, normalized size = 1.55 \begin {gather*} -\frac {1}{16} \, \log \left (\sqrt {-x^{4} + 16} + 4\right ) + \frac {1}{16} \, \log \left (-\sqrt {-x^{4} + 16} + 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.28, size = 14, normalized size = 0.70 \begin {gather*} -\frac {\mathrm {atanh}\left (\frac {\sqrt {16-x^4}}{4}\right )}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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