Optimal. Leaf size=14 \[ \frac {1}{2} \tan ^{-1}\left (\sqrt {-1+x^4}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 14, 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 = {272, 65, 209}
\begin {gather*} \frac {1}{2} \text {ArcTan}\left (\sqrt {x^4-1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 209
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {-1+x^4}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^4\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^4}\right )\\ &=\frac {1}{2} \tan ^{-1}\left (\sqrt {-1+x^4}\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 14, normalized size = 1.00 \begin {gather*} \frac {1}{2} \tan ^{-1}\left (\sqrt {-1+x^4}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.31, size = 11, normalized size = 0.79
method | result | size |
default | \(-\frac {\arctan \left (\frac {1}{\sqrt {x^{4}-1}}\right )}{2}\) | \(11\) |
elliptic | \(-\frac {\arctan \left (\frac {1}{\sqrt {x^{4}-1}}\right )}{2}\) | \(11\) |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{4}-1}}{x^{2}}\right )}{2}\) | \(28\) |
meijerg | \(\frac {\sqrt {-\mathrm {signum}\left (x^{4}-1\right )}\, \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{4}+1}}{2}\right )+\left (-2 \ln \left (2\right )+4 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }\right )}{4 \sqrt {\pi }\, \sqrt {\mathrm {signum}\left (x^{4}-1\right )}}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 10, normalized size = 0.71 \begin {gather*} \frac {1}{2} \, \arctan \left (\sqrt {x^{4} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 10, normalized size = 0.71 \begin {gather*} \frac {1}{2} \, \arctan \left (\sqrt {x^{4} - 1}\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 = 0.41, size = 24, normalized size = 1.71 \begin {gather*} \begin {cases} \frac {i \operatorname {acosh}{\left (\frac {1}{x^{2}} \right )}}{2} & \text {for}\: \frac {1}{\left |{x^{4}}\right |} > 1 \\- \frac {\operatorname {asin}{\left (\frac {1}{x^{2}} \right )}}{2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.69, size = 10, normalized size = 0.71 \begin {gather*} \frac {1}{2} \, \arctan \left (\sqrt {x^{4} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.15, size = 10, normalized size = 0.71 \begin {gather*} \frac {\mathrm {atan}\left (\sqrt {x^4-1}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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