Optimal. Leaf size=35 \[ -\frac {\sqrt {1-x^4}}{4 x^4}-\frac {1}{4} \tanh ^{-1}\left (\sqrt {1-x^4}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {266, 51, 63, 206} \[ -\frac {\sqrt {1-x^4}}{4 x^4}-\frac {1}{4} \tanh ^{-1}\left (\sqrt {1-x^4}\right ) \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 206
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x^5 \sqrt {1-x^4}} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x^2} \, dx,x,x^4\right )\\ &=-\frac {\sqrt {1-x^4}}{4 x^4}+\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,x^4\right )\\ &=-\frac {\sqrt {1-x^4}}{4 x^4}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-x^4}\right )\\ &=-\frac {\sqrt {1-x^4}}{4 x^4}-\frac {1}{4} \tanh ^{-1}\left (\sqrt {1-x^4}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 35, normalized size = 1.00 \[ -\frac {\sqrt {1-x^4}}{4 x^4}-\frac {1}{4} \tanh ^{-1}\left (\sqrt {1-x^4}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 50, normalized size = 1.43 \[ -\frac {x^{4} \log \left (\sqrt {-x^{4} + 1} + 1\right ) - x^{4} \log \left (\sqrt {-x^{4} + 1} - 1\right ) + 2 \, \sqrt {-x^{4} + 1}}{8 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 45, normalized size = 1.29 \[ -\frac {\sqrt {-x^{4} + 1}}{4 \, x^{4}} - \frac {1}{8} \, \log \left (\sqrt {-x^{4} + 1} + 1\right ) + \frac {1}{8} \, \log \left (-\sqrt {-x^{4} + 1} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 28, normalized size = 0.80 \[ -\frac {\arctanh \left (\frac {1}{\sqrt {-x^{4}+1}}\right )}{4}-\frac {\sqrt {-x^{4}+1}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 43, normalized size = 1.23 \[ -\frac {\sqrt {-x^{4} + 1}}{4 \, x^{4}} - \frac {1}{8} \, \log \left (\sqrt {-x^{4} + 1} + 1\right ) + \frac {1}{8} \, \log \left (\sqrt {-x^{4} + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 27, normalized size = 0.77 \[ -\frac {\mathrm {atanh}\left (\sqrt {1-x^4}\right )}{4}-\frac {\sqrt {1-x^4}}{4\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.20, size = 73, normalized size = 2.09 \[ \begin {cases} - \frac {\operatorname {acosh}{\left (\frac {1}{x^{2}} \right )}}{4} - \frac {\sqrt {-1 + \frac {1}{x^{4}}}}{4 x^{2}} & \text {for}\: \frac {1}{\left |{x^{4}}\right |} > 1 \\\frac {i \operatorname {asin}{\left (\frac {1}{x^{2}} \right )}}{4} - \frac {i}{4 x^{2} \sqrt {1 - \frac {1}{x^{4}}}} + \frac {i}{4 x^{6} \sqrt {1 - \frac {1}{x^{4}}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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