Optimal. Leaf size=27 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a x^4+b}}{\sqrt {b}}\right )}{2 \sqrt {b}} \]
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Rubi [A] time = 0.02, antiderivative size = 27, 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 = {266, 63, 208} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {a x^4+b}}{\sqrt {b}}\right )}{2 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {b+a x^4}} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {b+a x}} \, dx,x,x^4\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {b}{a}+\frac {x^2}{a}} \, dx,x,\sqrt {b+a x^4}\right )}{2 a}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {b+a x^4}}{\sqrt {b}}\right )}{2 \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 1.00 \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {a x^4+b}}{\sqrt {b}}\right )}{2 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 27, normalized size = 1.00 \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {b+a x^4}}{\sqrt {b}}\right )}{2 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 63, normalized size = 2.33 \begin {gather*} \left [\frac {\log \left (\frac {a x^{4} - 2 \, \sqrt {a x^{4} + b} \sqrt {b} + 2 \, b}{x^{4}}\right )}{4 \, \sqrt {b}}, \frac {\sqrt {-b} \arctan \left (\frac {\sqrt {a x^{4} + b} \sqrt {-b}}{b}\right )}{2 \, b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 23, normalized size = 0.85 \begin {gather*} \frac {\arctan \left (\frac {\sqrt {a x^{4} + b}}{\sqrt {-b}}\right )}{2 \, \sqrt {-b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 29, normalized size = 1.07
method | result | size |
default | \(-\frac {\ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {a \,x^{4}+b}}{x^{2}}\right )}{2 \sqrt {b}}\) | \(29\) |
elliptic | \(-\frac {\ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {a \,x^{4}+b}}{x^{2}}\right )}{2 \sqrt {b}}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.80, size = 37, normalized size = 1.37 \begin {gather*} \frac {\log \left (\frac {\sqrt {a x^{4} + b} - \sqrt {b}}{\sqrt {a x^{4} + b} + \sqrt {b}}\right )}{4 \, \sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.39, size = 19, normalized size = 0.70 \begin {gather*} -\frac {\mathrm {atanh}\left (\frac {\sqrt {a\,x^4+b}}{\sqrt {b}}\right )}{2\,\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.87, size = 22, normalized size = 0.81 \begin {gather*} - \frac {\operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} x^{2}} \right )}}{2 \sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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