Optimal. Leaf size=29 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a x^4-b}}{\sqrt {b}}\right )}{2 \sqrt {b}} \]
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Rubi [A] time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {266, 63, 205} \begin {gather*} \frac {\tan ^{-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 205
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 {\tan ^{-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 = 29, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-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 = 0.04, size = 29, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {a x^4-b}}{\sqrt {b}}\right )}{2 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 64, normalized size = 2.21 \begin {gather*} \left [-\frac {\sqrt {-b} \log \left (\frac {a x^{4} - 2 \, \sqrt {a x^{4} - b} \sqrt {-b} - 2 \, b}{x^{4}}\right )}{4 \, b}, \frac {\arctan \left (\frac {\sqrt {a x^{4} - b}}{\sqrt {b}}\right )}{2 \, \sqrt {b}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 21, normalized size = 0.72 \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 = 0.01, size = 35, normalized size = 1.21 \begin {gather*} -\frac {\ln \left (\frac {-2 b +2 \sqrt {-b}\, \sqrt {a \,x^{4}-b}}{x^{2}}\right )}{2 \sqrt {-b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 21, normalized size = 0.72 \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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mupad [B] time = 0.38, size = 21, normalized size = 0.72 \begin {gather*} \frac {\mathrm {atan}\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 = 0.93, size = 53, normalized size = 1.83 \begin {gather*} \begin {cases} \frac {i \operatorname {acosh}{\left (\frac {\sqrt {b}}{\sqrt {a} x^{2}} \right )}}{2 \sqrt {b}} & \text {for}\: \left |{\frac {b}{a x^{4}}}\right | > 1 \\- \frac {\operatorname {asin}{\left (\frac {\sqrt {b}}{\sqrt {a} x^{2}} \right )}}{2 \sqrt {b}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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