Optimal. Leaf size=31 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}\right )}{2 \sqrt {b}} \]
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Rubi [A] time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {275, 217, 203} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}\right )}{2 \sqrt {b}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 217
Rule 275
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {a-b x^4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {x^2}{\sqrt {a-b x^4}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}\right )}{2 \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 31, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}\right )}{2 \sqrt {b}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 76, normalized size = 2.45 \[ \left [-\frac {\sqrt {-b} \log \left (2 \, b x^{4} - 2 \, \sqrt {-b x^{4} + a} \sqrt {-b} x^{2} - a\right )}{4 \, b}, -\frac {\arctan \left (\frac {\sqrt {-b x^{4} + a} \sqrt {b} x^{2}}{b x^{4} - a}\right )}{2 \, \sqrt {b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 30, normalized size = 0.97 \[ -\frac {\log \left ({\left | -\sqrt {-b} x^{2} + \sqrt {-b x^{4} + a} \right |}\right )}{2 \, \sqrt {-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 24, normalized size = 0.77 \[ \frac {\arctan \left (\frac {\sqrt {b}\, x^{2}}{\sqrt {-b \,x^{4}+a}}\right )}{2 \sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.04, size = 23, normalized size = 0.74 \[ -\frac {\arctan \left (\frac {\sqrt {-b x^{4} + a}}{\sqrt {b} x^{2}}\right )}{2 \, \sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x}{\sqrt {a-b\,x^4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.64, size = 53, normalized size = 1.71 \[ \begin {cases} - \frac {i \operatorname {acosh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{2 \sqrt {b}} & \text {for}\: \left |{\frac {b x^{4}}{a}}\right | > 1 \\\frac {\operatorname {asin}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{2 \sqrt {b}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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