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 = {281, 223, 209}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}\right )}{2 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 223
Rule 281
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {a-b x^4}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {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 = 0.11, size = 31, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {a-b x^4}}{\sqrt {b} x^2}\right )}{2 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 24, normalized size = 0.77
method | result | size |
default | \(\frac {\arctan \left (\frac {x^{2} \sqrt {b}}{\sqrt {-b \,x^{4}+a}}\right )}{2 \sqrt {b}}\) | \(24\) |
elliptic | \(\frac {\arctan \left (\frac {x^{2} \sqrt {b}}{\sqrt {-b \,x^{4}+a}}\right )}{2 \sqrt {b}}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 23, normalized size = 0.74 \begin {gather*} -\frac {\arctan \left (\frac {\sqrt {-b x^{4} + a}}{\sqrt {b} x^{2}}\right )}{2 \, \sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 76, normalized size = 2.45 \begin {gather*} \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 ] \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 = 0.47, size = 53, normalized size = 1.71 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.40, size = 30, normalized size = 0.97 \begin {gather*} -\frac {\log \left ({\left | -\sqrt {-b} x^{2} + \sqrt {-b x^{4} + a} \right |}\right )}{2 \, \sqrt {-b}} \end {gather*}
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 \begin {gather*} \int \frac {x}{\sqrt {a-b\,x^4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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