Optimal. Leaf size=33 \[ \frac {\log \left (\sqrt {a x^4-b}+\sqrt {a} x^2\right )}{2 \sqrt {a}} \]
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Rubi [A] time = 0.02, antiderivative size = 32, normalized size of antiderivative = 0.97, 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 = {275, 217, 206} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {a} x^2}{\sqrt {a x^4-b}}\right )}{2 \sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 275
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {-b+a x^4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b+a x^2}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x^2}{\sqrt {-b+a x^4}}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {a} x^2}{\sqrt {-b+a x^4}}\right )}{2 \sqrt {a}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 0.97 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {a} x^2}{\sqrt {a x^4-b}}\right )}{2 \sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.16, size = 33, normalized size = 1.00 \begin {gather*} \frac {\log \left (\sqrt {a x^4-b}+\sqrt {a} x^2\right )}{2 \sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 67, normalized size = 2.03 \begin {gather*} \left [\frac {\log \left (2 \, a x^{4} + 2 \, \sqrt {a x^{4} - b} \sqrt {a} x^{2} - b\right )}{4 \, \sqrt {a}}, -\frac {\sqrt {-a} \arctan \left (\frac {\sqrt {-a} x^{2}}{\sqrt {a x^{4} - b}}\right )}{2 \, a}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 27, normalized size = 0.82 \begin {gather*} -\frac {\log \left ({\left | -\sqrt {a} x^{2} + \sqrt {a x^{4} - b} \right |}\right )}{2 \, \sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 26, normalized size = 0.79 \begin {gather*} \frac {\ln \left (\sqrt {a}\, x^{2}+\sqrt {a \,x^{4}-b}\right )}{2 \sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.79, size = 49, normalized size = 1.48 \begin {gather*} -\frac {\log \left (-\frac {\sqrt {a} - \frac {\sqrt {a x^{4} - b}}{x^{2}}}{\sqrt {a} + \frac {\sqrt {a x^{4} - b}}{x^{2}}}\right )}{4 \, \sqrt {a}} \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\,x^4-b}} \,d x \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 = 1.61 \begin {gather*} \begin {cases} \frac {\operatorname {acosh}{\left (\frac {\sqrt {a} x^{2}}{\sqrt {b}} \right )}}{2 \sqrt {a}} & \text {for}\: \left |{\frac {a x^{4}}{b}}\right | > 1 \\- \frac {i \operatorname {asin}{\left (\frac {\sqrt {a} x^{2}}{\sqrt {b}} \right )}}{2 \sqrt {a}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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