Optimal. Leaf size=30 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{2 \sqrt {c}} \]
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Rubi [A] time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {4, 275, 217, 206} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{2 \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 4
Rule 206
Rule 217
Rule 275
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {a+(2+2 b-2 (1+b)) x^2+c x^4}} \, dx &=\int \frac {x}{\sqrt {a+c x^4}} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+c x^2}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x^2}{\sqrt {a+c x^4}}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{2 \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 1.00 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{2 \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 32, normalized size = 1.07 \begin {gather*} -\frac {\log \left (\sqrt {a+c x^4}-\sqrt {c} x^2\right )}{2 \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 2.24, size = 63, normalized size = 2.10 \begin {gather*} \left [\frac {\log \left (-2 \, c x^{4} - 2 \, \sqrt {c x^{4} + a} \sqrt {c} x^{2} - a\right )}{4 \, \sqrt {c}}, -\frac {\sqrt {-c} \arctan \left (\frac {\sqrt {-c} x^{2}}{\sqrt {c x^{4} + a}}\right )}{2 \, c}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 25, normalized size = 0.83 \begin {gather*} -\frac {\log \left ({\left | -\sqrt {c} x^{2} + \sqrt {c x^{4} + a} \right |}\right )}{2 \, \sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 24, normalized size = 0.80 \begin {gather*} \frac {\ln \left (\sqrt {c}\, x^{2}+\sqrt {c \,x^{4}+a}\right )}{2 \sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.43, size = 45, normalized size = 1.50 \begin {gather*} -\frac {\log \left (-\frac {\sqrt {c} - \frac {\sqrt {c x^{4} + a}}{x^{2}}}{\sqrt {c} + \frac {\sqrt {c x^{4} + a}}{x^{2}}}\right )}{4 \, \sqrt {c}} \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 {c\,x^4+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.10, size = 20, normalized size = 0.67 \begin {gather*} \frac {\operatorname {asinh}{\left (\frac {\sqrt {c} x^{2}}{\sqrt {a}} \right )}}{2 \sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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