Optimal. Leaf size=44 \[ -\frac {\tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {a}} \]
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Rubi [A] time = 0.04, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1114, 724, 206} \[ -\frac {\tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {a}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 724
Rule 1114
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {a+b x^2+c x^4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )\\ &=-\operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x^2}{\sqrt {a+b x^2+c x^4}}\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {a}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 44, normalized size = 1.00 \[ -\frac {\tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {a}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 124, normalized size = 2.82 \[ \left [\frac {\log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} - 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{4}}\right )}{4 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{4} + a b x^{2} + a^{2}\right )}}\right )}{2 \, a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 38, normalized size = 0.86 \[ \frac {\arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 39, normalized size = 0.89 \[ -\frac {\ln \left (\frac {b \,x^{2}+2 a +2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {a}}{x^{2}}\right )}{2 \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.44, size = 44, normalized size = 1.00 \[ -\frac {\ln \left (\frac {1}{x^2}\right )}{2\,\sqrt {a}}-\frac {\ln \left (2\,a+2\,\sqrt {a}\,\sqrt {c\,x^4+b\,x^2+a}+b\,x^2\right )}{2\,\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sqrt {a + b x^{2} + c x^{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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