Optimal. Leaf size=21 \[ x \sqrt{x^2+1} \left (-\sqrt{\frac{a}{x^4}}\right ) \]
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Rubi [A] time = 0.0042075, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {15, 264} \[ x \sqrt{x^2+1} \left (-\sqrt{\frac{a}{x^4}}\right ) \]
Antiderivative was successfully verified.
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Rule 15
Rule 264
Rubi steps
\begin{align*} \int \frac{\sqrt{\frac{a}{x^4}}}{\sqrt{1+x^2}} \, dx &=\left (\sqrt{\frac{a}{x^4}} x^2\right ) \int \frac{1}{x^2 \sqrt{1+x^2}} \, dx\\ &=-\sqrt{\frac{a}{x^4}} x \sqrt{1+x^2}\\ \end{align*}
Mathematica [A] time = 0.0038213, size = 21, normalized size = 1. \[ x \sqrt{x^2+1} \left (-\sqrt{\frac{a}{x^4}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 18, normalized size = 0.9 \begin{align*} -x\sqrt{{\frac{a}{{x}^{4}}}}\sqrt{{x}^{2}+1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.5862, size = 31, normalized size = 1.48 \begin{align*} -\frac{\sqrt{a} x^{2} + \sqrt{a}}{\sqrt{x^{2} + 1} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.982961, size = 65, normalized size = 3.1 \begin{align*} -x^{2} \sqrt{\frac{a}{x^{4}}} - \sqrt{x^{2} + 1} x \sqrt{\frac{a}{x^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{a}{x^{4}}}}{\sqrt{x^{2} + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15102, size = 30, normalized size = 1.43 \begin{align*} \frac{2 \, \sqrt{a}}{{\left (x - \sqrt{x^{2} + 1}\right )}^{2} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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