Optimal. Leaf size=40 \[ \frac{\Pi \left (-\frac{2 b}{a \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt{2}}\right )\right |-1\right )}{\sqrt{2} a \sqrt [4]{d}} \]
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Rubi [A] time = 0.0179991, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {1218} \[ \frac{\Pi \left (-\frac{2 b}{a \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt{2}}\right )\right |-1\right )}{\sqrt{2} a \sqrt [4]{d}} \]
Antiderivative was successfully verified.
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Rule 1218
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b x^2\right ) \sqrt{4-d x^4}} \, dx &=\frac{\Pi \left (-\frac{2 b}{a \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt{2}}\right )\right |-1\right )}{\sqrt{2} a \sqrt [4]{d}}\\ \end{align*}
Mathematica [C] time = 0.125321, size = 59, normalized size = 1.48 \[ -\frac{i \Pi \left (-\frac{2 b}{a \sqrt{d}};\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{d}} x}{\sqrt{2}}\right )\right |-1\right )}{\sqrt{2} a \sqrt{-\sqrt{d}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.095, size = 78, normalized size = 2. \begin{align*}{\frac{\sqrt{2}}{a}\sqrt{1-{\frac{{x}^{2}}{2}\sqrt{d}}}\sqrt{1+{\frac{{x}^{2}}{2}\sqrt{d}}}{\it EllipticPi} \left ({\frac{x\sqrt{2}}{2}\sqrt [4]{d}},-2\,{\frac{b}{a\sqrt{d}}},{\sqrt{2}\sqrt{-{\frac{1}{2}\sqrt{d}}}{\frac{1}{\sqrt [4]{d}}}} \right ){\frac{1}{\sqrt [4]{d}}}{\frac{1}{\sqrt{-d{x}^{4}+4}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-d x^{4} + 4}{\left (b x^{2} + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{1}{\left (a + b x^{2}\right ) \sqrt{- d x^{4} + 4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-d x^{4} + 4}{\left (b x^{2} + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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