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 = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, 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*}
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Mathematica [C] time = 0.12, 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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fricas [F] time = 177.33, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-d x^{4} + 4}}{b d x^{6} + a d x^{4} - 4 \, b x^{2} - 4 \, a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-d x^{4} + 4} {\left (b x^{2} + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 78, normalized size = 1.95 \[ \frac {\sqrt {2}\, \sqrt {-\frac {\sqrt {d}\, x^{2}}{2}+1}\, \sqrt {\frac {\sqrt {d}\, x^{2}}{2}+1}\, \EllipticPi \left (\frac {\sqrt {2}\, d^{\frac {1}{4}} x}{2}, -\frac {2 b}{a \sqrt {d}}, \frac {\sqrt {-\frac {\sqrt {d}}{2}}\, \sqrt {2}}{d^{\frac {1}{4}}}\right )}{\sqrt {-d \,x^{4}+4}\, a \,d^{\frac {1}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-d x^{4} + 4} {\left (b x^{2} + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\left (b\,x^2+a\right )\,\sqrt {4-d\,x^4}} \,d x \]
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 \[ \int \frac {1}{\left (a + b x^{2}\right ) \sqrt {- d x^{4} + 4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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