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 = 0.01, 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 = {1232}
\begin {gather*} \frac {\Pi \left (-\frac {2 b}{a \sqrt {d}};\left .\text {ArcSin}\left (\frac {\sqrt [4]{d} x}{\sqrt {2}}\right )\right |-1\right )}{\sqrt {2} a \sqrt [4]{d}} \end {gather*}
Antiderivative was successfully verified.
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Rule 1232
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] Result contains complex when optimal does not.
time = 10.09, size = 59, normalized size = 1.48 \begin {gather*} -\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}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(77\) vs.
\(2(32)=64\).
time = 0.12, size = 78, normalized size = 1.95
method | result | size |
default | \(\frac {\sqrt {2}\, \sqrt {1-\frac {x^{2} \sqrt {d}}{2}}\, \sqrt {1+\frac {x^{2} \sqrt {d}}{2}}\, \EllipticPi \left (\frac {d^{\frac {1}{4}} x \sqrt {2}}{2}, -\frac {2 b}{a \sqrt {d}}, \frac {\sqrt {-\frac {\sqrt {d}}{2}}\, \sqrt {2}}{d^{\frac {1}{4}}}\right )}{a \,d^{\frac {1}{4}} \sqrt {-d \,x^{4}+4}}\) | \(78\) |
elliptic | \(\frac {\sqrt {2}\, \sqrt {1-\frac {x^{2} \sqrt {d}}{2}}\, \sqrt {1+\frac {x^{2} \sqrt {d}}{2}}\, \EllipticPi \left (\frac {d^{\frac {1}{4}} x \sqrt {2}}{2}, -\frac {2 b}{a \sqrt {d}}, \frac {\sqrt {-\frac {\sqrt {d}}{2}}\, \sqrt {2}}{d^{\frac {1}{4}}}\right )}{a \,d^{\frac {1}{4}} \sqrt {-d \,x^{4}+4}}\) | \(78\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (a + b x^{2}\right ) \sqrt {- d x^{4} + 4}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 = -0.02 \begin {gather*} \int \frac {1}{\left (b\,x^2+a\right )\,\sqrt {4-d\,x^4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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