Optimal. Leaf size=120 \[ \frac {\tan ^{-1}\left (\frac {2-\sqrt {2} \sqrt {2-3 x^2}}{\sqrt [4]{2} \sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{2\ 2^{3/4} \sqrt {3}}+\frac {\tanh ^{-1}\left (\frac {2+\sqrt {2} \sqrt {2-3 x^2}}{\sqrt [4]{2} \sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{2\ 2^{3/4} \sqrt {3}} \]
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Rubi [A]
time = 0.01, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {406}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {2-\sqrt {2} \sqrt {2-3 x^2}}{\sqrt [4]{2} \sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{2\ 2^{3/4} \sqrt {3}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {2-3 x^2}+2}{\sqrt [4]{2} \sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{2\ 2^{3/4} \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 406
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx &=\frac {\tan ^{-1}\left (\frac {2-\sqrt {2} \sqrt {2-3 x^2}}{\sqrt [4]{2} \sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{2\ 2^{3/4} \sqrt {3}}+\frac {\tanh ^{-1}\left (\frac {2+\sqrt {2} \sqrt {2-3 x^2}}{\sqrt [4]{2} \sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{2\ 2^{3/4} \sqrt {3}}\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 119, normalized size = 0.99 \begin {gather*} \frac {\tan ^{-1}\left (\frac {3 \sqrt {2} x^2-4 \sqrt {2-3 x^2}}{2\ 2^{3/4} \sqrt {3} x \sqrt [4]{2-3 x^2}}\right )+\tanh ^{-1}\left (\frac {2\ 2^{3/4} \sqrt {3} x \sqrt [4]{2-3 x^2}}{3 \sqrt {2} x^2+4 \sqrt {2-3 x^2}}\right )}{4\ 2^{3/4} \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.52, size = 186, normalized size = 1.55
method | result | size |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+72\right )^{2}\right ) \ln \left (\frac {6 \left (-3 x^{2}+2\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+72\right )^{2}\right )+\left (-3 x^{2}+2\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+72\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+72\right )^{2}\right )+18 \sqrt {-3 x^{2}+2}\, x +3 \RootOf \left (\textit {\_Z}^{4}+72\right )^{2} x}{3 x^{2}-4}\right )}{24}+\frac {\RootOf \left (\textit {\_Z}^{4}+72\right ) \ln \left (\frac {6 \left (-3 x^{2}+2\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{4}+72\right )-\left (-3 x^{2}+2\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+72\right )^{3}+18 \sqrt {-3 x^{2}+2}\, x -3 \RootOf \left (\textit {\_Z}^{4}+72\right )^{2} x}{3 x^{2}-4}\right )}{24}\) | \(186\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 553 vs.
\(2 (89) = 178\).
time = 2.42, size = 553, normalized size = 4.61 \begin {gather*} \frac {1}{72} \cdot 18^{\frac {3}{4}} \sqrt {2} \arctan \left (-\frac {6 \cdot 18^{\frac {3}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} x^{3} + 54 \, x^{4} + 24 \cdot 18^{\frac {1}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}} x + 12 \, \sqrt {2} {\left (3 \, x^{2} - 4\right )} \sqrt {-3 \, x^{2} + 2} - 72 \, x^{2} + {\left (18^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{3} + 4 \, x\right )} \sqrt {-3 \, x^{2} + 2} - 72 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} x^{2} - 6 \cdot 18^{\frac {1}{4}} \sqrt {2} {\left (3 \, x^{3} - 4 \, x\right )} - 48 \, \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}}\right )} \sqrt {-\frac {3 \, \sqrt {2} x^{2} + 2 \cdot 18^{\frac {1}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} x + 4 \, \sqrt {-3 \, x^{2} + 2}}{3 \, x^{2} - 4}}}{6 \, {\left (9 \, x^{4} + 24 \, x^{2} - 16\right )}}\right ) - \frac {1}{72} \cdot 18^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {6 \cdot 18^{\frac {3}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} x^{3} - 54 \, x^{4} + 24 \cdot 18^{\frac {1}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}} x - 12 \, \sqrt {2} {\left (3 \, x^{2} - 4\right )} \sqrt {-3 \, x^{2} + 2} + 72 \, x^{2} + {\left (18^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{3} + 4 \, x\right )} \sqrt {-3 \, x^{2} + 2} + 72 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} x^{2} - 6 \cdot 18^{\frac {1}{4}} \sqrt {2} {\left (3 \, x^{3} - 4 \, x\right )} + 48 \, \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}}\right )} \sqrt {-\frac {3 \, \sqrt {2} x^{2} - 2 \cdot 18^{\frac {1}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} x + 4 \, \sqrt {-3 \, x^{2} + 2}}{3 \, x^{2} - 4}}}{6 \, {\left (9 \, x^{4} + 24 \, x^{2} - 16\right )}}\right ) + \frac {1}{288} \cdot 18^{\frac {3}{4}} \sqrt {2} \log \left (-\frac {36 \, {\left (3 \, \sqrt {2} x^{2} + 2 \cdot 18^{\frac {1}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} x + 4 \, \sqrt {-3 \, x^{2} + 2}\right )}}{3 \, x^{2} - 4}\right ) - \frac {1}{288} \cdot 18^{\frac {3}{4}} \sqrt {2} \log \left (-\frac {36 \, {\left (3 \, \sqrt {2} x^{2} - 2 \cdot 18^{\frac {1}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} x + 4 \, \sqrt {-3 \, x^{2} + 2}\right )}}{3 \, x^{2} - 4}\right ) \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}{3 x^{2} \sqrt [4]{2 - 3 x^{2}} - 4 \sqrt [4]{2 - 3 x^{2}}}\, 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 = -0.01 \begin {gather*} -\int \frac {1}{{\left (2-3\,x^2\right )}^{1/4}\,\left (3\,x^2-4\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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