Optimal. Leaf size=143 \[ -\frac {\tan ^{-1}\left (1+\sqrt [4]{4-6 x^2}\right )}{6 \sqrt [4]{2}}+\frac {\tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )}{6 \sqrt [4]{2}}+\frac {\log \left (\sqrt {2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2-3 x^2}\right )}{12 \sqrt [4]{2}}-\frac {\log \left (\sqrt {2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2-3 x^2}\right )}{12 \sqrt [4]{2}} \]
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Rubi [A]
time = 0.08, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {455, 65, 217,
1179, 642, 1176, 631, 210} \begin {gather*} -\frac {\text {ArcTan}\left (\sqrt [4]{4-6 x^2}+1\right )}{6 \sqrt [4]{2}}+\frac {\text {ArcTan}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )}{6 \sqrt [4]{2}}+\frac {\log \left (\sqrt {2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2}\right )}{12 \sqrt [4]{2}}-\frac {\log \left (\sqrt {2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2}\right )}{12 \sqrt [4]{2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 210
Rule 217
Rule 455
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {x}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{(2-3 x)^{3/4} (4-3 x)} \, dx,x,x^2\right )\\ &=-\left (\frac {2}{3} \text {Subst}\left (\int \frac {1}{2+x^4} \, dx,x,\sqrt [4]{2-3 x^2}\right )\right )\\ &=-\frac {\text {Subst}\left (\int \frac {\sqrt {2}-x^2}{2+x^4} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{3 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {\sqrt {2}+x^2}{2+x^4} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{3 \sqrt {2}}\\ &=-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2}-2^{3/4} x+x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{6 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2}+2^{3/4} x+x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{6 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {2^{3/4}+2 x}{-\sqrt {2}-2^{3/4} x-x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{12 \sqrt [4]{2}}+\frac {\text {Subst}\left (\int \frac {2^{3/4}-2 x}{-\sqrt {2}+2^{3/4} x-x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{12 \sqrt [4]{2}}\\ &=\frac {\log \left (\sqrt {2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2-3 x^2}\right )}{12 \sqrt [4]{2}}-\frac {\log \left (\sqrt {2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2-3 x^2}\right )}{12 \sqrt [4]{2}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt [4]{4-6 x^2}\right )}{6 \sqrt [4]{2}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt [4]{4-6 x^2}\right )}{6 \sqrt [4]{2}}\\ &=-\frac {\tan ^{-1}\left (1+\sqrt [4]{4-6 x^2}\right )}{6 \sqrt [4]{2}}+\frac {\tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )}{6 \sqrt [4]{2}}+\frac {\log \left (\sqrt {2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2-3 x^2}\right )}{12 \sqrt [4]{2}}-\frac {\log \left (\sqrt {2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2-3 x^2}\right )}{12 \sqrt [4]{2}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 78, normalized size = 0.55 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )-\tanh ^{-1}\left (\frac {2 \sqrt [4]{4-6 x^2}}{2+\sqrt {4-6 x^2}}\right )}{6 \sqrt [4]{2}} \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 = 1.14, size = 190, normalized size = 1.33
method | result | size |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{4}+2\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{4}+2\right ) \left (-3 x^{2}+2\right )^{\frac {3}{4}}+2 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \sqrt {-3 x^{2}+2}+2 \RootOf \left (\textit {\_Z}^{4}+2\right )^{3} \left (-3 x^{2}+2\right )^{\frac {1}{4}}-3 x^{2}}{3 x^{2}-4}\right )}{12}-\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \sqrt {-3 x^{2}+2}+2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \left (-3 x^{2}+2\right )^{\frac {1}{4}}-2 \left (-3 x^{2}+2\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right )+3 x^{2}}{3 x^{2}-4}\right )}{12}\) | \(190\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 118, normalized size = 0.83 \begin {gather*} -\frac {1}{12} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{12} \cdot 2^{\frac {3}{4}} \arctan \left (-\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} - 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{24} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) + \frac {1}{24} \cdot 2^{\frac {3}{4}} \log \left (-2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) \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. 230 vs.
\(2 (104) = 208\).
time = 0.71, size = 230, normalized size = 1.61 \begin {gather*} \frac {1}{24} \cdot 8^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {1}{4} \cdot 8^{\frac {1}{4}} \sqrt {2} \sqrt {8^{\frac {3}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 4 \, \sqrt {2} + 4 \, \sqrt {-3 \, x^{2} + 2}} - \frac {1}{2} \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} - 1\right ) + \frac {1}{24} \cdot 8^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {1}{16} \cdot 8^{\frac {1}{4}} \sqrt {2} \sqrt {-16 \cdot 8^{\frac {3}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 64 \, \sqrt {2} + 64 \, \sqrt {-3 \, x^{2} + 2}} - \frac {1}{2} \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 1\right ) - \frac {1}{96} \cdot 8^{\frac {3}{4}} \sqrt {2} \log \left (16 \cdot 8^{\frac {3}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 64 \, \sqrt {2} + 64 \, \sqrt {-3 \, x^{2} + 2}\right ) + \frac {1}{96} \cdot 8^{\frac {3}{4}} \sqrt {2} \log \left (-16 \cdot 8^{\frac {3}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 64 \, \sqrt {2} + 64 \, \sqrt {-3 \, x^{2} + 2}\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 {x}{3 x^{2} \left (2 - 3 x^{2}\right )^{\frac {3}{4}} - 4 \left (2 - 3 x^{2}\right )^{\frac {3}{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.22, size = 118, normalized size = 0.83 \begin {gather*} -\frac {1}{12} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{12} \cdot 2^{\frac {3}{4}} \arctan \left (-\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} - 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{24} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) + \frac {1}{24} \cdot 2^{\frac {3}{4}} \log \left (-2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.49, size = 49, normalized size = 0.34 \begin {gather*} 2^{3/4}\,\mathrm {atan}\left (2^{1/4}\,{\left (2-3\,x^2\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{12}-\frac {1}{12}{}\mathrm {i}\right )+2^{3/4}\,\mathrm {atan}\left (2^{1/4}\,{\left (2-3\,x^2\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{12}+\frac {1}{12}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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