Optimal. Leaf size=197 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}}-\frac {\tan ^{-1}\left (1+\sqrt [4]{4-6 x^2}\right )}{8 \sqrt [4]{2}}+\frac {\tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )}{8 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}}+\frac {\log \left (\sqrt {2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2-3 x^2}\right )}{16 \sqrt [4]{2}}-\frac {\log \left (\sqrt {2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2-3 x^2}\right )}{16 \sqrt [4]{2}} \]
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Rubi [A]
time = 0.13, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps
used = 18, number of rules used = 13, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {454, 272, 65,
218, 212, 209, 455, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}}-\frac {\text {ArcTan}\left (\sqrt [4]{4-6 x^2}+1\right )}{8 \sqrt [4]{2}}+\frac {\text {ArcTan}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )}{8 \sqrt [4]{2}}+\frac {\log \left (\sqrt {2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2}\right )}{16 \sqrt [4]{2}}-\frac {\log \left (\sqrt {2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2}\right )}{16 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 209
Rule 210
Rule 212
Rule 217
Rule 218
Rule 272
Rule 454
Rule 455
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {1}{x \left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx &=\int \left (\frac {1}{4 x \left (2-3 x^2\right )^{3/4}}-\frac {3 x}{4 \left (2-3 x^2\right )^{3/4} \left (-4+3 x^2\right )}\right ) \, dx\\ &=\frac {1}{4} \int \frac {1}{x \left (2-3 x^2\right )^{3/4}} \, dx-\frac {3}{4} \int \frac {x}{\left (2-3 x^2\right )^{3/4} \left (-4+3 x^2\right )} \, dx\\ &=\frac {1}{8} \text {Subst}\left (\int \frac {1}{(2-3 x)^{3/4} x} \, dx,x,x^2\right )-\frac {3}{8} \text {Subst}\left (\int \frac {1}{(2-3 x)^{3/4} (-4+3 x)} \, dx,x,x^2\right )\\ &=-\left (\frac {1}{6} \text {Subst}\left (\int \frac {1}{\frac {2}{3}-\frac {x^4}{3}} \, dx,x,\sqrt [4]{2-3 x^2}\right )\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{-2-x^4} \, dx,x,\sqrt [4]{2-3 x^2}\right )\\ &=-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2}-x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{4 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2}+x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{4 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {\sqrt {2}-x^2}{-2-x^4} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{4 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {\sqrt {2}+x^2}{-2-x^4} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{4 \sqrt {2}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2}-2^{3/4} x+x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{8 \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 )}{8 \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 )}{16 \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 )}{16 \sqrt [4]{2}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}}+\frac {\log \left (\sqrt {2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2-3 x^2}\right )}{16 \sqrt [4]{2}}-\frac {\log \left (\sqrt {2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2-3 x^2}\right )}{16 \sqrt [4]{2}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt [4]{4-6 x^2}\right )}{8 \sqrt [4]{2}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt [4]{4-6 x^2}\right )}{8 \sqrt [4]{2}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}}-\frac {\tan ^{-1}\left (1+\sqrt [4]{4-6 x^2}\right )}{8 \sqrt [4]{2}}+\frac {\tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )}{8 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}}+\frac {\log \left (\sqrt {2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2-3 x^2}\right )}{16 \sqrt [4]{2}}-\frac {\log \left (\sqrt {2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2-3 x^2}\right )}{16 \sqrt [4]{2}}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 121, normalized size = 0.61 \begin {gather*} \frac {-2 \tan ^{-1}\left (\sqrt [4]{1-\frac {3 x^2}{2}}\right )+\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )-2 \tanh ^{-1}\left (\sqrt [4]{1-\frac {3 x^2}{2}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {2 \sqrt [4]{4-6 x^2}}{2+\sqrt {4-6 x^2}}\right )}{8\ 2^{3/4}} \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 = 9.98, size = 562, normalized size = 2.85
method | result | size |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (-\frac {3 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{2}-4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right )+4 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \sqrt {-3 x^{2}+2}-4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+4 \left (-3 x^{2}+2\right )^{\frac {3}{4}}}{x^{2}}\right )}{16}-\frac {\RootOf \left (\textit {\_Z}^{4}-2\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{2}-4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3}-4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (-3 x^{2}+2\right )^{\frac {1}{4}}-4 \sqrt {-3 x^{2}+2}\, \RootOf \left (\textit {\_Z}^{4}-2\right )-4 \left (-3 x^{2}+2\right )^{\frac {3}{4}}}{x^{2}}\right )}{16}+\frac {\ln \left (-\frac {4 \sqrt {-3 x^{2}+2}\, \RootOf \left (\textit {\_Z}^{4}-2\right )^{3}-4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (-3 x^{2}+2\right )^{\frac {1}{4}}-3 \RootOf \left (\textit {\_Z}^{4}-2\right ) x^{2}-4 \left (-3 x^{2}+2\right )^{\frac {3}{4}}+4 \RootOf \left (\textit {\_Z}^{4}-2\right )}{3 x^{2}-4}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{3}}{32}+\frac {\ln \left (-\frac {4 \sqrt {-3 x^{2}+2}\, \RootOf \left (\textit {\_Z}^{4}-2\right )^{3}-4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (-3 x^{2}+2\right )^{\frac {1}{4}}-3 \RootOf \left (\textit {\_Z}^{4}-2\right ) x^{2}-4 \left (-3 x^{2}+2\right )^{\frac {3}{4}}+4 \RootOf \left (\textit {\_Z}^{4}-2\right )}{3 x^{2}-4}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right )}{32}-\frac {\RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \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}\, \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right )-2 \sqrt {-3 x^{2}+2}\, \RootOf \left (\textit {\_Z}^{4}-2\right )^{3}-4 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right ) \left (-3 x^{2}+2\right )^{\frac {1}{4}}+3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{2}+3 \RootOf \left (\textit {\_Z}^{4}-2\right ) x^{2}+4 \left (-3 x^{2}+2\right )^{\frac {3}{4}}}{3 x^{2}-4}\right )}{16}\) | \(562\) |
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. 315 vs.
\(2 (144) = 288\).
time = 2.59, size = 315, normalized size = 1.60 \begin {gather*} \frac {1}{32} \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}{32} \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}{128} \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}{128} \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}{16} \cdot 8^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 8^{\frac {1}{4}} \sqrt {\sqrt {2} + \sqrt {-3 \, x^{2} + 2}} - \frac {1}{2} \cdot 8^{\frac {1}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) - \frac {1}{64} \cdot 8^{\frac {3}{4}} \log \left (8^{\frac {3}{4}} + 4 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) + \frac {1}{64} \cdot 8^{\frac {3}{4}} \log \left (-8^{\frac {3}{4}} + 4 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{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^{3} \left (2 - 3 x^{2}\right )^{\frac {3}{4}} - 4 x \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 = 0.96, size = 210, normalized size = 1.07 \begin {gather*} -\frac {1}{16} \cdot 4^{\frac {1}{8}} \sqrt {2} \arctan \left (\frac {1}{8} \cdot 4^{\frac {7}{8}} \sqrt {2} {\left (4^{\frac {1}{8}} \sqrt {2} + 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{16} \cdot 4^{\frac {1}{8}} \sqrt {2} \arctan \left (-\frac {1}{8} \cdot 4^{\frac {7}{8}} \sqrt {2} {\left (4^{\frac {1}{8}} \sqrt {2} - 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{32} \cdot 4^{\frac {1}{8}} \sqrt {2} \log \left (4^{\frac {1}{8}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {-3 \, x^{2} + 2} + 4^{\frac {1}{4}}\right ) + \frac {1}{32} \cdot 4^{\frac {1}{8}} \sqrt {2} \log \left (-4^{\frac {1}{8}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {-3 \, x^{2} + 2} + 4^{\frac {1}{4}}\right ) - \frac {1}{8} \cdot 4^{\frac {1}{8}} \arctan \left (\frac {1}{4} \cdot 4^{\frac {7}{8}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) - \frac {1}{16} \cdot 4^{\frac {1}{8}} \log \left ({\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 4^{\frac {1}{8}}\right ) + \frac {1}{16} \cdot 4^{\frac {1}{8}} \log \left (-{\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 4^{\frac {1}{8}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 91, normalized size = 0.46 \begin {gather*} -\frac {2^{1/4}\,\mathrm {atan}\left (\frac {2^{3/4}\,{\left (2-3\,x^2\right )}^{1/4}}{2}\right )}{8}+\frac {2^{1/4}\,\mathrm {atan}\left (\frac {2^{3/4}\,{\left (2-3\,x^2\right )}^{1/4}\,1{}\mathrm {i}}{2}\right )\,1{}\mathrm {i}}{8}+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}{16}-\frac {1}{16}{}\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}{16}+\frac {1}{16}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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