Optimal. Leaf size=246 \[ \frac {3 \sqrt [4]{3 x^2-1} x}{2 \left (\sqrt {3 x^2-1}+1\right )}-\frac {\left (3 x^2-1\right )^{3/4}}{2 x}-\frac {1}{4} \sqrt {\frac {3}{2}} \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )-\frac {1}{4} \sqrt {\frac {3}{2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )+\frac {\sqrt {3} \sqrt {\frac {x^2}{\left (\sqrt {3 x^2-1}+1\right )^2}} \left (\sqrt {3 x^2-1}+1\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )|\frac {1}{2}\right )}{4 x}-\frac {\sqrt {3} \sqrt {\frac {x^2}{\left (\sqrt {3 x^2-1}+1\right )^2}} \left (\sqrt {3 x^2-1}+1\right ) E\left (2 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )|\frac {1}{2}\right )}{2 x} \]
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Rubi [A] time = 0.12, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {440, 325, 230, 305, 220, 1196, 398} \[ \frac {3 \sqrt [4]{3 x^2-1} x}{2 \left (\sqrt {3 x^2-1}+1\right )}-\frac {\left (3 x^2-1\right )^{3/4}}{2 x}-\frac {1}{4} \sqrt {\frac {3}{2}} \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )-\frac {1}{4} \sqrt {\frac {3}{2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )+\frac {\sqrt {3} \sqrt {\frac {x^2}{\left (\sqrt {3 x^2-1}+1\right )^2}} \left (\sqrt {3 x^2-1}+1\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )|\frac {1}{2}\right )}{4 x}-\frac {\sqrt {3} \sqrt {\frac {x^2}{\left (\sqrt {3 x^2-1}+1\right )^2}} \left (\sqrt {3 x^2-1}+1\right ) E\left (2 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )|\frac {1}{2}\right )}{2 x} \]
Antiderivative was successfully verified.
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Rule 220
Rule 230
Rule 305
Rule 325
Rule 398
Rule 440
Rule 1196
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx &=\int \left (-\frac {1}{2 x^2 \sqrt [4]{-1+3 x^2}}+\frac {3}{2 \left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {1}{x^2 \sqrt [4]{-1+3 x^2}} \, dx\right )+\frac {3}{2} \int \frac {1}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx\\ &=-\frac {\left (-1+3 x^2\right )^{3/4}}{2 x}-\frac {1}{4} \sqrt {\frac {3}{2}} \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac {1}{4} \sqrt {\frac {3}{2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )+\frac {3}{4} \int \frac {1}{\sqrt [4]{-1+3 x^2}} \, dx\\ &=-\frac {\left (-1+3 x^2\right )^{3/4}}{2 x}-\frac {1}{4} \sqrt {\frac {3}{2}} \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac {1}{4} \sqrt {\frac {3}{2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )+\frac {\left (\sqrt {3} \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )}{2 x}\\ &=-\frac {\left (-1+3 x^2\right )^{3/4}}{2 x}-\frac {1}{4} \sqrt {\frac {3}{2}} \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac {1}{4} \sqrt {\frac {3}{2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )+\frac {\left (\sqrt {3} \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )}{2 x}-\frac {\left (\sqrt {3} \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )}{2 x}\\ &=-\frac {\left (-1+3 x^2\right )^{3/4}}{2 x}+\frac {3 x \sqrt [4]{-1+3 x^2}}{2 \left (1+\sqrt {-1+3 x^2}\right )}-\frac {1}{4} \sqrt {\frac {3}{2}} \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac {1}{4} \sqrt {\frac {3}{2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac {\sqrt {3} \sqrt {\frac {x^2}{\left (1+\sqrt {-1+3 x^2}\right )^2}} \left (1+\sqrt {-1+3 x^2}\right ) E\left (2 \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )|\frac {1}{2}\right )}{2 x}+\frac {\sqrt {3} \sqrt {\frac {x^2}{\left (1+\sqrt {-1+3 x^2}\right )^2}} \left (1+\sqrt {-1+3 x^2}\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )|\frac {1}{2}\right )}{4 x}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 64, normalized size = 0.26 \[ \frac {-3 \sqrt [4]{1-3 x^2} x^4 F_1\left (\frac {3}{2};\frac {1}{4},1;\frac {5}{2};3 x^2,\frac {3 x^2}{2}\right )-12 x^2+4}{8 x \sqrt [4]{3 x^2-1}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 7.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (3 \, x^{2} - 1\right )}^{\frac {3}{4}}}{9 \, x^{6} - 9 \, x^{4} + 2 \, x^{2}}, x\right ) \]
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 \[ \int \frac {1}{{\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} {\left (3 \, x^{2} - 2\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.19, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (3 x^{2}-2\right ) \left (3 x^{2}-1\right )^{\frac {1}{4}} x^{2}}\, dx \]
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 \[ \int \frac {1}{{\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} {\left (3 \, x^{2} - 2\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^2\,{\left (3\,x^2-1\right )}^{1/4}\,\left (3\,x^2-2\right )} \,d x \]
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 \[ \int \frac {1}{x^{2} \left (3 x^{2} - 2\right ) \sqrt [4]{3 x^{2} - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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