Optimal. Leaf size=165 \[ -\frac {2 \sqrt [4]{3 x^2-1}}{x}+\frac {3}{8} \sqrt {\frac {3}{2}} \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )-\frac {3}{8} \sqrt {\frac {3}{2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )-\frac {11 \sqrt {3} \sqrt {\frac {x^2}{\left (\sqrt {3 x^2-1}+1\right )^2}} \left (\sqrt {3 x^2-1}+1\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )|\frac {1}{2}\right )}{8 x}-\frac {\sqrt [4]{3 x^2-1}}{6 x^3} \]
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Rubi [A] time = 0.17, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {443, 325, 234, 220, 400, 442} \[ -\frac {2 \sqrt [4]{3 x^2-1}}{x}-\frac {\sqrt [4]{3 x^2-1}}{6 x^3}+\frac {3}{8} \sqrt {\frac {3}{2}} \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )-\frac {3}{8} \sqrt {\frac {3}{2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )-\frac {11 \sqrt {3} \sqrt {\frac {x^2}{\left (\sqrt {3 x^2-1}+1\right )^2}} \left (\sqrt {3 x^2-1}+1\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )|\frac {1}{2}\right )}{8 x} \]
Antiderivative was successfully verified.
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Rule 220
Rule 234
Rule 325
Rule 400
Rule 442
Rule 443
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx &=\int \left (-\frac {1}{2 x^4 \left (-1+3 x^2\right )^{3/4}}-\frac {3}{4 x^2 \left (-1+3 x^2\right )^{3/4}}+\frac {9}{4 \left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {1}{x^4 \left (-1+3 x^2\right )^{3/4}} \, dx\right )-\frac {3}{4} \int \frac {1}{x^2 \left (-1+3 x^2\right )^{3/4}} \, dx+\frac {9}{4} \int \frac {1}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx\\ &=-\frac {\sqrt [4]{-1+3 x^2}}{6 x^3}-\frac {3 \sqrt [4]{-1+3 x^2}}{4 x}-2 \left (\frac {9}{8} \int \frac {1}{\left (-1+3 x^2\right )^{3/4}} \, dx\right )-\frac {5}{4} \int \frac {1}{x^2 \left (-1+3 x^2\right )^{3/4}} \, dx+\frac {27}{8} \int \frac {x^2}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx\\ &=-\frac {\sqrt [4]{-1+3 x^2}}{6 x^3}-\frac {2 \sqrt [4]{-1+3 x^2}}{x}+\frac {3}{8} \sqrt {\frac {3}{2}} \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac {3}{8} \sqrt {\frac {3}{2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac {15}{8} \int \frac {1}{\left (-1+3 x^2\right )^{3/4}} \, dx-2 \frac {\left (3 \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 )}{4 x}\\ &=-\frac {\sqrt [4]{-1+3 x^2}}{6 x^3}-\frac {2 \sqrt [4]{-1+3 x^2}}{x}+\frac {3}{8} \sqrt {\frac {3}{2}} \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac {3}{8} \sqrt {\frac {3}{2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac {3 \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}-\frac {\left (5 \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 )}{4 x}\\ &=-\frac {\sqrt [4]{-1+3 x^2}}{6 x^3}-\frac {2 \sqrt [4]{-1+3 x^2}}{x}+\frac {3}{8} \sqrt {\frac {3}{2}} \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac {3}{8} \sqrt {\frac {3}{2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac {11 \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 )}{8 x}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 52, normalized size = 0.32 \[ \frac {\left (1-3 x^2\right )^{3/4} F_1\left (-\frac {3}{2};\frac {3}{4},1;-\frac {1}{2};3 x^2,\frac {3 x^2}{2}\right )}{6 x^3 \left (3 x^2-1\right )^{3/4}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 5.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}}{9 \, x^{8} - 9 \, x^{6} + 2 \, x^{4}}, x\right ) \]
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 \[ \int \frac {1}{{\left (3 \, x^{2} - 1\right )}^{\frac {3}{4}} {\left (3 \, x^{2} - 2\right )} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.76, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (3 x^{2}-2\right ) \left (3 x^{2}-1\right )^{\frac {3}{4}} x^{4}}\, dx \]
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 \[ \int \frac {1}{{\left (3 \, x^{2} - 1\right )}^{\frac {3}{4}} {\left (3 \, x^{2} - 2\right )} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^4\,{\left (3\,x^2-1\right )}^{3/4}\,\left (3\,x^2-2\right )} \,d x \]
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 \[ \int \frac {1}{x^{4} \left (3 x^{2} - 2\right ) \left (3 x^{2} - 1\right )^{\frac {3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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