Optimal. Leaf size=165 \[ -\frac {\sqrt [4]{-1+3 x^2}}{6 x^3}-\frac {2 \sqrt [4]{-1+3 x^2}}{x}+\frac {3}{8} \sqrt {\frac {3}{2}} \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac {3}{8} \sqrt {\frac {3}{2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac {11 \sqrt {3} \sqrt {\frac {x^2}{\left (1+\sqrt {-1+3 x^2}\right )^2}} \left (1+\sqrt {-1+3 x^2}\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )|\frac {1}{2}\right )}{8 x} \]
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Rubi [A]
time = 0.11, 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 = {454, 331, 240,
226, 409, 453} \begin {gather*} \frac {3}{8} \sqrt {\frac {3}{2}} \text {ArcTan}\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 \text {ArcTan}\left (\sqrt [4]{3 x^2-1}\right )|\frac {1}{2}\right )}{8 x}-\frac {2 \sqrt [4]{3 x^2-1}}{x}-\frac {3}{8} \sqrt {\frac {3}{2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )-\frac {\sqrt [4]{3 x^2-1}}{6 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 240
Rule 331
Rule 409
Rule 453
Rule 454
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 ) \text {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 ) \text {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] Result contains higher order function than in optimal. Order 6 vs. order 4 in
optimal.
time = 10.05, size = 52, normalized size = 0.32 \begin {gather*} \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 (-1+3 x^2\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, 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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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 [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{4} \cdot \left (3 x^{2} - 2\right ) \left (3 x^{2} - 1\right )^{\frac {3}{4}}}\, 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}{x^4\,{\left (3\,x^2-1\right )}^{3/4}\,\left (3\,x^2-2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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