Optimal. Leaf size=556 \[ \frac {2 x}{27 \left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )}-\frac {2 \left (1-x^2\right )^{2/3}}{27 x}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{18\ 2^{2/3} \sqrt {3}}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{18\ 2^{2/3}}-\frac {2 \sqrt {2} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{27 \sqrt [4]{3} \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} x}+\frac {\sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{9\ 3^{3/4} \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} x}-\frac {\left (1-x^2\right )^{2/3}}{9 x^3}+\frac {\tan ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{18\ 2^{2/3} \sqrt {3}}-\frac {\tanh ^{-1}(x)}{54\ 2^{2/3}} \]
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Rubi [A] time = 0.31, antiderivative size = 556, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {480, 583, 530, 235, 304, 219, 1879, 393} \[ \frac {2 x}{27 \left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )}-\frac {2 \left (1-x^2\right )^{2/3}}{27 x}-\frac {\left (1-x^2\right )^{2/3}}{9 x^3}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{18\ 2^{2/3} \sqrt {3}}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{18\ 2^{2/3}}-\frac {2 \sqrt {2} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{27 \sqrt [4]{3} \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} x}+\frac {\sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{9\ 3^{3/4} \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} x}+\frac {\tan ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{18\ 2^{2/3} \sqrt {3}}-\frac {\tanh ^{-1}(x)}{54\ 2^{2/3}} \]
Antiderivative was successfully verified.
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Rule 219
Rule 235
Rule 304
Rule 393
Rule 480
Rule 530
Rule 583
Rule 1879
Rubi steps
\begin {align*} \int \frac {1}{x^4 \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx &=-\frac {\left (1-x^2\right )^{2/3}}{9 x^3}+\frac {1}{9} \int \frac {2+\frac {5 x^2}{3}}{x^2 \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx\\ &=-\frac {\left (1-x^2\right )^{2/3}}{9 x^3}-\frac {2 \left (1-x^2\right )^{2/3}}{27 x}-\frac {1}{27} \int \frac {-1+\frac {2 x^2}{3}}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx\\ &=-\frac {\left (1-x^2\right )^{2/3}}{9 x^3}-\frac {2 \left (1-x^2\right )^{2/3}}{27 x}-\frac {2}{81} \int \frac {1}{\sqrt [3]{1-x^2}} \, dx+\frac {1}{9} \int \frac {1}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx\\ &=-\frac {\left (1-x^2\right )^{2/3}}{9 x^3}-\frac {2 \left (1-x^2\right )^{2/3}}{27 x}+\frac {\tan ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{18\ 2^{2/3} \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{18\ 2^{2/3} \sqrt {3}}-\frac {\tanh ^{-1}(x)}{54\ 2^{2/3}}+\frac {\tanh ^{-1}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{18\ 2^{2/3}}+\frac {\sqrt {-x^2} \operatorname {Subst}\left (\int \frac {x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1-x^2}\right )}{27 x}\\ &=-\frac {\left (1-x^2\right )^{2/3}}{9 x^3}-\frac {2 \left (1-x^2\right )^{2/3}}{27 x}+\frac {\tan ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{18\ 2^{2/3} \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{18\ 2^{2/3} \sqrt {3}}-\frac {\tanh ^{-1}(x)}{54\ 2^{2/3}}+\frac {\tanh ^{-1}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{18\ 2^{2/3}}-\frac {\sqrt {-x^2} \operatorname {Subst}\left (\int \frac {1+\sqrt {3}-x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1-x^2}\right )}{27 x}+\frac {\left (\sqrt {2 \left (2+\sqrt {3}\right )} \sqrt {-x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1-x^2}\right )}{27 x}\\ &=-\frac {\left (1-x^2\right )^{2/3}}{9 x^3}-\frac {2 \left (1-x^2\right )^{2/3}}{27 x}+\frac {2 x}{27 \left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{18\ 2^{2/3} \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{18\ 2^{2/3} \sqrt {3}}-\frac {\tanh ^{-1}(x)}{54\ 2^{2/3}}+\frac {\tanh ^{-1}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{18\ 2^{2/3}}+\frac {\sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{1-x^2}}{1-\sqrt {3}-\sqrt [3]{1-x^2}}\right )|-7+4 \sqrt {3}\right )}{9\ 3^{3/4} x \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}}}-\frac {2 \sqrt {2} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{1-x^2}}{1-\sqrt {3}-\sqrt [3]{1-x^2}}\right )|-7+4 \sqrt {3}\right )}{27 \sqrt [4]{3} x \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}}}\\ \end {align*}
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Mathematica [C] time = 0.11, size = 166, normalized size = 0.30 \[ \frac {-\frac {9 x^4 F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};x^2,-\frac {x^2}{3}\right )}{\left (x^2+3\right ) \left (2 x^2 \left (F_1\left (\frac {3}{2};\frac {1}{3},2;\frac {5}{2};x^2,-\frac {x^2}{3}\right )-F_1\left (\frac {3}{2};\frac {4}{3},1;\frac {5}{2};x^2,-\frac {x^2}{3}\right )\right )-9 F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};x^2,-\frac {x^2}{3}\right )\right )}+2 x^4+x^2-3}{27 x^3 \sqrt [3]{1-x^2}}-\frac {2}{729} x^3 F_1\left (\frac {3}{2};\frac {1}{3},1;\frac {5}{2};x^2,-\frac {x^2}{3}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 3.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (-x^{2} + 1\right )}^{\frac {2}{3}}}{x^{8} + 2 \, x^{6} - 3 \, x^{4}}, x\right ) \]
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 \[ \int \frac {1}{{\left (x^{2} + 3\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-x^{2}+1\right )^{\frac {1}{3}} \left (x^{2}+3\right ) x^{4}}\, dx \]
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 \[ \int \frac {1}{{\left (x^{2} + 3\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^4\,{\left (1-x^2\right )}^{1/3}\,\left (x^2+3\right )} \,d x \]
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 \[ \int \frac {1}{x^{4} \sqrt [3]{- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 3\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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