Optimal. Leaf size=167 \[ -\sqrt [3]{2} 3^{2/3} \log \left (\sqrt [3]{2} 3^{2/3} \sqrt [3]{x^4-1}+3 x\right )+3 \sqrt [3]{2} \sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{\sqrt [3]{3} x-2 \sqrt [3]{2} \sqrt [3]{x^4-1}}\right )+\left (\frac {3}{2}\right )^{2/3} \log \left (-\sqrt [3]{2} 3^{2/3} \sqrt [3]{x^4-1} x+2^{2/3} \sqrt [3]{3} \left (x^4-1\right )^{2/3}+3 x^2\right )+\frac {3 \left (x^4-1\right )^{2/3} \left (x^4-5 x^3-1\right )}{5 x^5} \]
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Rubi [F] time = 1.49, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-2-x^3+2 x^4\right )}{x^6 \left (-2+3 x^3+2 x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-2-x^3+2 x^4\right )}{x^6 \left (-2+3 x^3+2 x^4\right )} \, dx &=\int \left (\frac {3 \left (-1+x^4\right )^{2/3}}{x^6}+\frac {6 \left (-1+x^4\right )^{2/3}}{x^3}+\frac {\left (-1+x^4\right )^{2/3}}{x^2}-\frac {2 (9+8 x) \left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4}\right ) \, dx\\ &=-\left (2 \int \frac {(9+8 x) \left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4} \, dx\right )+3 \int \frac {\left (-1+x^4\right )^{2/3}}{x^6} \, dx+6 \int \frac {\left (-1+x^4\right )^{2/3}}{x^3} \, dx+\int \frac {\left (-1+x^4\right )^{2/3}}{x^2} \, dx\\ &=-\left (2 \int \left (\frac {9 \left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4}+\frac {8 x \left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4}\right ) \, dx\right )+3 \operatorname {Subst}\left (\int \frac {\left (-1+x^2\right )^{2/3}}{x^2} \, dx,x,x^2\right )+\frac {\left (-1+x^4\right )^{2/3} \int \frac {\left (1-x^4\right )^{2/3}}{x^2} \, dx}{\left (1-x^4\right )^{2/3}}+\frac {\left (3 \left (-1+x^4\right )^{2/3}\right ) \int \frac {\left (1-x^4\right )^{2/3}}{x^6} \, dx}{\left (1-x^4\right )^{2/3}}\\ &=-\frac {3 \left (-1+x^4\right )^{2/3}}{x^2}-\frac {3 \left (-1+x^4\right )^{2/3} \, _2F_1\left (-\frac {5}{4},-\frac {2}{3};-\frac {1}{4};x^4\right )}{5 x^5 \left (1-x^4\right )^{2/3}}-\frac {\left (-1+x^4\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{4};\frac {3}{4};x^4\right )}{x \left (1-x^4\right )^{2/3}}+4 \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^2}} \, dx,x,x^2\right )-16 \int \frac {x \left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4} \, dx-18 \int \frac {\left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4} \, dx\\ &=-\frac {3 \left (-1+x^4\right )^{2/3}}{x^2}-\frac {3 \left (-1+x^4\right )^{2/3} \, _2F_1\left (-\frac {5}{4},-\frac {2}{3};-\frac {1}{4};x^4\right )}{5 x^5 \left (1-x^4\right )^{2/3}}-\frac {\left (-1+x^4\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{4};\frac {3}{4};x^4\right )}{x \left (1-x^4\right )^{2/3}}-16 \int \frac {x \left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4} \, dx-18 \int \frac {\left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4} \, dx+\frac {\left (6 \sqrt {x^4}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1+x^3}} \, dx,x,\sqrt [3]{-1+x^4}\right )}{x^2}\\ &=-\frac {3 \left (-1+x^4\right )^{2/3}}{x^2}-\frac {3 \left (-1+x^4\right )^{2/3} \, _2F_1\left (-\frac {5}{4},-\frac {2}{3};-\frac {1}{4};x^4\right )}{5 x^5 \left (1-x^4\right )^{2/3}}-\frac {\left (-1+x^4\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{4};\frac {3}{4};x^4\right )}{x \left (1-x^4\right )^{2/3}}-16 \int \frac {x \left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4} \, dx-18 \int \frac {\left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4} \, dx+\frac {\left (6 \sqrt {x^4}\right ) \operatorname {Subst}\left (\int \frac {1-\sqrt {3}+x}{\sqrt {1+x^3}} \, dx,x,\sqrt [3]{-1+x^4}\right )}{x^2}+\frac {\left (6 \sqrt {2 \left (2-\sqrt {3}\right )} \sqrt {x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^3}} \, dx,x,\sqrt [3]{-1+x^4}\right )}{x^2}\\ &=-\frac {3 \left (-1+x^4\right )^{2/3}}{x^2}+\frac {12 x^2}{1+\sqrt {3}+\sqrt [3]{-1+x^4}}-\frac {6 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+\sqrt [3]{-1+x^4}\right ) \sqrt {\frac {1-\sqrt [3]{-1+x^4}+\left (-1+x^4\right )^{2/3}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^4}\right )^2}} E\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+\sqrt [3]{-1+x^4}}{1+\sqrt {3}+\sqrt [3]{-1+x^4}}\right )|-7-4 \sqrt {3}\right )}{x^2 \sqrt {\frac {1+\sqrt [3]{-1+x^4}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^4}\right )^2}}}+\frac {4 \sqrt {2} 3^{3/4} \left (1+\sqrt [3]{-1+x^4}\right ) \sqrt {\frac {1-\sqrt [3]{-1+x^4}+\left (-1+x^4\right )^{2/3}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^4}\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+\sqrt [3]{-1+x^4}}{1+\sqrt {3}+\sqrt [3]{-1+x^4}}\right )|-7-4 \sqrt {3}\right )}{x^2 \sqrt {\frac {1+\sqrt [3]{-1+x^4}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^4}\right )^2}}}-\frac {3 \left (-1+x^4\right )^{2/3} \, _2F_1\left (-\frac {5}{4},-\frac {2}{3};-\frac {1}{4};x^4\right )}{5 x^5 \left (1-x^4\right )^{2/3}}-\frac {\left (-1+x^4\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{4};\frac {3}{4};x^4\right )}{x \left (1-x^4\right )^{2/3}}-16 \int \frac {x \left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4} \, dx-18 \int \frac {\left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4} \, dx\\ \end {align*}
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Mathematica [F] time = 0.33, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-2-x^3+2 x^4\right )}{x^6 \left (-2+3 x^3+2 x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 3.52, size = 167, normalized size = 1.00 \begin {gather*} -\sqrt [3]{2} 3^{2/3} \log \left (\sqrt [3]{2} 3^{2/3} \sqrt [3]{x^4-1}+3 x\right )+3 \sqrt [3]{2} \sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{\sqrt [3]{3} x-2 \sqrt [3]{2} \sqrt [3]{x^4-1}}\right )+\left (\frac {3}{2}\right )^{2/3} \log \left (-\sqrt [3]{2} 3^{2/3} \sqrt [3]{x^4-1} x+2^{2/3} \sqrt [3]{3} \left (x^4-1\right )^{2/3}+3 x^2\right )+\frac {3 \left (x^4-1\right )^{2/3} \left (x^4-5 x^3-1\right )}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 124.26, size = 418, normalized size = 2.50 \begin {gather*} \frac {10 \, \sqrt {3} \left (-18\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {4 \, \sqrt {3} \left (-18\right )^{\frac {2}{3}} {\left (2 \, x^{9} - 3 \, x^{8} - 9 \, x^{7} - 4 \, x^{5} + 3 \, x^{4} + 2 \, x\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}} + 6 \, \sqrt {3} \left (-18\right )^{\frac {1}{3}} {\left (4 \, x^{10} - 42 \, x^{9} + 9 \, x^{8} - 8 \, x^{6} + 42 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (8 \, x^{12} - 180 \, x^{11} + 216 \, x^{10} + 27 \, x^{9} - 24 \, x^{8} + 360 \, x^{7} - 216 \, x^{6} + 24 \, x^{4} - 180 \, x^{3} - 8\right )}}{3 \, {\left (8 \, x^{12} + 36 \, x^{11} - 432 \, x^{10} + 27 \, x^{9} - 24 \, x^{8} - 72 \, x^{7} + 432 \, x^{6} + 24 \, x^{4} + 36 \, x^{3} - 8\right )}}\right ) + 10 \, \left (-18\right )^{\frac {1}{3}} x^{5} \log \left (\frac {3 \, \left (-18\right )^{\frac {2}{3}} {\left (x^{4} - 1\right )}^{\frac {1}{3}} x^{2} + 18 \, {\left (x^{4} - 1\right )}^{\frac {2}{3}} x - \left (-18\right )^{\frac {1}{3}} {\left (2 \, x^{4} + 3 \, x^{3} - 2\right )}}{2 \, x^{4} + 3 \, x^{3} - 2}\right ) - 5 \, \left (-18\right )^{\frac {1}{3}} x^{5} \log \left (\frac {36 \, \left (-18\right )^{\frac {1}{3}} {\left (x^{5} - 3 \, x^{4} - x\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}} + \left (-18\right )^{\frac {2}{3}} {\left (4 \, x^{8} - 42 \, x^{7} + 9 \, x^{6} - 8 \, x^{4} + 42 \, x^{3} + 4\right )} + 54 \, {\left (4 \, x^{6} - 3 \, x^{5} - 4 \, x^{2}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{3}}}{4 \, x^{8} + 12 \, x^{7} + 9 \, x^{6} - 8 \, x^{4} - 12 \, x^{3} + 4}\right ) + 18 \, {\left (x^{4} - 5 \, x^{3} - 1\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}}}{30 \, x^{5}} \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*} \int \frac {{\left (2 \, x^{4} - x^{3} - 2\right )} {\left (x^{4} + 3\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{4} + 3 \, x^{3} - 2\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 89.03, size = 808, normalized size = 4.84
result too large to display
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*} \int \frac {{\left (2 \, x^{4} - x^{3} - 2\right )} {\left (x^{4} + 3\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{4} + 3 \, x^{3} - 2\right )} x^{6}}\,{d x} \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 {{\left (x^4-1\right )}^{2/3}\,\left (x^4+3\right )\,\left (-2\,x^4+x^3+2\right )}{x^6\,\left (2\,x^4+3\,x^3-2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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