Optimal. Leaf size=143 \[ \frac {\log \left (\sqrt [3]{2} \sqrt [3]{2 x^4+1}-x\right )}{2\ 2^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2} \sqrt [3]{2 x^4+1}+x}\right )}{2\ 2^{2/3}}+\frac {3 \left (2 x^4+1\right )^{2/3}}{4 x^2}-\frac {\log \left (\sqrt [3]{2} \sqrt [3]{2 x^4+1} x+2^{2/3} \left (2 x^4+1\right )^{2/3}+x^2\right )}{4\ 2^{2/3}} \]
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Rubi [F] time = 1.17, 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 (-3+2 x^4\right ) \left (1+2 x^4\right )^{2/3}}{x^3 \left (2-x^3+4 x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {\left (-3+2 x^4\right ) \left (1+2 x^4\right )^{2/3}}{x^3 \left (2-x^3+4 x^4\right )} \, dx &=\int \left (-\frac {3 \left (1+2 x^4\right )^{2/3}}{2 x^3}+\frac {(-3+16 x) \left (1+2 x^4\right )^{2/3}}{2 \left (2-x^3+4 x^4\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {(-3+16 x) \left (1+2 x^4\right )^{2/3}}{2-x^3+4 x^4} \, dx-\frac {3}{2} \int \frac {\left (1+2 x^4\right )^{2/3}}{x^3} \, dx\\ &=\frac {1}{2} \int \left (-\frac {3 \left (1+2 x^4\right )^{2/3}}{2-x^3+4 x^4}+\frac {16 x \left (1+2 x^4\right )^{2/3}}{2-x^3+4 x^4}\right ) \, dx-\frac {3}{4} \operatorname {Subst}\left (\int \frac {\left (1+2 x^2\right )^{2/3}}{x^2} \, dx,x,x^2\right )\\ &=\frac {3 \left (1+2 x^4\right )^{2/3}}{4 x^2}-\frac {3}{2} \int \frac {\left (1+2 x^4\right )^{2/3}}{2-x^3+4 x^4} \, dx-2 \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+2 x^2}} \, dx,x,x^2\right )+8 \int \frac {x \left (1+2 x^4\right )^{2/3}}{2-x^3+4 x^4} \, dx\\ &=\frac {3 \left (1+2 x^4\right )^{2/3}}{4 x^2}-\frac {3}{2} \int \frac {\left (1+2 x^4\right )^{2/3}}{2-x^3+4 x^4} \, dx+8 \int \frac {x \left (1+2 x^4\right )^{2/3}}{2-x^3+4 x^4} \, dx-\frac {\left (3 \sqrt {x^4}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1+2 x^4}\right )}{\sqrt {2} x^2}\\ &=\frac {3 \left (1+2 x^4\right )^{2/3}}{4 x^2}-\frac {3}{2} \int \frac {\left (1+2 x^4\right )^{2/3}}{2-x^3+4 x^4} \, dx+8 \int \frac {x \left (1+2 x^4\right )^{2/3}}{2-x^3+4 x^4} \, dx+\frac {\left (3 \sqrt {x^4}\right ) \operatorname {Subst}\left (\int \frac {1+\sqrt {3}-x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1+2 x^4}\right )}{\sqrt {2} x^2}-\frac {\left (3 \sqrt {2+\sqrt {3}} \sqrt {x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1+2 x^4}\right )}{x^2}\\ &=\frac {3 \left (1+2 x^4\right )^{2/3}}{4 x^2}+\frac {6 x^2}{1-\sqrt {3}-\sqrt [3]{1+2 x^4}}-\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1+2 x^4}\right ) \sqrt {\frac {1+\sqrt [3]{1+2 x^4}+\left (1+2 x^4\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1+2 x^4}\right )^2}} E\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{1+2 x^4}}{1-\sqrt {3}-\sqrt [3]{1+2 x^4}}\right )|-7+4 \sqrt {3}\right )}{2 x^2 \sqrt {-\frac {1-\sqrt [3]{1+2 x^4}}{\left (1-\sqrt {3}-\sqrt [3]{1+2 x^4}\right )^2}}}+\frac {\sqrt {2} 3^{3/4} \left (1-\sqrt [3]{1+2 x^4}\right ) \sqrt {\frac {1+\sqrt [3]{1+2 x^4}+\left (1+2 x^4\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1+2 x^4}\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{1+2 x^4}}{1-\sqrt {3}-\sqrt [3]{1+2 x^4}}\right )|-7+4 \sqrt {3}\right )}{x^2 \sqrt {-\frac {1-\sqrt [3]{1+2 x^4}}{\left (1-\sqrt {3}-\sqrt [3]{1+2 x^4}\right )^2}}}-\frac {3}{2} \int \frac {\left (1+2 x^4\right )^{2/3}}{2-x^3+4 x^4} \, dx+8 \int \frac {x \left (1+2 x^4\right )^{2/3}}{2-x^3+4 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 (-3+2 x^4\right ) \left (1+2 x^4\right )^{2/3}}{x^3 \left (2-x^3+4 x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.11, size = 143, normalized size = 1.00 \begin {gather*} \frac {3 \left (1+2 x^4\right )^{2/3}}{4 x^2}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{1+2 x^4}}\right )}{2\ 2^{2/3}}+\frac {\log \left (-x+\sqrt [3]{2} \sqrt [3]{1+2 x^4}\right )}{2\ 2^{2/3}}-\frac {\log \left (x^2+\sqrt [3]{2} x \sqrt [3]{1+2 x^4}+2^{2/3} \left (1+2 x^4\right )^{2/3}\right )}{4\ 2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 107.94, size = 404, normalized size = 2.83 \begin {gather*} -\frac {4 \cdot 4^{\frac {1}{6}} \sqrt {3} x^{2} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (12 \cdot 4^{\frac {2}{3}} {\left (8 \, x^{9} + 2 \, x^{8} - x^{7} + 8 \, x^{5} + x^{4} + 2 \, x\right )} {\left (2 \, x^{4} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (64 \, x^{12} + 240 \, x^{11} + 48 \, x^{10} - x^{9} + 96 \, x^{8} + 240 \, x^{7} + 24 \, x^{6} + 48 \, x^{4} + 60 \, x^{3} + 8\right )} + 12 \, {\left (16 \, x^{10} + 28 \, x^{9} + x^{8} + 16 \, x^{6} + 14 \, x^{5} + 4 \, x^{2}\right )} {\left (2 \, x^{4} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (64 \, x^{12} - 48 \, x^{11} - 96 \, x^{10} - x^{9} + 96 \, x^{8} - 48 \, x^{7} - 48 \, x^{6} + 48 \, x^{4} - 12 \, x^{3} + 8\right )}}\right ) - 2 \cdot 4^{\frac {2}{3}} x^{2} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} {\left (2 \, x^{4} + 1\right )}^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} {\left (4 \, x^{4} - x^{3} + 2\right )} - 12 \, {\left (2 \, x^{4} + 1\right )}^{\frac {2}{3}} x}{4 \, x^{4} - x^{3} + 2}\right ) + 4^{\frac {2}{3}} x^{2} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (2 \, x^{5} + x^{4} + x\right )} {\left (2 \, x^{4} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (16 \, x^{8} + 28 \, x^{7} + x^{6} + 16 \, x^{4} + 14 \, x^{3} + 4\right )} + 6 \, {\left (8 \, x^{6} + x^{5} + 4 \, x^{2}\right )} {\left (2 \, x^{4} + 1\right )}^{\frac {1}{3}}}{16 \, x^{8} - 8 \, x^{7} + x^{6} + 16 \, x^{4} - 4 \, x^{3} + 4}\right ) - 36 \, {\left (2 \, x^{4} + 1\right )}^{\frac {2}{3}}}{48 \, x^{2}} \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} + 1\right )}^{\frac {2}{3}} {\left (2 \, x^{4} - 3\right )}}{{\left (4 \, x^{4} - x^{3} + 2\right )} x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 114.84, size = 664, normalized size = 4.64
method | result | size |
risch | \(\frac {3 \left (2 x^{4}+1\right )^{\frac {2}{3}}}{4 x^{2}}+\frac {\RootOf \left (\textit {\_Z}^{3}-2\right ) \ln \left (\frac {2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{3}-2\right )^{3} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{3}-2 \left (2 x^{4}+1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x -2 \left (2 x^{4}+1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{2}+8 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{4}+\left (2 x^{4}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{2}+4 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{4}+2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{3}+\RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}-4 \left (2 x^{4}+1\right )^{\frac {2}{3}} x +4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )+2 \RootOf \left (\textit {\_Z}^{3}-2\right )}{4 x^{4}-x^{3}+2}\right )}{4}+\frac {\RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) \ln \left (-\frac {2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{3}-2\right )^{3} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{3}-2 \left (2 x^{4}+1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x +4 \left (2 x^{4}+1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{2}-8 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{4}+\left (2 x^{4}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{2}-4 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{4}+2 \left (2 x^{4}+1\right )^{\frac {2}{3}} x -4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )-2 \RootOf \left (\textit {\_Z}^{3}-2\right )}{4 x^{4}-x^{3}+2}\right )}{2}\) | \(664\) |
trager | \(\text {Expression too large to display}\) | \(1485\) |
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} + 1\right )}^{\frac {2}{3}} {\left (2 \, x^{4} - 3\right )}}{{\left (4 \, x^{4} - x^{3} + 2\right )} x^{3}}\,{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 (2\,x^4+1\right )}^{2/3}\,\left (2\,x^4-3\right )}{x^3\,\left (4\,x^4-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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