Optimal. Leaf size=101 \[ \frac {4 \sqrt [4]{x^3+1}}{x}-2 \sqrt {2} \tan ^{-1}\left (\frac {\frac {\sqrt {x^3+1}}{\sqrt {2}}-\frac {x^2}{\sqrt {2}}}{x \sqrt [4]{x^3+1}}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^3+1}}{\sqrt {x^3+1}+x^2}\right ) \]
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Rubi [F] time = 1.50, 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 (4+x^3\right ) \left (-1-x^3+x^4\right )}{x^2 \left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {\left (4+x^3\right ) \left (-1-x^3+x^4\right )}{x^2 \left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx &=\int \left (-\frac {2}{\left (1+x^3\right )^{3/4}}-\frac {4}{x^2 \left (1+x^3\right )^{3/4}}+\frac {x}{\left (1+x^3\right )^{3/4}}+\frac {2 \left (1-x+4 x^2+x^3\right )}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )}\right ) \, dx\\ &=-\left (2 \int \frac {1}{\left (1+x^3\right )^{3/4}} \, dx\right )+2 \int \frac {1-x+4 x^2+x^3}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx-4 \int \frac {1}{x^2 \left (1+x^3\right )^{3/4}} \, dx+\int \frac {x}{\left (1+x^3\right )^{3/4}} \, dx\\ &=\frac {4 \, _2F_1\left (-\frac {1}{3},\frac {3}{4};\frac {2}{3};-x^3\right )}{x}-2 x \, _2F_1\left (\frac {1}{3},\frac {3}{4};\frac {4}{3};-x^3\right )+\frac {1}{2} x^2 \, _2F_1\left (\frac {2}{3},\frac {3}{4};\frac {5}{3};-x^3\right )+2 \int \left (\frac {1}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )}-\frac {x}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )}+\frac {4 x^2}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )}+\frac {x^3}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )}\right ) \, dx\\ &=\frac {4 \, _2F_1\left (-\frac {1}{3},\frac {3}{4};\frac {2}{3};-x^3\right )}{x}-2 x \, _2F_1\left (\frac {1}{3},\frac {3}{4};\frac {4}{3};-x^3\right )+\frac {1}{2} x^2 \, _2F_1\left (\frac {2}{3},\frac {3}{4};\frac {5}{3};-x^3\right )+2 \int \frac {1}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx-2 \int \frac {x}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx+2 \int \frac {x^3}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx+8 \int \frac {x^2}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx\\ \end {align*}
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Mathematica [F] time = 0.45, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (4+x^3\right ) \left (-1-x^3+x^4\right )}{x^2 \left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 4.12, size = 101, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt [4]{1+x^3}}{x}-2 \sqrt {2} \tan ^{-1}\left (\frac {-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {1+x^3}}{\sqrt {2}}}{x \sqrt [4]{1+x^3}}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{1+x^3}}{x^2+\sqrt {1+x^3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 25.98, size = 709, normalized size = 7.02 \begin {gather*} \frac {4 \, \sqrt {2} x \arctan \left (-\frac {x^{8} + 2 \, x^{7} + x^{6} + 2 \, x^{4} + 2 \, x^{3} + 2 \, \sqrt {2} {\left (3 \, x^{5} - x^{4} - x\right )} {\left (x^{3} + 1\right )}^{\frac {3}{4}} + 2 \, \sqrt {2} {\left (x^{7} - 3 \, x^{6} - 3 \, x^{3}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}} + 4 \, {\left (x^{6} + x^{5} + x^{2}\right )} \sqrt {x^{3} + 1} - {\left (16 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}} x^{5} + 2 \, \sqrt {2} {\left (3 \, x^{6} - x^{5} - x^{2}\right )} \sqrt {x^{3} + 1} + \sqrt {2} {\left (x^{8} + 8 \, x^{7} - x^{6} + 8 \, x^{4} - 2 \, x^{3} - 1\right )} + 4 \, {\left (x^{7} + x^{6} + x^{3}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{4} - 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 4 \, \sqrt {x^{3} + 1} x^{2} - 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {3}{4}} x + 1}{x^{4} + x^{3} + 1}} + 1}{x^{8} - 14 \, x^{7} + x^{6} - 14 \, x^{4} + 2 \, x^{3} + 1}\right ) - 4 \, \sqrt {2} x \arctan \left (-\frac {x^{8} + 2 \, x^{7} + x^{6} + 2 \, x^{4} + 2 \, x^{3} - 2 \, \sqrt {2} {\left (3 \, x^{5} - x^{4} - x\right )} {\left (x^{3} + 1\right )}^{\frac {3}{4}} - 2 \, \sqrt {2} {\left (x^{7} - 3 \, x^{6} - 3 \, x^{3}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}} + 4 \, {\left (x^{6} + x^{5} + x^{2}\right )} \sqrt {x^{3} + 1} - {\left (16 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}} x^{5} - 2 \, \sqrt {2} {\left (3 \, x^{6} - x^{5} - x^{2}\right )} \sqrt {x^{3} + 1} - \sqrt {2} {\left (x^{8} + 8 \, x^{7} - x^{6} + 8 \, x^{4} - 2 \, x^{3} - 1\right )} + 4 \, {\left (x^{7} + x^{6} + x^{3}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{4} + 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 4 \, \sqrt {x^{3} + 1} x^{2} + 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {3}{4}} x + 1}{x^{4} + x^{3} + 1}} + 1}{x^{8} - 14 \, x^{7} + x^{6} - 14 \, x^{4} + 2 \, x^{3} + 1}\right ) - \sqrt {2} x \log \left (\frac {4 \, {\left (x^{4} + 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 4 \, \sqrt {x^{3} + 1} x^{2} + 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {3}{4}} x + 1\right )}}{x^{4} + x^{3} + 1}\right ) + \sqrt {2} x \log \left (\frac {4 \, {\left (x^{4} - 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 4 \, \sqrt {x^{3} + 1} x^{2} - 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {3}{4}} x + 1\right )}}{x^{4} + x^{3} + 1}\right ) + 8 \, {\left (x^{3} + 1\right )}^{\frac {1}{4}}}{2 \, x} \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 (x^{4} - x^{3} - 1\right )} {\left (x^{3} + 4\right )}}{{\left (x^{4} + x^{3} + 1\right )} {\left (x^{3} + 1\right )}^{\frac {3}{4}} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 7.10, size = 222, normalized size = 2.20
method | result | size |
trager | \(\frac {4 \left (x^{3}+1\right )^{\frac {1}{4}}}{x}+2 \RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{4}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{3}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}+2 \RootOf \left (\textit {\_Z}^{4}+1\right ) \sqrt {x^{3}+1}\, x^{2}-2 \left (x^{3}+1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}}{x^{4}+x^{3}+1}\right )+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \sqrt {x^{3}+1}\, x^{2}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}+\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{4}-\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{3}-2 \left (x^{3}+1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}+1\right )}{x^{4}+x^{3}+1}\right )\) | \(222\) |
risch | \(\frac {4 \left (x^{3}+1\right )^{\frac {1}{4}}}{x}+\frac {\left (2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{10}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{9}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{9}+3 x^{6}+3 x^{3}+1\right )^{\frac {3}{4}} x^{3}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{7}-2 \RootOf \left (\textit {\_Z}^{4}+1\right ) \left (x^{9}+3 x^{6}+3 x^{3}+1\right )^{\frac {1}{4}} x^{7}-3 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{6}-2 \sqrt {x^{9}+3 x^{6}+3 x^{3}+1}\, x^{5}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{4}-4 \RootOf \left (\textit {\_Z}^{4}+1\right ) \left (x^{9}+3 x^{6}+3 x^{3}+1\right )^{\frac {1}{4}} x^{4}-3 x^{3} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2}-2 \sqrt {x^{9}+3 x^{6}+3 x^{3}+1}\, x^{2}-2 \RootOf \left (\textit {\_Z}^{4}+1\right ) \left (x^{9}+3 x^{6}+3 x^{3}+1\right )^{\frac {1}{4}} x -\RootOf \left (\textit {\_Z}^{4}+1\right )^{2}}{\left (x^{4}+x^{3}+1\right ) \left (1+x \right )^{2} \left (x^{2}-x +1\right )^{2}}\right )+2 \RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{10}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{9}+3 x^{6}+3 x^{3}+1\right )^{\frac {1}{4}} x^{7}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{9}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{7}+4 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{9}+3 x^{6}+3 x^{3}+1\right )^{\frac {1}{4}} x^{4}-3 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{6}-2 \RootOf \left (\textit {\_Z}^{4}+1\right ) \left (x^{9}+3 x^{6}+3 x^{3}+1\right )^{\frac {3}{4}} x^{3}+2 \sqrt {x^{9}+3 x^{6}+3 x^{3}+1}\, x^{5}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{4}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{9}+3 x^{6}+3 x^{3}+1\right )^{\frac {1}{4}} x -3 x^{3} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2}+2 \sqrt {x^{9}+3 x^{6}+3 x^{3}+1}\, x^{2}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{2}}{\left (x^{4}+x^{3}+1\right ) \left (1+x \right )^{2} \left (x^{2}-x +1\right )^{2}}\right )\right ) \left (\left (x^{3}+1\right )^{3}\right )^{\frac {1}{4}}}{\left (x^{3}+1\right )^{\frac {3}{4}}}\) | \(595\) |
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 (x^{4} - x^{3} - 1\right )} {\left (x^{3} + 4\right )}}{{\left (x^{4} + x^{3} + 1\right )} {\left (x^{3} + 1\right )}^{\frac {3}{4}} x^{2}}\,{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^3+4\right )\,\left (-x^4+x^3+1\right )}{x^2\,{\left (x^3+1\right )}^{3/4}\,\left (x^4+x^3+1\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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