Optimal. Leaf size=95 \[ \frac {4 \sqrt [4]{x^3-1}}{x}+2 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^3-1}}{\sqrt {x^3-1}-x^2}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^3-1}}{\sqrt {x^3-1}+x^2}\right ) \]
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Rubi [F] time = 1.41, 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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Rubi steps
\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\\ &=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 {\left (1-x^3\right )^{3/4} \int \frac {x}{\left (1-x^3\right )^{3/4}} \, dx}{\left (-1+x^3\right )^{3/4}}-\frac {\left (2 \left (1-x^3\right )^{3/4}\right ) \int \frac {1}{\left (1-x^3\right )^{3/4}} \, dx}{\left (-1+x^3\right )^{3/4}}+\frac {\left (4 \left (1-x^3\right )^{3/4}\right ) \int \frac {1}{x^2 \left (1-x^3\right )^{3/4}} \, dx}{\left (-1+x^3\right )^{3/4}}\\ &=-\frac {4 \left (1-x^3\right )^{3/4} \, _2F_1\left (-\frac {1}{3},\frac {3}{4};\frac {2}{3};x^3\right )}{x \left (-1+x^3\right )^{3/4}}-\frac {2 x \left (1-x^3\right )^{3/4} \, _2F_1\left (\frac {1}{3},\frac {3}{4};\frac {4}{3};x^3\right )}{\left (-1+x^3\right )^{3/4}}+\frac {x^2 \left (1-x^3\right )^{3/4} \, _2F_1\left (\frac {2}{3},\frac {3}{4};\frac {5}{3};x^3\right )}{2 \left (-1+x^3\right )^{3/4}}+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 = 0.46, 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 = 4.24, size = 95, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt [4]{-1+x^3}}{x}+2 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^3}}{-x^2+\sqrt {-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 = 24.71, size = 459, normalized size = 4.83 \begin {gather*} -\frac {4 \, \sqrt {2} x \arctan \left (\frac {\sqrt {2} {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} + \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {3}{4}} x - {\left (x^{4} - \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 2 \, \sqrt {x^{3} - 1} x^{2} - \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {3}{4}} x - 1\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}}}{x^{4} - x^{3} + 1}\right ) + 4 \, \sqrt {2} x \arctan \left (\frac {\sqrt {2} {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} + \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {3}{4}} x + {\left (x^{4} + \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 2 \, \sqrt {x^{3} - 1} x^{2} + \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {3}{4}} x - 1\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}}}{x^{4} - x^{3} + 1}\right ) + \sqrt {2} x \log \left (\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}\right ) - \sqrt {2} x \log \left (\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}\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 = 8.28, size = 221, normalized size = 2.33
method | result | size |
trager | \(\frac {4 \left (x^{3}-1\right )^{\frac {1}{4}}}{x}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {-2 \sqrt {x^{3}-1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-2 \left (x^{3}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}-\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{4}+2 \left (x^{3}-1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{3}-\RootOf \left (\textit {\_Z}^{4}+1\right )}{x^{4}+x^{3}-1}\right )+2 \RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+2 \left (x^{3}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{3}-2 \sqrt {x^{3}-1}\, \RootOf \left (\textit {\_Z}^{4}+1\right ) x^{2}+2 \left (x^{3}-1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}}{x^{4}+x^{3}-1}\right )\) | \(221\) |
risch | \(\frac {4 \left (x^{3}-1\right )^{\frac {1}{4}}}{x}+\frac {\left (-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 (-1+x \right )^{2} \left (x^{2}+x +1\right )^{2} \left (x^{4}+x^{3}-1\right )}\right )-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 \left (x^{9}-3 x^{6}+3 x^{3}-1\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{3}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{7}-2 \left (x^{9}-3 x^{6}+3 x^{3}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right ) 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 \left (x^{9}-3 x^{6}+3 x^{3}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right ) 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 \left (x^{9}-3 x^{6}+3 x^{3}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right ) x -\RootOf \left (\textit {\_Z}^{4}+1\right )^{2}}{\left (-1+x \right )^{2} \left (x^{2}+x +1\right )^{2} \left (x^{4}+x^{3}-1\right )}\right )\right ) \left (\left (x^{3}-1\right )^{3}\right )^{\frac {1}{4}}}{\left (x^{3}-1\right )^{\frac {3}{4}}}\) | \(593\) |
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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