Optimal. Leaf size=123 \[ \frac {4 \left (-1+x^3\right ) \left (-x+x^4\right )^{3/4}}{21 x^6}-\frac {1}{3} \sqrt {2} \text {ArcTan}\left (\frac {\sqrt {2} x \sqrt [4]{-x+x^4}}{-x^2+\sqrt {-x+x^4}}\right )-\frac {1}{3} \sqrt {2} \tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-x+x^4}}{\sqrt {2}}}{x \sqrt [4]{-x+x^4}}\right ) \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(338\) vs. \(2(123)=246\).
time = 0.52, antiderivative size = 338, normalized size of antiderivative = 2.75, number of steps
used = 20, number of rules used = 14, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {2081, 6857,
277, 270, 477, 476, 508, 472, 217, 1179, 642, 1176, 631, 210} \begin {gather*} \frac {\sqrt {2} \sqrt [4]{x} \sqrt [4]{x^3-1} \text {ArcTan}\left (1-\frac {\sqrt {2} x^{3/4}}{\sqrt [4]{x^3-1}}\right )}{3 \sqrt [4]{x^4-x}}-\frac {\sqrt {2} \sqrt [4]{x} \sqrt [4]{x^3-1} \text {ArcTan}\left (\frac {\sqrt {2} x^{3/4}}{\sqrt [4]{x^3-1}}+1\right )}{3 \sqrt [4]{x^4-x}}+\frac {\sqrt [4]{x} \sqrt [4]{x^3-1} \log \left (\frac {x^{3/2}}{\sqrt {x^3-1}}-\frac {\sqrt {2} x^{3/4}}{\sqrt [4]{x^3-1}}+1\right )}{3 \sqrt {2} \sqrt [4]{x^4-x}}-\frac {\sqrt [4]{x} \sqrt [4]{x^3-1} \log \left (\frac {x^{3/2}}{\sqrt {x^3-1}}+\frac {\sqrt {2} x^{3/4}}{\sqrt [4]{x^3-1}}+1\right )}{3 \sqrt {2} \sqrt [4]{x^4-x}}+\frac {\left (1-x^3\right )^2}{21 x^5 \sqrt [4]{x^4-x}}+\frac {1-x^3}{7 x^5 \sqrt [4]{x^4-x}}-\frac {1-x^3}{7 x^2 \sqrt [4]{x^4-x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 217
Rule 270
Rule 277
Rule 472
Rule 476
Rule 477
Rule 508
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 2081
Rule 6857
Rubi steps
\begin {align*} \int \frac {1-3 x^3+3 x^6}{x^6 \left (-1+2 x^3\right ) \sqrt [4]{-x+x^4}} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \int \frac {1-3 x^3+3 x^6}{x^{25/4} \sqrt [4]{-1+x^3} \left (-1+2 x^3\right )} \, dx}{\sqrt [4]{-x+x^4}}\\ &=\frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \int \left (-\frac {3}{4 x^{25/4} \sqrt [4]{-1+x^3}}+\frac {3}{2 x^{13/4} \sqrt [4]{-1+x^3}}+\frac {1}{4 x^{25/4} \sqrt [4]{-1+x^3} \left (-1+2 x^3\right )}\right ) \, dx}{\sqrt [4]{-x+x^4}}\\ &=\frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \int \frac {1}{x^{25/4} \sqrt [4]{-1+x^3} \left (-1+2 x^3\right )} \, dx}{4 \sqrt [4]{-x+x^4}}-\frac {\left (3 \sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \int \frac {1}{x^{25/4} \sqrt [4]{-1+x^3}} \, dx}{4 \sqrt [4]{-x+x^4}}+\frac {\left (3 \sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \int \frac {1}{x^{13/4} \sqrt [4]{-1+x^3}} \, dx}{2 \sqrt [4]{-x+x^4}}\\ &=\frac {1-x^3}{7 x^5 \sqrt [4]{-x+x^4}}-\frac {2 \left (1-x^3\right )}{3 x^2 \sqrt [4]{-x+x^4}}-\frac {\left (3 \sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \int \frac {1}{x^{13/4} \sqrt [4]{-1+x^3}} \, dx}{7 \sqrt [4]{-x+x^4}}+\frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{x^{22} \sqrt [4]{-1+x^{12}} \left (-1+2 x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x+x^4}}\\ &=\frac {1-x^3}{7 x^5 \sqrt [4]{-x+x^4}}-\frac {10 \left (1-x^3\right )}{21 x^2 \sqrt [4]{-x+x^4}}+\frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{x^8 \sqrt [4]{-1+x^4} \left (-1+2 x^4\right )} \, dx,x,x^{3/4}\right )}{3 \sqrt [4]{-x+x^4}}\\ &=\frac {1-x^3}{7 x^5 \sqrt [4]{-x+x^4}}-\frac {10 \left (1-x^3\right )}{21 x^2 \sqrt [4]{-x+x^4}}+\frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \text {Subst}\left (\int \frac {\left (1-x^4\right )^2}{x^8 \left (-1-x^4\right )} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt [4]{-x+x^4}}\\ &=\frac {1-x^3}{7 x^5 \sqrt [4]{-x+x^4}}-\frac {10 \left (1-x^3\right )}{21 x^2 \sqrt [4]{-x+x^4}}+\frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \text {Subst}\left (\int \left (-\frac {1}{x^8}+\frac {3}{x^4}-\frac {4}{1+x^4}\right ) \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt [4]{-x+x^4}}\\ &=\frac {1-x^3}{7 x^5 \sqrt [4]{-x+x^4}}-\frac {1-x^3}{7 x^2 \sqrt [4]{-x+x^4}}+\frac {\left (1-x^3\right )^2}{21 x^5 \sqrt [4]{-x+x^4}}-\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt [4]{-x+x^4}}\\ &=\frac {1-x^3}{7 x^5 \sqrt [4]{-x+x^4}}-\frac {1-x^3}{7 x^2 \sqrt [4]{-x+x^4}}+\frac {\left (1-x^3\right )^2}{21 x^5 \sqrt [4]{-x+x^4}}-\frac {\left (2 \sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt [4]{-x+x^4}}-\frac {\left (2 \sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt [4]{-x+x^4}}\\ &=\frac {1-x^3}{7 x^5 \sqrt [4]{-x+x^4}}-\frac {1-x^3}{7 x^2 \sqrt [4]{-x+x^4}}+\frac {\left (1-x^3\right )^2}{21 x^5 \sqrt [4]{-x+x^4}}-\frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt [4]{-x+x^4}}-\frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt [4]{-x+x^4}}+\frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt {2} \sqrt [4]{-x+x^4}}+\frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt {2} \sqrt [4]{-x+x^4}}\\ &=\frac {1-x^3}{7 x^5 \sqrt [4]{-x+x^4}}-\frac {1-x^3}{7 x^2 \sqrt [4]{-x+x^4}}+\frac {\left (1-x^3\right )^2}{21 x^5 \sqrt [4]{-x+x^4}}+\frac {\sqrt [4]{x} \sqrt [4]{-1+x^3} \log \left (1+\frac {x^{3/2}}{\sqrt {-1+x^3}}-\frac {\sqrt {2} x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt {2} \sqrt [4]{-x+x^4}}-\frac {\sqrt [4]{x} \sqrt [4]{-1+x^3} \log \left (1+\frac {x^{3/2}}{\sqrt {-1+x^3}}+\frac {\sqrt {2} x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt {2} \sqrt [4]{-x+x^4}}-\frac {\left (\sqrt {2} \sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt [4]{-x+x^4}}+\frac {\left (\sqrt {2} \sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt [4]{-x+x^4}}\\ &=\frac {1-x^3}{7 x^5 \sqrt [4]{-x+x^4}}-\frac {1-x^3}{7 x^2 \sqrt [4]{-x+x^4}}+\frac {\left (1-x^3\right )^2}{21 x^5 \sqrt [4]{-x+x^4}}+\frac {\sqrt {2} \sqrt [4]{x} \sqrt [4]{-1+x^3} \tan ^{-1}\left (1-\frac {\sqrt {2} x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt [4]{-x+x^4}}-\frac {\sqrt {2} \sqrt [4]{x} \sqrt [4]{-1+x^3} \tan ^{-1}\left (1+\frac {\sqrt {2} x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt [4]{-x+x^4}}+\frac {\sqrt [4]{x} \sqrt [4]{-1+x^3} \log \left (1+\frac {x^{3/2}}{\sqrt {-1+x^3}}-\frac {\sqrt {2} x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt {2} \sqrt [4]{-x+x^4}}-\frac {\sqrt [4]{x} \sqrt [4]{-1+x^3} \log \left (1+\frac {x^{3/2}}{\sqrt {-1+x^3}}+\frac {\sqrt {2} x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt {2} \sqrt [4]{-x+x^4}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 11.03, size = 192, normalized size = 1.56 \begin {gather*} \frac {-45+240 x^3-345 x^6+150 x^9-5 \left (3+13 x^3-144 x^6+128 x^9\right ) \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {x^3}{1-x^3}\right )-8 x^3 \, _2F_1\left (\frac {5}{4},2;\frac {9}{4};\frac {x^3}{1-x^3}\right )-80 x^6 \, _2F_1\left (\frac {5}{4},2;\frac {9}{4};\frac {x^3}{1-x^3}\right )+192 x^9 \, _2F_1\left (\frac {5}{4},2;\frac {9}{4};\frac {x^3}{1-x^3}\right )+16 x^3 \left (1-2 x^3\right )^2 \, _3F_2\left (\frac {5}{4},2,2;1,\frac {9}{4};\frac {x^3}{1-x^3}\right )}{315 x^4 \left (x \left (-1+x^3\right )\right )^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 4.88, size = 183, normalized size = 1.49
method | result | size |
trager | \(\frac {4 \left (x^{3}-1\right ) \left (x^{4}-x \right )^{\frac {3}{4}}}{21 x^{6}}-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {2 \left (x^{4}-x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-2 \sqrt {x^{4}-x}\, \RootOf \left (\textit {\_Z}^{4}+1\right ) x +2 \left (x^{4}-x \right )^{\frac {3}{4}}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}}{2 x^{3}-1}\right )}{3}+\frac {\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}-x}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x -2 \left (x^{4}-x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+2 \left (x^{4}-x \right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{4}+1\right )}{2 x^{3}-1}\right )}{3}\) | \(183\) |
risch | \(\frac {\frac {4}{21} x^{6}-\frac {8}{21} x^{3}+\frac {4}{21}}{x^{5} \left (x \left (x^{3}-1\right )\right )^{\frac {1}{4}}}+\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {2 \left (x^{4}-x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+2 \sqrt {x^{4}-x}\, \RootOf \left (\textit {\_Z}^{4}+1\right ) x +2 \left (x^{4}-x \right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}}{2 x^{3}-1}\right )}{3}-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {-2 \sqrt {x^{4}-x}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x -2 \left (x^{4}-x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+2 \left (x^{4}-x \right )^{\frac {3}{4}}-\RootOf \left (\textit {\_Z}^{4}+1\right )}{2 x^{3}-1}\right )}{3}\) | \(188\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 479 vs.
\(2 (99) = 198\).
time = 1.45, size = 479, normalized size = 3.89 \begin {gather*} \frac {28 \, \sqrt {2} x^{6} \arctan \left (-\frac {\sqrt {2} {\left (x^{4} - x\right )}^{\frac {3}{4}} x - \sqrt {2} {\left (x^{4} - x\right )}^{\frac {1}{4}} {\left (x^{3} - 1\right )} + {\left (2 \, x^{4} - \sqrt {2} {\left (x^{4} - x\right )}^{\frac {3}{4}} x - \sqrt {2} {\left (x^{4} - x\right )}^{\frac {1}{4}} {\left (x^{3} - 1\right )} - 2 \, x\right )} \sqrt {\frac {2 \, x^{3} + 2 \, \sqrt {2} {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {x^{4} - x} x + 2 \, \sqrt {2} {\left (x^{4} - x\right )}^{\frac {3}{4}} - 1}{2 \, x^{3} - 1}}}{2 \, {\left (x^{4} - x\right )}}\right ) + 28 \, \sqrt {2} x^{6} \arctan \left (-\frac {\sqrt {2} {\left (x^{4} - x\right )}^{\frac {3}{4}} x - \sqrt {2} {\left (x^{4} - x\right )}^{\frac {1}{4}} {\left (x^{3} - 1\right )} - {\left (2 \, x^{4} + \sqrt {2} {\left (x^{4} - x\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (x^{4} - x\right )}^{\frac {1}{4}} {\left (x^{3} - 1\right )} - 2 \, x\right )} \sqrt {\frac {2 \, x^{3} - 2 \, \sqrt {2} {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {x^{4} - x} x - 2 \, \sqrt {2} {\left (x^{4} - x\right )}^{\frac {3}{4}} - 1}{2 \, x^{3} - 1}}}{2 \, {\left (x^{4} - x\right )}}\right ) - 7 \, \sqrt {2} x^{6} \log \left (\frac {2 \, x^{3} + 2 \, \sqrt {2} {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {x^{4} - x} x + 2 \, \sqrt {2} {\left (x^{4} - x\right )}^{\frac {3}{4}} - 1}{2 \, x^{3} - 1}\right ) + 7 \, \sqrt {2} x^{6} \log \left (\frac {2 \, x^{3} - 2 \, \sqrt {2} {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {x^{4} - x} x - 2 \, \sqrt {2} {\left (x^{4} - x\right )}^{\frac {3}{4}} - 1}{2 \, x^{3} - 1}\right ) + 16 \, {\left (x^{4} - x\right )}^{\frac {3}{4}} {\left (x^{3} - 1\right )}}{84 \, x^{6}} \end {gather*}
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 \begin {gather*} \int \frac {3 x^{6} - 3 x^{3} + 1}{x^{6} \sqrt [4]{x \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (2 x^{3} - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 125, normalized size = 1.02 \begin {gather*} -\frac {4}{21} \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {7}{4}} - \frac {1}{3} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{3} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {1}{6} \, \sqrt {2} \log \left (\sqrt {2} {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x^{3}} + 1} + 1\right ) - \frac {1}{6} \, \sqrt {2} \log \left (-\sqrt {2} {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x^{3}} + 1} + 1\right ) \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 {3\,x^6-3\,x^3+1}{x^6\,{\left (x^4-x\right )}^{1/4}\,\left (2\,x^3-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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