Optimal. Leaf size=107 \[ \frac {\left (-x^2+x^4\right )^{3/4} \left (-85+2 x^2+67 x^4\right )}{80 x \left (-1+x^2\right )^2 \left (1+x^2\right )}+\frac {15 \text {ArcTan}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^2+x^4}}\right )}{32 \sqrt [4]{2}}+\frac {15 \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^2+x^4}}\right )}{32 \sqrt [4]{2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 194, normalized size of antiderivative = 1.81, number of steps
used = 11, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {2081, 1268,
477, 425, 541, 12, 385, 218, 212, 209} \begin {gather*} \frac {15 \sqrt [4]{x^2-1} \sqrt {x} \text {ArcTan}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{32 \sqrt [4]{2} \sqrt [4]{x^4-x^2}}-\frac {x}{40 \left (1-x^2\right ) \sqrt [4]{x^4-x^2}}+\frac {x}{4 \left (1-x^2\right ) \left (x^2+1\right ) \sqrt [4]{x^4-x^2}}+\frac {67 x}{80 \sqrt [4]{x^4-x^2}}+\frac {15 \sqrt [4]{x^2-1} \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{32 \sqrt [4]{2} \sqrt [4]{x^4-x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 209
Rule 212
Rule 218
Rule 385
Rule 425
Rule 477
Rule 541
Rule 1268
Rule 2081
Rubi steps
\begin {align*} \int \frac {1}{\left (-1+x^4\right )^2 \sqrt [4]{-x^2+x^4}} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt [4]{-1+x^2} \left (-1+x^4\right )^2} \, dx}{\sqrt [4]{-x^2+x^4}}\\ &=\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \int \frac {1}{\sqrt {x} \left (-1+x^2\right )^{9/4} \left (1+x^2\right )^2} \, dx}{\sqrt [4]{-x^2+x^4}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+x^4\right )^{9/4} \left (1+x^4\right )^2} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-x^2+x^4}}\\ &=\frac {x}{4 \left (1-x^2\right ) \left (1+x^2\right ) \sqrt [4]{-x^2+x^4}}+\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {7-8 x^4}{\left (-1+x^4\right )^{9/4} \left (1+x^4\right )} \, dx,x,\sqrt {x}\right )}{4 \sqrt [4]{-x^2+x^4}}\\ &=-\frac {x}{40 \left (1-x^2\right ) \sqrt [4]{-x^2+x^4}}+\frac {x}{4 \left (1-x^2\right ) \left (1+x^2\right ) \sqrt [4]{-x^2+x^4}}+\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {-71+4 x^4}{\left (-1+x^4\right )^{5/4} \left (1+x^4\right )} \, dx,x,\sqrt {x}\right )}{40 \sqrt [4]{-x^2+x^4}}\\ &=\frac {67 x}{80 \sqrt [4]{-x^2+x^4}}-\frac {x}{40 \left (1-x^2\right ) \sqrt [4]{-x^2+x^4}}+\frac {x}{4 \left (1-x^2\right ) \left (1+x^2\right ) \sqrt [4]{-x^2+x^4}}+\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {75}{\sqrt [4]{-1+x^4} \left (1+x^4\right )} \, dx,x,\sqrt {x}\right )}{80 \sqrt [4]{-x^2+x^4}}\\ &=\frac {67 x}{80 \sqrt [4]{-x^2+x^4}}-\frac {x}{40 \left (1-x^2\right ) \sqrt [4]{-x^2+x^4}}+\frac {x}{4 \left (1-x^2\right ) \left (1+x^2\right ) \sqrt [4]{-x^2+x^4}}+\frac {\left (15 \sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-1+x^4} \left (1+x^4\right )} \, dx,x,\sqrt {x}\right )}{16 \sqrt [4]{-x^2+x^4}}\\ &=\frac {67 x}{80 \sqrt [4]{-x^2+x^4}}-\frac {x}{40 \left (1-x^2\right ) \sqrt [4]{-x^2+x^4}}+\frac {x}{4 \left (1-x^2\right ) \left (1+x^2\right ) \sqrt [4]{-x^2+x^4}}+\frac {\left (15 \sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1-2 x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{16 \sqrt [4]{-x^2+x^4}}\\ &=\frac {67 x}{80 \sqrt [4]{-x^2+x^4}}-\frac {x}{40 \left (1-x^2\right ) \sqrt [4]{-x^2+x^4}}+\frac {x}{4 \left (1-x^2\right ) \left (1+x^2\right ) \sqrt [4]{-x^2+x^4}}+\frac {\left (15 \sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{32 \sqrt [4]{-x^2+x^4}}+\frac {\left (15 \sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{32 \sqrt [4]{-x^2+x^4}}\\ &=\frac {67 x}{80 \sqrt [4]{-x^2+x^4}}-\frac {x}{40 \left (1-x^2\right ) \sqrt [4]{-x^2+x^4}}+\frac {x}{4 \left (1-x^2\right ) \left (1+x^2\right ) \sqrt [4]{-x^2+x^4}}+\frac {15 \sqrt {x} \sqrt [4]{-1+x^2} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{32 \sqrt [4]{2} \sqrt [4]{-x^2+x^4}}+\frac {15 \sqrt {x} \sqrt [4]{-1+x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{32 \sqrt [4]{2} \sqrt [4]{-x^2+x^4}}\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 133, normalized size = 1.24 \begin {gather*} \frac {x^{5/2} \left (4 \sqrt {x} \left (-85+2 x^2+67 x^4\right )+75\ 2^{3/4} \sqrt [4]{-1+x^2} \left (-1+x^4\right ) \text {ArcTan}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )+75\ 2^{3/4} \sqrt [4]{-1+x^2} \left (-1+x^4\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )\right )}{320 \left (x^2 \left (-1+x^2\right )\right )^{5/4} \left (1+x^2\right )} \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 = 6.64, size = 275, normalized size = 2.57
method | result | size |
risch | \(\frac {x \left (67 x^{4}+2 x^{2}-85\right )}{80 \left (x^{2}-1\right ) \left (x^{2} \left (x^{2}-1\right )\right )^{\frac {1}{4}} \left (x^{2}+1\right )}+\frac {15 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (-\frac {\sqrt {x^{4}-x^{2}}\, \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x +2 \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} \left (x^{4}-x^{2}\right )^{\frac {1}{4}} x^{2}-3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{3}-4 \left (x^{4}-x^{2}\right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x}{x \left (x^{2}+1\right )}\right )}{128}-\frac {15 \RootOf \left (\textit {\_Z}^{4}-8\right ) \ln \left (-\frac {\sqrt {x^{4}-x^{2}}\, \RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x -2 \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} \left (x^{4}-x^{2}\right )^{\frac {1}{4}} x^{2}+3 \RootOf \left (\textit {\_Z}^{4}-8\right ) x^{3}-4 \left (x^{4}-x^{2}\right )^{\frac {3}{4}}-\RootOf \left (\textit {\_Z}^{4}-8\right ) x}{x \left (x^{2}+1\right )}\right )}{128}\) | \(275\) |
trager | \(\frac {\left (x^{4}-x^{2}\right )^{\frac {3}{4}} \left (67 x^{4}+2 x^{2}-85\right )}{80 x \left (x^{2}-1\right )^{2} \left (x^{2}+1\right )}+\frac {15 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (-\frac {\sqrt {x^{4}-x^{2}}\, \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x +2 \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} \left (x^{4}-x^{2}\right )^{\frac {1}{4}} x^{2}-3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{3}-4 \left (x^{4}-x^{2}\right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x}{x \left (x^{2}+1\right )}\right )}{128}-\frac {15 \RootOf \left (\textit {\_Z}^{4}-8\right ) \ln \left (-\frac {\sqrt {x^{4}-x^{2}}\, \RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x -2 \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} \left (x^{4}-x^{2}\right )^{\frac {1}{4}} x^{2}+3 \RootOf \left (\textit {\_Z}^{4}-8\right ) x^{3}-4 \left (x^{4}-x^{2}\right )^{\frac {3}{4}}-\RootOf \left (\textit {\_Z}^{4}-8\right ) x}{x \left (x^{2}+1\right )}\right )}{128}\) | \(277\) |
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. 331 vs.
\(2 (87) = 174\).
time = 1.42, size = 331, normalized size = 3.09 \begin {gather*} -\frac {300 \cdot 2^{\frac {3}{4}} {\left (x^{7} - x^{5} - x^{3} + x\right )} \arctan \left (\frac {4 \cdot 2^{\frac {3}{4}} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} x^{2} + 2^{\frac {3}{4}} {\left (2 \cdot 2^{\frac {3}{4}} \sqrt {x^{4} - x^{2}} x + 2^{\frac {1}{4}} {\left (3 \, x^{3} - x\right )}\right )} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}}{2 \, {\left (x^{3} + x\right )}}\right ) - 75 \cdot 2^{\frac {3}{4}} {\left (x^{7} - x^{5} - x^{3} + x\right )} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} x^{2} + 2^{\frac {3}{4}} {\left (3 \, x^{3} - x\right )} + 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{4} - x^{2}} x + 4 \, {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}}{x^{3} + x}\right ) + 75 \cdot 2^{\frac {3}{4}} {\left (x^{7} - x^{5} - x^{3} + x\right )} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} x^{2} - 2^{\frac {3}{4}} {\left (3 \, x^{3} - x\right )} - 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{4} - x^{2}} x + 4 \, {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}}{x^{3} + x}\right ) - 16 \, {\left (67 \, x^{4} + 2 \, x^{2} - 85\right )} {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}}{1280 \, {\left (x^{7} - x^{5} - x^{3} + x\right )}} \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 {1}{\sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right )} \left (x - 1\right )^{2} \left (x + 1\right )^{2} \left (x^{2} + 1\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 104, normalized size = 0.97 \begin {gather*} \frac {15}{64} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {15}{128} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {15}{128} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} - {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {\frac {10}{x^{2}} - 9}{10 \, {\left (\frac {1}{x^{2}} - 1\right )} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}} + \frac {{\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {3}{4}}}{16 \, {\left (\frac {1}{x^{2}} + 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 {1}{{\left (x^4-1\right )}^2\,{\left (x^4-x^2\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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