Optimal. Leaf size=94 \[ -\frac {3^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt {2} \sqrt [4]{x^4-1}}\right )}{8 \sqrt {2}}-\frac {3^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt {2} \sqrt [4]{x^4-1}}\right )}{8 \sqrt {2}}+\frac {\left (x^4-1\right )^{3/4} \left (x^4+6\right )}{42 x^7} \]
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Rubi [A] time = 0.12, antiderivative size = 105, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {580, 583, 12, 377, 212, 206, 203} \begin {gather*} -\frac {3^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt {2} \sqrt [4]{x^4-1}}\right )}{8 \sqrt {2}}-\frac {3^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt {2} \sqrt [4]{x^4-1}}\right )}{8 \sqrt {2}}+\frac {\left (x^4-1\right )^{3/4}}{7 x^7}+\frac {\left (x^4-1\right )^{3/4}}{42 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 203
Rule 206
Rule 212
Rule 377
Rule 580
Rule 583
Rubi steps
\begin {align*} \int \frac {\left (-1+x^4\right )^{3/4} \left (4+x^4\right )}{x^8 \left (-4+x^4\right )} \, dx &=\frac {\left (-1+x^4\right )^{3/4}}{7 x^7}-\frac {1}{28} \int \frac {8-44 x^4}{x^4 \left (-4+x^4\right ) \sqrt [4]{-1+x^4}} \, dx\\ &=\frac {\left (-1+x^4\right )^{3/4}}{7 x^7}+\frac {\left (-1+x^4\right )^{3/4}}{42 x^3}+\frac {1}{336} \int \frac {504}{\left (-4+x^4\right ) \sqrt [4]{-1+x^4}} \, dx\\ &=\frac {\left (-1+x^4\right )^{3/4}}{7 x^7}+\frac {\left (-1+x^4\right )^{3/4}}{42 x^3}+\frac {3}{2} \int \frac {1}{\left (-4+x^4\right ) \sqrt [4]{-1+x^4}} \, dx\\ &=\frac {\left (-1+x^4\right )^{3/4}}{7 x^7}+\frac {\left (-1+x^4\right )^{3/4}}{42 x^3}+\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{-4+3 x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=\frac {\left (-1+x^4\right )^{3/4}}{7 x^7}+\frac {\left (-1+x^4\right )^{3/4}}{42 x^3}-\frac {3}{8} \operatorname {Subst}\left (\int \frac {1}{2-\sqrt {3} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {3}{8} \operatorname {Subst}\left (\int \frac {1}{2+\sqrt {3} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=\frac {\left (-1+x^4\right )^{3/4}}{7 x^7}+\frac {\left (-1+x^4\right )^{3/4}}{42 x^3}-\frac {3^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt {2} \sqrt [4]{-1+x^4}}\right )}{8 \sqrt {2}}-\frac {3^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt {2} \sqrt [4]{-1+x^4}}\right )}{8 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 118, normalized size = 1.26 \begin {gather*} \frac {\left (x^4-1\right )^{3/4} \left (x^4+6\right )}{42 x^7}-\frac {3^{3/4} \left (-\log \left (2-\frac {\sqrt {2} \sqrt [4]{3} x}{\sqrt [4]{1-x^4}}\right )+\log \left (\frac {\sqrt {2} \sqrt [4]{3} x}{\sqrt [4]{1-x^4}}+2\right )+2 \tan ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt {2} \sqrt [4]{1-x^4}}\right )\right )}{16 \sqrt {2}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.48, size = 94, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^4\right )^{3/4} \left (6+x^4\right )}{42 x^7}-\frac {3^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt {2} \sqrt [4]{-1+x^4}}\right )}{8 \sqrt {2}}-\frac {3^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt {2} \sqrt [4]{-1+x^4}}\right )}{8 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 6.46, size = 277, normalized size = 2.95 \begin {gather*} \frac {84 \cdot 27^{\frac {1}{4}} \sqrt {2} x^{7} \arctan \left (-\frac {108 \cdot 27^{\frac {1}{4}} \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 24 \cdot 27^{\frac {3}{4}} \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {3}{4}} x - \sqrt {6} 3^{\frac {1}{4}} {\left (36 \cdot 27^{\frac {1}{4}} \sqrt {2} \sqrt {x^{4} - 1} x^{2} + 27^{\frac {3}{4}} \sqrt {2} {\left (7 \, x^{4} - 4\right )}\right )}}{54 \, {\left (x^{4} - 4\right )}}\right ) - 21 \cdot 27^{\frac {1}{4}} \sqrt {2} x^{7} \log \left (\frac {2 \, {\left (4 \cdot 27^{\frac {3}{4}} \sqrt {2} \sqrt {x^{4} - 1} x^{2} + 36 \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 3 \cdot 27^{\frac {1}{4}} \sqrt {2} {\left (7 \, x^{4} - 4\right )} + 72 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} x\right )}}{x^{4} - 4}\right ) + 21 \cdot 27^{\frac {1}{4}} \sqrt {2} x^{7} \log \left (-\frac {2 \, {\left (4 \cdot 27^{\frac {3}{4}} \sqrt {2} \sqrt {x^{4} - 1} x^{2} - 36 \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 3 \cdot 27^{\frac {1}{4}} \sqrt {2} {\left (7 \, x^{4} - 4\right )} - 72 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} x\right )}}{x^{4} - 4}\right ) + 32 \, {\left (x^{4} + 6\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}}}{1344 \, x^{7}} \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} + 4\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}}}{{\left (x^{4} - 4\right )} x^{8}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 11.23, size = 241, normalized size = 2.56
method | result | size |
trager | \(\frac {\left (x^{4}-1\right )^{\frac {3}{4}} \left (x^{4}+6\right )}{42 x^{7}}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-108\right )^{2}\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-108\right )^{2}\right ) \sqrt {x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}-108\right )^{2} x^{2}-6 \left (x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-108\right )^{2} x^{3}-21 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-108\right )^{2}\right ) x^{4}+72 \left (x^{4}-1\right )^{\frac {3}{4}} x +12 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-108\right )^{2}\right )}{\left (x^{2}-2\right ) \left (x^{2}+2\right )}\right )}{32}-\frac {\RootOf \left (\textit {\_Z}^{4}-108\right ) \ln \left (\frac {2 \sqrt {x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}-108\right )^{3} x^{2}+6 \left (x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-108\right )^{2} x^{3}+21 \RootOf \left (\textit {\_Z}^{4}-108\right ) x^{4}+72 \left (x^{4}-1\right )^{\frac {3}{4}} x -12 \RootOf \left (\textit {\_Z}^{4}-108\right )}{\left (x^{2}-2\right ) \left (x^{2}+2\right )}\right )}{32}\) | \(241\) |
risch | \(\frac {x^{8}+5 x^{4}-6}{42 x^{7} \left (x^{4}-1\right )^{\frac {1}{4}}}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-108\right )^{2}\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-108\right )^{2}\right ) \sqrt {x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}-108\right )^{2} x^{2}-6 \left (x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-108\right )^{2} x^{3}-21 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-108\right )^{2}\right ) x^{4}+72 \left (x^{4}-1\right )^{\frac {3}{4}} x +12 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-108\right )^{2}\right )}{\left (x^{2}-2\right ) \left (x^{2}+2\right )}\right )}{32}-\frac {\RootOf \left (\textit {\_Z}^{4}-108\right ) \ln \left (\frac {2 \sqrt {x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}-108\right )^{3} x^{2}+6 \left (x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-108\right )^{2} x^{3}+21 \RootOf \left (\textit {\_Z}^{4}-108\right ) x^{4}+72 \left (x^{4}-1\right )^{\frac {3}{4}} x -12 \RootOf \left (\textit {\_Z}^{4}-108\right )}{\left (x^{2}-2\right ) \left (x^{2}+2\right )}\right )}{32}\) | \(246\) |
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} + 4\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}}}{{\left (x^{4} - 4\right )} x^{8}}\,{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^4-1\right )}^{3/4}\,\left (x^4+4\right )}{x^8\,\left (x^4-4\right )} \,d x \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 {\left (\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )\right )^{\frac {3}{4}} \left (x^{2} - 2 x + 2\right ) \left (x^{2} + 2 x + 2\right )}{x^{8} \left (x^{2} - 2\right ) \left (x^{2} + 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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