Optimal. Leaf size=76 \[ \frac {3}{2} \tan ^{-1}\left (\frac {x \left (-x^4-1\right )^{3/4}}{x^4+1}\right )-\frac {3}{2} \tanh ^{-1}\left (\frac {x \left (-x^4-1\right )^{3/4}}{x^4+1}\right )+\frac {\sqrt [4]{-x^4-1} \left (1-14 x^4\right )}{5 x^5} \]
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Rubi [A] time = 0.10, antiderivative size = 73, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {580, 583, 12, 494, 298, 203, 206} \begin {gather*} -\frac {14 \sqrt [4]{-x^4-1}}{5 x}-\frac {3}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-x^4-1}}\right )+\frac {3}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-x^4-1}}\right )+\frac {\sqrt [4]{-x^4-1}}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 203
Rule 206
Rule 298
Rule 494
Rule 580
Rule 583
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{-1-x^4} \left (-1+x^4\right )}{x^6 \left (1+2 x^4\right )} \, dx &=\frac {\sqrt [4]{-1-x^4}}{5 x^5}+\frac {1}{5} \int \frac {-14-13 x^4}{x^2 \left (-1-x^4\right )^{3/4} \left (1+2 x^4\right )} \, dx\\ &=\frac {\sqrt [4]{-1-x^4}}{5 x^5}-\frac {14 \sqrt [4]{-1-x^4}}{5 x}+\frac {1}{5} \int \frac {15 x^2}{\left (-1-x^4\right )^{3/4} \left (1+2 x^4\right )} \, dx\\ &=\frac {\sqrt [4]{-1-x^4}}{5 x^5}-\frac {14 \sqrt [4]{-1-x^4}}{5 x}+3 \int \frac {x^2}{\left (-1-x^4\right )^{3/4} \left (1+2 x^4\right )} \, dx\\ &=\frac {\sqrt [4]{-1-x^4}}{5 x^5}-\frac {14 \sqrt [4]{-1-x^4}}{5 x}+3 \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{-1-x^4}}\right )\\ &=\frac {\sqrt [4]{-1-x^4}}{5 x^5}-\frac {14 \sqrt [4]{-1-x^4}}{5 x}+\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1-x^4}}\right )-\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1-x^4}}\right )\\ &=\frac {\sqrt [4]{-1-x^4}}{5 x^5}-\frac {14 \sqrt [4]{-1-x^4}}{5 x}-\frac {3}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-1-x^4}}\right )+\frac {3}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1-x^4}}\right )\\ \end {align*}
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Mathematica [C] time = 0.08, size = 78, normalized size = 1.03 \begin {gather*} \frac {\frac {5 \left (x^4+1\right )^{3/4} x^8 \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{2 x^4+1}\right )}{\left (2 x^4+1\right )^{3/4}}+14 x^8+13 x^4-1}{5 x^5 \left (-x^4-1\right )^{3/4}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.26, size = 76, normalized size = 1.00 \begin {gather*} \frac {\left (1-14 x^4\right ) \sqrt [4]{-1-x^4}}{5 x^5}+\frac {3}{2} \tan ^{-1}\left (\frac {x \left (-1-x^4\right )^{3/4}}{1+x^4}\right )-\frac {3}{2} \tanh ^{-1}\left (\frac {x \left (-1-x^4\right )^{3/4}}{1+x^4}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [C] time = 3.43, size = 201, normalized size = 2.64 \begin {gather*} \frac {30 \, x^{5} \log \left (-\frac {2 \, {\left (2 \, {\left (-x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {-x^{4} - 1} x^{2} + 2 \, {\left (-x^{4} - 1\right )}^{\frac {3}{4}} x - 1\right )}}{2 \, x^{4} + 1}\right ) - 15 i \, x^{5} \log \left (-\frac {2 \, {\left (2 i \, {\left (-x^{4} - 1\right )}^{\frac {1}{4}} x^{3} - 2 \, \sqrt {-x^{4} - 1} x^{2} - 2 i \, {\left (-x^{4} - 1\right )}^{\frac {3}{4}} x - 1\right )}}{2 \, x^{4} + 1}\right ) + 15 i \, x^{5} \log \left (-\frac {2 \, {\left (-2 i \, {\left (-x^{4} - 1\right )}^{\frac {1}{4}} x^{3} - 2 \, \sqrt {-x^{4} - 1} x^{2} + 2 i \, {\left (-x^{4} - 1\right )}^{\frac {3}{4}} x - 1\right )}}{2 \, x^{4} + 1}\right ) - 8 \, {\left (14 \, x^{4} - 1\right )} {\left (-x^{4} - 1\right )}^{\frac {1}{4}}}{40 \, x^{5}} \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} - 1\right )} {\left (-x^{4} - 1\right )}^{\frac {1}{4}}}{{\left (2 \, x^{4} + 1\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.10, size = 152, normalized size = 2.00
method | result | size |
trager | \(-\frac {\left (14 x^{4}-1\right ) \left (-x^{4}-1\right )^{\frac {1}{4}}}{5 x^{5}}-\frac {3 \ln \left (\frac {2 \left (-x^{4}-1\right )^{\frac {3}{4}} x -2 \sqrt {-x^{4}-1}\, x^{2}+2 \left (-x^{4}-1\right )^{\frac {1}{4}} x^{3}+1}{2 x^{4}+1}\right )}{4}-\frac {3 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-2 \sqrt {-x^{4}-1}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \left (-x^{4}-1\right )^{\frac {3}{4}} x -2 \left (-x^{4}-1\right )^{\frac {1}{4}} x^{3}-\RootOf \left (\textit {\_Z}^{2}+1\right )}{2 x^{4}+1}\right )}{4}\) | \(152\) |
risch | \(\frac {14 x^{8}+13 x^{4}-1}{5 x^{5} \left (-x^{4}-1\right )^{\frac {3}{4}}}+\frac {\left (-\frac {3 \ln \left (\frac {2 \left (-x^{12}-3 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} x^{9}+2 \sqrt {-x^{12}-3 x^{8}-3 x^{4}-1}\, x^{6}+x^{8}+2 \left (-x^{12}-3 x^{8}-3 x^{4}-1\right )^{\frac {3}{4}} x^{3}+4 \left (-x^{12}-3 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} x^{5}+2 \sqrt {-x^{12}-3 x^{8}-3 x^{4}-1}\, x^{2}+2 x^{4}+2 \left (-x^{12}-3 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} x +1}{\left (x^{4}+1\right )^{2} \left (2 x^{4}+1\right )}\right )}{4}-\frac {3 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (-x^{12}-3 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} x^{9}+2 \sqrt {-x^{12}-3 x^{8}-3 x^{4}-1}\, x^{6}-x^{8}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (-x^{12}-3 x^{8}-3 x^{4}-1\right )^{\frac {3}{4}} x^{3}-4 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (-x^{12}-3 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} x^{5}+2 \sqrt {-x^{12}-3 x^{8}-3 x^{4}-1}\, x^{2}-2 x^{4}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (-x^{12}-3 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} x -1}{\left (x^{4}+1\right )^{2} \left (2 x^{4}+1\right )}\right )}{4}\right ) \left (-\left (x^{4}+1\right )^{3}\right )^{\frac {1}{4}}}{\left (-x^{4}-1\right )^{\frac {3}{4}}}\) | \(426\) |
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} - 1\right )} {\left (-x^{4} - 1\right )}^{\frac {1}{4}}}{{\left (2 \, x^{4} + 1\right )} x^{6}}\,{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 )\,{\left (-x^4-1\right )}^{1/4}}{x^6\,\left (2\,x^4+1\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 (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \sqrt [4]{- x^{4} - 1}}{x^{6} \left (2 x^{4} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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