Optimal. Leaf size=141 \[ \frac {9}{8} \sqrt [4]{\frac {3}{2}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{2 x^4+1}}\right )-\sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{2 x^4+1}}\right )-\frac {9}{8} \sqrt [4]{\frac {3}{2}} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{2 x^4+1}}\right )+\sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{2 x^4+1}}\right )+\frac {\sqrt [4]{2 x^4+1} \left (9 x^4+2\right )}{20 x^5} \]
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Rubi [C] time = 0.37, antiderivative size = 123, normalized size of antiderivative = 0.87, number of steps used = 9, number of rules used = 8, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {6725, 264, 277, 331, 298, 203, 206, 510} \begin {gather*} \frac {3}{8} x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {x^4}{2},-2 x^4\right )+\frac {\sqrt [4]{2 x^4+1}}{4 x}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{2 x^4+1}}\right )}{4\ 2^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{2 x^4+1}}\right )}{4\ 2^{3/4}}+\frac {\left (2 x^4+1\right )^{5/4}}{10 x^5} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 203
Rule 206
Rule 264
Rule 277
Rule 298
Rule 331
Rule 510
Rule 6725
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{1+2 x^4} \left (-1-x^4+2 x^8\right )}{x^6 \left (2+x^4\right )} \, dx &=\int \left (-\frac {\sqrt [4]{1+2 x^4}}{2 x^6}-\frac {\sqrt [4]{1+2 x^4}}{4 x^2}+\frac {9 x^2 \sqrt [4]{1+2 x^4}}{4 \left (2+x^4\right )}\right ) \, dx\\ &=-\left (\frac {1}{4} \int \frac {\sqrt [4]{1+2 x^4}}{x^2} \, dx\right )-\frac {1}{2} \int \frac {\sqrt [4]{1+2 x^4}}{x^6} \, dx+\frac {9}{4} \int \frac {x^2 \sqrt [4]{1+2 x^4}}{2+x^4} \, dx\\ &=\frac {\sqrt [4]{1+2 x^4}}{4 x}+\frac {\left (1+2 x^4\right )^{5/4}}{10 x^5}+\frac {3}{8} x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {x^4}{2},-2 x^4\right )-\frac {1}{2} \int \frac {x^2}{\left (1+2 x^4\right )^{3/4}} \, dx\\ &=\frac {\sqrt [4]{1+2 x^4}}{4 x}+\frac {\left (1+2 x^4\right )^{5/4}}{10 x^5}+\frac {3}{8} x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {x^4}{2},-2 x^4\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{1-2 x^4} \, dx,x,\frac {x}{\sqrt [4]{1+2 x^4}}\right )\\ &=\frac {\sqrt [4]{1+2 x^4}}{4 x}+\frac {\left (1+2 x^4\right )^{5/4}}{10 x^5}+\frac {3}{8} x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {x^4}{2},-2 x^4\right )-\frac {\operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+2 x^4}}\right )}{4 \sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+2 x^4}}\right )}{4 \sqrt {2}}\\ &=\frac {\sqrt [4]{1+2 x^4}}{4 x}+\frac {\left (1+2 x^4\right )^{5/4}}{10 x^5}+\frac {3}{8} x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {x^4}{2},-2 x^4\right )+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+2 x^4}}\right )}{4\ 2^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+2 x^4}}\right )}{4\ 2^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.14, size = 99, normalized size = 0.70 \begin {gather*} \frac {2}{7} x^7 F_1\left (\frac {7}{4};\frac {3}{4},1;\frac {11}{4};-2 x^4,-\frac {x^4}{2}\right )+\frac {5 x^3 \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};-\frac {3 x^4}{x^4+2}\right )}{12 \sqrt [4]{2} \left (x^4+2\right )^{3/4}}+\frac {\sqrt [4]{2 x^4+1} \left (9 x^4+2\right )}{20 x^5} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.50, size = 141, normalized size = 1.00 \begin {gather*} \frac {9}{8} \sqrt [4]{\frac {3}{2}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{2 x^4+1}}\right )-\sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{2 x^4+1}}\right )-\frac {9}{8} \sqrt [4]{\frac {3}{2}} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{2 x^4+1}}\right )+\sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{2 x^4+1}}\right )+\frac {\sqrt [4]{2 x^4+1} \left (9 x^4+2\right )}{20 x^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 32.41, size = 519, normalized size = 3.68 \begin {gather*} -\frac {180 \cdot 3^{\frac {1}{4}} 2^{\frac {3}{4}} x^{5} \arctan \left (-\frac {12 \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 12 \cdot 3^{\frac {1}{4}} 2^{\frac {3}{4}} {\left (2 \, x^{4} + 1\right )}^{\frac {3}{4}} x - \sqrt {3} {\left (4 \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} \sqrt {2 \, x^{4} + 1} x^{2} + 3^{\frac {1}{4}} 2^{\frac {3}{4}} {\left (7 \, x^{4} + 2\right )}\right )} \sqrt {\sqrt {3} \sqrt {2}}}{6 \, {\left (x^{4} + 2\right )}}\right ) + 45 \cdot 3^{\frac {1}{4}} 2^{\frac {3}{4}} x^{5} \log \left (\frac {6 \, \sqrt {3} \sqrt {2} {\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 6 \cdot 3^{\frac {1}{4}} 2^{\frac {3}{4}} \sqrt {2 \, x^{4} + 1} x^{2} + 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (7 \, x^{4} + 2\right )} + 12 \, {\left (2 \, x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} + 2}\right ) - 45 \cdot 3^{\frac {1}{4}} 2^{\frac {3}{4}} x^{5} \log \left (\frac {6 \, \sqrt {3} \sqrt {2} {\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 6 \cdot 3^{\frac {1}{4}} 2^{\frac {3}{4}} \sqrt {2 \, x^{4} + 1} x^{2} - 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (7 \, x^{4} + 2\right )} + 12 \, {\left (2 \, x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} + 2}\right ) - 320 \cdot 2^{\frac {1}{4}} x^{5} \arctan \left (-2 \cdot 2^{\frac {3}{4}} {\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 2 \cdot 2^{\frac {1}{4}} {\left (2 \, x^{4} + 1\right )}^{\frac {3}{4}} x + \frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (2 \cdot 2^{\frac {3}{4}} \sqrt {2 \, x^{4} + 1} x^{2} + 2^{\frac {1}{4}} {\left (4 \, x^{4} + 1\right )}\right )}\right ) - 80 \cdot 2^{\frac {1}{4}} x^{5} \log \left (4 \, \sqrt {2} {\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \cdot 2^{\frac {1}{4}} \sqrt {2 \, x^{4} + 1} x^{2} + 2^{\frac {3}{4}} {\left (4 \, x^{4} + 1\right )} + 4 \, {\left (2 \, x^{4} + 1\right )}^{\frac {3}{4}} x\right ) + 80 \cdot 2^{\frac {1}{4}} x^{5} \log \left (4 \, \sqrt {2} {\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 4 \cdot 2^{\frac {1}{4}} \sqrt {2 \, x^{4} + 1} x^{2} - 2^{\frac {3}{4}} {\left (4 \, x^{4} + 1\right )} + 4 \, {\left (2 \, x^{4} + 1\right )}^{\frac {3}{4}} x\right ) - 16 \, {\left (9 \, x^{4} + 2\right )} {\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}}}{320 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 177, normalized size = 1.26 \begin {gather*} -\frac {1}{16} \cdot 54^{\frac {3}{4}} \arctan \left (\frac {24^{\frac {3}{4}} {\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}}}{12 \, x}\right ) - \frac {1}{32} \cdot 54^{\frac {3}{4}} \log \left (\frac {1}{2} \cdot 24^{\frac {1}{4}} + \frac {{\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{32} \cdot 54^{\frac {3}{4}} \log \left (-\frac {1}{2} \cdot 24^{\frac {1}{4}} + \frac {{\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) + 2^{\frac {1}{4}} \arctan \left (\frac {2^{\frac {3}{4}} {\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}}}{2 \, x}\right ) + \frac {1}{2} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + \frac {{\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \cdot 2^{\frac {1}{4}} \log \left (-2^{\frac {1}{4}} + \frac {{\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {{\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}} {\left (\frac {1}{x^{4}} + 2\right )}}{10 \, x} + \frac {{\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}}}{4 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.14, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (2 x^{4}+1\right )^{\frac {1}{4}} \left (2 x^{8}-x^{4}-1\right )}{x^{6} \left (x^{4}+2\right )}\, dx \end {gather*}
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 (2 \, x^{8} - x^{4} - 1\right )} {\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}}}{{\left (x^{4} + 2\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 (2\,x^4+1\right )}^{1/4}\,\left (-2\,x^8+x^4+1\right )}{x^6\,\left (x^4+2\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 ) \left (2 x^{4} + 1\right )^{\frac {5}{4}}}{x^{6} \left (x^{4} + 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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