Optimal. Leaf size=117 \[ -\frac {7}{128} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+x^2}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+x^2}}\right )}{2^{3/4}}+\frac {7}{128} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+x^2}}\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+x^2}}\right )}{2^{3/4}}+\frac {1}{192} \sqrt [4]{x^4+x^2} \left (32 x^5+4 x^3-7 x\right ) \]
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Rubi [B] time = 0.58, antiderivative size = 248, normalized size of antiderivative = 2.12, number of steps used = 24, number of rules used = 11, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {2056, 1586, 6725, 321, 329, 331, 298, 203, 206, 466, 494} \begin {gather*} -\frac {7}{192} \sqrt [4]{x^4+x^2} x-\frac {7 \sqrt [4]{x^4+x^2} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )}{128 \sqrt [4]{x^2+1} \sqrt {x}}-\frac {\sqrt [4]{x^4+x^2} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2+1}}\right )}{2^{3/4} \sqrt [4]{x^2+1} \sqrt {x}}+\frac {7 \sqrt [4]{x^4+x^2} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )}{128 \sqrt [4]{x^2+1} \sqrt {x}}+\frac {\sqrt [4]{x^4+x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2+1}}\right )}{2^{3/4} \sqrt [4]{x^2+1} \sqrt {x}}+\frac {1}{6} \sqrt [4]{x^4+x^2} x^5+\frac {1}{48} \sqrt [4]{x^4+x^2} x^3 \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 298
Rule 321
Rule 329
Rule 331
Rule 466
Rule 494
Rule 1586
Rule 2056
Rule 6725
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{x^2+x^4} \left (-1-x^4+x^8\right )}{-1+x^4} \, dx &=\frac {\sqrt [4]{x^2+x^4} \int \frac {\sqrt {x} \sqrt [4]{1+x^2} \left (-1-x^4+x^8\right )}{-1+x^4} \, dx}{\sqrt {x} \sqrt [4]{1+x^2}}\\ &=\frac {\sqrt [4]{x^2+x^4} \int \frac {\sqrt {x} \left (-1-x^4+x^8\right )}{\left (-1+x^2\right ) \left (1+x^2\right )^{3/4}} \, dx}{\sqrt {x} \sqrt [4]{1+x^2}}\\ &=\frac {\sqrt [4]{x^2+x^4} \int \left (\frac {x^{9/2}}{\left (1+x^2\right )^{3/4}}+\frac {x^{13/2}}{\left (1+x^2\right )^{3/4}}-\frac {\sqrt {x}}{\left (-1+x^2\right ) \left (1+x^2\right )^{3/4}}\right ) \, dx}{\sqrt {x} \sqrt [4]{1+x^2}}\\ &=\frac {\sqrt [4]{x^2+x^4} \int \frac {x^{9/2}}{\left (1+x^2\right )^{3/4}} \, dx}{\sqrt {x} \sqrt [4]{1+x^2}}+\frac {\sqrt [4]{x^2+x^4} \int \frac {x^{13/2}}{\left (1+x^2\right )^{3/4}} \, dx}{\sqrt {x} \sqrt [4]{1+x^2}}-\frac {\sqrt [4]{x^2+x^4} \int \frac {\sqrt {x}}{\left (-1+x^2\right ) \left (1+x^2\right )^{3/4}} \, dx}{\sqrt {x} \sqrt [4]{1+x^2}}\\ &=\frac {1}{4} x^3 \sqrt [4]{x^2+x^4}+\frac {1}{6} x^5 \sqrt [4]{x^2+x^4}-\frac {\left (7 \sqrt [4]{x^2+x^4}\right ) \int \frac {x^{5/2}}{\left (1+x^2\right )^{3/4}} \, dx}{8 \sqrt {x} \sqrt [4]{1+x^2}}-\frac {\left (11 \sqrt [4]{x^2+x^4}\right ) \int \frac {x^{9/2}}{\left (1+x^2\right )^{3/4}} \, dx}{12 \sqrt {x} \sqrt [4]{1+x^2}}-\frac {\left (2 \sqrt [4]{x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1+x^2}}\\ &=-\frac {7}{16} x \sqrt [4]{x^2+x^4}+\frac {1}{48} x^3 \sqrt [4]{x^2+x^4}+\frac {1}{6} x^5 \sqrt [4]{x^2+x^4}+\frac {\left (21 \sqrt [4]{x^2+x^4}\right ) \int \frac {\sqrt {x}}{\left (1+x^2\right )^{3/4}} \, dx}{32 \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\left (77 \sqrt [4]{x^2+x^4}\right ) \int \frac {x^{5/2}}{\left (1+x^2\right )^{3/4}} \, dx}{96 \sqrt {x} \sqrt [4]{1+x^2}}-\frac {\left (2 \sqrt [4]{x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-1+2 x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {x} \sqrt [4]{1+x^2}}\\ &=-\frac {7}{192} x \sqrt [4]{x^2+x^4}+\frac {1}{48} x^3 \sqrt [4]{x^2+x^4}+\frac {1}{6} x^5 \sqrt [4]{x^2+x^4}-\frac {\left (77 \sqrt [4]{x^2+x^4}\right ) \int \frac {\sqrt {x}}{\left (1+x^2\right )^{3/4}} \, dx}{128 \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\left (21 \sqrt [4]{x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (1+x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{16 \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\sqrt [4]{x^2+x^4} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {2} \sqrt {x} \sqrt [4]{1+x^2}}-\frac {\sqrt [4]{x^2+x^4} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {2} \sqrt {x} \sqrt [4]{1+x^2}}\\ &=-\frac {7}{192} x \sqrt [4]{x^2+x^4}+\frac {1}{48} x^3 \sqrt [4]{x^2+x^4}+\frac {1}{6} x^5 \sqrt [4]{x^2+x^4}-\frac {\sqrt [4]{x^2+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{2^{3/4} \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\sqrt [4]{x^2+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{2^{3/4} \sqrt {x} \sqrt [4]{1+x^2}}-\frac {\left (77 \sqrt [4]{x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (1+x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{64 \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\left (21 \sqrt [4]{x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{16 \sqrt {x} \sqrt [4]{1+x^2}}\\ &=-\frac {7}{192} x \sqrt [4]{x^2+x^4}+\frac {1}{48} x^3 \sqrt [4]{x^2+x^4}+\frac {1}{6} x^5 \sqrt [4]{x^2+x^4}-\frac {\sqrt [4]{x^2+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{2^{3/4} \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\sqrt [4]{x^2+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{2^{3/4} \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\left (21 \sqrt [4]{x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{32 \sqrt {x} \sqrt [4]{1+x^2}}-\frac {\left (21 \sqrt [4]{x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{32 \sqrt {x} \sqrt [4]{1+x^2}}-\frac {\left (77 \sqrt [4]{x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{64 \sqrt {x} \sqrt [4]{1+x^2}}\\ &=-\frac {7}{192} x \sqrt [4]{x^2+x^4}+\frac {1}{48} x^3 \sqrt [4]{x^2+x^4}+\frac {1}{6} x^5 \sqrt [4]{x^2+x^4}-\frac {21 \sqrt [4]{x^2+x^4} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{32 \sqrt {x} \sqrt [4]{1+x^2}}-\frac {\sqrt [4]{x^2+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{2^{3/4} \sqrt {x} \sqrt [4]{1+x^2}}+\frac {21 \sqrt [4]{x^2+x^4} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{32 \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\sqrt [4]{x^2+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{2^{3/4} \sqrt {x} \sqrt [4]{1+x^2}}-\frac {\left (77 \sqrt [4]{x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{128 \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\left (77 \sqrt [4]{x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{128 \sqrt {x} \sqrt [4]{1+x^2}}\\ &=-\frac {7}{192} x \sqrt [4]{x^2+x^4}+\frac {1}{48} x^3 \sqrt [4]{x^2+x^4}+\frac {1}{6} x^5 \sqrt [4]{x^2+x^4}-\frac {7 \sqrt [4]{x^2+x^4} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{128 \sqrt {x} \sqrt [4]{1+x^2}}-\frac {\sqrt [4]{x^2+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{2^{3/4} \sqrt {x} \sqrt [4]{1+x^2}}+\frac {7 \sqrt [4]{x^2+x^4} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{128 \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\sqrt [4]{x^2+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{2^{3/4} \sqrt {x} \sqrt [4]{1+x^2}}\\ \end {align*}
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Mathematica [A] time = 0.59, size = 117, normalized size = 1.00 \begin {gather*} \frac {1}{384} x \sqrt [4]{x^4+x^2} \left (64 x^4+8 x^2+\frac {3 \left (\frac {1}{x^2}+1\right )^{3/4} \left (7 \tan ^{-1}\left (\sqrt [4]{\frac {1}{x^2}+1}\right )+64 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{x^2}+1}}{\sqrt [4]{2}}\right )+7 \tanh ^{-1}\left (\sqrt [4]{\frac {1}{x^2}+1}\right )+64 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{x^2}+1}}{\sqrt [4]{2}}\right )\right )}{x^2+1}-14\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.50, size = 117, normalized size = 1.00 \begin {gather*} \frac {1}{192} \sqrt [4]{x^2+x^4} \left (-7 x+4 x^3+32 x^5\right )-\frac {7}{128} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^2+x^4}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^4}}\right )}{2^{3/4}}+\frac {7}{128} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^2+x^4}}\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^4}}\right )}{2^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 7.46, size = 330, normalized size = 2.82 \begin {gather*} \frac {1}{8} \cdot 8^{\frac {3}{4}} \arctan \left (\frac {16 \cdot 8^{\frac {1}{4}} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2^{\frac {3}{4}} {\left (8^{\frac {3}{4}} {\left (3 \, x^{3} + x\right )} + 8 \cdot 8^{\frac {1}{4}} \sqrt {x^{4} + x^{2}} x\right )} + 4 \cdot 8^{\frac {3}{4}} {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{8 \, {\left (x^{3} - x\right )}}\right ) + \frac {1}{32} \cdot 8^{\frac {3}{4}} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 8^{\frac {3}{4}} \sqrt {x^{4} + x^{2}} x + 8^{\frac {1}{4}} {\left (3 \, x^{3} + x\right )} + 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{x^{3} - x}\right ) - \frac {1}{32} \cdot 8^{\frac {3}{4}} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} - 8^{\frac {3}{4}} \sqrt {x^{4} + x^{2}} x - 8^{\frac {1}{4}} {\left (3 \, x^{3} + x\right )} + 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{x^{3} - x}\right ) + \frac {1}{192} \, {\left (32 \, x^{5} + 4 \, x^{3} - 7 \, x\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} - \frac {7}{256} \, \arctan \left (\frac {2 \, {\left ({\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} + {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}\right )}}{x}\right ) + \frac {7}{256} \, \log \left (\frac {2 \, x^{3} + 2 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2 \, \sqrt {x^{4} + x^{2}} x + x + 2 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 123, normalized size = 1.05 \begin {gather*} -\frac {1}{192} \, {\left (7 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {9}{4}} - 18 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {5}{4}} - 21 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right )} x^{6} + \frac {1}{2} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{4} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{4} \cdot 2^{\frac {1}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} \right |}\right ) + \frac {7}{128} \, \arctan \left ({\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {7}{256} \, \log \left ({\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {7}{256} \, \log \left ({\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{4}+x^{2}\right )^{\frac {1}{4}} \left (x^{8}-x^{4}-1\right )}{x^{4}-1}\, dx\]
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^{8} - x^{4} - 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}}}{x^{4} - 1}\,{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+x^2\right )}^{1/4}\,\left (-x^8+x^4+1\right )}{x^4-1} \,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 {\sqrt [4]{x^{2} \left (x^{2} + 1\right )} \left (x^{8} - x^{4} - 1\right )}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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