Optimal. Leaf size=125 \[ \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )-\sqrt [4]{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )-\sqrt [4]{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}} \]
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Rubi [A] time = 0.35, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6728, 240, 212, 206, 203, 377} \begin {gather*} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )-\sqrt [4]{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )-\sqrt [4]{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 240
Rule 377
Rule 6728
Rubi steps
\begin {align*} \int \frac {2-x^4+2 x^8}{\sqrt [4]{1+x^4} \left (-2+x^4+x^8\right )} \, dx &=\int \left (\frac {2}{\sqrt [4]{1+x^4}}+\frac {3 \left (2-x^4\right )}{\sqrt [4]{1+x^4} \left (-2+x^4+x^8\right )}\right ) \, dx\\ &=2 \int \frac {1}{\sqrt [4]{1+x^4}} \, dx+3 \int \frac {2-x^4}{\sqrt [4]{1+x^4} \left (-2+x^4+x^8\right )} \, dx\\ &=2 \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+3 \int \left (\frac {2}{3 \sqrt [4]{1+x^4} \left (-2+2 x^4\right )}-\frac {8}{3 \sqrt [4]{1+x^4} \left (4+2 x^4\right )}\right ) \, dx\\ &=2 \int \frac {1}{\sqrt [4]{1+x^4} \left (-2+2 x^4\right )} \, dx-8 \int \frac {1}{\sqrt [4]{1+x^4} \left (4+2 x^4\right )} \, dx+\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\tan ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )+2 \operatorname {Subst}\left (\int \frac {1}{-2+4 x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-8 \operatorname {Subst}\left (\int \frac {1}{4-2 x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\tan ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\sqrt {2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {2}-x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\sqrt {2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {2}+x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\tan ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\sqrt [4]{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{1+x^4}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{2}}+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\sqrt [4]{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{1+x^4}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 125, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )-\sqrt [4]{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )-\sqrt [4]{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.44, size = 125, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\sqrt [4]{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{1+x^4}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{2}}+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\sqrt [4]{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{1+x^4}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 255, normalized size = 2.04 \begin {gather*} -\frac {1}{2} \cdot 8^{\frac {3}{4}} \arctan \left (\frac {8^{\frac {3}{4}} \sqrt {2} x \sqrt {\frac {\sqrt {2} x^{2} + 2 \, \sqrt {x^{4} + 1}}{x^{2}}} - 2 \cdot 8^{\frac {3}{4}} {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{8 \, x}\right ) - \frac {1}{2} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {2^{\frac {3}{4}} x \sqrt {\frac {\sqrt {2} x^{2} + \sqrt {x^{4} + 1}}{x^{2}}} - 2^{\frac {3}{4}} {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{2 \, x}\right ) - \frac {1}{8} \cdot 8^{\frac {3}{4}} \log \left (\frac {8^{\frac {1}{4}} x + 2 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{8} \cdot 8^{\frac {3}{4}} \log \left (-\frac {8^{\frac {1}{4}} x - 2 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{8} \cdot 2^{\frac {3}{4}} \log \left (\frac {2^{\frac {1}{4}} x + {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{8} \cdot 2^{\frac {3}{4}} \log \left (-\frac {2^{\frac {1}{4}} x - {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) - \arctan \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \log \left (\frac {x + {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \log \left (-\frac {x - {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) \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 {2 \, x^{8} - x^{4} + 2}{{\left (x^{8} + x^{4} - 2\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {2 x^{8}-x^{4}+2}{\left (x^{4}+1\right )^{\frac {1}{4}} \left (x^{8}+x^{4}-2\right )}\, dx\]
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 {2 \, x^{8} - x^{4} + 2}{{\left (x^{8} + x^{4} - 2\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}}}\,{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 {2\,x^8-x^4+2}{{\left (x^4+1\right )}^{1/4}\,\left (x^8+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 {2 x^{8} - x^{4} + 2}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \sqrt [4]{x^{4} + 1} \left (x^{4} + 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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