Optimal. Leaf size=54 \[ \frac {\left (\left (x^4+1\right )^2\right )^{7/8} \left (\frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )\right )}{\left (x^4+1\right )^{7/4}} \]
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Rubi [A] time = 0.03, antiderivative size = 79, normalized size of antiderivative = 1.46, number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {1343, 240, 212, 206, 203} \begin {gather*} \frac {\sqrt [4]{x^4+1} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [8]{x^8+2 x^4+1}}+\frac {\sqrt [4]{x^4+1} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [8]{x^8+2 x^4+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 240
Rule 1343
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [8]{1+2 x^4+x^8}} \, dx &=\frac {\sqrt [4]{2+2 x^4} \int \frac {1}{\sqrt [4]{2+2 x^4}} \, dx}{\sqrt [8]{1+2 x^4+x^8}}\\ &=\frac {\sqrt [4]{2+2 x^4} \operatorname {Subst}\left (\int \frac {1}{1-2 x^4} \, dx,x,\frac {x}{\sqrt [4]{2+2 x^4}}\right )}{\sqrt [8]{1+2 x^4+x^8}}\\ &=\frac {\sqrt [4]{2+2 x^4} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{2+2 x^4}}\right )}{2 \sqrt [8]{1+2 x^4+x^8}}+\frac {\sqrt [4]{2+2 x^4} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{2+2 x^4}}\right )}{2 \sqrt [8]{1+2 x^4+x^8}}\\ &=\frac {\sqrt [4]{1+x^4} \tan ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [8]{1+2 x^4+x^8}}+\frac {\sqrt [4]{1+x^4} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [8]{1+2 x^4+x^8}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 70, normalized size = 1.30 \begin {gather*} \frac {\sqrt [4]{x^4+1} \left (-\log \left (1-\frac {x}{\sqrt [4]{x^4+1}}\right )+\log \left (\frac {x}{\sqrt [4]{x^4+1}}+1\right )+2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )\right )}{4 \sqrt [8]{\left (x^4+1\right )^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 16.24, size = 54, normalized size = 1.00 \begin {gather*} \frac {\left (\left (x^4+1\right )^2\right )^{7/8} \left (\frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )\right )}{\left (x^4+1\right )^{7/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 65, normalized size = 1.20 \begin {gather*} -\frac {1}{2} \, \arctan \left (\frac {{\left (x^{8} + 2 \, x^{4} + 1\right )}^{\frac {1}{8}}}{x}\right ) + \frac {1}{4} \, \log \left (\frac {x + {\left (x^{8} + 2 \, x^{4} + 1\right )}^{\frac {1}{8}}}{x}\right ) - \frac {1}{4} \, \log \left (-\frac {x - {\left (x^{8} + 2 \, x^{4} + 1\right )}^{\frac {1}{8}}}{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 {1}{{\left (x^{8} + 2 \, x^{4} + 1\right )}^{\frac {1}{8}}}\,{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 \begin {gather*} \int \frac {1}{\left (x^{8}+2 x^{4}+1\right )^{\frac {1}{8}}}\, 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*} \frac {x}{{\left (x^{4} + 1\right )}^{\frac {1}{4}}} + \int \frac {x^{4}}{{\left (x^{4} + 1\right )}^{\frac {5}{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 = -0.02 \begin {gather*} \int \frac {1}{{\left (x^8+2\,x^4+1\right )}^{1/8}} \,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 {1}{\sqrt [8]{x^{8} + 2 x^{4} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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