Optimal. Leaf size=141 \[ -\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} x}{\sqrt [4]{x^4+2}}\right )}{2 \sqrt {2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{x^4+2}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (-\frac {\sqrt {2} x}{\sqrt [4]{x^4+2}}+\frac {x^2}{\sqrt {x^4+2}}+1\right )}{4 \sqrt {2}}+\frac {\log \left (\frac {\sqrt {2} x}{\sqrt [4]{x^4+2}}+\frac {x^2}{\sqrt {x^4+2}}+1\right )}{4 \sqrt {2}} \]
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Rubi [A] time = 0.06, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {377, 211, 1165, 628, 1162, 617, 204} \[ -\frac {\log \left (\frac {x^2}{\sqrt {x^4+2}}-\frac {\sqrt {2} x}{\sqrt [4]{x^4+2}}+1\right )}{4 \sqrt {2}}+\frac {\log \left (\frac {x^2}{\sqrt {x^4+2}}+\frac {\sqrt {2} x}{\sqrt [4]{x^4+2}}+1\right )}{4 \sqrt {2}}-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} x}{\sqrt [4]{x^4+2}}\right )}{2 \sqrt {2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{x^4+2}}+1\right )}{2 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 377
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {1}{\left (1+x^4\right ) \sqrt [4]{2+x^4}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {2}}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {2}}\\ &=-\frac {\log \left (1+\frac {x^2}{\sqrt {2+x^4}}-\frac {\sqrt {2} x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {2}}+\frac {\log \left (1+\frac {x^2}{\sqrt {2+x^4}}+\frac {\sqrt {2} x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} x}{\sqrt [4]{2+x^4}}\right )}{2 \sqrt {2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} x}{\sqrt [4]{2+x^4}}\right )}{2 \sqrt {2}}\\ &=-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} x}{\sqrt [4]{2+x^4}}\right )}{2 \sqrt {2}}+\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} x}{\sqrt [4]{2+x^4}}\right )}{2 \sqrt {2}}-\frac {\log \left (1+\frac {x^2}{\sqrt {2+x^4}}-\frac {\sqrt {2} x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {2}}+\frac {\log \left (1+\frac {x^2}{\sqrt {2+x^4}}+\frac {\sqrt {2} x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 120, normalized size = 0.85 \[ \frac {-2 \tan ^{-1}\left (1-\frac {\sqrt {2} x}{\sqrt [4]{x^4+2}}\right )+2 \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{x^4+2}}+1\right )-\log \left (-\frac {\sqrt {2} x}{\sqrt [4]{x^4+2}}+\frac {x^2}{\sqrt {x^4+2}}+1\right )+\log \left (\frac {\sqrt {2} x}{\sqrt [4]{x^4+2}}+\frac {x^2}{\sqrt {x^4+2}}+1\right )}{4 \sqrt {2}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.21, size = 85, normalized size = 0.60 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^4+2}}{\sqrt {x^4+2}-x^2}\right )}{2 \sqrt {2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^4+2}}{\sqrt {x^4+2}+x^2}\right )}{2 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 6.77, size = 388, normalized size = 2.75 \[ \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (x^{4} + 2\right )}^{\frac {3}{4}} x^{2} - \sqrt {2} {\left (x^{4} + 2\right )}^{\frac {5}{4}} - {\left (2 \, x^{5} - \sqrt {2} {\left (x^{4} + 2\right )}^{\frac {3}{4}} x^{2} - \sqrt {2} {\left (x^{4} + 2\right )}^{\frac {5}{4}} + 4 \, x\right )} \sqrt {\frac {x^{4} + \sqrt {2} {\left (x^{4} + 2\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {x^{4} + 2} x^{2} + \sqrt {2} {\left (x^{4} + 2\right )}^{\frac {3}{4}} x + 1}{x^{4} + 1}}}{2 \, {\left (x^{5} + 2 \, x\right )}}\right ) + \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (x^{4} + 2\right )}^{\frac {3}{4}} x^{2} - \sqrt {2} {\left (x^{4} + 2\right )}^{\frac {5}{4}} + {\left (2 \, x^{5} + \sqrt {2} {\left (x^{4} + 2\right )}^{\frac {3}{4}} x^{2} + \sqrt {2} {\left (x^{4} + 2\right )}^{\frac {5}{4}} + 4 \, x\right )} \sqrt {\frac {x^{4} - \sqrt {2} {\left (x^{4} + 2\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {x^{4} + 2} x^{2} - \sqrt {2} {\left (x^{4} + 2\right )}^{\frac {3}{4}} x + 1}{x^{4} + 1}}}{2 \, {\left (x^{5} + 2 \, x\right )}}\right ) + \frac {1}{16} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{4} + \sqrt {2} {\left (x^{4} + 2\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {x^{4} + 2} x^{2} + \sqrt {2} {\left (x^{4} + 2\right )}^{\frac {3}{4}} x + 1\right )}}{x^{4} + 1}\right ) - \frac {1}{16} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{4} - \sqrt {2} {\left (x^{4} + 2\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {x^{4} + 2} x^{2} - \sqrt {2} {\left (x^{4} + 2\right )}^{\frac {3}{4}} x + 1\right )}}{x^{4} + 1}\right ) \]
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 \[ \int \frac {1}{{\left (x^{4} + 2\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.34, size = 150, normalized size = 1.06
method | result | size |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {\sqrt {x^{4}+2}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-\left (x^{4}+2\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+\left (x^{4}+2\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}+1\right )}{x^{4}+1}\right )}{4}+\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {\left (x^{4}+2\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}-\sqrt {x^{4}+2}\, \RootOf \left (\textit {\_Z}^{4}+1\right ) x^{2}+\left (x^{4}+2\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}}{x^{4}+1}\right )}{4}\) | \(150\) |
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 \[ \int \frac {1}{{\left (x^{4} + 2\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}}\,{d x} \]
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 \[ \int \frac {1}{\left (x^4+1\right )\,{\left (x^4+2\right )}^{1/4}} \,d x \]
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 \[ \int \frac {1}{\left (x^{4} + 1\right ) \sqrt [4]{x^{4} + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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