Optimal. Leaf size=225 \[ \frac {1}{8} \sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{x^4+1}}{\sqrt {x^4+1}-x^2}\right )+\frac {1}{8} \sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{x^4+1}}{\sqrt {x^4+1}-x^2}\right )+\frac {1}{8} \sqrt {2-\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{x^4+1}}{\sqrt {x^4+1}+x^2}\right )+\frac {1}{8} \sqrt {2+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{x^4+1}}{\sqrt {x^4+1}+x^2}\right ) \]
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Rubi [A] time = 0.19, antiderivative size = 242, normalized size of antiderivative = 1.08, number of steps used = 25, number of rules used = 13, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {1428, 408, 240, 212, 206, 203, 377, 211, 1165, 628, 1162, 617, 204} \begin {gather*} -\frac {1}{4} \sqrt [8]{-1} \tan ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4+1}}\right )+\frac {\sqrt [8]{-1} \tan ^{-1}\left (1-\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4+1}}\right )}{4 \sqrt {2}}-\frac {\sqrt [8]{-1} \tan ^{-1}\left (\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4+1}}+1\right )}{4 \sqrt {2}}-\frac {1}{4} \sqrt [8]{-1} \tanh ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4+1}}\right )+\frac {\sqrt [8]{-1} \log \left (\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{x^4+1}}+\frac {x^2}{\sqrt {x^4+1}}+\sqrt [4]{-1}\right )}{8 \sqrt {2}}-\frac {\sqrt [8]{-1} \log \left (\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4+1}}-\frac {(-1)^{3/4} x^2}{\sqrt {x^4+1}}+1\right )}{8 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 204
Rule 206
Rule 211
Rule 212
Rule 240
Rule 377
Rule 408
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1428
Rubi steps
\begin {align*} \int \frac {\left (1+x^4\right )^{3/4}}{1+2 x^4+2 x^8} \, dx &=-\left (2 i \int \frac {\left (1+x^4\right )^{3/4}}{(2-2 i)+4 x^4} \, dx\right )+2 i \int \frac {\left (1+x^4\right )^{3/4}}{(2+2 i)+4 x^4} \, dx\\ &=-\left ((-1+i) \int \frac {1}{\sqrt [4]{1+x^4} \left ((2-2 i)+4 x^4\right )} \, dx\right )+(1+i) \int \frac {1}{\sqrt [4]{1+x^4} \left ((2+2 i)+4 x^4\right )} \, dx\\ &=-\left ((-1+i) \operatorname {Subst}\left (\int \frac {1}{(2-2 i)+(2+2 i) x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\right )+(1+i) \operatorname {Subst}\left (\int \frac {1}{(2+2 i)+(2-2 i) x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=-\left (-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1}-x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {2}}\right )--\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1}+x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt [4]{-1}-x^2}{(2+2 i)+(2-2 i) x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt [4]{-1}+x^2}{(2+2 i)+(2-2 i) x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {2}}\\ &=\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (-1)^{7/8} \tan ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {2}}+\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (-1)^{7/8} \tanh ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {2}}+\left (\left (\frac {1}{16}+\frac {i}{16}\right ) (-1)^{7/8}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [8]{-1} \sqrt {2}+2 x}{-\sqrt [4]{-1}-\sqrt [8]{-1} \sqrt {2} x-x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\left (\left (\frac {1}{16}+\frac {i}{16}\right ) (-1)^{7/8}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [8]{-1} \sqrt {2}-2 x}{-\sqrt [4]{-1}+\sqrt [8]{-1} \sqrt {2} x-x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1}-\sqrt [8]{-1} \sqrt {2} x+x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {2}}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1}+\sqrt [8]{-1} \sqrt {2} x+x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {2}}\\ &=\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (-1)^{7/8} \tan ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {2}}+\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (-1)^{7/8} \tanh ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {2}}-\left (\frac {1}{16}+\frac {i}{16}\right ) (-1)^{7/8} \log \left (\sqrt [4]{-1}+\frac {x^2}{\sqrt {1+x^4}}+\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{1+x^4}}\right )+\left (\frac {1}{16}+\frac {i}{16}\right ) (-1)^{7/8} \log \left (1-\frac {(-1)^{3/4} x^2}{\sqrt {1+x^4}}+\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{1+x^4}}\right )+\left (\left (-\frac {1}{8}-\frac {i}{8}\right ) (-1)^{7/8}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{1+x^4}}\right )+\left (\left (\frac {1}{8}+\frac {i}{8}\right ) (-1)^{7/8}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{1+x^4}}\right )\\ &=\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (-1)^{7/8} \tan ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {2}}-\left (\frac {1}{8}+\frac {i}{8}\right ) (-1)^{7/8} \tan ^{-1}\left (1-\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{1+x^4}}\right )+\left (\frac {1}{8}+\frac {i}{8}\right ) (-1)^{7/8} \tan ^{-1}\left (1+\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{1+x^4}}\right )+\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (-1)^{7/8} \tanh ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {2}}-\left (\frac {1}{16}+\frac {i}{16}\right ) (-1)^{7/8} \log \left (\sqrt [4]{-1}+\frac {x^2}{\sqrt {1+x^4}}+\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{1+x^4}}\right )+\left (\frac {1}{16}+\frac {i}{16}\right ) (-1)^{7/8} \log \left (1-\frac {(-1)^{3/4} x^2}{\sqrt {1+x^4}}+\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{1+x^4}}\right )\\ \end {align*}
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Mathematica [F] time = 0.04, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+x^4\right )^{3/4}}{1+2 x^4+2 x^8} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.79, size = 225, normalized size = 1.00 \begin {gather*} \frac {1}{8} \sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{x^4+1}}{\sqrt {x^4+1}-x^2}\right )+\frac {1}{8} \sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{x^4+1}}{\sqrt {x^4+1}-x^2}\right )+\frac {1}{8} \sqrt {2-\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{x^4+1}}{\sqrt {x^4+1}+x^2}\right )+\frac {1}{8} \sqrt {2+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{x^4+1}}{\sqrt {x^4+1}+x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {{\left (x^{4} + 1\right )}^{\frac {3}{4}}}{2 \, x^{8} + 2 \, x^{4} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 5.30, size = 463, normalized size = 2.06 \begin {gather*} \frac {\RootOf \left (\textit {\_Z}^{8}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{8}+1\right )^{7} x^{2}+2 \left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{8}+1\right )^{6} x^{3}+\RootOf \left (\textit {\_Z}^{8}+1\right )^{5} x^{4}-\RootOf \left (\textit {\_Z}^{8}+1\right ) x^{4}-2 \left (x^{4}+1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{8}+1\right )}{\RootOf \left (\textit {\_Z}^{8}+1\right )^{4} x^{4}+x^{4}+1}\right )}{8}+\frac {\RootOf \left (\textit {\_Z}^{8}+1\right )^{3} \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{8}+1\right )^{7} x^{4}-2 \sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{8}+1\right )^{5} x^{2}+\RootOf \left (\textit {\_Z}^{8}+1\right )^{3} x^{4}-2 \left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{8}+1\right )^{2} x^{3}+2 \left (x^{4}+1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{8}+1\right )^{3}}{\RootOf \left (\textit {\_Z}^{8}+1\right )^{4} x^{4}-x^{4}-1}\right )}{8}+\frac {\RootOf \left (\textit {\_Z}^{8}+1\right )^{7} \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{8}+1\right )^{11} x^{4}+\RootOf \left (\textit {\_Z}^{8}+1\right )^{7} x^{4}+\RootOf \left (\textit {\_Z}^{8}+1\right )^{7}+2 \left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{8}+1\right )^{2} x^{3}-2 \sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{8}+1\right ) x^{2}+2 \left (x^{4}+1\right )^{\frac {3}{4}} x}{\RootOf \left (\textit {\_Z}^{8}+1\right )^{4} x^{4}-x^{4}-1}\right )}{8}-\frac {\RootOf \left (\textit {\_Z}^{8}+1\right )^{5} \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{8}+1\right )^{9} x^{4}+2 \left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{8}+1\right )^{6} x^{3}-\RootOf \left (\textit {\_Z}^{8}+1\right )^{5} x^{4}+2 \sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{8}+1\right )^{3} x^{2}-\RootOf \left (\textit {\_Z}^{8}+1\right )^{5}+2 \left (x^{4}+1\right )^{\frac {3}{4}} x}{\RootOf \left (\textit {\_Z}^{8}+1\right )^{4} x^{4}+x^{4}+1}\right )}{8} \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 (x^{4} + 1\right )}^{\frac {3}{4}}}{2 \, x^{8} + 2 \, 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 = -0.00 \begin {gather*} \int \frac {{\left (x^4+1\right )}^{3/4}}{2\,x^8+2\,x^4+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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