Optimal. Leaf size=59 \[ \tan ^{-1}\left (2 x^2+1\right )+\frac {2 x^2+1}{2 \left (2 x^4+2 x^2+1\right )}+\frac {4 x^2+3}{16 \left (2 x^4+2 x^2+1\right )^2} \]
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Rubi [A] time = 0.06, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1593, 1663, 1660, 12, 614, 617, 204} \begin {gather*} \frac {2 x^2+1}{2 \left (2 x^4+2 x^2+1\right )}+\frac {4 x^2+3}{16 \left (2 x^4+2 x^2+1\right )^2}+\tan ^{-1}\left (2 x^2+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 204
Rule 614
Rule 617
Rule 1593
Rule 1660
Rule 1663
Rubi steps
\begin {align*} \int \frac {x+x^5}{\left (1+2 x^2+2 x^4\right )^3} \, dx &=\int \frac {x \left (1+x^4\right )}{\left (1+2 x^2+2 x^4\right )^3} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1+x^2}{\left (1+2 x+2 x^2\right )^3} \, dx,x,x^2\right )\\ &=\frac {3+4 x^2}{16 \left (1+2 x^2+2 x^4\right )^2}+\frac {1}{16} \operatorname {Subst}\left (\int \frac {16}{\left (1+2 x+2 x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac {3+4 x^2}{16 \left (1+2 x^2+2 x^4\right )^2}+\operatorname {Subst}\left (\int \frac {1}{\left (1+2 x+2 x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac {3+4 x^2}{16 \left (1+2 x^2+2 x^4\right )^2}+\frac {1+2 x^2}{2 \left (1+2 x^2+2 x^4\right )}+\operatorname {Subst}\left (\int \frac {1}{1+2 x+2 x^2} \, dx,x,x^2\right )\\ &=\frac {3+4 x^2}{16 \left (1+2 x^2+2 x^4\right )^2}+\frac {1+2 x^2}{2 \left (1+2 x^2+2 x^4\right )}-\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+2 x^2\right )\\ &=\frac {3+4 x^2}{16 \left (1+2 x^2+2 x^4\right )^2}+\frac {1+2 x^2}{2 \left (1+2 x^2+2 x^4\right )}+\tan ^{-1}\left (1+2 x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 0.75 \begin {gather*} \tan ^{-1}\left (2 x^2+1\right )+\frac {32 x^6+48 x^4+36 x^2+11}{16 \left (2 x^4+2 x^2+1\right )^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x+x^5}{\left (1+2 x^2+2 x^4\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.89, size = 75, normalized size = 1.27 \begin {gather*} \frac {32 \, x^{6} + 48 \, x^{4} + 36 \, x^{2} + 16 \, {\left (4 \, x^{8} + 8 \, x^{6} + 8 \, x^{4} + 4 \, x^{2} + 1\right )} \arctan \left (2 \, x^{2} + 1\right ) + 11}{16 \, {\left (4 \, x^{8} + 8 \, x^{6} + 8 \, x^{4} + 4 \, x^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.80, size = 42, normalized size = 0.71 \begin {gather*} \frac {32 \, x^{6} + 48 \, x^{4} + 36 \, x^{2} + 11}{16 \, {\left (2 \, x^{4} + 2 \, x^{2} + 1\right )}^{2}} + \arctan \left (2 \, x^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 41, normalized size = 0.69 \begin {gather*} \arctan \left (2 x^{2}+1\right )+\frac {2 x^{6}+3 x^{4}+\frac {9}{4} x^{2}+\frac {11}{16}}{\left (2 x^{4}+2 x^{2}+1\right )^{2}} \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 {32 \, x^{6} + 48 \, x^{4} + 36 \, x^{2} + 11}{16 \, {\left (4 \, x^{8} + 8 \, x^{6} + 8 \, x^{4} + 4 \, x^{2} + 1\right )}} + 2 \, \int \frac {x}{2 \, x^{4} + 2 \, x^{2} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 47, normalized size = 0.80 \begin {gather*} \mathrm {atan}\left (2\,x^2+1\right )+\frac {\frac {x^6}{2}+\frac {3\,x^4}{4}+\frac {9\,x^2}{16}+\frac {11}{64}}{x^8+2\,x^6+2\,x^4+x^2+\frac {1}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 46, normalized size = 0.78 \begin {gather*} \frac {32 x^{6} + 48 x^{4} + 36 x^{2} + 11}{64 x^{8} + 128 x^{6} + 128 x^{4} + 64 x^{2} + 16} + \operatorname {atan}{\left (2 x^{2} + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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