Optimal. Leaf size=37 \[ \frac{1}{4 x^6 \left (x^4+1\right )}+\frac{5}{4 x^2}-\frac{5}{12 x^6}+\frac{5}{4} \tan ^{-1}\left (x^2\right ) \]
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Rubi [A] time = 0.0157296, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {28, 275, 290, 325, 203} \[ \frac{1}{4 x^6 \left (x^4+1\right )}+\frac{5}{4 x^2}-\frac{5}{12 x^6}+\frac{5}{4} \tan ^{-1}\left (x^2\right ) \]
Antiderivative was successfully verified.
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Rule 28
Rule 275
Rule 290
Rule 325
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{x^7 \left (1+2 x^4+x^8\right )} \, dx &=\int \frac{1}{x^7 \left (1+x^4\right )^2} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^4 \left (1+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac{1}{4 x^6 \left (1+x^4\right )}+\frac{5}{4} \operatorname{Subst}\left (\int \frac{1}{x^4 \left (1+x^2\right )} \, dx,x,x^2\right )\\ &=-\frac{5}{12 x^6}+\frac{1}{4 x^6 \left (1+x^4\right )}-\frac{5}{4} \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1+x^2\right )} \, dx,x,x^2\right )\\ &=-\frac{5}{12 x^6}+\frac{5}{4 x^2}+\frac{1}{4 x^6 \left (1+x^4\right )}+\frac{5}{4} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,x^2\right )\\ &=-\frac{5}{12 x^6}+\frac{5}{4 x^2}+\frac{1}{4 x^6 \left (1+x^4\right )}+\frac{5}{4} \tan ^{-1}\left (x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0107823, size = 33, normalized size = 0.89 \[ \frac{x^2}{4 \left (x^4+1\right )}+\frac{1}{x^2}-\frac{1}{6 x^6}-\frac{5}{4} \tan ^{-1}\left (\frac{1}{x^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 28, normalized size = 0.8 \begin{align*} -{\frac{1}{6\,{x}^{6}}}+{x}^{-2}+{\frac{{x}^{2}}{4\,{x}^{4}+4}}+{\frac{5\,\arctan \left ({x}^{2} \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4855, size = 41, normalized size = 1.11 \begin{align*} \frac{15 \, x^{8} + 10 \, x^{4} - 2}{12 \,{\left (x^{10} + x^{6}\right )}} + \frac{5}{4} \, \arctan \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42714, size = 96, normalized size = 2.59 \begin{align*} \frac{15 \, x^{8} + 10 \, x^{4} + 15 \,{\left (x^{10} + x^{6}\right )} \arctan \left (x^{2}\right ) - 2}{12 \,{\left (x^{10} + x^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.18952, size = 29, normalized size = 0.78 \begin{align*} \frac{5 \operatorname{atan}{\left (x^{2} \right )}}{4} + \frac{15 x^{8} + 10 x^{4} - 2}{12 x^{10} + 12 x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14105, size = 42, normalized size = 1.14 \begin{align*} \frac{x^{2}}{4 \,{\left (x^{4} + 1\right )}} + \frac{6 \, x^{4} - 1}{6 \, x^{6}} + \frac{5}{4} \, \arctan \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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