Optimal. Leaf size=52 \[ -\frac{2}{x^2+1}-\frac{3}{x^2}-\frac{1}{4 \left (x^2+1\right )^2}+\frac{3}{4 x^4}-\frac{1}{6 x^6}+5 \log \left (x^2+1\right )-10 \log (x) \]
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Rubi [A] time = 0.0280003, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {266, 44} \[ -\frac{2}{x^2+1}-\frac{3}{x^2}-\frac{1}{4 \left (x^2+1\right )^2}+\frac{3}{4 x^4}-\frac{1}{6 x^6}+5 \log \left (x^2+1\right )-10 \log (x) \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^7 \left (1+x^2\right )^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^4 (1+x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{x^4}-\frac{3}{x^3}+\frac{6}{x^2}-\frac{10}{x}+\frac{1}{(1+x)^3}+\frac{4}{(1+x)^2}+\frac{10}{1+x}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{6 x^6}+\frac{3}{4 x^4}-\frac{3}{x^2}-\frac{1}{4 \left (1+x^2\right )^2}-\frac{2}{1+x^2}-10 \log (x)+5 \log \left (1+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0262952, size = 49, normalized size = 0.94 \[ -\frac{60 x^8+90 x^6+20 x^4-5 x^2+2}{12 x^6 \left (x^2+1\right )^2}+5 \log \left (x^2+1\right )-10 \log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 47, normalized size = 0.9 \begin{align*} -{\frac{1}{6\,{x}^{6}}}+{\frac{3}{4\,{x}^{4}}}-3\,{x}^{-2}-{\frac{1}{4\, \left ({x}^{2}+1 \right ) ^{2}}}-2\, \left ({x}^{2}+1 \right ) ^{-1}-10\,\ln \left ( x \right ) +5\,\ln \left ({x}^{2}+1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.942041, size = 72, normalized size = 1.38 \begin{align*} -\frac{60 \, x^{8} + 90 \, x^{6} + 20 \, x^{4} - 5 \, x^{2} + 2}{12 \,{\left (x^{10} + 2 \, x^{8} + x^{6}\right )}} + 5 \, \log \left (x^{2} + 1\right ) - 5 \, \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.2616, size = 189, normalized size = 3.63 \begin{align*} -\frac{60 \, x^{8} + 90 \, x^{6} + 20 \, x^{4} - 5 \, x^{2} - 60 \,{\left (x^{10} + 2 \, x^{8} + x^{6}\right )} \log \left (x^{2} + 1\right ) + 120 \,{\left (x^{10} + 2 \, x^{8} + x^{6}\right )} \log \left (x\right ) + 2}{12 \,{\left (x^{10} + 2 \, x^{8} + x^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.165768, size = 49, normalized size = 0.94 \begin{align*} - 10 \log{\left (x \right )} + 5 \log{\left (x^{2} + 1 \right )} - \frac{60 x^{8} + 90 x^{6} + 20 x^{4} - 5 x^{2} + 2}{12 x^{10} + 24 x^{8} + 12 x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07521, size = 78, normalized size = 1.5 \begin{align*} -\frac{30 \, x^{4} + 68 \, x^{2} + 39}{4 \,{\left (x^{2} + 1\right )}^{2}} + \frac{110 \, x^{6} - 36 \, x^{4} + 9 \, x^{2} - 2}{12 \, x^{6}} + 5 \, \log \left (x^{2} + 1\right ) - 5 \, \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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