Optimal. Leaf size=86 \[ \frac{5 x^8}{8}-\frac{17 x^6}{6}+\frac{19 x^4}{4}+19 x^2-\frac{25 \left (7 x^2+15\right )}{8 \left (x^4+2 x^2+3\right )}-\frac{183}{4} \log \left (x^4+2 x^2+3\right )+\frac{201 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{8 \sqrt{2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.135313, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {1663, 1660, 1657, 634, 618, 204, 628} \[ \frac{5 x^8}{8}-\frac{17 x^6}{6}+\frac{19 x^4}{4}+19 x^2-\frac{25 \left (7 x^2+15\right )}{8 \left (x^4+2 x^2+3\right )}-\frac{183}{4} \log \left (x^4+2 x^2+3\right )+\frac{201 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{8 \sqrt{2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1663
Rule 1660
Rule 1657
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^9 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^4 \left (4+x+3 x^2+5 x^3\right )}{\left (3+2 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=-\frac{25 \left (15+7 x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac{1}{16} \operatorname{Subst}\left (\int \frac{-150-400 x+200 x^2-56 x^4+40 x^5}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=-\frac{25 \left (15+7 x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac{1}{16} \operatorname{Subst}\left (\int \left (304+152 x-136 x^2+40 x^3-\frac{6 (177+244 x)}{3+2 x+x^2}\right ) \, dx,x,x^2\right )\\ &=19 x^2+\frac{19 x^4}{4}-\frac{17 x^6}{6}+\frac{5 x^8}{8}-\frac{25 \left (15+7 x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac{3}{8} \operatorname{Subst}\left (\int \frac{177+244 x}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=19 x^2+\frac{19 x^4}{4}-\frac{17 x^6}{6}+\frac{5 x^8}{8}-\frac{25 \left (15+7 x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac{201}{8} \operatorname{Subst}\left (\int \frac{1}{3+2 x+x^2} \, dx,x,x^2\right )-\frac{183}{4} \operatorname{Subst}\left (\int \frac{2+2 x}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=19 x^2+\frac{19 x^4}{4}-\frac{17 x^6}{6}+\frac{5 x^8}{8}-\frac{25 \left (15+7 x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac{183}{4} \log \left (3+2 x^2+x^4\right )-\frac{201}{4} \operatorname{Subst}\left (\int \frac{1}{-8-x^2} \, dx,x,2 \left (1+x^2\right )\right )\\ &=19 x^2+\frac{19 x^4}{4}-\frac{17 x^6}{6}+\frac{5 x^8}{8}-\frac{25 \left (15+7 x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac{201 \tan ^{-1}\left (\frac{1+x^2}{\sqrt{2}}\right )}{8 \sqrt{2}}-\frac{183}{4} \log \left (3+2 x^2+x^4\right )\\ \end{align*}
Mathematica [A] time = 0.0472626, size = 78, normalized size = 0.91 \[ \frac{1}{48} \left (30 x^8-136 x^6+228 x^4+912 x^2-\frac{150 \left (7 x^2+15\right )}{x^4+2 x^2+3}-2196 \log \left (x^4+2 x^2+3\right )+603 \sqrt{2} \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.011, size = 74, normalized size = 0.9 \begin{align*}{\frac{5\,{x}^{8}}{8}}-{\frac{17\,{x}^{6}}{6}}+{\frac{19\,{x}^{4}}{4}}+19\,{x}^{2}-{\frac{1}{2\,{x}^{4}+4\,{x}^{2}+6} \left ({\frac{175\,{x}^{2}}{4}}+{\frac{375}{4}} \right ) }-{\frac{183\,\ln \left ({x}^{4}+2\,{x}^{2}+3 \right ) }{4}}+{\frac{201\,\sqrt{2}}{16}\arctan \left ({\frac{ \left ( 2\,{x}^{2}+2 \right ) \sqrt{2}}{4}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.47227, size = 96, normalized size = 1.12 \begin{align*} \frac{5}{8} \, x^{8} - \frac{17}{6} \, x^{6} + \frac{19}{4} \, x^{4} + 19 \, x^{2} + \frac{201}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) - \frac{25 \,{\left (7 \, x^{2} + 15\right )}}{8 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} - \frac{183}{4} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.50026, size = 270, normalized size = 3.14 \begin{align*} \frac{30 \, x^{12} - 76 \, x^{10} + 46 \, x^{8} + 960 \, x^{6} + 2508 \, x^{4} + 603 \, \sqrt{2}{\left (x^{4} + 2 \, x^{2} + 3\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) + 1686 \, x^{2} - 2196 \,{\left (x^{4} + 2 \, x^{2} + 3\right )} \log \left (x^{4} + 2 \, x^{2} + 3\right ) - 2250}{48 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.171973, size = 85, normalized size = 0.99 \begin{align*} \frac{5 x^{8}}{8} - \frac{17 x^{6}}{6} + \frac{19 x^{4}}{4} + 19 x^{2} - \frac{175 x^{2} + 375}{8 x^{4} + 16 x^{2} + 24} - \frac{183 \log{\left (x^{4} + 2 x^{2} + 3 \right )}}{4} + \frac{201 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x^{2}}{2} + \frac{\sqrt{2}}{2} \right )}}{16} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.13086, size = 103, normalized size = 1.2 \begin{align*} \frac{5}{8} \, x^{8} - \frac{17}{6} \, x^{6} + \frac{19}{4} \, x^{4} + 19 \, x^{2} + \frac{201}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) + \frac{366 \, x^{4} + 557 \, x^{2} + 723}{8 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} - \frac{183}{4} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]