Optimal. Leaf size=238 \[ \frac{x \left (238-59 x^2\right )}{64 \left (x^4+2 x^2+3\right )}-\frac{25 x \left (x^2+3\right )}{16 \left (x^4+2 x^2+3\right )^2}+\frac{1}{512} \sqrt{3 \left (48835+32827 \sqrt{3}\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{1}{512} \sqrt{3 \left (48835+32827 \sqrt{3}\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{1}{256} \sqrt{3 \left (32827 \sqrt{3}-48835\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{1}{256} \sqrt{3 \left (32827 \sqrt{3}-48835\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]
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Rubi [A] time = 0.290093, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {1668, 1678, 1169, 634, 618, 204, 628} \[ \frac{x \left (238-59 x^2\right )}{64 \left (x^4+2 x^2+3\right )}-\frac{25 x \left (x^2+3\right )}{16 \left (x^4+2 x^2+3\right )^2}+\frac{1}{512} \sqrt{3 \left (48835+32827 \sqrt{3}\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{1}{512} \sqrt{3 \left (48835+32827 \sqrt{3}\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{1}{256} \sqrt{3 \left (32827 \sqrt{3}-48835\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{1}{256} \sqrt{3 \left (32827 \sqrt{3}-48835\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]
Antiderivative was successfully verified.
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Rule 1668
Rule 1678
Rule 1169
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^4 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^3} \, dx &=-\frac{25 x \left (3+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{1}{96} \int \frac{450-750 x^2-672 x^4+480 x^6}{\left (3+2 x^2+x^4\right )^2} \, dx\\ &=-\frac{25 x \left (3+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (238-59 x^2\right )}{64 \left (3+2 x^2+x^4\right )}+\frac{\int \frac{-9936+18792 x^2}{3+2 x^2+x^4} \, dx}{4608}\\ &=-\frac{25 x \left (3+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (238-59 x^2\right )}{64 \left (3+2 x^2+x^4\right )}+\frac{\int \frac{-9936 \sqrt{2 \left (-1+\sqrt{3}\right )}-\left (-9936-18792 \sqrt{3}\right ) x}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{9216 \sqrt{6 \left (-1+\sqrt{3}\right )}}+\frac{\int \frac{-9936 \sqrt{2 \left (-1+\sqrt{3}\right )}+\left (-9936-18792 \sqrt{3}\right ) x}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{9216 \sqrt{6 \left (-1+\sqrt{3}\right )}}\\ &=-\frac{25 x \left (3+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (238-59 x^2\right )}{64 \left (3+2 x^2+x^4\right )}+\frac{1}{256} \left (261-46 \sqrt{3}\right ) \int \frac{1}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx+\frac{1}{256} \left (261-46 \sqrt{3}\right ) \int \frac{1}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx+\frac{1}{256} \left (\sqrt{\frac{3}{2 \left (-1+\sqrt{3}\right )}} \left (46+87 \sqrt{3}\right )\right ) \int \frac{-\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx-\frac{1}{256} \left (\sqrt{\frac{3}{2 \left (-1+\sqrt{3}\right )}} \left (46+87 \sqrt{3}\right )\right ) \int \frac{\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx\\ &=-\frac{25 x \left (3+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (238-59 x^2\right )}{64 \left (3+2 x^2+x^4\right )}+\frac{1}{512} \sqrt{146505+98481 \sqrt{3}} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )-\frac{1}{512} \sqrt{146505+98481 \sqrt{3}} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )+\frac{1}{128} \left (-261+46 \sqrt{3}\right ) \operatorname{Subst}\left (\int \frac{1}{-2 \left (1+\sqrt{3}\right )-x^2} \, dx,x,-\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x\right )+\frac{1}{128} \left (-261+46 \sqrt{3}\right ) \operatorname{Subst}\left (\int \frac{1}{-2 \left (1+\sqrt{3}\right )-x^2} \, dx,x,\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x\right )\\ &=-\frac{25 x \left (3+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (238-59 x^2\right )}{64 \left (3+2 x^2+x^4\right )}-\frac{1}{256} \sqrt{3 \left (-48835+32827 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (-1+\sqrt{3}\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{1}{256} \sqrt{3 \left (-48835+32827 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{1}{512} \sqrt{146505+98481 \sqrt{3}} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )-\frac{1}{512} \sqrt{146505+98481 \sqrt{3}} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )\\ \end{align*}
Mathematica [C] time = 0.303537, size = 129, normalized size = 0.54 \[ \frac{1}{256} \left (\frac{4 x \left (-59 x^6+120 x^4+199 x^2+414\right )}{\left (x^4+2 x^2+3\right )^2}+\frac{3 \left (174+133 i \sqrt{2}\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{\sqrt{1-i \sqrt{2}}}+\frac{3 \left (174-133 i \sqrt{2}\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{\sqrt{1+i \sqrt{2}}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.019, size = 418, normalized size = 1.8 \begin{align*}{\frac{1}{ \left ({x}^{4}+2\,{x}^{2}+3 \right ) ^{2}} \left ( -{\frac{59\,{x}^{7}}{64}}+{\frac{15\,{x}^{5}}{8}}+{\frac{199\,{x}^{3}}{64}}+{\frac{207\,x}{32}} \right ) }+{\frac{307\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{1024}}+{\frac{399\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{1024}}+{\frac{ \left ( -614+614\,\sqrt{3} \right ) \sqrt{3}}{512\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{-798+798\,\sqrt{3}}{512\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{23\,\sqrt{3}}{32\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{307\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{1024}}-{\frac{399\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{1024}}+{\frac{ \left ( -614+614\,\sqrt{3} \right ) \sqrt{3}}{512\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{-798+798\,\sqrt{3}}{512\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{23\,\sqrt{3}}{32\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{59 \, x^{7} - 120 \, x^{5} - 199 \, x^{3} - 414 \, x}{64 \,{\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}} + \frac{3}{64} \, \int \frac{87 \, x^{2} - 46}{x^{4} + 2 \, x^{2} + 3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.8353, size = 2402, normalized size = 10.09 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.591851, size = 68, normalized size = 0.29 \begin{align*} - \frac{59 x^{7} - 120 x^{5} - 199 x^{3} - 414 x}{64 x^{8} + 256 x^{6} + 640 x^{4} + 768 x^{2} + 576} + \operatorname{RootSum}{\left (17179869184 t^{4} - 38405406720 t^{2} + 29095522083, \left ( t \mapsto t \log{\left (\frac{10301210624 t^{3}}{6083466813} - \frac{4322999552 t}{2027822271} + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (5 \, x^{6} + 3 \, x^{4} + x^{2} + 4\right )} x^{4}}{{\left (x^{4} + 2 \, x^{2} + 3\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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