Optimal. Leaf size=235 \[ \frac{7 \left (58 x^2+11\right ) x}{64 \left (x^4+2 x^2+3\right )}+\frac{25 \left (3-x^2\right ) x}{16 \left (x^4+2 x^2+3\right )^2}-\frac{1}{512} \sqrt{1176531 \sqrt{3}-827621} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{1}{512} \sqrt{1176531 \sqrt{3}-827621} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+5 x+\frac{1}{256} \sqrt{827621+1176531 \sqrt{3}} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )-\frac{1}{256} \sqrt{827621+1176531 \sqrt{3}} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]
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Rubi [A] time = 0.300497, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.258, Rules used = {1668, 1678, 1676, 1169, 634, 618, 204, 628} \[ \frac{7 \left (58 x^2+11\right ) x}{64 \left (x^4+2 x^2+3\right )}+\frac{25 \left (3-x^2\right ) x}{16 \left (x^4+2 x^2+3\right )^2}-\frac{1}{512} \sqrt{1176531 \sqrt{3}-827621} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{1}{512} \sqrt{1176531 \sqrt{3}-827621} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+5 x+\frac{1}{256} \sqrt{827621+1176531 \sqrt{3}} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )-\frac{1}{256} \sqrt{827621+1176531 \sqrt{3}} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]
Antiderivative was successfully verified.
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Rule 1668
Rule 1678
Rule 1676
Rule 1169
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^6 \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+1650 x^2-672 x^6+480 x^8}{\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{7 x \left (11+58 x^2\right )}{64 \left (3+2 x^2+x^4\right )}+\frac{\int \frac{-12744-49104 x^2+23040 x^4}{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{7 x \left (11+58 x^2\right )}{64 \left (3+2 x^2+x^4\right )}+\frac{\int \left (23040-\frac{72 \left (1137+1322 x^2\right )}{3+2 x^2+x^4}\right ) \, dx}{4608}\\ &=5 x+\frac{25 x \left (3-x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{7 x \left (11+58 x^2\right )}{64 \left (3+2 x^2+x^4\right )}-\frac{1}{64} \int \frac{1137+1322 x^2}{3+2 x^2+x^4} \, dx\\ &=5 x+\frac{25 x \left (3-x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{7 x \left (11+58 x^2\right )}{64 \left (3+2 x^2+x^4\right )}-\frac{\int \frac{1137 \sqrt{2 \left (-1+\sqrt{3}\right )}-\left (1137-1322 \sqrt{3}\right ) x}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{128 \sqrt{6 \left (-1+\sqrt{3}\right )}}-\frac{\int \frac{1137 \sqrt{2 \left (-1+\sqrt{3}\right )}+\left (1137-1322 \sqrt{3}\right ) x}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{128 \sqrt{6 \left (-1+\sqrt{3}\right )}}\\ &=5 x+\frac{25 x \left (3-x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{7 x \left (11+58 x^2\right )}{64 \left (3+2 x^2+x^4\right )}-\frac{1}{256} \left (1322+379 \sqrt{3}\right ) \int \frac{1}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx-\frac{1}{256} \left (1322+379 \sqrt{3}\right ) \int \frac{1}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx-\frac{1}{512} \sqrt{-827621+1176531 \sqrt{3}} \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}{512} \sqrt{-827621+1176531 \sqrt{3}} \int \frac{\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx\\ &=5 x+\frac{25 x \left (3-x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{7 x \left (11+58 x^2\right )}{64 \left (3+2 x^2+x^4\right )}-\frac{1}{512} \sqrt{-827621+1176531 \sqrt{3}} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )+\frac{1}{512} \sqrt{-827621+1176531 \sqrt{3}} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )-\frac{1}{128} \left (-1322-379 \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 (-1322-379 \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 )\\ &=5 x+\frac{25 x \left (3-x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{7 x \left (11+58 x^2\right )}{64 \left (3+2 x^2+x^4\right )}+\frac{1}{256} \sqrt{827621+1176531 \sqrt{3}} \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{827621+1176531 \sqrt{3}} \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{-827621+1176531 \sqrt{3}} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )+\frac{1}{512} \sqrt{-827621+1176531 \sqrt{3}} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )\\ \end{align*}
Mathematica [C] time = 0.330087, size = 138, normalized size = 0.59 \[ \frac{1}{256} \left (\frac{4 x \left (320 x^8+1686 x^6+4089 x^4+5112 x^2+3411\right )}{\left (x^4+2 x^2+3\right )^2}-\frac{i \left (185 \sqrt{2}-2644 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{\sqrt{1-i \sqrt{2}}}+\frac{i \left (185 \sqrt{2}+2644 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{\sqrt{1+i \sqrt{2}}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.022, size = 422, normalized size = 1.8 \begin{align*} 5\,x-{\frac{1}{ \left ({x}^{4}+2\,{x}^{2}+3 \right ) ^{2}} \left ( -{\frac{203\,{x}^{7}}{32}}-{\frac{889\,{x}^{5}}{64}}-{\frac{159\,{x}^{3}}{8}}-{\frac{531\,x}{64}} \right ) }-{\frac{943\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{1024}}-{\frac{185\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{1024}}-{\frac{ \left ( -1886+1886\,\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{-370+370\,\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{379\,\sqrt{3}}{64\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{943\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{1024}}+{\frac{185\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{1024}}-{\frac{ \left ( -1886+1886\,\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{-370+370\,\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{379\,\sqrt{3}}{64\,\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*} 5 \, x + \frac{406 \, x^{7} + 889 \, x^{5} + 1272 \, x^{3} + 531 \, x}{64 \,{\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}} - \frac{1}{64} \, \int \frac{1322 \, x^{2} + 1137}{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 = 1.77273, size = 2670, normalized size = 11.36 \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 = 0.586753, size = 71, normalized size = 0.3 \begin{align*} 5 x + \frac{406 x^{7} + 889 x^{5} + 1272 x^{3} + 531 x}{64 x^{8} + 256 x^{6} + 640 x^{4} + 768 x^{2} + 576} + \operatorname{RootSum}{\left (17179869184 t^{4} + 216955879424 t^{2} + 4152675581883, \left ( t \mapsto t \log{\left (- \frac{31641829376 t^{3}}{1549210136091} - \frac{455309168896 t}{1549210136091} + 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^{6}}{{\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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