Optimal. Leaf size=246 \[ \frac{25 x \left (x^2+1\right )}{16 \left (x^4+2 x^2+3\right )^2}-\frac{x \left (88 x^2+353\right )}{192 \left (x^4+2 x^2+3\right )}-\frac{11 \sqrt{\frac{1}{3} \left (1825+1089 \sqrt{3}\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{1536}+\frac{11 \sqrt{\frac{1}{3} \left (1825+1089 \sqrt{3}\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{1536}-\frac{11}{768} \sqrt{\frac{1}{3} \left (1089 \sqrt{3}-1825\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{11}{768} \sqrt{\frac{1}{3} \left (1089 \sqrt{3}-1825\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]
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Rubi [A] time = 0.284363, antiderivative size = 246, 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} \[ \frac{25 x \left (x^2+1\right )}{16 \left (x^4+2 x^2+3\right )^2}-\frac{x \left (88 x^2+353\right )}{192 \left (x^4+2 x^2+3\right )}-\frac{11 \sqrt{\frac{1}{3} \left (1825+1089 \sqrt{3}\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{1536}+\frac{11 \sqrt{\frac{1}{3} \left (1825+1089 \sqrt{3}\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{1536}-\frac{11}{768} \sqrt{\frac{1}{3} \left (1089 \sqrt{3}-1825\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{11}{768} \sqrt{\frac{1}{3} \left (1089 \sqrt{3}-1825\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]
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^2 \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 (1+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{1}{96} \int \frac{-150+78 x^2+480 x^4}{\left (3+2 x^2+x^4\right )^2} \, dx\\ &=\frac{25 x \left (1+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}-\frac{x \left (353+88 x^2\right )}{192 \left (3+2 x^2+x^4\right )}+\frac{\int \frac{6072-2112 x^2}{3+2 x^2+x^4} \, dx}{4608}\\ &=\frac{25 x \left (1+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}-\frac{x \left (353+88 x^2\right )}{192 \left (3+2 x^2+x^4\right )}+\frac{\int \frac{6072 \sqrt{2 \left (-1+\sqrt{3}\right )}-\left (6072+2112 \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{6072 \sqrt{2 \left (-1+\sqrt{3}\right )}+\left (6072+2112 \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 (1+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}-\frac{x \left (353+88 x^2\right )}{192 \left (3+2 x^2+x^4\right )}-\frac{\left (11 \left (24-23 \sqrt{3}\right )\right ) \int \frac{1}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{2304}-\frac{\left (11 \left (24-23 \sqrt{3}\right )\right ) \int \frac{1}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{2304}-\frac{\left (11 \left (23+8 \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}{768 \sqrt{6 \left (-1+\sqrt{3}\right )}}+\frac{\left (11 \left (23+8 \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}{768 \sqrt{6 \left (-1+\sqrt{3}\right )}}\\ &=\frac{25 x \left (1+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}-\frac{x \left (353+88 x^2\right )}{192 \left (3+2 x^2+x^4\right )}-\frac{11}{768} \sqrt{\frac{1825}{12}+\frac{363 \sqrt{3}}{4}} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )+\frac{11}{768} \sqrt{\frac{1825}{12}+\frac{363 \sqrt{3}}{4}} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )+\frac{\left (11 \left (24-23 \sqrt{3}\right )\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 )}{1152}+\frac{\left (11 \left (24-23 \sqrt{3}\right )\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 )}{1152}\\ &=\frac{25 x \left (1+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}-\frac{x \left (353+88 x^2\right )}{192 \left (3+2 x^2+x^4\right )}-\frac{11}{768} \sqrt{\frac{1}{3} \left (-1825+1089 \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{11}{768} \sqrt{\frac{1}{3} \left (-1825+1089 \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{11}{768} \sqrt{\frac{1825}{12}+\frac{363 \sqrt{3}}{4}} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )+\frac{11}{768} \sqrt{\frac{1825}{12}+\frac{363 \sqrt{3}}{4}} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )\\ \end{align*}
Mathematica [C] time = 0.299272, size = 133, normalized size = 0.54 \[ \frac{1}{768} \left (-\frac{4 x \left (88 x^6+529 x^4+670 x^2+759\right )}{\left (x^4+2 x^2+3\right )^2}-\frac{11 i \left (31 \sqrt{2}-16 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{\sqrt{1-i \sqrt{2}}}+\frac{11 i \left (31 \sqrt{2}+16 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{\sqrt{1+i \sqrt{2}}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.022, size = 418, normalized size = 1.7 \begin{align*}{\frac{1}{ \left ({x}^{4}+2\,{x}^{2}+3 \right ) ^{2}} \left ( -{\frac{11\,{x}^{7}}{24}}-{\frac{529\,{x}^{5}}{192}}-{\frac{335\,{x}^{3}}{96}}-{\frac{253\,x}{64}} \right ) }-{\frac{517\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{9216}}-{\frac{341\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{3072}}-{\frac{ \left ( -1034+1034\,\sqrt{3} \right ) \sqrt{3}}{4608\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{-682+682\,\sqrt{3}}{1536\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{253\,\sqrt{3}}{576\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{517\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{9216}}+{\frac{341\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{3072}}-{\frac{ \left ( -1034+1034\,\sqrt{3} \right ) \sqrt{3}}{4608\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{-682+682\,\sqrt{3}}{1536\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{253\,\sqrt{3}}{576\,\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{88 \, x^{7} + 529 \, x^{5} + 670 \, x^{3} + 759 \, x}{192 \,{\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}} - \frac{11}{192} \, \int \frac{8 \, x^{2} - 23}{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 = 1.73407, size = 2074, normalized size = 8.43 \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 = 0.592378, size = 68, normalized size = 0.28 \begin{align*} - \frac{88 x^{7} + 529 x^{5} + 670 x^{3} + 759 x}{192 x^{8} + 768 x^{6} + 1920 x^{4} + 2304 x^{2} + 1728} + \operatorname{RootSum}{\left (463856467968 t^{4} - 57887948800 t^{2} + 1929229929, \left ( t \mapsto t \log{\left (\frac{14193524736 t^{3}}{54274187} - \frac{17989888 t}{1345641} + 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^{2}}{{\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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