Optimal. Leaf size=185 \[ \frac{d x \left (2-7 x^2\right )}{24 \left (x^4+x^2+1\right )}+\frac{d x \left (1-x^2\right )}{12 \left (x^4+x^2+1\right )^2}-\frac{9}{32} d \log \left (x^2-x+1\right )+\frac{9}{32} d \log \left (x^2+x+1\right )-\frac{13 d \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{48 \sqrt{3}}+\frac{13 d \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{48 \sqrt{3}}+\frac{e \left (2 x^2+1\right )}{6 \left (x^4+x^2+1\right )}+\frac{e \left (2 x^2+1\right )}{12 \left (x^4+x^2+1\right )^2}+\frac{2 e \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
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Rubi [A] time = 0.117298, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 11, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.688, Rules used = {1673, 12, 1092, 1178, 1169, 634, 618, 204, 628, 1107, 614} \[ \frac{d x \left (2-7 x^2\right )}{24 \left (x^4+x^2+1\right )}+\frac{d x \left (1-x^2\right )}{12 \left (x^4+x^2+1\right )^2}-\frac{9}{32} d \log \left (x^2-x+1\right )+\frac{9}{32} d \log \left (x^2+x+1\right )-\frac{13 d \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{48 \sqrt{3}}+\frac{13 d \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{48 \sqrt{3}}+\frac{e \left (2 x^2+1\right )}{6 \left (x^4+x^2+1\right )}+\frac{e \left (2 x^2+1\right )}{12 \left (x^4+x^2+1\right )^2}+\frac{2 e \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1673
Rule 12
Rule 1092
Rule 1178
Rule 1169
Rule 634
Rule 618
Rule 204
Rule 628
Rule 1107
Rule 614
Rubi steps
\begin{align*} \int \frac{d+e x}{\left (1+x^2+x^4\right )^3} \, dx &=\int \frac{d}{\left (1+x^2+x^4\right )^3} \, dx+\int \frac{e x}{\left (1+x^2+x^4\right )^3} \, dx\\ &=d \int \frac{1}{\left (1+x^2+x^4\right )^3} \, dx+e \int \frac{x}{\left (1+x^2+x^4\right )^3} \, dx\\ &=\frac{d x \left (1-x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac{1}{12} d \int \frac{11-5 x^2}{\left (1+x^2+x^4\right )^2} \, dx+\frac{1}{2} e \operatorname{Subst}\left (\int \frac{1}{\left (1+x+x^2\right )^3} \, dx,x,x^2\right )\\ &=\frac{d x \left (1-x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac{e \left (1+2 x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac{d x \left (2-7 x^2\right )}{24 \left (1+x^2+x^4\right )}+\frac{1}{72} d \int \frac{60-21 x^2}{1+x^2+x^4} \, dx+\frac{1}{2} e \operatorname{Subst}\left (\int \frac{1}{\left (1+x+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac{d x \left (1-x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac{e \left (1+2 x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac{d x \left (2-7 x^2\right )}{24 \left (1+x^2+x^4\right )}+\frac{e \left (1+2 x^2\right )}{6 \left (1+x^2+x^4\right )}+\frac{1}{144} d \int \frac{60-81 x}{1-x+x^2} \, dx+\frac{1}{144} d \int \frac{60+81 x}{1+x+x^2} \, dx+\frac{1}{3} e \operatorname{Subst}\left (\int \frac{1}{1+x+x^2} \, dx,x,x^2\right )\\ &=\frac{d x \left (1-x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac{e \left (1+2 x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac{d x \left (2-7 x^2\right )}{24 \left (1+x^2+x^4\right )}+\frac{e \left (1+2 x^2\right )}{6 \left (1+x^2+x^4\right )}+\frac{1}{96} (13 d) \int \frac{1}{1-x+x^2} \, dx+\frac{1}{96} (13 d) \int \frac{1}{1+x+x^2} \, dx-\frac{1}{32} (9 d) \int \frac{-1+2 x}{1-x+x^2} \, dx+\frac{1}{32} (9 d) \int \frac{1+2 x}{1+x+x^2} \, dx-\frac{1}{3} (2 e) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 x^2\right )\\ &=\frac{d x \left (1-x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac{e \left (1+2 x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac{d x \left (2-7 x^2\right )}{24 \left (1+x^2+x^4\right )}+\frac{e \left (1+2 x^2\right )}{6 \left (1+x^2+x^4\right )}+\frac{2 e \tan ^{-1}\left (\frac{1+2 x^2}{\sqrt{3}}\right )}{3 \sqrt{3}}-\frac{9}{32} d \log \left (1-x+x^2\right )+\frac{9}{32} d \log \left (1+x+x^2\right )-\frac{1}{48} (13 d) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 x\right )-\frac{1}{48} (13 d) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 x\right )\\ &=\frac{d x \left (1-x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac{e \left (1+2 x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac{d x \left (2-7 x^2\right )}{24 \left (1+x^2+x^4\right )}+\frac{e \left (1+2 x^2\right )}{6 \left (1+x^2+x^4\right )}-\frac{13 d \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{48 \sqrt{3}}+\frac{13 d \tan ^{-1}\left (\frac{1+2 x}{\sqrt{3}}\right )}{48 \sqrt{3}}+\frac{2 e \tan ^{-1}\left (\frac{1+2 x^2}{\sqrt{3}}\right )}{3 \sqrt{3}}-\frac{9}{32} d \log \left (1-x+x^2\right )+\frac{9}{32} d \log \left (1+x+x^2\right )\\ \end{align*}
Mathematica [C] time = 0.753721, size = 186, normalized size = 1.01 \[ \frac{1}{144} \left (\frac{6 \left (d x \left (2-7 x^2\right )+e \left (8 x^2+4\right )\right )}{x^4+x^2+1}+\frac{12 \left (d \left (x-x^3\right )+2 e x^2+e\right )}{\left (x^4+x^2+1\right )^2}-\frac{\left (7 \sqrt{3}-47 i\right ) d \tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}-i\right ) x\right )}{\sqrt{\frac{1}{6} \left (1+i \sqrt{3}\right )}}-\frac{\left (7 \sqrt{3}+47 i\right ) d \tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}+i\right ) x\right )}{\sqrt{\frac{1}{6} \left (1-i \sqrt{3}\right )}}-32 \sqrt{3} e \tan ^{-1}\left (\frac{\sqrt{3}}{2 x^2+1}\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.017, size = 180, normalized size = 1. \begin{align*}{\frac{1}{16\, \left ({x}^{2}+x+1 \right ) ^{2}} \left ( \left ( -{\frac{7\,d}{3}}-{\frac{4\,e}{3}} \right ){x}^{3}-6\,{x}^{2}d+ \left ( -{\frac{20\,d}{3}}+{\frac{e}{3}} \right ) x-4\,d+2\,e \right ) }+{\frac{9\,d\ln \left ({x}^{2}+x+1 \right ) }{32}}+{\frac{13\,d\sqrt{3}}{144}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{2\,\sqrt{3}e}{9}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{1}{16\, \left ({x}^{2}-x+1 \right ) ^{2}} \left ( \left ({\frac{7\,d}{3}}-{\frac{4\,e}{3}} \right ){x}^{3}-6\,{x}^{2}d+ \left ({\frac{20\,d}{3}}+{\frac{e}{3}} \right ) x-4\,d-2\,e \right ) }-{\frac{9\,d\ln \left ({x}^{2}-x+1 \right ) }{32}}+{\frac{13\,d\sqrt{3}}{144}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{2\,\sqrt{3}e}{9}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.45449, size = 185, normalized size = 1. \begin{align*} \frac{1}{144} \, \sqrt{3}{\left (13 \, d - 32 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{144} \, \sqrt{3}{\left (13 \, d + 32 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{9}{32} \, d \log \left (x^{2} + x + 1\right ) - \frac{9}{32} \, d \log \left (x^{2} - x + 1\right ) - \frac{7 \, d x^{7} - 8 \, e x^{6} + 5 \, d x^{5} - 12 \, e x^{4} + 7 \, d x^{3} - 16 \, e x^{2} - 4 \, d x - 6 \, e}{24 \,{\left (x^{8} + 2 \, x^{6} + 3 \, x^{4} + 2 \, x^{2} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67298, size = 726, normalized size = 3.92 \begin{align*} -\frac{84 \, d x^{7} - 96 \, e x^{6} + 60 \, d x^{5} - 144 \, e x^{4} + 84 \, d x^{3} - 192 \, e x^{2} - 2 \, \sqrt{3}{\left ({\left (13 \, d - 32 \, e\right )} x^{8} + 2 \,{\left (13 \, d - 32 \, e\right )} x^{6} + 3 \,{\left (13 \, d - 32 \, e\right )} x^{4} + 2 \,{\left (13 \, d - 32 \, e\right )} x^{2} + 13 \, d - 32 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - 2 \, \sqrt{3}{\left ({\left (13 \, d + 32 \, e\right )} x^{8} + 2 \,{\left (13 \, d + 32 \, e\right )} x^{6} + 3 \,{\left (13 \, d + 32 \, e\right )} x^{4} + 2 \,{\left (13 \, d + 32 \, e\right )} x^{2} + 13 \, d + 32 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - 48 \, d x - 81 \,{\left (d x^{8} + 2 \, d x^{6} + 3 \, d x^{4} + 2 \, d x^{2} + d\right )} \log \left (x^{2} + x + 1\right ) + 81 \,{\left (d x^{8} + 2 \, d x^{6} + 3 \, d x^{4} + 2 \, d x^{2} + d\right )} \log \left (x^{2} - x + 1\right ) - 72 \, e}{288 \,{\left (x^{8} + 2 \, x^{6} + 3 \, x^{4} + 2 \, x^{2} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.63088, size = 1103, normalized size = 5.96 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09851, size = 177, normalized size = 0.96 \begin{align*} \frac{1}{144} \, \sqrt{3}{\left (13 \, d - 32 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{144} \, \sqrt{3}{\left (13 \, d + 32 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{9}{32} \, d \log \left (x^{2} + x + 1\right ) - \frac{9}{32} \, d \log \left (x^{2} - x + 1\right ) - \frac{7 \, d x^{7} - 8 \, x^{6} e + 5 \, d x^{5} - 12 \, x^{4} e + 7 \, d x^{3} - 16 \, x^{2} e - 4 \, d x - 6 \, e}{24 \,{\left (x^{4} + x^{2} + 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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