Optimal. Leaf size=263 \[ \frac{x \left (x^2 (-(7 d-7 f+4 h))+2 d+3 f-h\right )}{24 \left (x^4+x^2+1\right )}+\frac{x \left (x^2 (-(d-2 f+h))+d+f-2 h\right )}{12 \left (x^4+x^2+1\right )^2}-\frac{1}{32} \log \left (x^2-x+1\right ) (9 d-4 f+3 h)+\frac{1}{32} \log \left (x^2+x+1\right ) (9 d-4 f+3 h)-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right ) (13 d+2 f+h)}{48 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) (13 d+2 f+h)}{48 \sqrt{3}}+\frac{\left (2 x^2+1\right ) (2 e-g)}{12 \left (x^4+x^2+1\right )}+\frac{x^2 (2 e-g)+e-2 g}{12 \left (x^4+x^2+1\right )^2}+\frac{(2 e-g) \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
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Rubi [A] time = 0.262926, antiderivative size = 263, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 11, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.355, Rules used = {1673, 1678, 1178, 1169, 634, 618, 204, 628, 1247, 638, 614} \[ \frac{x \left (x^2 (-(7 d-7 f+4 h))+2 d+3 f-h\right )}{24 \left (x^4+x^2+1\right )}+\frac{x \left (x^2 (-(d-2 f+h))+d+f-2 h\right )}{12 \left (x^4+x^2+1\right )^2}-\frac{1}{32} \log \left (x^2-x+1\right ) (9 d-4 f+3 h)+\frac{1}{32} \log \left (x^2+x+1\right ) (9 d-4 f+3 h)-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right ) (13 d+2 f+h)}{48 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) (13 d+2 f+h)}{48 \sqrt{3}}+\frac{\left (2 x^2+1\right ) (2 e-g)}{12 \left (x^4+x^2+1\right )}+\frac{x^2 (2 e-g)+e-2 g}{12 \left (x^4+x^2+1\right )^2}+\frac{(2 e-g) \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1673
Rule 1678
Rule 1178
Rule 1169
Rule 634
Rule 618
Rule 204
Rule 628
Rule 1247
Rule 638
Rule 614
Rubi steps
\begin{align*} \int \frac{d+e x+f x^2+g x^3+h x^4}{\left (1+x^2+x^4\right )^3} \, dx &=\int \frac{x \left (e+g x^2\right )}{\left (1+x^2+x^4\right )^3} \, dx+\int \frac{d+f x^2+h x^4}{\left (1+x^2+x^4\right )^3} \, dx\\ &=\frac{x \left (d+f-2 h-(d-2 f+h) x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac{1}{12} \int \frac{11 d-f+2 h-5 (d-2 f+h) x^2}{\left (1+x^2+x^4\right )^2} \, dx+\frac{1}{2} \operatorname{Subst}\left (\int \frac{e+g x}{\left (1+x+x^2\right )^3} \, dx,x,x^2\right )\\ &=\frac{e-2 g+(2 e-g) x^2}{12 \left (1+x^2+x^4\right )^2}+\frac{x \left (d+f-2 h-(d-2 f+h) x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac{x \left (2 d+3 f-h-(7 d-7 f+4 h) x^2\right )}{24 \left (1+x^2+x^4\right )}+\frac{1}{72} \int \frac{15 (4 d-f+h)-3 (7 d-7 f+4 h) x^2}{1+x^2+x^4} \, dx+\frac{1}{4} (2 e-g) \operatorname{Subst}\left (\int \frac{1}{\left (1+x+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac{e-2 g+(2 e-g) x^2}{12 \left (1+x^2+x^4\right )^2}+\frac{x \left (d+f-2 h-(d-2 f+h) x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac{(2 e-g) \left (1+2 x^2\right )}{12 \left (1+x^2+x^4\right )}+\frac{x \left (2 d+3 f-h-(7 d-7 f+4 h) x^2\right )}{24 \left (1+x^2+x^4\right )}+\frac{1}{144} \int \frac{15 (4 d-f+h)-(15 (4 d-f+h)+3 (7 d-7 f+4 h)) x}{1-x+x^2} \, dx+\frac{1}{144} \int \frac{15 (4 d-f+h)+(15 (4 d-f+h)+3 (7 d-7 f+4 h)) x}{1+x+x^2} \, dx+\frac{1}{6} (2 e-g) \operatorname{Subst}\left (\int \frac{1}{1+x+x^2} \, dx,x,x^2\right )\\ &=\frac{e-2 g+(2 e-g) x^2}{12 \left (1+x^2+x^4\right )^2}+\frac{x \left (d+f-2 h-(d-2 f+h) x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac{(2 e-g) \left (1+2 x^2\right )}{12 \left (1+x^2+x^4\right )}+\frac{x \left (2 d+3 f-h-(7 d-7 f+4 h) x^2\right )}{24 \left (1+x^2+x^4\right )}+\frac{1}{3} (-2 e+g) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 x^2\right )+\frac{1}{32} (-9 d+4 f-3 h) \int \frac{-1+2 x}{1-x+x^2} \, dx+\frac{1}{96} (13 d+2 f+h) \int \frac{1}{1-x+x^2} \, dx+\frac{1}{96} (13 d+2 f+h) \int \frac{1}{1+x+x^2} \, dx+\frac{1}{32} (9 d-4 f+3 h) \int \frac{1+2 x}{1+x+x^2} \, dx\\ &=\frac{e-2 g+(2 e-g) x^2}{12 \left (1+x^2+x^4\right )^2}+\frac{x \left (d+f-2 h-(d-2 f+h) x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac{(2 e-g) \left (1+2 x^2\right )}{12 \left (1+x^2+x^4\right )}+\frac{x \left (2 d+3 f-h-(7 d-7 f+4 h) x^2\right )}{24 \left (1+x^2+x^4\right )}+\frac{(2 e-g) \tan ^{-1}\left (\frac{1+2 x^2}{\sqrt{3}}\right )}{3 \sqrt{3}}-\frac{1}{32} (9 d-4 f+3 h) \log \left (1-x+x^2\right )+\frac{1}{32} (9 d-4 f+3 h) \log \left (1+x+x^2\right )+\frac{1}{48} (-13 d-2 f-h) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 x\right )+\frac{1}{48} (-13 d-2 f-h) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 x\right )\\ &=\frac{e-2 g+(2 e-g) x^2}{12 \left (1+x^2+x^4\right )^2}+\frac{x \left (d+f-2 h-(d-2 f+h) x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac{(2 e-g) \left (1+2 x^2\right )}{12 \left (1+x^2+x^4\right )}+\frac{x \left (2 d+3 f-h-(7 d-7 f+4 h) x^2\right )}{24 \left (1+x^2+x^4\right )}-\frac{(13 d+2 f+h) \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{48 \sqrt{3}}+\frac{(13 d+2 f+h) \tan ^{-1}\left (\frac{1+2 x}{\sqrt{3}}\right )}{48 \sqrt{3}}+\frac{(2 e-g) \tan ^{-1}\left (\frac{1+2 x^2}{\sqrt{3}}\right )}{3 \sqrt{3}}-\frac{1}{32} (9 d-4 f+3 h) \log \left (1-x+x^2\right )+\frac{1}{32} (9 d-4 f+3 h) \log \left (1+x+x^2\right )\\ \end{align*}
Mathematica [C] time = 0.942372, size = 303, normalized size = 1.15 \[ \frac{1}{144} \left (-\frac{6 \left (x \left (7 d x^2-2 d-7 f x^2-3 f+4 h x^2+h\right )-4 e \left (2 x^2+1\right )+g \left (4 x^2+2\right )\right )}{x^4+x^2+1}+\frac{12 \left (x \left (-d x^2+d+2 f x^2+f-h \left (x^2+2\right )\right )+2 e x^2+e-g \left (x^2+2\right )\right )}{\left (x^4+x^2+1\right )^2}-\frac{\tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}-i\right ) x\right ) \left (\left (7 \sqrt{3}-47 i\right ) d+\left (-7 \sqrt{3}+17 i\right ) f+2 \left (2 \sqrt{3}-7 i\right ) h\right )}{\sqrt{\frac{1}{6} \left (1+i \sqrt{3}\right )}}-\frac{\tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}+i\right ) x\right ) \left (\left (7 \sqrt{3}+47 i\right ) d-\left (7 \sqrt{3}+17 i\right ) f+2 \left (2 \sqrt{3}+7 i\right ) h\right )}{\sqrt{\frac{1}{6} \left (1-i \sqrt{3}\right )}}-16 \sqrt{3} (2 e-g) \tan ^{-1}\left (\frac{\sqrt{3}}{2 x^2+1}\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.017, size = 396, normalized size = 1.5 \begin{align*}{\frac{1}{16\, \left ({x}^{2}+x+1 \right ) ^{2}} \left ( \left ( -{\frac{7\,d}{3}}+{\frac{7\,f}{3}}-{\frac{4\,h}{3}}-{\frac{4\,e}{3}}-{\frac{g}{3}} \right ){x}^{3}+ \left ( -6\,d+4\,f-2\,h-2\,g \right ){x}^{2}+ \left ( -{\frac{20\,d}{3}}+{\frac{13\,f}{3}}-{\frac{5\,h}{3}}+{\frac{e}{3}}-{\frac{8\,g}{3}} \right ) x-4\,d+{\frac{4\,f}{3}}+2\,e-2\,g \right ) }+{\frac{9\,d\ln \left ({x}^{2}+x+1 \right ) }{32}}-{\frac{\ln \left ({x}^{2}+x+1 \right ) f}{8}}+{\frac{3\,\ln \left ({x}^{2}+x+1 \right ) h}{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{\sqrt{3}f}{72}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}g}{9}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}h}{144}\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{7\,f}{3}}+{\frac{4\,h}{3}}-{\frac{4\,e}{3}}-{\frac{g}{3}} \right ){x}^{3}+ \left ( -6\,d+4\,f-2\,h+2\,g \right ){x}^{2}+ \left ({\frac{20\,d}{3}}-{\frac{13\,f}{3}}+{\frac{5\,h}{3}}+{\frac{e}{3}}-{\frac{8\,g}{3}} \right ) x-4\,d+{\frac{4\,f}{3}}-2\,e+2\,g \right ) }-{\frac{9\,d\ln \left ({x}^{2}-x+1 \right ) }{32}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) f}{8}}-{\frac{3\,\ln \left ({x}^{2}-x+1 \right ) h}{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 ) }+{\frac{\sqrt{3}f}{72}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }-{\frac{\sqrt{3}g}{9}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}h}{144}\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 = 1.44078, size = 293, normalized size = 1.11 \begin{align*} \frac{1}{144} \, \sqrt{3}{\left (13 \, d - 32 \, e + 2 \, f + 16 \, g + h\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{144} \, \sqrt{3}{\left (13 \, d + 32 \, e + 2 \, f - 16 \, g + h\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{32} \,{\left (9 \, d - 4 \, f + 3 \, h\right )} \log \left (x^{2} + x + 1\right ) - \frac{1}{32} \,{\left (9 \, d - 4 \, f + 3 \, h\right )} \log \left (x^{2} - x + 1\right ) - \frac{{\left (7 \, d - 7 \, f + 4 \, h\right )} x^{7} - 4 \,{\left (2 \, e - g\right )} x^{6} + 5 \,{\left (d - 2 \, f + h\right )} x^{5} - 6 \,{\left (2 \, e - g\right )} x^{4} + 7 \,{\left (d - 2 \, f + h\right )} x^{3} - 8 \,{\left (2 \, e - g\right )} x^{2} -{\left (4 \, d + 5 \, f - 5 \, h\right )} x - 6 \, e + 6 \, g}{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 [B] time = 13.0623, size = 1278, normalized size = 4.86 \begin{align*} -\frac{12 \,{\left (7 \, d - 7 \, f + 4 \, h\right )} x^{7} - 48 \,{\left (2 \, e - g\right )} x^{6} + 60 \,{\left (d - 2 \, f + h\right )} x^{5} - 72 \,{\left (2 \, e - g\right )} x^{4} + 84 \,{\left (d - 2 \, f + h\right )} x^{3} - 96 \,{\left (2 \, e - g\right )} x^{2} - 2 \, \sqrt{3}{\left ({\left (13 \, d - 32 \, e + 2 \, f + 16 \, g + h\right )} x^{8} + 2 \,{\left (13 \, d - 32 \, e + 2 \, f + 16 \, g + h\right )} x^{6} + 3 \,{\left (13 \, d - 32 \, e + 2 \, f + 16 \, g + h\right )} x^{4} + 2 \,{\left (13 \, d - 32 \, e + 2 \, f + 16 \, g + h\right )} x^{2} + 13 \, d - 32 \, e + 2 \, f + 16 \, g + h\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - 2 \, \sqrt{3}{\left ({\left (13 \, d + 32 \, e + 2 \, f - 16 \, g + h\right )} x^{8} + 2 \,{\left (13 \, d + 32 \, e + 2 \, f - 16 \, g + h\right )} x^{6} + 3 \,{\left (13 \, d + 32 \, e + 2 \, f - 16 \, g + h\right )} x^{4} + 2 \,{\left (13 \, d + 32 \, e + 2 \, f - 16 \, g + h\right )} x^{2} + 13 \, d + 32 \, e + 2 \, f - 16 \, g + h\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - 12 \,{\left (4 \, d + 5 \, f - 5 \, h\right )} x - 9 \,{\left ({\left (9 \, d - 4 \, f + 3 \, h\right )} x^{8} + 2 \,{\left (9 \, d - 4 \, f + 3 \, h\right )} x^{6} + 3 \,{\left (9 \, d - 4 \, f + 3 \, h\right )} x^{4} + 2 \,{\left (9 \, d - 4 \, f + 3 \, h\right )} x^{2} + 9 \, d - 4 \, f + 3 \, h\right )} \log \left (x^{2} + x + 1\right ) + 9 \,{\left ({\left (9 \, d - 4 \, f + 3 \, h\right )} x^{8} + 2 \,{\left (9 \, d - 4 \, f + 3 \, h\right )} x^{6} + 3 \,{\left (9 \, d - 4 \, f + 3 \, h\right )} x^{4} + 2 \,{\left (9 \, d - 4 \, f + 3 \, h\right )} x^{2} + 9 \, d - 4 \, f + 3 \, h\right )} \log \left (x^{2} - x + 1\right ) - 72 \, e + 72 \, g}{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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12826, size = 308, normalized size = 1.17 \begin{align*} \frac{1}{144} \, \sqrt{3}{\left (13 \, d + 2 \, f + 16 \, g + h - 32 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{144} \, \sqrt{3}{\left (13 \, d + 2 \, f - 16 \, g + h + 32 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{32} \,{\left (9 \, d - 4 \, f + 3 \, h\right )} \log \left (x^{2} + x + 1\right ) - \frac{1}{32} \,{\left (9 \, d - 4 \, f + 3 \, h\right )} \log \left (x^{2} - x + 1\right ) - \frac{7 \, d x^{7} - 7 \, f x^{7} + 4 \, h x^{7} + 4 \, g x^{6} - 8 \, x^{6} e + 5 \, d x^{5} - 10 \, f x^{5} + 5 \, h x^{5} + 6 \, g x^{4} - 12 \, x^{4} e + 7 \, d x^{3} - 14 \, f x^{3} + 7 \, h x^{3} + 8 \, g x^{2} - 16 \, x^{2} e - 4 \, d x - 5 \, f x + 5 \, h x + 6 \, g - 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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