Optimal. Leaf size=187 \[ \frac{x \left (x^2 (-(d-2 f+h))+d+f-2 h\right )}{6 \left (x^4+x^2+1\right )}-\frac{1}{8} \log \left (x^2-x+1\right ) (2 d-f+h)+\frac{1}{8} \log \left (x^2+x+1\right ) (2 d-f+h)-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right ) (4 d+f+h)}{12 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) (4 d+f+h)}{12 \sqrt{3}}+\frac{x^2 (2 e-g)+e-2 g}{6 \left (x^4+x^2+1\right )}+\frac{(2 e-g) \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
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Rubi [A] time = 0.167095, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.29, Rules used = {1673, 1678, 1169, 634, 618, 204, 628, 1247, 638} \[ \frac{x \left (x^2 (-(d-2 f+h))+d+f-2 h\right )}{6 \left (x^4+x^2+1\right )}-\frac{1}{8} \log \left (x^2-x+1\right ) (2 d-f+h)+\frac{1}{8} \log \left (x^2+x+1\right ) (2 d-f+h)-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right ) (4 d+f+h)}{12 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) (4 d+f+h)}{12 \sqrt{3}}+\frac{x^2 (2 e-g)+e-2 g}{6 \left (x^4+x^2+1\right )}+\frac{(2 e-g) \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1673
Rule 1678
Rule 1169
Rule 634
Rule 618
Rule 204
Rule 628
Rule 1247
Rule 638
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 )^2} \, dx &=\int \frac{x \left (e+g x^2\right )}{\left (1+x^2+x^4\right )^2} \, dx+\int \frac{d+f x^2+h x^4}{\left (1+x^2+x^4\right )^2} \, dx\\ &=\frac{x \left (d+f-2 h-(d-2 f+h) x^2\right )}{6 \left (1+x^2+x^4\right )}+\frac{1}{6} \int \frac{5 d-f+2 h+(-d+2 f-h) x^2}{1+x^2+x^4} \, dx+\frac{1}{2} \operatorname{Subst}\left (\int \frac{e+g x}{\left (1+x+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac{e-2 g+(2 e-g) x^2}{6 \left (1+x^2+x^4\right )}+\frac{x \left (d+f-2 h-(d-2 f+h) x^2\right )}{6 \left (1+x^2+x^4\right )}+\frac{1}{12} \int \frac{5 d-f+2 h-(6 d-3 f+3 h) x}{1-x+x^2} \, dx+\frac{1}{12} \int \frac{5 d-f+2 h+(6 d-3 f+3 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}{6 \left (1+x^2+x^4\right )}+\frac{x \left (d+f-2 h-(d-2 f+h) x^2\right )}{6 \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}{8} (-2 d+f-h) \int \frac{-1+2 x}{1-x+x^2} \, dx+\frac{1}{8} (2 d-f+h) \int \frac{1+2 x}{1+x+x^2} \, dx+\frac{1}{24} (4 d+f+h) \int \frac{1}{1-x+x^2} \, dx+\frac{1}{24} (4 d+f+h) \int \frac{1}{1+x+x^2} \, dx\\ &=\frac{e-2 g+(2 e-g) x^2}{6 \left (1+x^2+x^4\right )}+\frac{x \left (d+f-2 h-(d-2 f+h) x^2\right )}{6 \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}{8} (2 d-f+h) \log \left (1-x+x^2\right )+\frac{1}{8} (2 d-f+h) \log \left (1+x+x^2\right )+\frac{1}{12} (-4 d-f-h) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 x\right )+\frac{1}{12} (-4 d-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}{6 \left (1+x^2+x^4\right )}+\frac{x \left (d+f-2 h-(d-2 f+h) x^2\right )}{6 \left (1+x^2+x^4\right )}-\frac{(4 d+f+h) \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{12 \sqrt{3}}+\frac{(4 d+f+h) \tan ^{-1}\left (\frac{1+2 x}{\sqrt{3}}\right )}{12 \sqrt{3}}+\frac{(2 e-g) \tan ^{-1}\left (\frac{1+2 x^2}{\sqrt{3}}\right )}{3 \sqrt{3}}-\frac{1}{8} (2 d-f+h) \log \left (1-x+x^2\right )+\frac{1}{8} (2 d-f+h) \log \left (1+x+x^2\right )\\ \end{align*}
Mathematica [C] time = 0.628847, size = 234, normalized size = 1.25 \[ \frac{1}{36} \left (-\frac{6 \left (x \left (d \left (x^2-1\right )-f \left (2 x^2+1\right )+h \left (x^2+2\right )\right )-e \left (2 x^2+1\right )+g \left (x^2+2\right )\right )}{x^4+x^2+1}-\frac{\tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}-i\right ) x\right ) \left (\left (\sqrt{3}-11 i\right ) d-2 \left (\sqrt{3}-2 i\right ) f+\left (\sqrt{3}-5 i\right ) h\right )}{\sqrt{\frac{1}{6} \left (1+i \sqrt{3}\right )}}-\frac{\tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}+i\right ) x\right ) \left (\left (\sqrt{3}+11 i\right ) d-2 \left (\sqrt{3}+2 i\right ) f+\left (\sqrt{3}+5 i\right ) h\right )}{\sqrt{\frac{1}{6} \left (1-i \sqrt{3}\right )}}-4 \sqrt{3} (2 e-g) \tan ^{-1}\left (\frac{\sqrt{3}}{2 x^2+1}\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.014, size = 328, normalized size = 1.8 \begin{align*}{\frac{1}{4\,{x}^{2}+4\,x+4} \left ( \left ( -{\frac{d}{3}}+{\frac{2\,f}{3}}-{\frac{g}{3}}-{\frac{e}{3}}-{\frac{h}{3}} \right ) x-{\frac{2\,d}{3}}+{\frac{f}{3}}-{\frac{2\,g}{3}}+{\frac{e}{3}}+{\frac{h}{3}} \right ) }+{\frac{d\ln \left ({x}^{2}+x+1 \right ) }{4}}-{\frac{\ln \left ({x}^{2}+x+1 \right ) f}{8}}+{\frac{\ln \left ({x}^{2}+x+1 \right ) h}{8}}+{\frac{d\sqrt{3}}{9}\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}{36}\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}{36}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{1}{4\,{x}^{2}-4\,x+4} \left ( \left ({\frac{d}{3}}-{\frac{2\,f}{3}}-{\frac{g}{3}}-{\frac{e}{3}}+{\frac{h}{3}} \right ) x-{\frac{2\,d}{3}}+{\frac{f}{3}}+{\frac{2\,g}{3}}-{\frac{e}{3}}+{\frac{h}{3}} \right ) }-{\frac{d\ln \left ({x}^{2}-x+1 \right ) }{4}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) f}{8}}-{\frac{\ln \left ({x}^{2}-x+1 \right ) h}{8}}+{\frac{d\sqrt{3}}{9}\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}{36}\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}{36}\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 = 1.50396, size = 193, normalized size = 1.03 \begin{align*} \frac{1}{36} \, \sqrt{3}{\left (4 \, d - 8 \, e + f + 4 \, g + h\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{36} \, \sqrt{3}{\left (4 \, d + 8 \, e + f - 4 \, g + h\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{8} \,{\left (2 \, d - f + h\right )} \log \left (x^{2} + x + 1\right ) - \frac{1}{8} \,{\left (2 \, d - f + h\right )} \log \left (x^{2} - x + 1\right ) - \frac{{\left (d - 2 \, f + h\right )} x^{3} -{\left (2 \, e - g\right )} x^{2} -{\left (d + f - 2 \, h\right )} x - e + 2 \, g}{6 \,{\left (x^{4} + x^{2} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 10.4954, size = 694, normalized size = 3.71 \begin{align*} -\frac{12 \,{\left (d - 2 \, f + h\right )} x^{3} - 12 \,{\left (2 \, e - g\right )} x^{2} - 2 \, \sqrt{3}{\left ({\left (4 \, d - 8 \, e + f + 4 \, g + h\right )} x^{4} +{\left (4 \, d - 8 \, e + f + 4 \, g + h\right )} x^{2} + 4 \, d - 8 \, e + f + 4 \, g + h\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - 2 \, \sqrt{3}{\left ({\left (4 \, d + 8 \, e + f - 4 \, g + h\right )} x^{4} +{\left (4 \, d + 8 \, e + f - 4 \, g + h\right )} x^{2} + 4 \, d + 8 \, e + f - 4 \, g + h\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - 12 \,{\left (d + f - 2 \, h\right )} x - 9 \,{\left ({\left (2 \, d - f + h\right )} x^{4} +{\left (2 \, d - f + h\right )} x^{2} + 2 \, d - f + h\right )} \log \left (x^{2} + x + 1\right ) + 9 \,{\left ({\left (2 \, d - f + h\right )} x^{4} +{\left (2 \, d - f + h\right )} x^{2} + 2 \, d - f + h\right )} \log \left (x^{2} - x + 1\right ) - 12 \, e + 24 \, g}{72 \,{\left (x^{4} + 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 = 1.07795, size = 209, normalized size = 1.12 \begin{align*} \frac{1}{36} \, \sqrt{3}{\left (4 \, d + f + 4 \, g + h - 8 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{36} \, \sqrt{3}{\left (4 \, d + f - 4 \, g + h + 8 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{8} \,{\left (2 \, d - f + h\right )} \log \left (x^{2} + x + 1\right ) - \frac{1}{8} \,{\left (2 \, d - f + h\right )} \log \left (x^{2} - x + 1\right ) - \frac{d x^{3} - 2 \, f x^{3} + h x^{3} + g x^{2} - 2 \, x^{2} e - d x - f x + 2 \, h x + 2 \, g - e}{6 \,{\left (x^{4} + x^{2} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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