Optimal. Leaf size=136 \[ -\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right ) (d+f-2 h)}{2 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) (d+f-2 h)}{2 \sqrt{3}}-\frac{1}{4} (d-f) \log \left (x^2-x+1\right )+\frac{1}{4} (d-f) \log \left (x^2+x+1\right )+\frac{(2 e-g) \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{1}{4} g \log \left (x^4+x^2+1\right )+h x \]
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Rubi [A] time = 0.139987, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 8, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.258, Rules used = {1673, 1676, 1169, 634, 618, 204, 628, 1247} \[ -\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right ) (d+f-2 h)}{2 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) (d+f-2 h)}{2 \sqrt{3}}-\frac{1}{4} (d-f) \log \left (x^2-x+1\right )+\frac{1}{4} (d-f) \log \left (x^2+x+1\right )+\frac{(2 e-g) \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{1}{4} g \log \left (x^4+x^2+1\right )+h x \]
Antiderivative was successfully verified.
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Rule 1673
Rule 1676
Rule 1169
Rule 634
Rule 618
Rule 204
Rule 628
Rule 1247
Rubi steps
\begin{align*} \int \frac{d+e x+f x^2+g x^3+h x^4}{1+x^2+x^4} \, dx &=\int \frac{x \left (e+g x^2\right )}{1+x^2+x^4} \, dx+\int \frac{d+f x^2+h x^4}{1+x^2+x^4} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{e+g x}{1+x+x^2} \, dx,x,x^2\right )+\int \left (h+\frac{d-h+(f-h) x^2}{1+x^2+x^4}\right ) \, dx\\ &=h x+\frac{1}{4} (2 e-g) \operatorname{Subst}\left (\int \frac{1}{1+x+x^2} \, dx,x,x^2\right )+\frac{1}{4} g \operatorname{Subst}\left (\int \frac{1+2 x}{1+x+x^2} \, dx,x,x^2\right )+\int \frac{d-h+(f-h) x^2}{1+x^2+x^4} \, dx\\ &=h x+\frac{1}{4} g \log \left (1+x^2+x^4\right )+\frac{1}{2} \int \frac{d-h-(d-f) x}{1-x+x^2} \, dx+\frac{1}{2} \int \frac{d-h+(d-f) x}{1+x+x^2} \, dx+\frac{1}{2} (-2 e+g) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 x^2\right )\\ &=h x+\frac{(2 e-g) \tan ^{-1}\left (\frac{1+2 x^2}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{1}{4} g \log \left (1+x^2+x^4\right )+\frac{1}{4} (d-f) \int \frac{1+2 x}{1+x+x^2} \, dx+\frac{1}{4} (-d+f) \int \frac{-1+2 x}{1-x+x^2} \, dx+\frac{1}{4} (d+f-2 h) \int \frac{1}{1-x+x^2} \, dx+\frac{1}{4} (d+f-2 h) \int \frac{1}{1+x+x^2} \, dx\\ &=h x+\frac{(2 e-g) \tan ^{-1}\left (\frac{1+2 x^2}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{1}{4} (d-f) \log \left (1-x+x^2\right )+\frac{1}{4} (d-f) \log \left (1+x+x^2\right )+\frac{1}{4} g \log \left (1+x^2+x^4\right )+\frac{1}{2} (-d-f+2 h) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 x\right )+\frac{1}{2} (-d-f+2 h) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 x\right )\\ &=h x-\frac{(d+f-2 h) \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{(d+f-2 h) \tan ^{-1}\left (\frac{1+2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{(2 e-g) \tan ^{-1}\left (\frac{1+2 x^2}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{1}{4} (d-f) \log \left (1-x+x^2\right )+\frac{1}{4} (d-f) \log \left (1+x+x^2\right )+\frac{1}{4} g \log \left (1+x^2+x^4\right )\\ \end{align*}
Mathematica [C] time = 0.601022, size = 165, normalized size = 1.21 \[ \frac{1}{24} \left (4 \tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}-i\right ) x\right ) \left (\left (\sqrt{3}+3 i\right ) d+\left (\sqrt{3}-3 i\right ) f-2 \sqrt{3} h\right )+4 \tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}+i\right ) x\right ) \left (\left (\sqrt{3}-3 i\right ) d+\left (\sqrt{3}+3 i\right ) f-2 \sqrt{3} h\right )-8 \sqrt{3} e \tan ^{-1}\left (\frac{\sqrt{3}}{2 x^2+1}\right )+6 g \log \left (x^4+x^2+1\right )+4 \sqrt{3} g \tan ^{-1}\left (\frac{\sqrt{3}}{2 x^2+1}\right )+24 h x\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.004, size = 241, normalized size = 1.8 \begin{align*} hx+{\frac{d\ln \left ({x}^{2}+x+1 \right ) }{4}}-{\frac{\ln \left ({x}^{2}+x+1 \right ) f}{4}}+{\frac{\ln \left ({x}^{2}+x+1 \right ) g}{4}}+{\frac{d\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{\sqrt{3}e}{3}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}f}{6}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}g}{6}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{\sqrt{3}h}{3}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{\ln \left ({x}^{2}-x+1 \right ) f}{4}}-{\frac{d\ln \left ({x}^{2}-x+1 \right ) }{4}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) g}{4}}+{\frac{d\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}e}{3}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}f}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }-{\frac{\sqrt{3}g}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }-{\frac{\sqrt{3}h}{3}\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 = 1.48999, size = 124, normalized size = 0.91 \begin{align*} \frac{1}{6} \, \sqrt{3}{\left (d - 2 \, e + f + g - 2 \, h\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{6} \, \sqrt{3}{\left (d + 2 \, e + f - g - 2 \, h\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + h x + \frac{1}{4} \,{\left (d - f + g\right )} \log \left (x^{2} + x + 1\right ) - \frac{1}{4} \,{\left (d - f - g\right )} \log \left (x^{2} - x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 11.1041, size = 285, normalized size = 2.1 \begin{align*} \frac{1}{6} \, \sqrt{3}{\left (d - 2 \, e + f + g - 2 \, h\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{6} \, \sqrt{3}{\left (d + 2 \, e + f - g - 2 \, h\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + h x + \frac{1}{4} \,{\left (d - f + g\right )} \log \left (x^{2} + x + 1\right ) - \frac{1}{4} \,{\left (d - f - g\right )} \log \left (x^{2} - x + 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 = 1.10218, size = 127, normalized size = 0.93 \begin{align*} \frac{1}{6} \, \sqrt{3}{\left (d + f + g - 2 \, h - 2 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{6} \, \sqrt{3}{\left (d + f - g - 2 \, h + 2 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + h x + \frac{1}{4} \,{\left (d - f + g\right )} \log \left (x^{2} + x + 1\right ) - \frac{1}{4} \,{\left (d - f - g\right )} \log \left (x^{2} - x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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