Optimal. Leaf size=263 \[ -\frac{1}{32} \log \left (x^2-x+1\right ) (9 d-4 f+3 h)+\frac{1}{32} \log \left (x^2+x+1\right ) (9 d-4 f+3 h)+\frac{x \left (x^2 (-(7 d-7 f+4 h))+2 d+3 f-h\right )}{24 \left (x^4+x^2+1\right )}+\frac{x \left (x^2 (-(d-2 f+h))+d+f-2 h\right )}{12 \left (x^4+x^2+1\right )^2}-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right ) (13 d+2 f+h)}{48 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) (13 d+2 f+h)}{48 \sqrt{3}}+\frac{(2 e-g) \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{\left (2 x^2+1\right ) (2 e-g)}{12 \left (x^4+x^2+1\right )}+\frac{x^2 (2 e-g)+e-2 g}{12 \left (x^4+x^2+1\right )^2} \]
[Out]
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Rubi [A] time = 0.614234, 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 \[ -\frac{1}{32} \log \left (x^2-x+1\right ) (9 d-4 f+3 h)+\frac{1}{32} \log \left (x^2+x+1\right ) (9 d-4 f+3 h)+\frac{x \left (x^2 (-(7 d-7 f+4 h))+2 d+3 f-h\right )}{24 \left (x^4+x^2+1\right )}+\frac{x \left (x^2 (-(d-2 f+h))+d+f-2 h\right )}{12 \left (x^4+x^2+1\right )^2}-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right ) (13 d+2 f+h)}{48 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) (13 d+2 f+h)}{48 \sqrt{3}}+\frac{(2 e-g) \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{\left (2 x^2+1\right ) (2 e-g)}{12 \left (x^4+x^2+1\right )}+\frac{x^2 (2 e-g)+e-2 g}{12 \left (x^4+x^2+1\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x + f*x^2 + g*x^3 + h*x^4)/(1 + x^2 + x^4)^3,x]
[Out]
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Rubi in Sympy [A] time = 102.35, size = 233, normalized size = 0.89 \[ \frac{x \left (6 d + 9 f - 3 h - x^{3} \left (18 e - 18 g\right ) - x^{2} \left (21 d - 21 f + 12 h\right ) + x \left (6 e + 6 g\right )\right )}{72 \left (x^{4} + x^{2} + 1\right )} + \frac{x \left (d + f - 2 h - x^{3} \left (e - 2 g\right ) - x^{2} \left (d - 2 f + h\right ) + x \left (e + g\right )\right )}{12 \left (x^{4} + x^{2} + 1\right )^{2}} + \frac{\sqrt{3} \left (2 e - g\right ) \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{2}}{3} + \frac{1}{3}\right ) \right )}}{9} - \left (\frac{9 d}{32} - \frac{f}{8} + \frac{3 h}{32}\right ) \log{\left (x^{2} - x + 1 \right )} + \left (\frac{9 d}{32} - \frac{f}{8} + \frac{3 h}{32}\right ) \log{\left (x^{2} + x + 1 \right )} + \frac{\sqrt{3} \left (13 d + 2 f + h\right ) \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{144} + \frac{\sqrt{3} \left (13 d + 2 f + h\right ) \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{144} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((h*x**4+g*x**3+f*x**2+e*x+d)/(x**4+x**2+1)**3,x)
[Out]
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Mathematica [C] time = 1.83384, 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.
[In] Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4)/(1 + x^2 + x^4)^3,x]
[Out]
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Maple [A] time = 0.023, size = 396, normalized size = 1.5 \[{\frac{1}{16\, \left ({x}^{2}+x+1 \right ) ^{2}} \left ( \left ( -{\frac{7\,d}{3}}+{\frac{7\,f}{3}}-{\frac{4\,h}{3}}-{\frac{4\,e}{3}}-{\frac{g}{3}} \right ){x}^{3}+ \left ( -6\,d+4\,f-2\,h-2\,g \right ){x}^{2}+ \left ( -{\frac{20\,d}{3}}+{\frac{13\,f}{3}}-{\frac{5\,h}{3}}+{\frac{e}{3}}-{\frac{8\,g}{3}} \right ) x-4\,d+{\frac{4\,f}{3}}+2\,e-2\,g \right ) }+{\frac{9\,d\ln \left ({x}^{2}+x+1 \right ) }{32}}-{\frac{\ln \left ({x}^{2}+x+1 \right ) f}{8}}+{\frac{3\,\ln \left ({x}^{2}+x+1 \right ) h}{32}}+{\frac{13\,d\sqrt{3}}{144}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{2\,\sqrt{3}e}{9}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}f}{72}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}g}{9}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}h}{144}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{1}{16\, \left ({x}^{2}-x+1 \right ) ^{2}} \left ( \left ({\frac{7\,d}{3}}-{\frac{7\,f}{3}}+{\frac{4\,h}{3}}-{\frac{4\,e}{3}}-{\frac{g}{3}} \right ){x}^{3}+ \left ( -6\,d+4\,f-2\,h+2\,g \right ){x}^{2}+ \left ({\frac{20\,d}{3}}-{\frac{13\,f}{3}}+{\frac{5\,h}{3}}+{\frac{e}{3}}-{\frac{8\,g}{3}} \right ) x-4\,d+{\frac{4\,f}{3}}-2\,e+2\,g \right ) }-{\frac{9\,d\ln \left ({x}^{2}-x+1 \right ) }{32}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) f}{8}}-{\frac{3\,\ln \left ({x}^{2}-x+1 \right ) h}{32}}+{\frac{13\,d\sqrt{3}}{144}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{2\,\sqrt{3}e}{9}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}f}{72}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }-{\frac{\sqrt{3}g}{9}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}h}{144}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((h*x^4+g*x^3+f*x^2+e*x+d)/(x^4+x^2+1)^3,x)
[Out]
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Maxima [A] time = 0.785515, size = 293, normalized size = 1.11 \[ \frac{1}{144} \, \sqrt{3}{\left (13 \, d - 32 \, e + 2 \, f + 16 \, g + h\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{144} \, \sqrt{3}{\left (13 \, d + 32 \, e + 2 \, f - 16 \, g + h\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{32} \,{\left (9 \, d - 4 \, f + 3 \, h\right )} \log \left (x^{2} + x + 1\right ) - \frac{1}{32} \,{\left (9 \, d - 4 \, f + 3 \, h\right )} \log \left (x^{2} - x + 1\right ) - \frac{{\left (7 \, d - 7 \, f + 4 \, h\right )} x^{7} - 4 \,{\left (2 \, e - g\right )} x^{6} + 5 \,{\left (d - 2 \, f + h\right )} x^{5} - 6 \,{\left (2 \, e - g\right )} x^{4} + 7 \,{\left (d - 2 \, f + h\right )} x^{3} - 8 \,{\left (2 \, e - g\right )} x^{2} -{\left (4 \, d + 5 \, f - 5 \, h\right )} x - 6 \, e + 6 \, g}{24 \,{\left (x^{8} + 2 \, x^{6} + 3 \, x^{4} + 2 \, x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((h*x^4 + g*x^3 + f*x^2 + e*x + d)/(x^4 + x^2 + 1)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 1.51778, size = 666, normalized size = 2.53 \[ \frac{\sqrt{3}{\left (3 \, \sqrt{3}{\left ({\left (9 \, d - 4 \, f + 3 \, h\right )} x^{8} + 2 \,{\left (9 \, d - 4 \, f + 3 \, h\right )} x^{6} + 3 \,{\left (9 \, d - 4 \, f + 3 \, h\right )} x^{4} + 2 \,{\left (9 \, d - 4 \, f + 3 \, h\right )} x^{2} + 9 \, d - 4 \, f + 3 \, h\right )} \log \left (x^{2} + x + 1\right ) - 3 \, \sqrt{3}{\left ({\left (9 \, d - 4 \, f + 3 \, h\right )} x^{8} + 2 \,{\left (9 \, d - 4 \, f + 3 \, h\right )} x^{6} + 3 \,{\left (9 \, d - 4 \, f + 3 \, h\right )} x^{4} + 2 \,{\left (9 \, d - 4 \, f + 3 \, h\right )} x^{2} + 9 \, d - 4 \, f + 3 \, h\right )} \log \left (x^{2} - x + 1\right ) + 2 \,{\left ({\left (13 \, d - 32 \, e + 2 \, f + 16 \, g + h\right )} x^{8} + 2 \,{\left (13 \, d - 32 \, e + 2 \, f + 16 \, g + h\right )} x^{6} + 3 \,{\left (13 \, d - 32 \, e + 2 \, f + 16 \, g + h\right )} x^{4} + 2 \,{\left (13 \, d - 32 \, e + 2 \, f + 16 \, g + h\right )} x^{2} + 13 \, d - 32 \, e + 2 \, f + 16 \, g + h\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + 2 \,{\left ({\left (13 \, d + 32 \, e + 2 \, f - 16 \, g + h\right )} x^{8} + 2 \,{\left (13 \, d + 32 \, e + 2 \, f - 16 \, g + h\right )} x^{6} + 3 \,{\left (13 \, d + 32 \, e + 2 \, f - 16 \, g + h\right )} x^{4} + 2 \,{\left (13 \, d + 32 \, e + 2 \, f - 16 \, g + h\right )} x^{2} + 13 \, d + 32 \, e + 2 \, f - 16 \, g + h\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - 4 \, \sqrt{3}{\left ({\left (7 \, d - 7 \, f + 4 \, h\right )} x^{7} - 4 \,{\left (2 \, e - g\right )} x^{6} + 5 \,{\left (d - 2 \, f + h\right )} x^{5} - 6 \,{\left (2 \, e - g\right )} x^{4} + 7 \,{\left (d - 2 \, f + h\right )} x^{3} - 8 \,{\left (2 \, e - g\right )} x^{2} -{\left (4 \, d + 5 \, f - 5 \, h\right )} x - 6 \, e + 6 \, g\right )}\right )}}{288 \,{\left (x^{8} + 2 \, x^{6} + 3 \, x^{4} + 2 \, x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((h*x^4 + g*x^3 + f*x^2 + e*x + d)/(x^4 + x^2 + 1)^3,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((h*x**4+g*x**3+f*x**2+e*x+d)/(x**4+x**2+1)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.267982, size = 308, normalized size = 1.17 \[ \frac{1}{144} \, \sqrt{3}{\left (13 \, d + 2 \, f + 16 \, g + h - 32 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{144} \, \sqrt{3}{\left (13 \, d + 2 \, f - 16 \, g + h + 32 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{32} \,{\left (9 \, d - 4 \, f + 3 \, h\right )}{\rm ln}\left (x^{2} + x + 1\right ) - \frac{1}{32} \,{\left (9 \, d - 4 \, f + 3 \, h\right )}{\rm ln}\left (x^{2} - x + 1\right ) - \frac{7 \, d x^{7} - 7 \, f x^{7} + 4 \, h x^{7} + 4 \, g x^{6} - 8 \, x^{6} e + 5 \, d x^{5} - 10 \, f x^{5} + 5 \, h x^{5} + 6 \, g x^{4} - 12 \, x^{4} e + 7 \, d x^{3} - 14 \, f x^{3} + 7 \, h x^{3} + 8 \, g x^{2} - 16 \, x^{2} e - 4 \, d x - 5 \, f x + 5 \, h x + 6 \, g - 6 \, e}{24 \,{\left (x^{4} + x^{2} + 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((h*x^4 + g*x^3 + f*x^2 + e*x + d)/(x^4 + x^2 + 1)^3,x, algorithm="giac")
[Out]