Optimal. Leaf size=248 \[ \frac{5 x^7}{7}-\frac{17 x^5}{5}+\frac{19 x^3}{3}-\frac{1}{32} \sqrt{\frac{1}{2} \left (618291 \sqrt{3}-262771\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{1}{32} \sqrt{\frac{1}{2} \left (618291 \sqrt{3}-262771\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{25 \left (5 x^2+3\right ) x}{8 \left (x^4+2 x^2+3\right )}+38 x+\frac{1}{16} \sqrt{\frac{1}{2} \left (262771+618291 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )-\frac{1}{16} \sqrt{\frac{1}{2} \left (262771+618291 \sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]
[Out]
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Rubi [A] time = 0.731425, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226 \[ \frac{5 x^7}{7}-\frac{17 x^5}{5}+\frac{19 x^3}{3}-\frac{1}{32} \sqrt{\frac{1}{2} \left (618291 \sqrt{3}-262771\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{1}{32} \sqrt{\frac{1}{2} \left (618291 \sqrt{3}-262771\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{25 \left (5 x^2+3\right ) x}{8 \left (x^4+2 x^2+3\right )}+38 x+\frac{1}{16} \sqrt{\frac{1}{2} \left (262771+618291 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )-\frac{1}{16} \sqrt{\frac{1}{2} \left (262771+618291 \sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]
Antiderivative was successfully verified.
[In] Int[(x^8*(4 + x^2 + 3*x^4 + 5*x^6))/(3 + 2*x^2 + x^4)^2,x]
[Out]
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Rubi in Sympy [A] time = 62.7685, size = 352, normalized size = 1.42 \[ \frac{5 x^{7}}{7} - \frac{17 x^{5}}{5} + \frac{19 x^{3}}{3} + \frac{x \left (96000 x^{2} + 57600\right )}{6144 \left (x^{4} + 2 x^{2} + 3\right )} + 38 x + \frac{\sqrt{6} \left (- 514176 \sqrt{3} + 379008\right ) \log{\left (x^{2} - \sqrt{2} x \sqrt{-1 + \sqrt{3}} + \sqrt{3} \right )}}{73728 \sqrt{-1 + \sqrt{3}}} - \frac{\sqrt{6} \left (- 514176 \sqrt{3} + 379008\right ) \log{\left (x^{2} + \sqrt{2} x \sqrt{-1 + \sqrt{3}} + \sqrt{3} \right )}}{73728 \sqrt{-1 + \sqrt{3}}} - \frac{\sqrt{3} \left (- \frac{\sqrt{2} \sqrt{-1 + \sqrt{3}} \left (- 1028352 \sqrt{3} + 758016\right )}{2} + 758016 \sqrt{2} \sqrt{-1 + \sqrt{3}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (x - \frac{\sqrt{-2 + 2 \sqrt{3}}}{2}\right )}{\sqrt{1 + \sqrt{3}}} \right )}}{36864 \sqrt{-1 + \sqrt{3}} \sqrt{1 + \sqrt{3}}} - \frac{\sqrt{3} \left (- \frac{\sqrt{2} \sqrt{-1 + \sqrt{3}} \left (- 1028352 \sqrt{3} + 758016\right )}{2} + 758016 \sqrt{2} \sqrt{-1 + \sqrt{3}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (x + \frac{\sqrt{-2 + 2 \sqrt{3}}}{2}\right )}{\sqrt{1 + \sqrt{3}}} \right )}}{36864 \sqrt{-1 + \sqrt{3}} \sqrt{1 + \sqrt{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**8*(5*x**6+3*x**4+x**2+4)/(x**4+2*x**2+3)**2,x)
[Out]
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Mathematica [C] time = 0.337006, size = 145, normalized size = 0.58 \[ \frac{5 x^7}{7}-\frac{17 x^5}{5}+\frac{19 x^3}{3}+\frac{25 \left (5 x^2+3\right ) x}{8 \left (x^4+2 x^2+3\right )}+38 x-\frac{\left (1339 \sqrt{2}+352 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{16 \sqrt{2-2 i \sqrt{2}}}-\frac{\left (1339 \sqrt{2}-352 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{16 \sqrt{2+2 i \sqrt{2}}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^8*(4 + x^2 + 3*x^4 + 5*x^6))/(3 + 2*x^2 + x^4)^2,x]
[Out]
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Maple [B] time = 0.106, size = 427, normalized size = 1.7 \[{\frac{5\,{x}^{7}}{7}}-{\frac{17\,{x}^{5}}{5}}+{\frac{19\,{x}^{3}}{3}}+38\,x-{\frac{1}{{x}^{4}+2\,{x}^{2}+3} \left ( -{\frac{125\,{x}^{3}}{8}}-{\frac{75\,x}{8}} \right ) }+{\frac{505\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{64}}+{\frac{11\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{4}}-{\frac{ \left ( -1010+1010\,\sqrt{3} \right ) \sqrt{3}}{32\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{-22+22\,\sqrt{3}}{2\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{329\,\sqrt{3}}{8\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{505\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{64}}-{\frac{11\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{4}}-{\frac{ \left ( -1010+1010\,\sqrt{3} \right ) \sqrt{3}}{32\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{-22+22\,\sqrt{3}}{2\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{329\,\sqrt{3}}{8\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^8*(5*x^6+3*x^4+x^2+4)/(x^4+2*x^2+3)^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \frac{5}{7} \, x^{7} - \frac{17}{5} \, x^{5} + \frac{19}{3} \, x^{3} + 38 \, x + \frac{25 \,{\left (5 \, x^{3} + 3 \, x\right )}}{8 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} - \frac{1}{8} \, \int \frac{1339 \, x^{2} + 987}{x^{4} + 2 \, x^{2} + 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^8/(x^4 + 2*x^2 + 3)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.312552, size = 1040, normalized size = 4.19 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^8/(x^4 + 2*x^2 + 3)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.95431, size = 71, normalized size = 0.29 \[ \frac{5 x^{7}}{7} - \frac{17 x^{5}}{5} + \frac{19 x^{3}}{3} + 38 x + \frac{125 x^{3} + 75 x}{8 x^{4} + 16 x^{2} + 24} + \operatorname{RootSum}{\left (1048576 t^{4} + 538155008 t^{2} + 1146851282043, \left ( t \mapsto t \log{\left (- \frac{16547840 t^{3}}{453886804809} - \frac{11974973632 t}{453886804809} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**8*(5*x**6+3*x**4+x**2+4)/(x**4+2*x**2+3)**2,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x^{6} + 3 \, x^{4} + x^{2} + 4\right )} x^{8}}{{\left (x^{4} + 2 \, x^{2} + 3\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^8/(x^4 + 2*x^2 + 3)^2,x, algorithm="giac")
[Out]