Optimal. Leaf size=75 \[ -\frac{\left (15 x^2+244\right ) x}{8 \left (x^4+3 x^2+2\right )}+\frac{\left (103 x^2+102\right ) x}{4 \left (x^4+3 x^2+2\right )^2}+5 x+\frac{413}{8} \tan ^{-1}(x)-\frac{191 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{2 \sqrt{2}} \]
[Out]
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Rubi [A] time = 0.156005, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161 \[ -\frac{\left (15 x^2+244\right ) x}{8 \left (x^4+3 x^2+2\right )}+\frac{\left (103 x^2+102\right ) x}{4 \left (x^4+3 x^2+2\right )^2}+5 x+\frac{413}{8} \tan ^{-1}(x)-\frac{191 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{2 \sqrt{2}} \]
Antiderivative was successfully verified.
[In] Int[(x^6*(4 + x^2 + 3*x^4 + 5*x^6))/(2 + 3*x^2 + x^4)^3,x]
[Out]
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Rubi in Sympy [A] time = 27.4393, size = 70, normalized size = 0.93 \[ \frac{x \left (150174 x^{2} + 148716\right )}{5832 \left (x^{4} + 3 x^{2} + 2\right )^{2}} - \frac{x \left (3542940 x^{2} + 57631824\right )}{1889568 \left (x^{4} + 3 x^{2} + 2\right )} + 5 x + \frac{413 \operatorname{atan}{\left (x \right )}}{8} - \frac{191 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**6*(5*x**6+3*x**4+x**2+4)/(x**4+3*x**2+2)**3,x)
[Out]
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Mathematica [A] time = 0.112113, size = 60, normalized size = 0.8 \[ \frac{1}{8} \left (\frac{x \left (40 x^8+225 x^6+231 x^4-76 x^2-124\right )}{\left (x^4+3 x^2+2\right )^2}+413 \tan ^{-1}(x)-382 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(x^6*(4 + x^2 + 3*x^4 + 5*x^6))/(2 + 3*x^2 + x^4)^3,x]
[Out]
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Maple [A] time = 0.018, size = 56, normalized size = 0.8 \[ 5\,x-16\,{\frac{1}{ \left ({x}^{2}+2 \right ) ^{2}} \left ( -1/32\,{x}^{3}+{\frac{25\,x}{16}} \right ) }-{\frac{191\,\sqrt{2}}{4}\arctan \left ({\frac{\sqrt{2}x}{2}} \right ) }+{\frac{1}{ \left ({x}^{2}+1 \right ) ^{2}} \left ( -{\frac{19\,{x}^{3}}{8}}-{\frac{21\,x}{8}} \right ) }+{\frac{413\,\arctan \left ( x \right ) }{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^6*(5*x^6+3*x^4+x^2+4)/(x^4+3*x^2+2)^3,x)
[Out]
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Maxima [A] time = 0.785785, size = 85, normalized size = 1.13 \[ -\frac{191}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + 5 \, x - \frac{15 \, x^{7} + 289 \, x^{5} + 556 \, x^{3} + 284 \, x}{8 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} + \frac{413}{8} \, \arctan \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^6/(x^4 + 3*x^2 + 2)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.269459, size = 151, normalized size = 2.01 \[ \frac{\sqrt{2}{\left (413 \, \sqrt{2}{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \left (x\right ) - 764 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + \sqrt{2}{\left (40 \, x^{9} + 225 \, x^{7} + 231 \, x^{5} - 76 \, x^{3} - 124 \, x\right )}\right )}}{16 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^6/(x^4 + 3*x^2 + 2)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.79104, size = 68, normalized size = 0.91 \[ 5 x - \frac{15 x^{7} + 289 x^{5} + 556 x^{3} + 284 x}{8 x^{8} + 48 x^{6} + 104 x^{4} + 96 x^{2} + 32} + \frac{413 \operatorname{atan}{\left (x \right )}}{8} - \frac{191 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**6*(5*x**6+3*x**4+x**2+4)/(x**4+3*x**2+2)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.270034, size = 72, normalized size = 0.96 \[ -\frac{191}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + 5 \, x - \frac{15 \, x^{7} + 289 \, x^{5} + 556 \, x^{3} + 284 \, x}{8 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}^{2}} + \frac{413}{8} \, \arctan \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^6/(x^4 + 3*x^2 + 2)^3,x, algorithm="giac")
[Out]