3.108 \(\int \frac{4+x^2+3 x^4+5 x^6}{x^7 \left (3+2 x^2+x^4\right )^2} \, dx\)

Optimal. Leaf size=87 \[ -\frac{2}{27 x^6}+\frac{13}{108 x^4}-\frac{13}{54 x^2}-\frac{1237 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{1944 \sqrt{2}}+\frac{25 \left (1-7 x^2\right )}{648 \left (x^4+2 x^2+3\right )}-\frac{61}{972} \log \left (x^4+2 x^2+3\right )+\frac{61 \log (x)}{243} \]

[Out]

-2/(27*x^6) + 13/(108*x^4) - 13/(54*x^2) + (25*(1 - 7*x^2))/(648*(3 + 2*x^2 + x^
4)) - (1237*ArcTan[(1 + x^2)/Sqrt[2]])/(1944*Sqrt[2]) + (61*Log[x])/243 - (61*Lo
g[3 + 2*x^2 + x^4])/972

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Rubi [A]  time = 0.243756, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226 \[ -\frac{2}{27 x^6}+\frac{13}{108 x^4}-\frac{13}{54 x^2}-\frac{1237 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{1944 \sqrt{2}}+\frac{25 \left (1-7 x^2\right )}{648 \left (x^4+2 x^2+3\right )}-\frac{61}{972} \log \left (x^4+2 x^2+3\right )+\frac{61 \log (x)}{243} \]

Antiderivative was successfully verified.

[In]  Int[(4 + x^2 + 3*x^4 + 5*x^6)/(x^7*(3 + 2*x^2 + x^4)^2),x]

[Out]

-2/(27*x^6) + 13/(108*x^4) - 13/(54*x^2) + (25*(1 - 7*x^2))/(648*(3 + 2*x^2 + x^
4)) - (1237*ArcTan[(1 + x^2)/Sqrt[2]])/(1944*Sqrt[2]) + (61*Log[x])/243 - (61*Lo
g[3 + 2*x^2 + x^4])/972

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Rubi in Sympy [A]  time = 27.6736, size = 102, normalized size = 1.17 \[ \frac{5 \left (- 68 x^{2} + 56\right )}{2592 \left (x^{4} + 2 x^{2} + 3\right )} + \frac{61 \log{\left (x^{2} \right )}}{486} - \frac{61 \log{\left (x^{4} + 2 x^{2} + 3 \right )}}{972} - \frac{1237 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \left (\frac{x^{2}}{2} + \frac{1}{2}\right ) \right )}}{3888} - \frac{41}{108 x^{2}} + \frac{71}{216 x^{4}} - \frac{5}{8 x^{4} \left (x^{4} + 2 x^{2} + 3\right )} - \frac{2}{27 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5*x**6+3*x**4+x**2+4)/x**7/(x**4+2*x**2+3)**2,x)

[Out]

5*(-68*x**2 + 56)/(2592*(x**4 + 2*x**2 + 3)) + 61*log(x**2)/486 - 61*log(x**4 +
2*x**2 + 3)/972 - 1237*sqrt(2)*atan(sqrt(2)*(x**2/2 + 1/2))/3888 - 41/(108*x**2)
+ 71/(216*x**4) - 5/(8*x**4*(x**4 + 2*x**2 + 3)) - 2/(27*x**6)

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Mathematica [C]  time = 0.111973, size = 110, normalized size = 1.26 \[ \frac{-\frac{576}{x^6}+\frac{936}{x^4}-\frac{1872}{x^2}+\sqrt{2} \left (-244 \sqrt{2}+1237 i\right ) \log \left (x^2-i \sqrt{2}+1\right )-\sqrt{2} \left (244 \sqrt{2}+1237 i\right ) \log \left (x^2+i \sqrt{2}+1\right )-\frac{300 \left (7 x^2-1\right )}{x^4+2 x^2+3}+1952 \log (x)}{7776} \]

Antiderivative was successfully verified.

[In]  Integrate[(4 + x^2 + 3*x^4 + 5*x^6)/(x^7*(3 + 2*x^2 + x^4)^2),x]

[Out]

(-576/x^6 + 936/x^4 - 1872/x^2 - (300*(-1 + 7*x^2))/(3 + 2*x^2 + x^4) + 1952*Log
[x] + Sqrt[2]*(1237*I - 244*Sqrt[2])*Log[1 - I*Sqrt[2] + x^2] - Sqrt[2]*(1237*I
+ 244*Sqrt[2])*Log[1 + I*Sqrt[2] + x^2])/7776

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Maple [A]  time = 0.02, size = 73, normalized size = 0.8 \[ -{\frac{2}{27\,{x}^{6}}}+{\frac{13}{108\,{x}^{4}}}-{\frac{13}{54\,{x}^{2}}}+{\frac{61\,\ln \left ( x \right ) }{243}}-{\frac{1}{486\,{x}^{4}+972\,{x}^{2}+1458} \left ({\frac{525\,{x}^{2}}{4}}-{\frac{75}{4}} \right ) }-{\frac{61\,\ln \left ({x}^{4}+2\,{x}^{2}+3 \right ) }{972}}-{\frac{1237\,\sqrt{2}}{3888}\arctan \left ({\frac{ \left ( 2\,{x}^{2}+2 \right ) \sqrt{2}}{4}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5*x^6+3*x^4+x^2+4)/x^7/(x^4+2*x^2+3)^2,x)

[Out]

-2/27/x^6+13/108/x^4-13/54/x^2+61/243*ln(x)-1/486*(525/4*x^2-75/4)/(x^4+2*x^2+3)
-61/972*ln(x^4+2*x^2+3)-1237/3888*2^(1/2)*arctan(1/4*(2*x^2+2)*2^(1/2))

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Maxima [A]  time = 0.789975, size = 103, normalized size = 1.18 \[ -\frac{1237}{3888} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) - \frac{331 \, x^{8} + 209 \, x^{6} + 360 \, x^{4} - 138 \, x^{2} + 144}{648 \,{\left (x^{10} + 2 \, x^{8} + 3 \, x^{6}\right )}} - \frac{61}{972} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) + \frac{61}{486} \, \log \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^6 + 3*x^4 + x^2 + 4)/((x^4 + 2*x^2 + 3)^2*x^7),x, algorithm="maxima")

[Out]

-1237/3888*sqrt(2)*arctan(1/2*sqrt(2)*(x^2 + 1)) - 1/648*(331*x^8 + 209*x^6 + 36
0*x^4 - 138*x^2 + 144)/(x^10 + 2*x^8 + 3*x^6) - 61/972*log(x^4 + 2*x^2 + 3) + 61
/486*log(x^2)

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Fricas [A]  time = 0.257951, size = 171, normalized size = 1.97 \[ -\frac{\sqrt{2}{\left (122 \, \sqrt{2}{\left (x^{10} + 2 \, x^{8} + 3 \, x^{6}\right )} \log \left (x^{4} + 2 \, x^{2} + 3\right ) - 488 \, \sqrt{2}{\left (x^{10} + 2 \, x^{8} + 3 \, x^{6}\right )} \log \left (x\right ) + 1237 \,{\left (x^{10} + 2 \, x^{8} + 3 \, x^{6}\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) + 3 \, \sqrt{2}{\left (331 \, x^{8} + 209 \, x^{6} + 360 \, x^{4} - 138 \, x^{2} + 144\right )}\right )}}{3888 \,{\left (x^{10} + 2 \, x^{8} + 3 \, x^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^6 + 3*x^4 + x^2 + 4)/((x^4 + 2*x^2 + 3)^2*x^7),x, algorithm="fricas")

[Out]

-1/3888*sqrt(2)*(122*sqrt(2)*(x^10 + 2*x^8 + 3*x^6)*log(x^4 + 2*x^2 + 3) - 488*s
qrt(2)*(x^10 + 2*x^8 + 3*x^6)*log(x) + 1237*(x^10 + 2*x^8 + 3*x^6)*arctan(1/2*sq
rt(2)*(x^2 + 1)) + 3*sqrt(2)*(331*x^8 + 209*x^6 + 360*x^4 - 138*x^2 + 144))/(x^1
0 + 2*x^8 + 3*x^6)

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Sympy [A]  time = 0.725467, size = 85, normalized size = 0.98 \[ \frac{61 \log{\left (x \right )}}{243} - \frac{61 \log{\left (x^{4} + 2 x^{2} + 3 \right )}}{972} - \frac{1237 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x^{2}}{2} + \frac{\sqrt{2}}{2} \right )}}{3888} - \frac{331 x^{8} + 209 x^{6} + 360 x^{4} - 138 x^{2} + 144}{648 x^{10} + 1296 x^{8} + 1944 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x**6+3*x**4+x**2+4)/x**7/(x**4+2*x**2+3)**2,x)

[Out]

61*log(x)/243 - 61*log(x**4 + 2*x**2 + 3)/972 - 1237*sqrt(2)*atan(sqrt(2)*x**2/2
+ sqrt(2)/2)/3888 - (331*x**8 + 209*x**6 + 360*x**4 - 138*x**2 + 144)/(648*x**1
0 + 1296*x**8 + 1944*x**6)

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GIAC/XCAS [A]  time = 0.269813, size = 113, normalized size = 1.3 \[ -\frac{1237}{3888} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) + \frac{122 \, x^{4} - 281 \, x^{2} + 441}{1944 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} - \frac{671 \, x^{6} + 702 \, x^{4} - 351 \, x^{2} + 216}{2916 \, x^{6}} - \frac{61}{972} \,{\rm ln}\left (x^{4} + 2 \, x^{2} + 3\right ) + \frac{61}{486} \,{\rm ln}\left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^6 + 3*x^4 + x^2 + 4)/((x^4 + 2*x^2 + 3)^2*x^7),x, algorithm="giac")

[Out]

-1237/3888*sqrt(2)*arctan(1/2*sqrt(2)*(x^2 + 1)) + 1/1944*(122*x^4 - 281*x^2 + 4
41)/(x^4 + 2*x^2 + 3) - 1/2916*(671*x^6 + 702*x^4 - 351*x^2 + 216)/x^6 - 61/972*
ln(x^4 + 2*x^2 + 3) + 61/486*ln(x^2)