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

Optimal. Leaf size=71 \[ -\frac{2}{9 x^2}-\frac{71 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{216 \sqrt{2}}-\frac{25 \left (x^2+5\right )}{72 \left (x^4+2 x^2+3\right )}+\frac{13}{108} \log \left (x^4+2 x^2+3\right )-\frac{13 \log (x)}{27} \]

[Out]

-2/(9*x^2) - (25*(5 + x^2))/(72*(3 + 2*x^2 + x^4)) - (71*ArcTan[(1 + x^2)/Sqrt[2
]])/(216*Sqrt[2]) - (13*Log[x])/27 + (13*Log[3 + 2*x^2 + x^4])/108

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Rubi [A]  time = 0.219128, antiderivative size = 71, 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 \[ -\frac{2}{9 x^2}-\frac{71 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{216 \sqrt{2}}-\frac{25 \left (x^2+5\right )}{72 \left (x^4+2 x^2+3\right )}+\frac{13}{108} \log \left (x^4+2 x^2+3\right )-\frac{13 \log (x)}{27} \]

Antiderivative was successfully verified.

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

[Out]

-2/(9*x^2) - (25*(5 + x^2))/(72*(3 + 2*x^2 + x^4)) - (71*ArcTan[(1 + x^2)/Sqrt[2
]])/(216*Sqrt[2]) - (13*Log[x])/27 + (13*Log[3 + 2*x^2 + x^4])/108

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Rubi in Sympy [A]  time = 26.4935, size = 85, normalized size = 1.2 \[ - \frac{5 \left (10 x^{2} + 14\right )}{144 \left (x^{4} + 2 x^{2} + 3\right )} - \frac{13 \log{\left (x^{2} \right )}}{54} + \frac{13 \log{\left (x^{4} + 2 x^{2} + 3 \right )}}{108} - \frac{71 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \left (\frac{x^{2}}{2} + \frac{1}{2}\right ) \right )}}{432} - \frac{5}{4 \left (x^{4} + 2 x^{2} + 3\right )} - \frac{2}{9 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*(10*x**2 + 14)/(144*(x**4 + 2*x**2 + 3)) - 13*log(x**2)/54 + 13*log(x**4 + 2*
x**2 + 3)/108 - 71*sqrt(2)*atan(sqrt(2)*(x**2/2 + 1/2))/432 - 5/(4*(x**4 + 2*x**
2 + 3)) - 2/(9*x**2)

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Mathematica [C]  time = 0.0856761, size = 97, normalized size = 1.37 \[ \frac{1}{864} \left (-\frac{192}{x^2}+\sqrt{2} \left (52 \sqrt{2}+71 i\right ) \log \left (x^2-i \sqrt{2}+1\right )+\sqrt{2} \left (52 \sqrt{2}-71 i\right ) \log \left (x^2+i \sqrt{2}+1\right )-\frac{300 \left (x^2+5\right )}{x^4+2 x^2+3}-416 \log (x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-192/x^2 - (300*(5 + x^2))/(3 + 2*x^2 + x^4) - 416*Log[x] + Sqrt[2]*(71*I + 52*
Sqrt[2])*Log[1 - I*Sqrt[2] + x^2] + Sqrt[2]*(-71*I + 52*Sqrt[2])*Log[1 + I*Sqrt[
2] + x^2])/864

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Maple [A]  time = 0.02, size = 63, normalized size = 0.9 \[ -{\frac{2}{9\,{x}^{2}}}-{\frac{13\,\ln \left ( x \right ) }{27}}+{\frac{1}{54\,{x}^{4}+108\,{x}^{2}+162} \left ( -{\frac{75\,{x}^{2}}{4}}-{\frac{375}{4}} \right ) }+{\frac{13\,\ln \left ({x}^{4}+2\,{x}^{2}+3 \right ) }{108}}-{\frac{71\,\sqrt{2}}{432}\arctan \left ({\frac{ \left ( 2\,{x}^{2}+2 \right ) \sqrt{2}}{4}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9/x^2-13/27*ln(x)+1/54*(-75/4*x^2-375/4)/(x^4+2*x^2+3)+13/108*ln(x^4+2*x^2+3)
-71/432*2^(1/2)*arctan(1/4*(2*x^2+2)*2^(1/2))

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Maxima [A]  time = 0.798573, size = 89, normalized size = 1.25 \[ -\frac{71}{432} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) - \frac{41 \, x^{4} + 157 \, x^{2} + 48}{72 \,{\left (x^{6} + 2 \, x^{4} + 3 \, x^{2}\right )}} + \frac{13}{108} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) - \frac{13}{54} \, \log \left (x^{2}\right ) \]

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^3),x, algorithm="maxima")

[Out]

-71/432*sqrt(2)*arctan(1/2*sqrt(2)*(x^2 + 1)) - 1/72*(41*x^4 + 157*x^2 + 48)/(x^
6 + 2*x^4 + 3*x^2) + 13/108*log(x^4 + 2*x^2 + 3) - 13/54*log(x^2)

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Fricas [A]  time = 0.271167, size = 158, normalized size = 2.23 \[ \frac{\sqrt{2}{\left (26 \, \sqrt{2}{\left (x^{6} + 2 \, x^{4} + 3 \, x^{2}\right )} \log \left (x^{4} + 2 \, x^{2} + 3\right ) - 104 \, \sqrt{2}{\left (x^{6} + 2 \, x^{4} + 3 \, x^{2}\right )} \log \left (x\right ) - 71 \,{\left (x^{6} + 2 \, x^{4} + 3 \, x^{2}\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) - 3 \, \sqrt{2}{\left (41 \, x^{4} + 157 \, x^{2} + 48\right )}\right )}}{432 \,{\left (x^{6} + 2 \, x^{4} + 3 \, x^{2}\right )}} \]

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^3),x, algorithm="fricas")

[Out]

1/432*sqrt(2)*(26*sqrt(2)*(x^6 + 2*x^4 + 3*x^2)*log(x^4 + 2*x^2 + 3) - 104*sqrt(
2)*(x^6 + 2*x^4 + 3*x^2)*log(x) - 71*(x^6 + 2*x^4 + 3*x^2)*arctan(1/2*sqrt(2)*(x
^2 + 1)) - 3*sqrt(2)*(41*x^4 + 157*x^2 + 48))/(x^6 + 2*x^4 + 3*x^2)

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Sympy [A]  time = 0.583051, size = 75, normalized size = 1.06 \[ - \frac{41 x^{4} + 157 x^{2} + 48}{72 x^{6} + 144 x^{4} + 216 x^{2}} - \frac{13 \log{\left (x \right )}}{27} + \frac{13 \log{\left (x^{4} + 2 x^{2} + 3 \right )}}{108} - \frac{71 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x^{2}}{2} + \frac{\sqrt{2}}{2} \right )}}{432} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(41*x**4 + 157*x**2 + 48)/(72*x**6 + 144*x**4 + 216*x**2) - 13*log(x)/27 + 13*l
og(x**4 + 2*x**2 + 3)/108 - 71*sqrt(2)*atan(sqrt(2)*x**2/2 + sqrt(2)/2)/432

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GIAC/XCAS [A]  time = 0.271676, size = 89, normalized size = 1.25 \[ -\frac{71}{432} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) - \frac{41 \, x^{4} + 157 \, x^{2} + 48}{72 \,{\left (x^{6} + 2 \, x^{4} + 3 \, x^{2}\right )}} + \frac{13}{108} \,{\rm ln}\left (x^{4} + 2 \, x^{2} + 3\right ) - \frac{13}{54} \,{\rm ln}\left (x^{2}\right ) \]

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^3),x, algorithm="giac")

[Out]

-71/432*sqrt(2)*arctan(1/2*sqrt(2)*(x^2 + 1)) - 1/72*(41*x^4 + 157*x^2 + 48)/(x^
6 + 2*x^4 + 3*x^2) + 13/108*ln(x^4 + 2*x^2 + 3) - 13/54*ln(x^2)