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

Optimal. Leaf size=224 \[ \frac{1}{96} \sqrt{\frac{1}{6} \left (11567+12897 \sqrt{3}\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{1}{96} \sqrt{\frac{1}{6} \left (11567+12897 \sqrt{3}\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{25 x \left (1-x^2\right )}{24 \left (x^4+2 x^2+3\right )}-\frac{1}{48} \sqrt{\frac{1}{6} \left (12897 \sqrt{3}-11567\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{1}{48} \sqrt{\frac{1}{6} \left (12897 \sqrt{3}-11567\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]

[Out]

(25*x*(1 - x^2))/(24*(3 + 2*x^2 + x^4)) - (Sqrt[(-11567 + 12897*Sqrt[3])/6]*ArcT
an[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/48 + (Sqrt[(-11567 + 1
2897*Sqrt[3])/6]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/4
8 + (Sqrt[(11567 + 12897*Sqrt[3])/6]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^
2])/96 - (Sqrt[(11567 + 12897*Sqrt[3])/6]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x
+ x^2])/96

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Rubi [A]  time = 0.617768, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{1}{96} \sqrt{\frac{1}{6} \left (11567+12897 \sqrt{3}\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{1}{96} \sqrt{\frac{1}{6} \left (11567+12897 \sqrt{3}\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{25 x \left (1-x^2\right )}{24 \left (x^4+2 x^2+3\right )}-\frac{1}{48} \sqrt{\frac{1}{6} \left (12897 \sqrt{3}-11567\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{1}{48} \sqrt{\frac{1}{6} \left (12897 \sqrt{3}-11567\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(25*x*(1 - x^2))/(24*(3 + 2*x^2 + x^4)) - (Sqrt[(-11567 + 12897*Sqrt[3])/6]*ArcT
an[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/48 + (Sqrt[(-11567 + 1
2897*Sqrt[3])/6]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/4
8 + (Sqrt[(11567 + 12897*Sqrt[3])/6]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^
2])/96 - (Sqrt[(11567 + 12897*Sqrt[3])/6]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x
+ x^2])/96

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Rubi in Sympy [A]  time = 27.5498, size = 328, normalized size = 1.46 \[ \frac{x \left (- 400 x^{2} + 400\right )}{384 \left (x^{4} + 2 x^{2} + 3\right )} - \frac{\sqrt{6} \left (- 760 \sqrt{3} + 56\right ) \log{\left (x^{2} - \sqrt{2} x \sqrt{-1 + \sqrt{3}} + \sqrt{3} \right )}}{4608 \sqrt{-1 + \sqrt{3}}} + \frac{\sqrt{6} \left (- 760 \sqrt{3} + 56\right ) \log{\left (x^{2} + \sqrt{2} x \sqrt{-1 + \sqrt{3}} + \sqrt{3} \right )}}{4608 \sqrt{-1 + \sqrt{3}}} + \frac{\sqrt{3} \left (112 \sqrt{2} \sqrt{-1 + \sqrt{3}} - \frac{\sqrt{2} \sqrt{-1 + \sqrt{3}} \left (- 1520 \sqrt{3} + 112\right )}{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (x - \frac{\sqrt{-2 + 2 \sqrt{3}}}{2}\right )}{\sqrt{1 + \sqrt{3}}} \right )}}{2304 \sqrt{-1 + \sqrt{3}} \sqrt{1 + \sqrt{3}}} + \frac{\sqrt{3} \left (112 \sqrt{2} \sqrt{-1 + \sqrt{3}} - \frac{\sqrt{2} \sqrt{-1 + \sqrt{3}} \left (- 1520 \sqrt{3} + 112\right )}{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (x + \frac{\sqrt{-2 + 2 \sqrt{3}}}{2}\right )}{\sqrt{1 + \sqrt{3}}} \right )}}{2304 \sqrt{-1 + \sqrt{3}} \sqrt{1 + \sqrt{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(-400*x**2 + 400)/(384*(x**4 + 2*x**2 + 3)) - sqrt(6)*(-760*sqrt(3) + 56)*log(
x**2 - sqrt(2)*x*sqrt(-1 + sqrt(3)) + sqrt(3))/(4608*sqrt(-1 + sqrt(3))) + sqrt(
6)*(-760*sqrt(3) + 56)*log(x**2 + sqrt(2)*x*sqrt(-1 + sqrt(3)) + sqrt(3))/(4608*
sqrt(-1 + sqrt(3))) + sqrt(3)*(112*sqrt(2)*sqrt(-1 + sqrt(3)) - sqrt(2)*sqrt(-1
+ sqrt(3))*(-1520*sqrt(3) + 112)/2)*atan(sqrt(2)*(x - sqrt(-2 + 2*sqrt(3))/2)/sq
rt(1 + sqrt(3)))/(2304*sqrt(-1 + sqrt(3))*sqrt(1 + sqrt(3))) + sqrt(3)*(112*sqrt
(2)*sqrt(-1 + sqrt(3)) - sqrt(2)*sqrt(-1 + sqrt(3))*(-1520*sqrt(3) + 112)/2)*ata
n(sqrt(2)*(x + sqrt(-2 + 2*sqrt(3))/2)/sqrt(1 + sqrt(3)))/(2304*sqrt(-1 + sqrt(3
))*sqrt(1 + sqrt(3)))

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Mathematica [C]  time = 0.524882, size = 115, normalized size = 0.51 \[ \frac{1}{48} \left (-\frac{50 x \left (x^2-1\right )}{x^4+2 x^2+3}+\frac{\left (95+44 i \sqrt{2}\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{\sqrt{1-i \sqrt{2}}}+\frac{\left (95-44 i \sqrt{2}\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{\sqrt{1+i \sqrt{2}}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-50*x*(-1 + x^2))/(3 + 2*x^2 + x^4) + ((95 + (44*I)*Sqrt[2])*ArcTan[x/Sqrt[1 -
I*Sqrt[2]]])/Sqrt[1 - I*Sqrt[2]] + ((95 - (44*I)*Sqrt[2])*ArcTan[x/Sqrt[1 + I*S
qrt[2]]])/Sqrt[1 + I*Sqrt[2]])/48

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Maple [B]  time = 0.036, size = 408, normalized size = 1.8 \[{\frac{1}{{x}^{4}+2\,{x}^{2}+3} \left ( -{\frac{25\,{x}^{3}}{24}}+{\frac{25\,x}{24}} \right ) }-{\frac{139\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{576}}-{\frac{11\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{48}}+{\frac{ \left ( -278+278\,\sqrt{3} \right ) \sqrt{3}}{288\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{-22+22\,\sqrt{3}}{24\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{7\,\sqrt{3}}{72\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{139\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{576}}+{\frac{11\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{48}}+{\frac{ \left ( -278+278\,\sqrt{3} \right ) \sqrt{3}}{288\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{-22+22\,\sqrt{3}}{24\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{7\,\sqrt{3}}{72\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-25/24*x^3+25/24*x)/(x^4+2*x^2+3)-139/576*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2)
)*(-2+2*3^(1/2))^(1/2)*3^(1/2)-11/48*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(-2+
2*3^(1/2))^(1/2)+139/288/(2+2*3^(1/2))^(1/2)*arctan((2*x+(-2+2*3^(1/2))^(1/2))/(
2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))*3^(1/2)+11/24/(2+2*3^(1/2))^(1/2)*arctan((2*x
+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))+7/72/(2+2*3^(1/2))^(1
/2)*arctan((2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*3^(1/2)+139/576*ln(x^
2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)*3^(1/2)+11/48*ln(x^2+3^(1
/2)-x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)+139/288/(2+2*3^(1/2))^(1/2)*arc
tan((2*x-(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))*3^(1/2)+11/24
/(2+2*3^(1/2))^(1/2)*arctan((2*x-(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*(-2+
2*3^(1/2))+7/72/(2+2*3^(1/2))^(1/2)*arctan((2*x-(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/
2))^(1/2))*3^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{25 \,{\left (x^{3} - x\right )}}{24 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} + \frac{1}{24} \, \int \frac{95 \, x^{2} + 7}{x^{4} + 2 \, x^{2} + 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25/24*(x^3 - x)/(x^4 + 2*x^2 + 3) + 1/24*integrate((95*x^2 + 7)/(x^4 + 2*x^2 +
3), x)

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Fricas [A]  time = 0.315086, size = 1038, normalized size = 4.63 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/412704*sqrt(1433)*3^(3/4)*(108104*2053489^(1/4)*sqrt(3)*(x^4 + 2*x^2 + 3)*arc
tan(6*2053489^(1/4)*(44*sqrt(3) + 139)/(sqrt(1433)*3^(1/4)*sqrt(1/4299)*(11567*s
qrt(3)*sqrt(2) + 38691*sqrt(2))*sqrt(sqrt(3)*(2*2053489^(1/4)*sqrt(1433)*3^(1/4)
*(794362419051925*sqrt(3)*x + 1283048345880768*x)*sqrt((11567*sqrt(3) + 38691)/(
149179599*sqrt(3) + 316396658)) + 1433*sqrt(3)*(9431668007995*sqrt(3)*x^2 + 1741
8384359577*x^2) + 40546740766370505*sqrt(3) + 74881634361821523)/(9431668007995*
sqrt(3) + 17418384359577))*sqrt((11567*sqrt(3) + 38691)/(149179599*sqrt(3) + 316
396658)) + sqrt(1433)*3^(1/4)*(11567*sqrt(3)*sqrt(2)*x + 38691*sqrt(2)*x)*sqrt((
11567*sqrt(3) + 38691)/(149179599*sqrt(3) + 316396658)) + 3*2053489^(1/4)*(95*sq
rt(3)*sqrt(2) - 7*sqrt(2)))) + 108104*2053489^(1/4)*sqrt(3)*(x^4 + 2*x^2 + 3)*ar
ctan(6*2053489^(1/4)*(44*sqrt(3) + 139)/(sqrt(1433)*3^(1/4)*sqrt(1/4299)*(11567*
sqrt(3)*sqrt(2) + 38691*sqrt(2))*sqrt(-sqrt(3)*(2*2053489^(1/4)*sqrt(1433)*3^(1/
4)*(794362419051925*sqrt(3)*x + 1283048345880768*x)*sqrt((11567*sqrt(3) + 38691)
/(149179599*sqrt(3) + 316396658)) - 1433*sqrt(3)*(9431668007995*sqrt(3)*x^2 + 17
418384359577*x^2) - 40546740766370505*sqrt(3) - 74881634361821523)/(943166800799
5*sqrt(3) + 17418384359577))*sqrt((11567*sqrt(3) + 38691)/(149179599*sqrt(3) + 3
16396658)) + sqrt(1433)*3^(1/4)*(11567*sqrt(3)*sqrt(2)*x + 38691*sqrt(2)*x)*sqrt
((11567*sqrt(3) + 38691)/(149179599*sqrt(3) + 316396658)) - 3*2053489^(1/4)*(95*
sqrt(3)*sqrt(2) - 7*sqrt(2)))) + 100*sqrt(1433)*3^(1/4)*(11567*sqrt(3)*sqrt(2)*(
x^3 - x) + 38691*sqrt(2)*(x^3 - x))*sqrt((11567*sqrt(3) + 38691)/(149179599*sqrt
(3) + 316396658)) + 2053489^(1/4)*(11567*sqrt(3)*sqrt(2)*(x^4 + 2*x^2 + 3) + 386
91*sqrt(2)*(x^4 + 2*x^2 + 3))*log(6*2053489^(1/4)*sqrt(1433)*3^(1/4)*(7943624190
51925*sqrt(3)*x + 1283048345880768*x)*sqrt((11567*sqrt(3) + 38691)/(149179599*sq
rt(3) + 316396658)) + 4299*sqrt(3)*(9431668007995*sqrt(3)*x^2 + 17418384359577*x
^2) + 121640222299111515*sqrt(3) + 224644903085464569) - 2053489^(1/4)*(11567*sq
rt(3)*sqrt(2)*(x^4 + 2*x^2 + 3) + 38691*sqrt(2)*(x^4 + 2*x^2 + 3))*log(-6*205348
9^(1/4)*sqrt(1433)*3^(1/4)*(794362419051925*sqrt(3)*x + 1283048345880768*x)*sqrt
((11567*sqrt(3) + 38691)/(149179599*sqrt(3) + 316396658)) + 4299*sqrt(3)*(943166
8007995*sqrt(3)*x^2 + 17418384359577*x^2) + 121640222299111515*sqrt(3) + 2246449
03085464569))/((11567*sqrt(3)*sqrt(2)*(x^4 + 2*x^2 + 3) + 38691*sqrt(2)*(x^4 + 2
*x^2 + 3))*sqrt((11567*sqrt(3) + 38691)/(149179599*sqrt(3) + 316396658)))

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Sympy [A]  time = 1.92792, size = 48, normalized size = 0.21 \[ - \frac{25 x^{3} - 25 x}{24 x^{4} + 48 x^{2} + 72} + \operatorname{RootSum}{\left (28311552 t^{4} - 23689216 t^{2} + 18481401, \left ( t \mapsto t \log{\left (\frac{40992768 t^{3}}{19364129} - \frac{48423104 t}{58092387} + x \right )} \right )\right )} \]

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)

[Out]

-(25*x**3 - 25*x)/(24*x**4 + 48*x**2 + 72) + RootSum(28311552*_t**4 - 23689216*_
t**2 + 18481401, Lambda(_t, _t*log(40992768*_t**3/19364129 - 48423104*_t/5809238
7 + x)))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{5 \, x^{6} + 3 \, x^{4} + x^{2} + 4}{{\left (x^{4} + 2 \, x^{2} + 3\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((5*x^6 + 3*x^4 + x^2 + 4)/(x^4 + 2*x^2 + 3)^2, x)