3.56 \(\int \frac{1}{\left (1+4 x+4 x^2+4 x^4\right )^2} \, dx\)

Optimal. Leaf size=317 \[ -\frac{17-\left (\frac{1}{x}+1\right )^2}{2 \left (\left (\frac{1}{x}+1\right )^4-2 \left (\frac{1}{x}+1\right )^2+5\right )}+\frac{\left (59-17 \left (\frac{1}{x}+1\right )^2\right ) \left (\frac{1}{x}+1\right )}{10 \left (\left (\frac{1}{x}+1\right )^4-2 \left (\frac{1}{x}+1\right )^2+5\right )}+\frac{1}{40} \sqrt{\frac{1}{10} \left (2665 \sqrt{5}-5959\right )} \log \left (\left (\frac{1}{x}+1\right )^2-\sqrt{2 \left (1+\sqrt{5}\right )} \left (\frac{1}{x}+1\right )+\sqrt{5}\right )-\frac{1}{40} \sqrt{\frac{1}{10} \left (2665 \sqrt{5}-5959\right )} \log \left (\left (\frac{1}{x}+1\right )^2+\sqrt{2 \left (1+\sqrt{5}\right )} \left (\frac{1}{x}+1\right )+\sqrt{5}\right )+\frac{7}{4} \tan ^{-1}\left (\frac{1}{2} \left (\left (\frac{1}{x}+1\right )^2-1\right )\right )-\frac{1}{20} \sqrt{\frac{1}{10} \left (5959+2665 \sqrt{5}\right )} \tan ^{-1}\left (\frac{\frac{2}{x}-\sqrt{2 \left (1+\sqrt{5}\right )}+2}{\sqrt{2 \left (\sqrt{5}-1\right )}}\right )-\frac{1}{20} \sqrt{\frac{1}{10} \left (5959+2665 \sqrt{5}\right )} \tan ^{-1}\left (\frac{\frac{2}{x}+\sqrt{2 \left (1+\sqrt{5}\right )}+2}{\sqrt{2 \left (\sqrt{5}-1\right )}}\right ) \]

[Out]

-(17 - (1 + x^(-1))^2)/(2*(5 - 2*(1 + x^(-1))^2 + (1 + x^(-1))^4)) + ((59 - 17*(
1 + x^(-1))^2)*(1 + x^(-1)))/(10*(5 - 2*(1 + x^(-1))^2 + (1 + x^(-1))^4)) + (7*A
rcTan[(-1 + (1 + x^(-1))^2)/2])/4 - (Sqrt[(5959 + 2665*Sqrt[5])/10]*ArcTan[(2 -
Sqrt[2*(1 + Sqrt[5])] + 2/x)/Sqrt[2*(-1 + Sqrt[5])]])/20 - (Sqrt[(5959 + 2665*Sq
rt[5])/10]*ArcTan[(2 + Sqrt[2*(1 + Sqrt[5])] + 2/x)/Sqrt[2*(-1 + Sqrt[5])]])/20
+ (Sqrt[(-5959 + 2665*Sqrt[5])/10]*Log[Sqrt[5] - Sqrt[2*(1 + Sqrt[5])]*(1 + x^(-
1)) + (1 + x^(-1))^2])/40 - (Sqrt[(-5959 + 2665*Sqrt[5])/10]*Log[Sqrt[5] + Sqrt[
2*(1 + Sqrt[5])]*(1 + x^(-1)) + (1 + x^(-1))^2])/40

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Rubi [A]  time = 0.870629, antiderivative size = 317, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 11, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.647 \[ -\frac{17-\left (\frac{1}{x}+1\right )^2}{2 \left (\left (\frac{1}{x}+1\right )^4-2 \left (\frac{1}{x}+1\right )^2+5\right )}+\frac{\left (59-17 \left (\frac{1}{x}+1\right )^2\right ) \left (\frac{1}{x}+1\right )}{10 \left (\left (\frac{1}{x}+1\right )^4-2 \left (\frac{1}{x}+1\right )^2+5\right )}+\frac{1}{40} \sqrt{\frac{1}{10} \left (2665 \sqrt{5}-5959\right )} \log \left (\left (\frac{1}{x}+1\right )^2-\sqrt{2 \left (1+\sqrt{5}\right )} \left (\frac{1}{x}+1\right )+\sqrt{5}\right )-\frac{1}{40} \sqrt{\frac{1}{10} \left (2665 \sqrt{5}-5959\right )} \log \left (\left (\frac{1}{x}+1\right )^2+\sqrt{2 \left (1+\sqrt{5}\right )} \left (\frac{1}{x}+1\right )+\sqrt{5}\right )+\frac{7}{4} \tan ^{-1}\left (\frac{1}{2} \left (\left (\frac{1}{x}+1\right )^2-1\right )\right )-\frac{1}{20} \sqrt{\frac{1}{10} \left (5959+2665 \sqrt{5}\right )} \tan ^{-1}\left (\frac{\frac{2}{x}-\sqrt{2 \left (1+\sqrt{5}\right )}+2}{\sqrt{2 \left (\sqrt{5}-1\right )}}\right )-\frac{1}{20} \sqrt{\frac{1}{10} \left (5959+2665 \sqrt{5}\right )} \tan ^{-1}\left (\frac{\frac{2}{x}+\sqrt{2 \left (1+\sqrt{5}\right )}+2}{\sqrt{2 \left (\sqrt{5}-1\right )}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + 4*x + 4*x^2 + 4*x^4)^(-2),x]

[Out]

-(17 - (1 + x^(-1))^2)/(2*(5 - 2*(1 + x^(-1))^2 + (1 + x^(-1))^4)) + ((59 - 17*(
1 + x^(-1))^2)*(1 + x^(-1)))/(10*(5 - 2*(1 + x^(-1))^2 + (1 + x^(-1))^4)) + (7*A
rcTan[(-1 + (1 + x^(-1))^2)/2])/4 - (Sqrt[(5959 + 2665*Sqrt[5])/10]*ArcTan[(2 -
Sqrt[2*(1 + Sqrt[5])] + 2/x)/Sqrt[2*(-1 + Sqrt[5])]])/20 - (Sqrt[(5959 + 2665*Sq
rt[5])/10]*ArcTan[(2 + Sqrt[2*(1 + Sqrt[5])] + 2/x)/Sqrt[2*(-1 + Sqrt[5])]])/20
+ (Sqrt[(-5959 + 2665*Sqrt[5])/10]*Log[Sqrt[5] - Sqrt[2*(1 + Sqrt[5])]*(1 + x^(-
1)) + (1 + x^(-1))^2])/40 - (Sqrt[(-5959 + 2665*Sqrt[5])/10]*Log[Sqrt[5] + Sqrt[
2*(1 + Sqrt[5])]*(1 + x^(-1)) + (1 + x^(-1))^2])/40

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Rubi in Sympy [A]  time = 77.5827, size = 393, normalized size = 1.24 \[ \frac{\left (1 + \frac{1}{x}\right ) \left (164413911467589567760039936 \left (1 + \frac{1}{x}\right )^{3} - 164413911467589567760039936 \left (1 + \frac{1}{x}\right )^{2} + 290142196707511001929482240 - \frac{280470790150593968531832832}{x}\right )}{377789318629571617095680 \left (256 \left (1 + \frac{1}{x}\right )^{4} - 512 \left (1 + \frac{1}{x}\right )^{2} + 1280\right )} + \frac{\sqrt{10} \left (- 130563988518379950868267008 \sqrt{5} + 294977899985969518628306944\right ) \log{\left (\sqrt{2} \left (-1 - \frac{1}{x}\right ) \sqrt{1 + \sqrt{5}} + \left (1 + \frac{1}{x}\right )^{2} + \sqrt{5} \right )}}{1934281311383406679529881600 \sqrt{1 + \sqrt{5}}} - \frac{\sqrt{10} \left (- 130563988518379950868267008 \sqrt{5} + 294977899985969518628306944\right ) \log{\left (\left (1 + \frac{1}{x}\right )^{2} + \sqrt{2} \sqrt{1 + \sqrt{5}} \left (1 + \frac{1}{x}\right ) + \sqrt{5} \right )}}{1934281311383406679529881600 \sqrt{1 + \sqrt{5}}} - \frac{\sqrt{5} \left (- \frac{\sqrt{2} \sqrt{1 + \sqrt{5}} \left (- 261127977036759901736534016 \sqrt{5} + 589955799971939037256613888\right )}{2} + 589955799971939037256613888 \sqrt{2} \sqrt{1 + \sqrt{5}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (1 + \frac{\sqrt{2 + 2 \sqrt{5}}}{2} + \frac{1}{x}\right )}{\sqrt{-1 + \sqrt{5}}} \right )}}{967140655691703339764940800 \sqrt{-1 + \sqrt{5}} \sqrt{1 + \sqrt{5}}} - \frac{\sqrt{5} \left (- \frac{\sqrt{2} \sqrt{1 + \sqrt{5}} \left (- 261127977036759901736534016 \sqrt{5} + 589955799971939037256613888\right )}{2} + 589955799971939037256613888 \sqrt{2} \sqrt{1 + \sqrt{5}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (- \frac{\sqrt{2 + 2 \sqrt{5}}}{2} + 1 + \frac{1}{x}\right )}{\sqrt{-1 + \sqrt{5}}} \right )}}{967140655691703339764940800 \sqrt{-1 + \sqrt{5}} \sqrt{1 + \sqrt{5}}} + \frac{7 \operatorname{atan}{\left (\frac{\left (1 + \frac{1}{x}\right )^{2}}{2} - \frac{1}{2} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(4*x**4+4*x**2+4*x+1)**2,x)

[Out]

(1 + 1/x)*(164413911467589567760039936*(1 + 1/x)**3 - 16441391146758956776003993
6*(1 + 1/x)**2 + 290142196707511001929482240 - 280470790150593968531832832/x)/(3
77789318629571617095680*(256*(1 + 1/x)**4 - 512*(1 + 1/x)**2 + 1280)) + sqrt(10)
*(-130563988518379950868267008*sqrt(5) + 294977899985969518628306944)*log(sqrt(2
)*(-1 - 1/x)*sqrt(1 + sqrt(5)) + (1 + 1/x)**2 + sqrt(5))/(1934281311383406679529
881600*sqrt(1 + sqrt(5))) - sqrt(10)*(-130563988518379950868267008*sqrt(5) + 294
977899985969518628306944)*log((1 + 1/x)**2 + sqrt(2)*sqrt(1 + sqrt(5))*(1 + 1/x)
+ sqrt(5))/(1934281311383406679529881600*sqrt(1 + sqrt(5))) - sqrt(5)*(-sqrt(2)
*sqrt(1 + sqrt(5))*(-261127977036759901736534016*sqrt(5) + 589955799971939037256
613888)/2 + 589955799971939037256613888*sqrt(2)*sqrt(1 + sqrt(5)))*atan(sqrt(2)*
(1 + sqrt(2 + 2*sqrt(5))/2 + 1/x)/sqrt(-1 + sqrt(5)))/(9671406556917033397649408
00*sqrt(-1 + sqrt(5))*sqrt(1 + sqrt(5))) - sqrt(5)*(-sqrt(2)*sqrt(1 + sqrt(5))*(
-261127977036759901736534016*sqrt(5) + 589955799971939037256613888)/2 + 58995579
9971939037256613888*sqrt(2)*sqrt(1 + sqrt(5)))*atan(sqrt(2)*(-sqrt(2 + 2*sqrt(5)
)/2 + 1 + 1/x)/sqrt(-1 + sqrt(5)))/(967140655691703339764940800*sqrt(-1 + sqrt(5
))*sqrt(1 + sqrt(5))) + 7*atan((1 + 1/x)**2/2 - 1/2)/4

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Mathematica [C]  time = 0.0399579, size = 108, normalized size = 0.34 \[ \frac{1}{40} \left (\text{RootSum}\left [4 \text{$\#$1}^4+4 \text{$\#$1}^2+4 \text{$\#$1}+1\&,\frac{18 \text{$\#$1}^2 \log (x-\text{$\#$1})-16 \text{$\#$1} \log (x-\text{$\#$1})+27 \log (x-\text{$\#$1})}{4 \text{$\#$1}^3+2 \text{$\#$1}+1}\&\right ]+\frac{72 x^3-32 x^2+84 x+38}{4 x^4+4 x^2+4 x+1}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + 4*x + 4*x^2 + 4*x^4)^(-2),x]

[Out]

((38 + 84*x - 32*x^2 + 72*x^3)/(1 + 4*x + 4*x^2 + 4*x^4) + RootSum[1 + 4*#1 + 4*
#1^2 + 4*#1^4 & , (27*Log[x - #1] - 16*Log[x - #1]*#1 + 18*Log[x - #1]*#1^2)/(1
+ 2*#1 + 4*#1^3) & ])/40

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Maple [C]  time = 0.013, size = 79, normalized size = 0.3 \[{1 \left ({\frac{9\,{x}^{3}}{20}}-{\frac{{x}^{2}}{5}}+{\frac{21\,x}{40}}+{\frac{19}{80}} \right ) \left ({x}^{4}+{x}^{2}+x+{\frac{1}{4}} \right ) ^{-1}}+{\frac{1}{40}\sum _{{\it \_R}={\it RootOf} \left ( 4\,{{\it \_Z}}^{4}+4\,{{\it \_Z}}^{2}+4\,{\it \_Z}+1 \right ) }{\frac{ \left ( 18\,{{\it \_R}}^{2}-16\,{\it \_R}+27 \right ) \ln \left ( x-{\it \_R} \right ) }{4\,{{\it \_R}}^{3}+2\,{\it \_R}+1}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(4*x^4+4*x^2+4*x+1)^2,x)

[Out]

(9/20*x^3-1/5*x^2+21/40*x+19/80)/(x^4+x^2+x+1/4)+1/40*sum((18*_R^2-16*_R+27)/(4*
_R^3+2*_R+1)*ln(x-_R),_R=RootOf(4*_Z^4+4*_Z^2+4*_Z+1))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{36 \, x^{3} - 16 \, x^{2} + 42 \, x + 19}{20 \,{\left (4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1\right )}} + \frac{1}{10} \, \int \frac{18 \, x^{2} - 16 \, x + 27}{4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^4 + 4*x^2 + 4*x + 1)^(-2),x, algorithm="maxima")

[Out]

1/20*(36*x^3 - 16*x^2 + 42*x + 19)/(4*x^4 + 4*x^2 + 4*x + 1) + 1/10*integrate((1
8*x^2 - 16*x + 27)/(4*x^4 + 4*x^2 + 4*x + 1), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^4 + 4*x^2 + 4*x + 1)^(-2),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 3.39001, size = 71, normalized size = 0.22 \[ \frac{36 x^{3} - 16 x^{2} + 42 x + 19}{80 x^{4} + 80 x^{2} + 80 x + 20} + \operatorname{RootSum}{\left (64000 t^{4} + 193344 t^{2} - 1064 t + 29, \left ( t \mapsto t \log{\left (- \frac{17084544000 t^{3}}{541735337} - \frac{188086000 t^{2}}{541735337} - \frac{51568487224 t}{541735337} + x - \frac{71080995}{541735337} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(4*x**4+4*x**2+4*x+1)**2,x)

[Out]

(36*x**3 - 16*x**2 + 42*x + 19)/(80*x**4 + 80*x**2 + 80*x + 20) + RootSum(64000*
_t**4 + 193344*_t**2 - 1064*_t + 29, Lambda(_t, _t*log(-17084544000*_t**3/541735
337 - 188086000*_t**2/541735337 - 51568487224*_t/541735337 + x - 71080995/541735
337)))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^4 + 4*x^2 + 4*x + 1)^(-2),x, algorithm="giac")

[Out]

integrate((4*x^4 + 4*x^2 + 4*x + 1)^(-2), x)