3.642 \(\int \frac{1}{\sqrt{1+4 x+4 x^2+4 x^4}} \, dx\)

Optimal. Leaf size=108 \[ -\frac{\left (\left (\frac{1}{x}+1\right )^2+\sqrt{5}\right ) \sqrt{\frac{\left (\frac{1}{x}+1\right )^4-2 \left (\frac{1}{x}+1\right )^2+5}{\left (\left (\frac{1}{x}+1\right )^2+\sqrt{5}\right )^2}} x^2 F\left (2 \tan ^{-1}\left (\frac{1+\frac{1}{x}}{\sqrt [4]{5}}\right )|\frac{1}{10} \left (5+\sqrt{5}\right )\right )}{2 \sqrt [4]{5} \sqrt{4 x^4+4 x^2+4 x+1}} \]

[Out]

-((Sqrt[5] + (1 + x^(-1))^2)*Sqrt[(5 - 2*(1 + x^(-1))^2 + (1 + x^(-1))^4)/(Sqrt[
5] + (1 + x^(-1))^2)^2]*x^2*EllipticF[2*ArcTan[(1 + x^(-1))/5^(1/4)], (5 + Sqrt[
5])/10])/(2*5^(1/4)*Sqrt[1 + 4*x + 4*x^2 + 4*x^4])

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Rubi [A]  time = 0.363689, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{\left (\left (\frac{1}{x}+1\right )^2+\sqrt{5}\right ) \sqrt{\frac{\left (\frac{1}{x}+1\right )^4-2 \left (\frac{1}{x}+1\right )^2+5}{\left (\left (\frac{1}{x}+1\right )^2+\sqrt{5}\right )^2}} x^2 F\left (2 \tan ^{-1}\left (\frac{1+\frac{1}{x}}{\sqrt [4]{5}}\right )|\frac{1}{10} \left (5+\sqrt{5}\right )\right )}{2 \sqrt [4]{5} \sqrt{4 x^4+4 x^2+4 x+1}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[1 + 4*x + 4*x^2 + 4*x^4],x]

[Out]

-((Sqrt[5] + (1 + x^(-1))^2)*Sqrt[(5 - 2*(1 + x^(-1))^2 + (1 + x^(-1))^4)/(Sqrt[
5] + (1 + x^(-1))^2)^2]*x^2*EllipticF[2*ArcTan[(1 + x^(-1))/5^(1/4)], (5 + Sqrt[
5])/10])/(2*5^(1/4)*Sqrt[1 + 4*x + 4*x^2 + 4*x^4])

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - 16 \int ^{1 + \frac{1}{x}} \frac{1}{\sqrt{\frac{256 x^{4} - 512 x^{2} + 1280}{\left (- 4 x + 4\right )^{4}}} \left (- 4 x + 4\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-16*Integral(1/(sqrt((256*x**4 - 512*x**2 + 1280)/(-4*x + 4)**4)*(-4*x + 4)**2),
(x, 1 + 1/x))

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Mathematica [C]  time = 0.970966, size = 249, normalized size = 2.31 \[ \frac{(2-i) \sqrt{-\frac{1}{10}+\frac{i}{5}} \sqrt{\frac{\left (2 i+\sqrt{-1-2 i}-\sqrt{-1+2 i}\right ) \left (-2 x+\sqrt{-1-2 i}-i\right )}{\left (-2 i+\sqrt{-1-2 i}+\sqrt{-1+2 i}\right ) \left (2 x+\sqrt{-1-2 i}+i\right )}} \left (2 i x^2+2 x+1\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\left (2 i+\sqrt{-1-2 i}+\sqrt{-1+2 i}\right ) \left (2 x+\sqrt{-1+2 i}-i\right )}{\sqrt{-1+2 i} \left (2 x+\sqrt{-1-2 i}+i\right )}}}{\sqrt{2}}\right )|\frac{1}{2} \left (5-\sqrt{5}\right )\right )}{\sqrt{\frac{(1+2 i) \left ((-1+i)+\sqrt{-1-2 i}\right ) \left (2 i x^2+2 x+1\right )}{\left (2 x+\sqrt{-1-2 i}+i\right )^2}} \sqrt{4 x^4+4 x^2+4 x+1}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/Sqrt[1 + 4*x + 4*x^2 + 4*x^4],x]

[Out]

((2 - I)*Sqrt[-1/10 + I/5]*Sqrt[((2*I + Sqrt[-1 - 2*I] - Sqrt[-1 + 2*I])*(-I + S
qrt[-1 - 2*I] - 2*x))/((-2*I + Sqrt[-1 - 2*I] + Sqrt[-1 + 2*I])*(I + Sqrt[-1 - 2
*I] + 2*x))]*(1 + 2*x + (2*I)*x^2)*EllipticF[ArcSin[Sqrt[((2*I + Sqrt[-1 - 2*I]
+ Sqrt[-1 + 2*I])*(-I + Sqrt[-1 + 2*I] + 2*x))/(Sqrt[-1 + 2*I]*(I + Sqrt[-1 - 2*
I] + 2*x))]/Sqrt[2]], (5 - Sqrt[5])/2])/(Sqrt[((1 + 2*I)*((-1 + I) + Sqrt[-1 - 2
*I])*(1 + 2*x + (2*I)*x^2))/(I + Sqrt[-1 - 2*I] + 2*x)^2]*Sqrt[1 + 4*x + 4*x^2 +
4*x^4])

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Maple [C]  time = 1.227, size = 961, normalized size = 8.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4))*((Ro
otOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2))*(x-Root
Of(4*_Z^4+4*_Z^2+4*_Z+1,index=1))/(RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4)-RootOf(4
*_Z^4+4*_Z^2+4*_Z+1,index=1))/(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2)))^(1/2)*(x
-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2))^2*((RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2)-
RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1))*(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=3))/(
RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=3)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1))/(x-Ro
otOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2)))^(1/2)*((RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2
)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1))*(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4))
/(RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1))/(x-
RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2)))^(1/2)/(RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=
4)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2))/(RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2)-R
ootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1))/((x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1))*(
x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2))*(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=3))
*(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4)))^(1/2)*EllipticF(((RootOf(4*_Z^4+4*_Z^
2+4*_Z+1,index=4)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2))*(x-RootOf(4*_Z^4+4*_Z^2+
4*_Z+1,index=1))/(RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4)-RootOf(4*_Z^4+4*_Z^2+4*_Z
+1,index=1))/(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2)))^(1/2),((RootOf(4*_Z^4+4*_
Z^2+4*_Z+1,index=2)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=3))*(RootOf(4*_Z^4+4*_Z^2+
4*_Z+1,index=1)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4))/(-RootOf(4*_Z^4+4*_Z^2+4*_
Z+1,index=3)+RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1))/(RootOf(4*_Z^4+4*_Z^2+4*_Z+1,
index=2)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4)))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/sqrt(4*x^4 + 4*x^2 + 4*x + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/sqrt(4*x^4 + 4*x^2 + 4*x + 1), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{4 x^{4} + 4 x^{2} + 4 x + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/sqrt(4*x**4 + 4*x**2 + 4*x + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/sqrt(4*x^4 + 4*x^2 + 4*x + 1), x)