\(\int \frac {1}{\sqrt {1+4 x+4 x^2+4 x^4}} \, dx\) [87]

Optimal result

Integrand size = 19, antiderivative size = 108 \[ \int \frac {1}{\sqrt {1+4 x+4 x^2+4 x^4}} \, dx=-\frac {\left (\sqrt {5}+\left (1+\frac {1}{x}\right )^2\right ) \sqrt {\frac {5-2 \left (1+\frac {1}{x}\right )^2+\left (1+\frac {1}{x}\right )^4}{\left (\sqrt {5}+\left (1+\frac {1}{x}\right )^2\right )^2}} x^2 \operatorname {EllipticF}\left (2 \arctan \left (\frac {1+\frac {1}{x}}{\sqrt [4]{5}}\right ),\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{2 \sqrt [4]{5} \sqrt {1+4 x+4 x^2+4 x^4}} \] Output:

-1/10*(5^(1/2)+(1+1/x)^2)*((5-2*(1+1/x)^2+(1+1/x)^4)/(5^(1/2)+(1+1/x)^2)^2 
)^(1/2)*x^2*InverseJacobiAM(2*arctan(1/5*(1+1/x)*5^(3/4)),1/10*(50+10*5^(1 
/2))^(1/2))*5^(3/4)/(4*x^4+4*x^2+4*x+1)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 10.67 (sec) , antiderivative size = 249, normalized size of antiderivative = 2.31 \[ \int \frac {1}{\sqrt {1+4 x+4 x^2+4 x^4}} \, dx=\frac {(2-i) \sqrt {-\frac {1}{10}+\frac {i}{5}} \sqrt {\frac {\left (2 i+\sqrt {-1-2 i}-\sqrt {-1+2 i}\right ) \left (-i+\sqrt {-1-2 i}-2 x\right )}{\left (-2 i+\sqrt {-1-2 i}+\sqrt {-1+2 i}\right ) \left (i+\sqrt {-1-2 i}+2 x\right )}} \left (1+2 x+2 i x^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {\left (2 i+\sqrt {-1-2 i}+\sqrt {-1+2 i}\right ) \left (-i+\sqrt {-1+2 i}+2 x\right )}{\sqrt {-1+2 i} \left (i+\sqrt {-1-2 i}+2 x\right )}}}{\sqrt {2}}\right ),\frac {1}{2} \left (5-\sqrt {5}\right )\right )}{\sqrt {\frac {(1+2 i) \left ((-1+i)+\sqrt {-1-2 i}\right ) \left (1+2 x+2 i x^2\right )}{\left (i+\sqrt {-1-2 i}+2 x\right )^2}} \sqrt {1+4 x+4 x^2+4 x^4}} \] Input:

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

Output:

((2 - I)*Sqrt[-1/10 + I/5]*Sqrt[((2*I + Sqrt[-1 - 2*I] - Sqrt[-1 + 2*I])*( 
-I + Sqrt[-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])
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2504, 27, 7270, 1416}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2504

\(\displaystyle -16 \int \frac {x^2}{16 \sqrt {\left (\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5\right ) x^4}}d\left (1+\frac {1}{x}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -\int \frac {x^2}{\sqrt {\left (\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5\right ) x^4}}d\left (1+\frac {1}{x}\right )\)

\(\Big \downarrow \) 7270

\(\displaystyle -\frac {\sqrt {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5} x^2 \int \frac {1}{\sqrt {\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5}}d\left (1+\frac {1}{x}\right )}{\sqrt {\left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right ) x^4}}\)

\(\Big \downarrow \) 1416

\(\displaystyle -\frac {\left (\left (\frac {1}{x}+1\right )^2+\sqrt {5}\right ) \sqrt {\frac {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5}{\left (\left (\frac {1}{x}+1\right )^2+\sqrt {5}\right )^2}} x^2 \operatorname {EllipticF}\left (2 \arctan \left (\frac {1+\frac {1}{x}}{\sqrt [4]{5}}\right ),\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{2 \sqrt [4]{5} \sqrt {\left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right ) x^4}}\)

Input:

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

Output:

-1/2*((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])/(5^(1/4)*Sqrt[(5 - 2*(1 + x^(-1))^2 + (1 + x^(-1) 
)^4)*x^4])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 2504

Int[(P4_)^(p_), x_Symbol] :> With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1] 
, c = Coeff[P4, x, 2], d = Coeff[P4, x, 3], e = Coeff[P4, x, 4]}, Simp[-16* 
a^2   Subst[Int[(1/(b - 4*a*x)^2)*(a*((-3*b^4 + 16*a*b^2*c - 64*a^2*b*d + 2 
56*a^3*e - 32*a^2*(3*b^2 - 8*a*c)*x^2 + 256*a^4*x^4)/(b - 4*a*x)^4))^p, x], 
 x, b/(4*a) + 1/x], x] /; NeQ[a, 0] && NeQ[b, 0] && EqQ[b^3 - 4*a*b*c + 8*a 
^2*d, 0]] /; FreeQ[p, x] && PolyQ[P4, x, 4] && IntegerQ[2*p] &&  !IGtQ[p, 0 
]
 

rule 7270

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p 
]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p])))   Int[u*v 
^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !Free 
Q[v, x] &&  !FreeQ[w, x]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(960\) vs. \(2(95)=190\).

Time = 1.06 (sec) , antiderivative size = 961, normalized size of antiderivative = 8.90

method	result	size
default	\(\text {Expression too large to display}\)	\(961\)
elliptic	\(\text {Expression too large to display}\)	\(961\)

Input:

int(1/(4*x^4+4*x^2+4*x+1)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

(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=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)*(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-Roo 
tOf(4*_Z^4+4*_Z^2+4*_Z+1,index=3))/(RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=3)-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= 
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)-RootO 
f(4*_Z^4+4*_Z^2+4*_Z+1,index=2))/(RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2)-Roo 
tOf(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(((Roo 
tOf(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,in 
dex=2)))^(1/2),((RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2)-RootOf(4*_Z^4+4*_...
 

Fricas [F]

\[ \int \frac {1}{\sqrt {1+4 x+4 x^2+4 x^4}} \, dx=\int { \frac {1}{\sqrt {4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1}} \,d x } \] Input:

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

Output:

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

Sympy [F]

\[ \int \frac {1}{\sqrt {1+4 x+4 x^2+4 x^4}} \, dx=\int \frac {1}{\sqrt {4 x^{4} + 4 x^{2} + 4 x + 1}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {1}{\sqrt {1+4 x+4 x^2+4 x^4}} \, dx=\int { \frac {1}{\sqrt {4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {1}{\sqrt {1+4 x+4 x^2+4 x^4}} \, dx=\int { \frac {1}{\sqrt {4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {1+4 x+4 x^2+4 x^4}} \, dx=\int \frac {1}{\sqrt {4\,x^4+4\,x^2+4\,x+1}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\sqrt {1+4 x+4 x^2+4 x^4}} \, dx=\int \frac {\sqrt {4 x^{4}+4 x^{2}+4 x +1}}{4 x^{4}+4 x^{2}+4 x +1}d x \] Input:

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

Output:

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