\(\int \frac {1}{\sqrt {1+x^4}} \, dx\) [929]

Optimal result

Integrand size = 9, antiderivative size = 43 \[ \int \frac {1}{\sqrt {1+x^4}} \, dx=\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{2 \sqrt {1+x^4}} \]

[Out]

1/2*(x^2+1)*(cos(2*arctan(x))^2)^(1/2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),1/2*2^(1/2))*((x^4+1)/(x^2+
1)^2)^(1/2)/(x^4+1)^(1/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {226} \[ \int \frac {1}{\sqrt {1+x^4}} \, dx=\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{2 \sqrt {x^4+1}} \]

[In]

Int[1/Sqrt[1 + x^4],x]

[Out]

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

Rule 226

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

Rubi steps \begin{align*} \text {integral}& = \frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {1+x^4}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.06 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.49 \[ \int \frac {1}{\sqrt {1+x^4}} \, dx=-\sqrt [4]{-1} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt [4]{-1} x\right ),-1\right ) \]

[In]

Integrate[1/Sqrt[1 + x^4],x]

[Out]

-((-1)^(1/4)*EllipticF[I*ArcSinh[(-1)^(1/4)*x], -1])

Maple [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4.

Time = 4.12 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.33

method	result	size
meijerg	\(x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-x^{4}\right )\)	\(14\)
default	\(\frac {\sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, F\left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\)	\(60\)
elliptic	\(\frac {\sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, F\left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\)	\(60\)

[In]

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

[Out]

x*hypergeom([1/4,1/2],[5/4],-x^4)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.08 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.30 \[ \int \frac {1}{\sqrt {1+x^4}} \, dx=-i \, \sqrt {i} F(\arcsin \left (\sqrt {i} x\right )\,|\,-1) \]

[In]

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

[Out]

-I*sqrt(I)*elliptic_f(arcsin(sqrt(I)*x), -1)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.32 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.63 \[ \int \frac {1}{\sqrt {1+x^4}} \, dx=\frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} \]

[In]

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

[Out]

x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), x**4*exp_polar(I*pi))/(4*gamma(5/4))

Maxima [F]

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

[In]

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

[Out]

integrate(1/sqrt(x^4 + 1), x)

Giac [F]

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

[In]

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

[Out]

integrate(1/sqrt(x^4 + 1), x)

Mupad [B] (verification not implemented)

Time = 5.44 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.28 \[ \int \frac {1}{\sqrt {1+x^4}} \, dx=x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{2};\ \frac {5}{4};\ -x^4\right ) \]

[In]

int(1/(x^4 + 1)^(1/2),x)

[Out]

x*hypergeom([1/4, 1/2], 5/4, -x^4)