\(\int \frac {x^4}{\sqrt {-1+x^4}} \, dx\) [977]

Optimal result

Integrand size = 13, antiderivative size = 72 \[ \int \frac {x^4}{\sqrt {-1+x^4}} \, dx=\frac {1}{3} x \sqrt {-1+x^4}+\frac {\sqrt {-1+x^2} \sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right ),\frac {1}{2}\right )}{3 \sqrt {2} \sqrt {-1+x^4}} \]

[Out]

1/6*EllipticF(x*2^(1/2)/(x^2-1)^(1/2),1/2*2^(1/2))*(x^2-1)^(1/2)*(x^2+1)^(1/2)*2^(1/2)/(x^4-1)^(1/2)+1/3*x*(x^
4-1)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {327, 228} \[ \int \frac {x^4}{\sqrt {-1+x^4}} \, dx=\frac {\sqrt {x^2-1} \sqrt {x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right ),\frac {1}{2}\right )}{3 \sqrt {2} \sqrt {x^4-1}}+\frac {1}{3} \sqrt {x^4-1} x \]

[In]

Int[x^4/Sqrt[-1 + x^4],x]

[Out]

(x*Sqrt[-1 + x^4])/3 + (Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[(Sqrt[2]*x)/Sqrt[-1 + x^2]], 1/2])/(3*Sq
rt[2]*Sqrt[-1 + x^4])

Rule 228

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

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x \sqrt {-1+x^4}+\frac {1}{3} \int \frac {1}{\sqrt {-1+x^4}} \, dx \\ & = \frac {1}{3} x \sqrt {-1+x^4}+\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{3 \sqrt {2} \sqrt {-1+x^4}} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 10.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.61 \[ \int \frac {x^4}{\sqrt {-1+x^4}} \, dx=\frac {x \left (-1+x^4+\sqrt {1-x^4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},x^4\right )\right )}{3 \sqrt {-1+x^4}} \]

[In]

Integrate[x^4/Sqrt[-1 + x^4],x]

[Out]

(x*(-1 + x^4 + Sqrt[1 - x^4]*Hypergeometric2F1[1/4, 1/2, 5/4, x^4]))/(3*Sqrt[-1 + x^4])

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 4.41 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.46

method	result	size
meijerg	\(\frac {\sqrt {-\operatorname {signum}\left (x^{4}-1\right )}\, x^{5} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},\frac {5}{4};\frac {9}{4};x^{4}\right )}{5 \sqrt {\operatorname {signum}\left (x^{4}-1\right )}}\)	\(33\)
default	\(\frac {x \sqrt {x^{4}-1}}{3}-\frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, F\left (i x , i\right )}{3 \sqrt {x^{4}-1}}\)	\(45\)
risch	\(\frac {x \sqrt {x^{4}-1}}{3}-\frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, F\left (i x , i\right )}{3 \sqrt {x^{4}-1}}\)	\(45\)
elliptic	\(\frac {x \sqrt {x^{4}-1}}{3}-\frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, F\left (i x , i\right )}{3 \sqrt {x^{4}-1}}\)	\(45\)

[In]

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

[Out]

1/5/signum(x^4-1)^(1/2)*(-signum(x^4-1))^(1/2)*x^5*hypergeom([1/2,5/4],[9/4],x^4)

Fricas [A] (verification not implemented)

none

Time = 0.09 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.26 \[ \int \frac {x^4}{\sqrt {-1+x^4}} \, dx=\frac {1}{3} \, \sqrt {x^{4} - 1} x - \frac {1}{3} \, F(\arcsin \left (\frac {1}{x}\right )\,|\,-1) \]

[In]

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

[Out]

1/3*sqrt(x^4 - 1)*x - 1/3*elliptic_f(arcsin(1/x), -1)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.38 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.38 \[ \int \frac {x^4}{\sqrt {-1+x^4}} \, dx=- \frac {i x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {x^{4}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} \]

[In]

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

[Out]

-I*x**5*gamma(5/4)*hyper((1/2, 5/4), (9/4,), x**4)/(4*gamma(9/4))

Maxima [F]

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

[In]

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

[Out]

integrate(x^4/sqrt(x^4 - 1), x)

Giac [F]

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

[In]

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

[Out]

integrate(x^4/sqrt(x^4 - 1), x)

Mupad [F(-1)]

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

[In]

int(x^4/(x^4 - 1)^(1/2),x)

[Out]

int(x^4/(x^4 - 1)^(1/2), x)