\(\int \frac {x^8}{\sqrt {1-x^4}} \, dx\) [884]

Optimal result

Integrand size = 15, antiderivative size = 43 \[ \int \frac {x^8}{\sqrt {1-x^4}} \, dx=-\frac {5}{21} x \sqrt {1-x^4}-\frac {1}{7} x^5 \sqrt {1-x^4}+\frac {5}{21} \operatorname {EllipticF}(\arcsin (x),-1) \]

[Out]

5/21*EllipticF(x,I)-5/21*x*(-x^4+1)^(1/2)-1/7*x^5*(-x^4+1)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {327, 227} \[ \int \frac {x^8}{\sqrt {1-x^4}} \, dx=\frac {5}{21} \operatorname {EllipticF}(\arcsin (x),-1)-\frac {5}{21} \sqrt {1-x^4} x-\frac {1}{7} \sqrt {1-x^4} x^5 \]

[In]

Int[x^8/Sqrt[1 - x^4],x]

[Out]

(-5*x*Sqrt[1 - x^4])/21 - (x^5*Sqrt[1 - x^4])/7 + (5*EllipticF[ArcSin[x], -1])/21

Rule 227

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[Rt[-b, 4]*(x/Rt[a, 4])], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 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}{7} x^5 \sqrt {1-x^4}+\frac {5}{7} \int \frac {x^4}{\sqrt {1-x^4}} \, dx \\ & = -\frac {5}{21} x \sqrt {1-x^4}-\frac {1}{7} x^5 \sqrt {1-x^4}+\frac {5}{21} \int \frac {1}{\sqrt {1-x^4}} \, dx \\ & = -\frac {5}{21} x \sqrt {1-x^4}-\frac {1}{7} x^5 \sqrt {1-x^4}+\frac {5}{21} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \\ \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 = 42, normalized size of antiderivative = 0.98 \[ \int \frac {x^8}{\sqrt {1-x^4}} \, dx=\frac {1}{21} \left (-x \sqrt {1-x^4} \left (5+3 x^4\right )+5 x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},x^4\right )\right ) \]

[In]

Integrate[x^8/Sqrt[1 - x^4],x]

[Out]

(-(x*Sqrt[1 - x^4]*(5 + 3*x^4)) + 5*x*Hypergeometric2F1[1/4, 1/2, 5/4, x^4])/21

Maple [C] (verified)

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

Time = 4.64 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.35

method	result	size
meijerg	\(\frac {x^{9} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},\frac {9}{4};\frac {13}{4};x^{4}\right )}{9}\)	\(15\)
risch	\(\frac {x \left (3 x^{4}+5\right ) \left (x^{4}-1\right )}{21 \sqrt {-x^{4}+1}}+\frac {5 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, F\left (x , i\right )}{21 \sqrt {-x^{4}+1}}\)	\(57\)
default	\(-\frac {x^{5} \sqrt {-x^{4}+1}}{7}-\frac {5 x \sqrt {-x^{4}+1}}{21}+\frac {5 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, F\left (x , i\right )}{21 \sqrt {-x^{4}+1}}\)	\(59\)
elliptic	\(-\frac {x^{5} \sqrt {-x^{4}+1}}{7}-\frac {5 x \sqrt {-x^{4}+1}}{21}+\frac {5 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, F\left (x , i\right )}{21 \sqrt {-x^{4}+1}}\)	\(59\)

[In]

int(x^8/(-x^4+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/9*x^9*hypergeom([1/2,9/4],[13/4],x^4)

Fricas [A] (verification not implemented)

none

Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.67 \[ \int \frac {x^8}{\sqrt {1-x^4}} \, dx=-\frac {1}{21} \, {\left (3 \, x^{5} + 5 \, x\right )} \sqrt {-x^{4} + 1} + \frac {5}{21} i \, F(\arcsin \left (\frac {1}{x}\right )\,|\,-1) \]

[In]

integrate(x^8/(-x^4+1)^(1/2),x, algorithm="fricas")

[Out]

-1/21*(3*x^5 + 5*x)*sqrt(-x^4 + 1) + 5/21*I*elliptic_f(arcsin(1/x), -1)

Sympy [A] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.72 \[ \int \frac {x^8}{\sqrt {1-x^4}} \, dx=\frac {x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac {13}{4}\right )} \]

[In]

integrate(x**8/(-x**4+1)**(1/2),x)

[Out]

x**9*gamma(9/4)*hyper((1/2, 9/4), (13/4,), x**4*exp_polar(2*I*pi))/(4*gamma(13/4))

Maxima [F]

\[ \int \frac {x^8}{\sqrt {1-x^4}} \, dx=\int { \frac {x^{8}}{\sqrt {-x^{4} + 1}} \,d x } \]

[In]

integrate(x^8/(-x^4+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^8/sqrt(-x^4 + 1), x)

Giac [F]

\[ \int \frac {x^8}{\sqrt {1-x^4}} \, dx=\int { \frac {x^{8}}{\sqrt {-x^{4} + 1}} \,d x } \]

[In]

integrate(x^8/(-x^4+1)^(1/2),x, algorithm="giac")

[Out]

integrate(x^8/sqrt(-x^4 + 1), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^8}{\sqrt {1-x^4}} \, dx=\int \frac {x^8}{\sqrt {1-x^4}} \,d x \]

[In]

int(x^8/(1 - x^4)^(1/2),x)

[Out]

int(x^8/(1 - x^4)^(1/2), x)