\(\int \frac {x^8}{\sqrt {1+x^8}} \, dx\) [1531]

Optimal result

Integrand size = 13, antiderivative size = 237 \[ \int \frac {x^8}{\sqrt {1+x^8}} \, dx=\frac {1}{5} x \sqrt {1+x^8}-\frac {x^3 \sqrt {\frac {\left (1+x^2\right )^2}{x^2}} \sqrt {-\frac {1+x^8}{x^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {-\frac {\sqrt {2}-2 x^2+\sqrt {2} x^4}{x^2}}\right ),-2 \left (1-\sqrt {2}\right )\right )}{10 \sqrt {2+\sqrt {2}} \left (1+x^2\right ) \sqrt {1+x^8}}+\frac {x^3 \sqrt {-\frac {\left (1-x^2\right )^2}{x^2}} \sqrt {-\frac {1+x^8}{x^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {\frac {\sqrt {2}+2 x^2+\sqrt {2} x^4}{x^2}}\right ),-2 \left (1-\sqrt {2}\right )\right )}{10 \sqrt {2+\sqrt {2}} \left (1-x^2\right ) \sqrt {1+x^8}} \]

[Out]

1/5*x*(x^8+1)^(1/2)+1/10*x^3*EllipticF(1/2*((2*x^2+2^(1/2)+x^4*2^(1/2))/x^2)^(1/2),(-2+2*2^(1/2))^(1/2))*(-(-x
^2+1)^2/x^2)^(1/2)*((-x^8-1)/x^4)^(1/2)/(-x^2+1)/(x^8+1)^(1/2)/(2+2^(1/2))^(1/2)-1/10*x^3*EllipticF(1/2*((2*x^
2-2^(1/2)-x^4*2^(1/2))/x^2)^(1/2),(-2+2*2^(1/2))^(1/2))*((x^2+1)^2/x^2)^(1/2)*((-x^8-1)/x^4)^(1/2)/(x^2+1)/(x^
8+1)^(1/2)/(2+2^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {327, 232, 1897} \[ \int \frac {x^8}{\sqrt {1+x^8}} \, dx=-\frac {\sqrt {\frac {\left (x^2+1\right )^2}{x^2}} \sqrt {-\frac {x^8+1}{x^4}} x^3 \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {-\frac {\sqrt {2} x^4-2 x^2+\sqrt {2}}{x^2}}\right ),-2 \left (1-\sqrt {2}\right )\right )}{10 \sqrt {2+\sqrt {2}} \left (x^2+1\right ) \sqrt {x^8+1}}+\frac {\sqrt {-\frac {\left (1-x^2\right )^2}{x^2}} \sqrt {-\frac {x^8+1}{x^4}} x^3 \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {\frac {\sqrt {2} x^4+2 x^2+\sqrt {2}}{x^2}}\right ),-2 \left (1-\sqrt {2}\right )\right )}{10 \sqrt {2+\sqrt {2}} \left (1-x^2\right ) \sqrt {x^8+1}}+\frac {1}{5} \sqrt {x^8+1} x \]

[In]

Int[x^8/Sqrt[1 + x^8],x]

[Out]

(x*Sqrt[1 + x^8])/5 - (x^3*Sqrt[(1 + x^2)^2/x^2]*Sqrt[-((1 + x^8)/x^4)]*EllipticF[ArcSin[Sqrt[-((Sqrt[2] - 2*x
^2 + Sqrt[2]*x^4)/x^2)]/2], -2*(1 - Sqrt[2])])/(10*Sqrt[2 + Sqrt[2]]*(1 + x^2)*Sqrt[1 + x^8]) + (x^3*Sqrt[-((1
- x^2)^2/x^2)]*Sqrt[-((1 + x^8)/x^4)]*EllipticF[ArcSin[Sqrt[(Sqrt[2] + 2*x^2 + Sqrt[2]*x^4)/x^2]/2], -2*(1 -
Sqrt[2])])/(10*Sqrt[2 + Sqrt[2]]*(1 - x^2)*Sqrt[1 + x^8])

Rule 232

Int[1/Sqrt[(a_) + (b_.)*(x_)^8], x_Symbol] :> Dist[1/2, Int[(1 - Rt[b/a, 4]*x^2)/Sqrt[a + b*x^8], x], x] + Dis
t[1/2, Int[(1 + Rt[b/a, 4]*x^2)/Sqrt[a + b*x^8], x], x] /; FreeQ[{a, b}, x]

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]

Rule 1897

Int[((c_) + (d_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^8], x_Symbol] :> Simp[(-c)*d*x^3*Sqrt[-(c - d*x^2)^2/(c*d*x^2
)]*(Sqrt[(-d^2)*((a + b*x^8)/(b*c^2*x^4))]/(Sqrt[2 + Sqrt[2]]*(c - d*x^2)*Sqrt[a + b*x^8]))*EllipticF[ArcSin[(
1/2)*Sqrt[(Sqrt[2]*c^2 + 2*c*d*x^2 + Sqrt[2]*d^2*x^4)/(c*d*x^2)]], -2*(1 - Sqrt[2])], x] /; FreeQ[{a, b, c, d}
, x] && EqQ[b*c^4 - a*d^4, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x \sqrt {1+x^8}-\frac {1}{5} \int \frac {1}{\sqrt {1+x^8}} \, dx \\ & = \frac {1}{5} x \sqrt {1+x^8}-\frac {1}{10} \int \frac {1-x^2}{\sqrt {1+x^8}} \, dx-\frac {1}{10} \int \frac {1+x^2}{\sqrt {1+x^8}} \, dx \\ & = \frac {1}{5} x \sqrt {1+x^8}-\frac {x^3 \sqrt {\frac {\left (1+x^2\right )^2}{x^2}} \sqrt {-\frac {1+x^8}{x^4}} F\left (\sin ^{-1}\left (\frac {1}{2} \sqrt {-\frac {\sqrt {2}-2 x^2+\sqrt {2} x^4}{x^2}}\right )|-2 \left (1-\sqrt {2}\right )\right )}{10 \sqrt {2+\sqrt {2}} \left (1+x^2\right ) \sqrt {1+x^8}}+\frac {x^3 \sqrt {-\frac {\left (1-x^2\right )^2}{x^2}} \sqrt {-\frac {1+x^8}{x^4}} F\left (\sin ^{-1}\left (\frac {1}{2} \sqrt {\frac {\sqrt {2}+2 x^2+\sqrt {2} x^4}{x^2}}\right )|-2 \left (1-\sqrt {2}\right )\right )}{10 \sqrt {2+\sqrt {2}} \left (1-x^2\right ) \sqrt {1+x^8}} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 10.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.14 \[ \int \frac {x^8}{\sqrt {1+x^8}} \, dx=\frac {1}{5} x \left (\sqrt {1+x^8}-\operatorname {Hypergeometric2F1}\left (\frac {1}{8},\frac {1}{2},\frac {9}{8},-x^8\right )\right ) \]

[In]

Integrate[x^8/Sqrt[1 + x^8],x]

[Out]

(x*(Sqrt[1 + x^8] - Hypergeometric2F1[1/8, 1/2, 9/8, -x^8]))/5

Maple [C] (verified)

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

Time = 4.21 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.07

method	result	size
meijerg	\(\frac {x^{9} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},\frac {9}{8};\frac {17}{8};-x^{8}\right )}{9}\)	\(17\)
risch	\(\frac {x \sqrt {x^{8}+1}}{5}-\frac {x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{8},\frac {1}{2};\frac {9}{8};-x^{8}\right )}{5}\)	\(26\)

[In]

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

[Out]

1/9*x^9*hypergeom([1/2,9/8],[17/8],-x^8)

Fricas [F]

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

[In]

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

[Out]

integral(x^8/sqrt(x^8 + 1), x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.47 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.12 \[ \int \frac {x^8}{\sqrt {1+x^8}} \, dx=\frac {x^{9} \Gamma \left (\frac {9}{8}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {9}{8} \\ \frac {17}{8} \end {matrix}\middle | {x^{8} e^{i \pi }} \right )}}{8 \Gamma \left (\frac {17}{8}\right )} \]

[In]

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

[Out]

x**9*gamma(9/8)*hyper((1/2, 9/8), (17/8,), x**8*exp_polar(I*pi))/(8*gamma(17/8))

Maxima [F]

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

[In]

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

[Out]

integrate(x^8/sqrt(x^8 + 1), x)

Giac [F]

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

[In]

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

[Out]

integrate(x^8/sqrt(x^8 + 1), x)

Mupad [F(-1)]

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

[In]

int(x^8/(x^8 + 1)^(1/2),x)

[Out]

int(x^8/(x^8 + 1)^(1/2), x)