\(\int \frac {1}{\sqrt {-x^2+x^8}} \, dx\) [189]

Optimal result

Integrand size = 13, antiderivative size = 20 \[ \int \frac {1}{\sqrt {-x^2+x^8}} \, dx=-\frac {1}{3} \arctan \left (\frac {x}{\sqrt {-x^2+x^8}}\right ) \]

[Out]

-1/3*arctan(x/(x^8-x^2)^(1/2))

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 20, 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 = {2033, 209} \[ \int \frac {1}{\sqrt {-x^2+x^8}} \, dx=-\frac {1}{3} \arctan \left (\frac {x}{\sqrt {x^8-x^2}}\right ) \]

[In]

Int[1/Sqrt[-x^2 + x^8],x]

[Out]

-1/3*ArcTan[x/Sqrt[-x^2 + x^8]]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 2033

Int[1/Sqrt[(a_.)*(x_)^2 + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[2/(2 - n), Subst[Int[1/(1 - a*x^2), x], x, x/Sq
rt[a*x^2 + b*x^n]], x] /; FreeQ[{a, b, n}, x] && NeQ[n, 2]

Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{3} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt {-x^2+x^8}}\right )\right ) \\ & = -\frac {1}{3} \arctan \left (\frac {x}{\sqrt {-x^2+x^8}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.85 \[ \int \frac {1}{\sqrt {-x^2+x^8}} \, dx=\frac {x \sqrt {-1+x^6} \arctan \left (\sqrt {-1+x^6}\right )}{3 \sqrt {x^2 \left (-1+x^6\right )}} \]

[In]

Integrate[1/Sqrt[-x^2 + x^8],x]

[Out]

(x*Sqrt[-1 + x^6]*ArcTan[Sqrt[-1 + x^6]])/(3*Sqrt[x^2*(-1 + x^6)])

Maple [A] (verified)

Time = 1.58 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.35

method	result	size
pseudoelliptic	\(-\frac {\arcsin \left (\frac {1}{x^{3}}\right )}{3}\)	\(7\)
trager	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +\sqrt {x^{8}-x^{2}}}{x^{4}}\right )}{3}\)	\(35\)
meijerg	\(\frac {\sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )+\left (-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }\right )}{6 \sqrt {\pi }\, \sqrt {\operatorname {signum}\left (x^{6}-1\right )}}\)	\(61\)

[In]

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

[Out]

-1/3*arcsin(1/x^3)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\sqrt {-x^2+x^8}} \, dx=\frac {1}{3} \, \arctan \left (\frac {\sqrt {x^{8} - x^{2}}}{x}\right ) \]

[In]

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

[Out]

1/3*arctan(sqrt(x^8 - x^2)/x)

Sympy [F]

\[ \int \frac {1}{\sqrt {-x^2+x^8}} \, dx=\int \frac {1}{\sqrt {x^{8} - x^{2}}}\, dx \]

[In]

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

[Out]

Integral(1/sqrt(x**8 - x**2), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {-x^2+x^8}} \, dx=\int { \frac {1}{\sqrt {x^{8} - x^{2}}} \,d x } \]

[In]

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

[Out]

integrate(1/sqrt(x^8 - x^2), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{\sqrt {-x^2+x^8}} \, dx=\text {Exception raised: NotImplementedError} \]

[In]

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

[Out]

Exception raised: NotImplementedError >> unable to parse Giac output: -atan(i)/3*sign(sageVARx)+1/3*atan(sqrt(
sageVARx^6-1))/sign(sageVARx)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {-x^2+x^8}} \, dx=\int \frac {1}{\sqrt {x^8-x^2}} \,d x \]

[In]

int(1/(x^8 - x^2)^(1/2),x)

[Out]

int(1/(x^8 - x^2)^(1/2), x)