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

Optimal. Leaf size=20 \[ -\frac {1}{3} \tan ^{-1}\left (\frac {x}{\sqrt {x^8-x^2}}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2008, 203} \begin {gather*} -\frac {1}{3} \tan ^{-1}\left (\frac {x}{\sqrt {x^8-x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-x^2 + x^8],x]

[Out]

-1/3*ArcTan[x/Sqrt[-x^2 + x^8]]

Rule 203

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

Rule 2008

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*} \int \frac {1}{\sqrt {-x^2+x^8}} \, dx &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt {-x^2+x^8}}\right )\right )\\ &=-\frac {1}{3} \tan ^{-1}\left (\frac {x}{\sqrt {-x^2+x^8}}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 37, normalized size = 1.85 \begin {gather*} \frac {x \sqrt {x^6-1} \tan ^{-1}\left (\sqrt {x^6-1}\right )}{3 \sqrt {x^2 \left (x^6-1\right )}} \end {gather*}

Antiderivative was successfully verified.

[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)])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.04, size = 20, normalized size = 1.00 \begin {gather*} -\frac {1}{3} \tan ^{-1}\left (\frac {x}{\sqrt {-x^2+x^8}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/Sqrt[-x^2 + x^8],x]

[Out]

-1/3*ArcTan[x/Sqrt[-x^2 + x^8]]

________________________________________________________________________________________

fricas [A]  time = 0.45, size = 18, normalized size = 0.90 \begin {gather*} \frac {1}{3} \, \arctan \left (\frac {\sqrt {x^{8} - x^{2}}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*arctan(sqrt(x^8 - x^2)/x)

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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(x)+1/3*atan(sqrt(x^6-1))
/sign(x)

________________________________________________________________________________________

maple [A]  time = 0.45, size = 26, normalized size = 1.30

method	result	size
default	\(-\frac {x \sqrt {x^{6}-1}\, \arcsin \left (\frac {1}{x^{3}}\right )}{3 \sqrt {x^{8}-x^{2}}}\)	\(26\)
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x +\sqrt {x^{8}-x^{2}}}{x^{4}}\right )}{3}\)	\(35\)
meijerg	\(\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \left (\left (-2 \ln \relax (2)+6 \ln \relax (x )+i \pi \right ) \sqrt {\pi }-2 \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right ) \sqrt {\pi }\right )}{6 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, \sqrt {\pi }}\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/(x^8-x^2)^(1/2)*x*(x^6-1)^(1/2)*arcsin(1/x^3)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {x^{8} - x^{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(x^8 - x^2), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {1}{\sqrt {x^8-x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^8 - x^2)^(1/2),x)

[Out]

int(1/(x^8 - x^2)^(1/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {x^{8} - x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(x**8 - x**2), x)

________________________________________________________________________________________