\(\int \frac {1}{(1+x^{2 n}) \sqrt {-x^2+(1+x^{2 n})^{\frac {1}{n}}}} \, dx\) [329]

Optimal result

Integrand size = 31, antiderivative size = 24 \[ \int \frac {1}{\left (1+x^{2 n}\right ) \sqrt {-x^2+\left (1+x^{2 n}\right )^{\frac {1}{n}}}} \, dx=\arctan \left (\frac {x}{\sqrt {-x^2+\left (1+x^{2 n}\right )^{\frac {1}{n}}}}\right ) \]

[Out]

arctan(x/(-x^2+(1+x^(2*n))^(1/n))^(1/2))

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2153, 209} \[ \int \frac {1}{\left (1+x^{2 n}\right ) \sqrt {-x^2+\left (1+x^{2 n}\right )^{\frac {1}{n}}}} \, dx=\arctan \left (\frac {x}{\sqrt {\left (x^{2 n}+1\right )^{\frac {1}{n}}-x^2}}\right ) \]

[In]

Int[1/((1 + x^(2*n))*Sqrt[-x^2 + (1 + x^(2*n))^n^(-1)]),x]

[Out]

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

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 2153

Int[1/(((a_) + (b_.)*(x_)^(n_.))*Sqrt[(c_.)*(x_)^2 + (d_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.)]), x_Symbol] :> Dis
t[1/a, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[c*x^2 + d*(a + b*x^n)^(2/n)]], x] /; FreeQ[{a, b, c, d, n}, x] &
& EqQ[p, 2/n]

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (1+x^{2 n}\right ) \sqrt {-x^2+\left (1+x^{2 n}\right )^{\frac {1}{n}}}} \, dx=\cot ^{-1}\left (\frac {\sqrt {-x^2+\left (1+x^{2 n}\right )^{\frac {1}{n}}}}{x}\right ) \]

[In]

Integrate[1/((1 + x^(2*n))*Sqrt[-x^2 + (1 + x^(2*n))^n^(-1)]),x]

[Out]

ArcCot[Sqrt[-x^2 + (1 + x^(2*n))^n^(-1)]/x]

Maple [F]

\[\int \frac {1}{\left (1+x^{2 n}\right ) \sqrt {-x^{2}+\left (1+x^{2 n}\right )^{\frac {1}{n}}}}d x\]

[In]

int(1/(1+x^(2*n))/(-x^2+(1+x^(2*n))^(1/n))^(1/2),x)

[Out]

int(1/(1+x^(2*n))/(-x^2+(1+x^(2*n))^(1/n))^(1/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\left (1+x^{2 n}\right ) \sqrt {-x^2+\left (1+x^{2 n}\right )^{\frac {1}{n}}}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/(1+x^(2*n))/(-x^2+(1+x^(2*n))^(1/n))^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F]

\[ \int \frac {1}{\left (1+x^{2 n}\right ) \sqrt {-x^2+\left (1+x^{2 n}\right )^{\frac {1}{n}}}} \, dx=\int \frac {1}{\sqrt {- x^{2} + \left (x^{2 n} + 1\right )^{\frac {1}{n}}} \left (x^{2 n} + 1\right )}\, dx \]

[In]

integrate(1/(1+x**(2*n))/(-x**2+(1+x**(2*n))**(1/n))**(1/2),x)

[Out]

Integral(1/(sqrt(-x**2 + (x**(2*n) + 1)**(1/n))*(x**(2*n) + 1)), x)

Maxima [F]

\[ \int \frac {1}{\left (1+x^{2 n}\right ) \sqrt {-x^2+\left (1+x^{2 n}\right )^{\frac {1}{n}}}} \, dx=\int { \frac {1}{\sqrt {-x^{2} + {\left (x^{2 \, n} + 1\right )}^{\left (\frac {1}{n}\right )}} {\left (x^{2 \, n} + 1\right )}} \,d x } \]

[In]

integrate(1/(1+x^(2*n))/(-x^2+(1+x^(2*n))^(1/n))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-x^2 + (x^(2*n) + 1)^(1/n))*(x^(2*n) + 1)), x)

Giac [F]

\[ \int \frac {1}{\left (1+x^{2 n}\right ) \sqrt {-x^2+\left (1+x^{2 n}\right )^{\frac {1}{n}}}} \, dx=\int { \frac {1}{\sqrt {-x^{2} + {\left (x^{2 \, n} + 1\right )}^{\left (\frac {1}{n}\right )}} {\left (x^{2 \, n} + 1\right )}} \,d x } \]

[In]

integrate(1/(1+x^(2*n))/(-x^2+(1+x^(2*n))^(1/n))^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-x^2 + (x^(2*n) + 1)^(1/n))*(x^(2*n) + 1)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (1+x^{2 n}\right ) \sqrt {-x^2+\left (1+x^{2 n}\right )^{\frac {1}{n}}}} \, dx=\int \frac {1}{\left (x^{2\,n}+1\right )\,\sqrt {{\left (x^{2\,n}+1\right )}^{1/n}-x^2}} \,d x \]

[In]

int(1/((x^(2*n) + 1)*((x^(2*n) + 1)^(1/n) - x^2)^(1/2)),x)

[Out]

int(1/((x^(2*n) + 1)*((x^(2*n) + 1)^(1/n) - x^2)^(1/2)), x)