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

Optimal. Leaf size=24 \[ \tan ^{-1}\left (\frac {x}{\sqrt {-x^2+\left (1+x^{2 n}\right )^{\frac {1}{n}}}}\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.04, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2153, 209} \begin {gather*} \text {ArcTan}\left (\frac {x}{\sqrt {\left (x^{2 n}+1\right )^{\frac {1}{n}}-x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

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

________________________________________________________________________________________

Mathematica [A]
time = 0.11, size = 26, normalized size = 1.08 \begin {gather*} \cot ^{-1}\left (\frac {\sqrt {-x^2+\left (1+x^{2 n}\right )^{\frac {1}{n}}}}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[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]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (1+x^{2 n}\right ) \sqrt {-x^{2}+\left (1+x^{2 n}\right )^{\frac {1}{n}}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- x^{2} + \left (x^{2 n} + 1\right )^{\frac {1}{n}}} \left (x^{2 n} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{\left (x^{2\,n}+1\right )\,\sqrt {{\left (x^{2\,n}+1\right )}^{1/n}-x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________