\(\int \frac {\arcsin (\sqrt {1+b x^2})^n}{\sqrt {1+b x^2}} \, dx\) [471]

Optimal result

Integrand size = 26, antiderivative size = 38 \[ \int \frac {\arcsin \left (\sqrt {1+b x^2}\right )^n}{\sqrt {1+b x^2}} \, dx=\frac {\sqrt {-b x^2} \arcsin \left (\sqrt {1+b x^2}\right )^{1+n}}{b (1+n) x} \]

[Out]

arcsin((b*x^2+1)^(1/2))^(1+n)*(-b*x^2)^(1/2)/b/(1+n)/x

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {4918, 4737} \[ \int \frac {\arcsin \left (\sqrt {1+b x^2}\right )^n}{\sqrt {1+b x^2}} \, dx=\frac {\sqrt {-b x^2} \arcsin \left (\sqrt {b x^2+1}\right )^{n+1}}{b (n+1) x} \]

[In]

Int[ArcSin[Sqrt[1 + b*x^2]]^n/Sqrt[1 + b*x^2],x]

[Out]

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

Rule 4737

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Si
mp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c
^2*d + e, 0] && NeQ[n, -1]

Rule 4918

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

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \frac {\arcsin \left (\sqrt {1+b x^2}\right )^n}{\sqrt {1+b x^2}} \, dx=\frac {\sqrt {-b x^2} \arcsin \left (\sqrt {1+b x^2}\right )^{1+n}}{b (1+n) x} \]

[In]

Integrate[ArcSin[Sqrt[1 + b*x^2]]^n/Sqrt[1 + b*x^2],x]

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.08 \[ \int \frac {\arcsin \left (\sqrt {1+b x^2}\right )^n}{\sqrt {1+b x^2}} \, dx=\frac {\sqrt {-b x^{2}} \arcsin \left (\sqrt {b x^{2} + 1}\right )^{n} \arcsin \left (\sqrt {b x^{2} + 1}\right )}{{\left (b n + b\right )} x} \]

[In]

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

[Out]

sqrt(-b*x^2)*arcsin(sqrt(b*x^2 + 1))^n*arcsin(sqrt(b*x^2 + 1))/((b*n + b)*x)

Sympy [F]

\[ \int \frac {\arcsin \left (\sqrt {1+b x^2}\right )^n}{\sqrt {1+b x^2}} \, dx=\begin {cases} \frac {2 x}{\pi } & \text {for}\: b = 0 \wedge n = -1 \\x \left (\frac {\pi }{2}\right )^{n} & \text {for}\: b = 0 \\\int \frac {1}{\sqrt {b x^{2} + 1} \operatorname {asin}{\left (\sqrt {b x^{2} + 1} \right )}}\, dx & \text {for}\: n = -1 \\\frac {\sqrt {- b x^{2}} \operatorname {asin}{\left (\sqrt {b x^{2} + 1} \right )} \operatorname {asin}^{n}{\left (\sqrt {b x^{2} + 1} \right )}}{b n x + b x} & \text {otherwise} \end {cases} \]

[In]

integrate(asin((b*x**2+1)**(1/2))**n/(b*x**2+1)**(1/2),x)

[Out]

Piecewise((2*x/pi, Eq(b, 0) & Eq(n, -1)), (x*(pi/2)**n, Eq(b, 0)), (Integral(1/(sqrt(b*x**2 + 1)*asin(sqrt(b*x
**2 + 1))), x), Eq(n, -1)), (sqrt(-b*x**2)*asin(sqrt(b*x**2 + 1))*asin(sqrt(b*x**2 + 1))**n/(b*n*x + b*x), Tru
e))

Maxima [F(-2)]

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

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: sign: argument cannot be imaginary; found sqrt(-_SAGE_VAR_b)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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