\(\int \arcsin (a x)^n \, dx\) [133]

Optimal result

Integrand size = 6, antiderivative size = 79 \[ \int \arcsin (a x)^n \, dx=-\frac {i (-i \arcsin (a x))^{-n} \arcsin (a x)^n \Gamma (1+n,-i \arcsin (a x))}{2 a}+\frac {i (i \arcsin (a x))^{-n} \arcsin (a x)^n \Gamma (1+n,i \arcsin (a x))}{2 a} \]

[Out]

-1/2*I*arcsin(a*x)^n*GAMMA(1+n,-I*arcsin(a*x))/a/((-I*arcsin(a*x))^n)+1/2*I*arcsin(a*x)^n*GAMMA(1+n,I*arcsin(a
*x))/a/((I*arcsin(a*x))^n)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4719, 3388, 2212} \[ \int \arcsin (a x)^n \, dx=\frac {i (i \arcsin (a x))^{-n} \arcsin (a x)^n \Gamma (n+1,i \arcsin (a x))}{2 a}-\frac {i (-i \arcsin (a x))^{-n} \arcsin (a x)^n \Gamma (n+1,-i \arcsin (a x))}{2 a} \]

[In]

Int[ArcSin[a*x]^n,x]

[Out]

((-1/2*I)*ArcSin[a*x]^n*Gamma[1 + n, (-I)*ArcSin[a*x]])/(a*((-I)*ArcSin[a*x])^n) + ((I/2)*ArcSin[a*x]^n*Gamma[
1 + n, I*ArcSin[a*x]])/(a*(I*ArcSin[a*x])^n)

Rule 2212

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-F^(g*(e - c*(f/d))))*((c
+ d*x)^FracPart[m]/(d*((-f)*g*(Log[F]/d))^(IntPart[m] + 1)*((-f)*g*Log[F]*((c + d*x)/d))^FracPart[m]))*Gamma[m
+ 1, ((-f)*g*(Log[F]/d))*(c + d*x)], x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rule 3388

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/(E^(
I*k*Pi)*E^(I*(e + f*x))), x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d
, e, f, m}, x] && IntegerQ[2*k]

Rule 4719

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

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.92 \[ \int \arcsin (a x)^n \, dx=\frac {i \arcsin (a x)^n \left (\arcsin (a x)^2\right )^{-n} \left (-(i \arcsin (a x))^n \Gamma (1+n,-i \arcsin (a x))+(-i \arcsin (a x))^n \Gamma (1+n,i \arcsin (a x))\right )}{2 a} \]

[In]

Integrate[ArcSin[a*x]^n,x]

[Out]

((I/2)*ArcSin[a*x]^n*(-((I*ArcSin[a*x])^n*Gamma[1 + n, (-I)*ArcSin[a*x]]) + ((-I)*ArcSin[a*x])^n*Gamma[1 + n,
I*ArcSin[a*x]]))/(a*(ArcSin[a*x]^2)^n)

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.07 (sec) , antiderivative size = 240, normalized size of antiderivative = 3.04

method	result	size
default	\(\frac {2^{n} \sqrt {\pi }\, \left (\frac {2^{-1-n} \arcsin \left (a x \right )^{n} \left (6+2 n \right ) a x}{\sqrt {\pi }\, \left (1+n \right ) \left (3+n \right )}+\frac {\arcsin \left (a x \right )^{n} 2^{-n} \sqrt {-a^{2} x^{2}+1}\, \left (a^{2} x^{2} \arcsin \left (a x \right )-\arcsin \left (a x \right )+a x \sqrt {-a^{2} x^{2}+1}\right )}{\sqrt {\pi }\, \left (1+n \right ) \left (a^{2} x^{2}-1\right )}+\frac {2^{-n} \sqrt {\arcsin \left (a x \right )}\, n \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {3}{2}, \arcsin \left (a x \right )\right ) a x}{\sqrt {\pi }\, \left (1+n \right )}-\frac {2^{-n} \sqrt {-a^{2} x^{2}+1}\, \left (a^{2} x^{2} \arcsin \left (a x \right )-\arcsin \left (a x \right )+a x \sqrt {-a^{2} x^{2}+1}\right ) \operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (a x \right )\right )}{\sqrt {\pi }\, \left (1+n \right ) \sqrt {\arcsin \left (a x \right )}\, \left (a^{2} x^{2}-1\right )}\right )}{a}\)	\(240\)

[In]

int(arcsin(a*x)^n,x,method=_RETURNVERBOSE)

[Out]

2^n*Pi^(1/2)/a*(2^(-1-n)/Pi^(1/2)/(1+n)*arcsin(a*x)^n*(6+2*n)/(3+n)*a*x+1/Pi^(1/2)/(1+n)*arcsin(a*x)^n*2^(-n)*
(-a^2*x^2+1)^(1/2)/(a^2*x^2-1)*(a^2*x^2*arcsin(a*x)-arcsin(a*x)+a*x*(-a^2*x^2+1)^(1/2))+2^(-n)/Pi^(1/2)/(1+n)*
arcsin(a*x)^(1/2)*n*LommelS1(n+1/2,3/2,arcsin(a*x))*a*x-2^(-n)/Pi^(1/2)/(1+n)/arcsin(a*x)^(1/2)*(-a^2*x^2+1)^(
1/2)/(a^2*x^2-1)*(a^2*x^2*arcsin(a*x)-arcsin(a*x)+a*x*(-a^2*x^2+1)^(1/2))*LommelS1(n+3/2,1/2,arcsin(a*x)))

Fricas [F]

\[ \int \arcsin (a x)^n \, dx=\int { \arcsin \left (a x\right )^{n} \,d x } \]

[In]

integrate(arcsin(a*x)^n,x, algorithm="fricas")

[Out]

integral(arcsin(a*x)^n, x)

Sympy [F]

\[ \int \arcsin (a x)^n \, dx=\int \operatorname {asin}^{n}{\left (a x \right )}\, dx \]

[In]

integrate(asin(a*x)**n,x)

[Out]

Integral(asin(a*x)**n, x)

Maxima [F(-2)]

Exception generated. \[ \int \arcsin (a x)^n \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(arcsin(a*x)^n,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [F]

\[ \int \arcsin (a x)^n \, dx=\int { \arcsin \left (a x\right )^{n} \,d x } \]

[In]

integrate(arcsin(a*x)^n,x, algorithm="giac")

[Out]

integrate(arcsin(a*x)^n, x)

Mupad [F(-1)]

Timed out. \[ \int \arcsin (a x)^n \, dx=\int {\mathrm {asin}\left (a\,x\right )}^n \,d x \]

[In]

int(asin(a*x)^n,x)

[Out]

int(asin(a*x)^n, x)