3.133 \(\int \sin ^{-1}(a x)^n \, dx\)

Optimal. Leaf size=79 \[ \frac{i \left (i \sin ^{-1}(a x)\right )^{-n} \sin ^{-1}(a x)^n \text{Gamma}\left (n+1,i \sin ^{-1}(a x)\right )}{2 a}-\frac{i \left (-i \sin ^{-1}(a x)\right )^{-n} \sin ^{-1}(a x)^n \text{Gamma}\left (n+1,-i \sin ^{-1}(a x)\right )}{2 a} \]

[Out]

((-I/2)*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)

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Rubi [A]  time = 0.0546235, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4623, 3307, 2181} \[ \frac{i \left (i \sin ^{-1}(a x)\right )^{-n} \sin ^{-1}(a x)^n \text{Gamma}\left (n+1,i \sin ^{-1}(a x)\right )}{2 a}-\frac{i \left (-i \sin ^{-1}(a x)\right )^{-n} \sin ^{-1}(a x)^n \text{Gamma}\left (n+1,-i \sin ^{-1}(a x)\right )}{2 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I/2)*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 4623

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]

Rule 3307

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 2181

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

Rubi steps

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

Mathematica [A]  time = 0.035468, size = 73, normalized size = 0.92 \[ \frac{i \sin ^{-1}(a x)^n \left (\sin ^{-1}(a x)^2\right )^{-n} \left (\left (-i \sin ^{-1}(a x)\right )^n \text{Gamma}\left (n+1,i \sin ^{-1}(a x)\right )-\left (i \sin ^{-1}(a x)\right )^n \text{Gamma}\left (n+1,-i \sin ^{-1}(a x)\right )\right )}{2 a} \]

Antiderivative was successfully verified.

[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)

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Maple [C]  time = 0.082, size = 240, normalized size = 3. \begin{align*}{\frac{{2}^{n}\sqrt{\pi }}{a} \left ({\frac{{2}^{-1-n} \left ( \arcsin \left ( ax \right ) \right ) ^{n} \left ( 6+2\,n \right ) ax}{\sqrt{\pi } \left ( 1+n \right ) \left ( 3+n \right ) }}+{\frac{ \left ( \arcsin \left ( ax \right ) \right ) ^{n}{2}^{-n}}{\sqrt{\pi } \left ( 1+n \right ) \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ({a}^{2}{x}^{2}\arcsin \left ( ax \right ) -\arcsin \left ( ax \right ) +ax\sqrt{-{a}^{2}{x}^{2}+1} \right ) }+{\frac{{2}^{-n}nax}{\sqrt{\pi } \left ( 1+n \right ) }\sqrt{\arcsin \left ( ax \right ) }{\it LommelS1} \left ( n+{\frac{1}{2}},{\frac{3}{2}},\arcsin \left ( ax \right ) \right ) }-{\frac{{2}^{-n}}{\sqrt{\pi } \left ( 1+n \right ) \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ({a}^{2}{x}^{2}\arcsin \left ( ax \right ) -\arcsin \left ( ax \right ) +ax\sqrt{-{a}^{2}{x}^{2}+1} \right ){\it LommelS1} \left ( n+{\frac{3}{2}},{\frac{1}{2}},\arcsin \left ( ax \right ) \right ){\frac{1}{\sqrt{\arcsin \left ( ax \right ) }}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\arcsin \left (a x\right )^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arcsin(a*x)^n, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{asin}^{n}{\left (a x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(asin(a*x)**n, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \arcsin \left (a x\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arcsin(a*x)^n, x)